Yann Bouchery
email: ybouchery@em-normandie

Asma Ghaffari

Zied Jemai

Jan Fransoo

Sustainable transportation and order quantity: Insights from multiobjective optimization

This article applies multiobjective optimization to show how the tradeoffs between cost and carbon emissions may be obtained in the context of sustainable operations. We formulate a model where transportation mode selection and order quantity decisions are considered jointly. We derive structural properties of the model and develop several insights. First, we show that switching to a greener mode of transportation while continuing to optimize the total logistics costs function may lead to a dominated solution. Second, we prove that the modal shift occurs only under strong carbon emissions reduction requirements. Third, we show that the efficient frontier is non-convex and we analyze some implications.

Finally, we analyze the impacts of an increase in truck capacity. The results are illustrated through an example of a French retailer.

Introduction

Environmental and social awareness has considerably increased since the Brundtland report's publication (World Commission on Environment and Development, 1987). Nowadays, many leading companies worldwide are committed in creating value for a broader set of stakeholders instead of focusing solely on creating profits for shareholders or owners. In line with this trend, the number of articles on sustainable operations has drastically increased these last years. We refer to [START_REF] Linton | Sustainable supply chains: An introduction[END_REF], [START_REF] Srivastava | Green supply-chain management: A state-of-the-art literature review[END_REF], [START_REF] Seuring | From a literature review to a conceptual framework for sustainable supply chain management[END_REF], [START_REF] Kleindorfer | Sustainable Operations Management[END_REF], [START_REF] Dekker | Operations Research for green logistics -An overview of aspects, issues, contributions and challenges[END_REF], [START_REF] Tang | Research advances in environmentally and socially sustainable operations[END_REF] and [START_REF] Govindan | Eco-efficiency based green supply chain management: Current status and opportunities[END_REF] for reviews. By acknowledging the different dimensions of sustainability, sustainable operations aim at optimizing several objectives. In this context, companies may identify situations where the objectives may be improved simultaneously, i.e. win-win situations. However, these 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, [START_REF] Caro | Double Counting in Supply Chain Carbon Footprinting[END_REF] assume that "firms have exhausted all their carbon abatement initiatives that are profitable in the absence of external incentives", i.e. that win-win situations are unavailable. An increasing number of companies also start thinking that "sustainability can only be attained by optimizing seemingly conflicting targets" [START_REF] Dhl | Delivering Tomorrow: Towards Sustainable Logistics[END_REF]. In this case, a company interested in sustainable operations seeks to identify the most favorable trade-off between the considered objectives. This leads [START_REF] Fransoo | Environmental sustainability in supply chains[END_REF] to consider that "this naturally leads to multiobjective considerations in order to analyze the tradeoff between economic and environmental performance measures" (p. 1). Accordingly, this article applies multiobjective optimization to a decision problem related to sustainable operations in order to show how to obtain the tradeoffs between different types of objectives. The aim of multiobjective optimization is to identify particular solutions such that, when attempting to improve an objective further, other objectives suffer as a result [START_REF] Ehrgott | Multicriteria Optimization[END_REF]. These solutions are called efficient or Pareto optimal and they correspond to the tradeoffs that are of interest for companies interested in sustainable operations.

The model developed consists of a joint transportation mode selection and order quantity optimization problem. We indeed acknowledge that inventory and transportation decisions are strongly interrelated.

However, there is a lack of articles on sustainable operations explicitly addressing this issue. We focus on two objective functions, i.e., cost and carbon emissions. Our multiobjective optimization results enable showing that switching to a greener mode of transportation while continuing to optimize the total logistics costs function may lead to a dominated solution (such that it is possible to reach the same level of carbon emissions with a lower cost by solely adjusting the order quantity). Second, we prove that a shift towards a more carbon-efficient mode of transportation is interesting only for strong carbon emissions reduction requirements (i.e., for high carbon emissions reduction target or for high carbon price). Otherwise, order quantity adjustments may enable efficiently greening the supply chain. Third, we show that the efficient frontier (i.e., the set of efficient solutions) is non-convex. This structural property is of great importance as this implies that some non-supported solutions exist (see e.g. [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF] for more details about non-supported solutions). This type of solutions cannot be generated by using a linear combination of the objectives. Thus, using carbon pricing when facing a non-convex efficient frontier may provide a misleading impression to the decision maker as the non-supported solutions would be hidden.

The article is organized as follows. The related literature is reviewed in Section 2. Section 3 is then devoted to the presentation of the model, to the multiobjective optimization results and to the presentation of an example. We also analyze in Section 3 the impacts of using carbon pricing and the impacts of extending vehicle capacity. Finally, Section 4 is devoted to the conclusion and to future research directions.

Literature review and contribution

This article is directly connected to two main fields of the literature on sustainable supply chains, i.e., order quantity models and transportation mode selection.

First, several articles on order quantity decisions with carbon emissions considerations have recently been proposed. The classical Economic Order Quantity (EOQ) model has been revisited by considering vehicle emissions costs [START_REF] Bonney | Environmentally responsible inventory models: Non-classical models for a non-classical era[END_REF], a constraint on carbon emissions [START_REF] Chen | The carbon-constrained EOQ[END_REF], the emissions trading scheme [START_REF] Hua | Managing carbon footprints in inventory management[END_REF] and four different regulatory policies [START_REF] Arslan | EOQ Revisited with Sustainability Considerations[END_REF]. [START_REF] Bouchery | Including sustainability criteria into inventory models[END_REF] include sustainability criteria into single and multiechelon EOQ models by using multiobjective optimization. [START_REF] Jaber | Supply chain coordination with emissions reduction incentives[END_REF] revisit the joint economic lot size problem by considering that the companies face different types of emissions trading schemes. [START_REF] Benjaafar | Carbon Footprint and the Management of Supply Chains: Insights From Simple Models[END_REF] include carbon emissions constraints on single and multi-stage lot-sizing models with a cost minimization objective and consider four regulatory policy settings. [START_REF] Battini | A sustainable EOQ model: Theoretical formulation and applications[END_REF] extend the EOQ model by explicitly including transportation modeling. They compare the results obtained with a pure cost model and the results obtained when additionally including environmental costs. [START_REF] Konur | Economic and environmental comparison of grouping strategies in coordinated multi-item inventory systems[END_REF] include carbon emissions into the joint replenishment problem. They apply multiobjective optimization and identify the set of efficient solutions for two types of strategies (namely direct and indirect grouping). Based on a set of numerical examples, they identify conditions such that a strategy outperforms the other one both in terms of costs and carbon emissions. The dynamic lot-sizing model has also been revisited by [START_REF] Absi | Lot sizing with carbon emission constraints[END_REF][START_REF] Velázquez-Martínez | The impact of carbon footprinting aggregation on realizing emission reduction targets[END_REF][START_REF] Retel Helmrich | The Economic Lot-Sizing Problem with an Emission Capacity Constraint[END_REF] by considering an additional constraint on carbon emissions. Except from Velázquez-Martínez et al.

(2014) and [START_REF] Battini | A sustainable EOQ model: Theoretical formulation and applications[END_REF], the articles cited above do not take explicitly transportation features into account. As a result, these articles do not account for the impact of order quantity decisions on the load factor even if this could strongly impacts the transportation emissions as shown by [START_REF] Velázquez-Martínez | The impact of carbon footprinting aggregation on realizing emission reduction targets[END_REF]. On the other hand, [START_REF] Velázquez-Martínez | The impact of carbon footprinting aggregation on realizing emission reduction targets[END_REF] and [START_REF] Battini | A sustainable EOQ model: Theoretical formulation and applications[END_REF] do not consider transportation mode selection issues.

The second field of literature related to this work focuses on including carbon emissions concerns into freight transportation mode selection problems. [START_REF] Winebrake | Assessing Energy, Environmental, and Economic Tradeoffs in Intermodal Freight Transportation[END_REF] present an energy and environmental analysis model to explore the tradeoffs in an intermodal transportation network. [START_REF] Bauer | Minimizing greenhouse gas emissions in intermodal freight transport: an application to rail service design[END_REF] especially focus on determining the optimal planning for intermodal rail transportation in order to minimize the carbon emissions from transportation. [START_REF] Cholette | The energy and carbon intensity of wine distribution: A study of logistical options for delivering wine to consumers[END_REF] present a case study where several modes of transportation are available in a wine supply chain context. Their analysis takes cost, carbon emissions and energy consumption into account. [START_REF] Pan | The reduction of greenhouse gas emissions from freight transport by pooling supply chains[END_REF] investigate how freight consolidation may help in decreasing the carbon emissions from transportation. They formulate a carbon emissions minimization model where both road and rail transportation are available. They apply their model for optimizing the carbon emissions of two large retail chains. [START_REF] Leal | Modal choice evaluation of transport alternatives for exporting bioethanol from Brazil[END_REF] consider the transportation mode selection decision in a case study based on a bio-ethanol supply chain.

Socio-environmental considerations are included into the model. Finally, [START_REF] Kopfer | Reducing greenhouse gas emissions of a heterogeneous vehicle fleet[END_REF] analyze the benefits of considering a heterogeneous fleet of vehicles for vehicle routing problems. The authors show that the fuel consumption (and therefore the carbon emissions) can be reduced by choosing vehicles of an adequate size for transport fulfillment. The impact of inventory decisions such as order quantity decisions are not considered in this stream of literature, as the order quantity are assumed to be given exogenously.

To our knowledge, six articles jointly consider transportation mode selection and inventory optimization problem while explicitly accounting for transportation features. [START_REF] Rosič | The economic and environmental performance of dual sourcing: A newsvendor approach[END_REF] extend the dual sourcing model based on the newsvendor framework by considering the environmental impact of transportation. They analyze two types of regulatory policies, i.e. the carbon tax and the carbon cap-and-trade mechanism. They prove that it is possible to reduce the carbon emissions from transportation without substantially affecting the economic performance of the system if the capand-trade mechanism is applied with appropriate carbon cap setting. In [START_REF] Hoen | Effect of carbon emission regulations on transport mode selection under stochastic demand[END_REF], a stochastic inventory model is extended to incorporate transport emissions costs. The transport mode and order-up-to level of a base-stock inventory policy are jointly optimized in a single product setting. [START_REF] Hoen | Switching Transport Modes to Meet Voluntary Carbon Emission Targets[END_REF] extend the previous model to a multiple products settings and study the portfolio effect of setting a global emissions target instead of a per product target. These three articles mainly focus on the relationship between transportation leadtime and safety stock levels by arguing that greener modes of transportation are often slower (and/or less flexible) and that this results in increasing the inventory levels. Our paper focuses on a different aspect of the interaction between transportation mode selection and inventory decisions. Indeed, we consider that order quantity decisions have a strong impact on the transportation mode selection decisions and on the carbon emissions levels. First, greener modes of transportation are often able to carry bigger volumes. Second, the order quantity decisions have an effect on load factors and thus on carbon emissions from transportation. Our model enables investigating such relationship and may be viewed as complementary to the literature focusing on the relationship between transportation leadtime and safety stock levels. To our knowledge, three articles focus on such interaction while explicitly accounting for transportation features. [START_REF] Konur | Carbon constrained integrated inventory control and truckload transportation with heterogeneous freight trucks[END_REF] extends the economic order quantity model to account for full truckload cost structure. The author proposes a carbon constrained model and focuses more particularly on the situations for which the optimal order quantity exceeds one full truckload. The article shows that considering a heterogeneous fleet of trucks may help reducing both cost and carbon emissions. [START_REF] Konur | Integrated inventory control and transportation decisions under carbon emissions regulations: LTL vs. TL carriers[END_REF] additionally consider less than truckload tariffs and they analyze the impact of four types of regulatory policies. Finally, [START_REF] Schaefer | Economic and environmental considerations in a continuous review inventory control system with integrated transportation decisions[END_REF] propose an integrated continuous review inventory model with explicit transportation decisions. The model accounts for both truckload and less than truckload transportation. The authors consider two objectives, i.e., the total cost and the total carbon emissions and they propose to approximate the efficient frontier by using linear combinations of the two objectives. The results show that an increase in demand variance and/or in lead time negatively affects both cost and emissions. Unlike [START_REF] Konur | Carbon constrained integrated inventory control and truckload transportation with heterogeneous freight trucks[END_REF] and [START_REF] Konur | Integrated inventory control and transportation decisions under carbon emissions regulations: LTL vs. TL carriers[END_REF], we focus here on situations for which the optimal order quantity does not exceed one full vehicle load as this is the case in a large variety of industrial situations. [START_REF] Schaefer | Economic and environmental considerations in a continuous review inventory control system with integrated transportation decisions[END_REF] is seminal in the sense that this is the first article that simultaneously considers the relationship between transportation leadtime and safety stock levels as well as the effect of order quantity levels on load factors. Due to the complexity of the problem, the authors approximate the efficient frontier by applying the weighted sum method. We focus here solely on the effect of order quantity levels on load factors. This enables us to analytically identify the efficient frontier. In addition, [START_REF] Schaefer | Economic and environmental considerations in a continuous review inventory control system with integrated transportation decisions[END_REF] solve the problem for each mode separately. They propose some examples with two modes of transportation and solve the integrated inventory and transportation mode selection problem numerically. We propose here analytical results that enable us identifying the efficient frontier when multiple transportation modes are available. Our model may consequently be viewed as complementary to the existing literature.

Our contribution is threefold. First, from a modeling perspective, this article is one of the first studying explicitly the relationship between order quantity and transportation mode decisions with cost and carbon emissions concerns. Our model may be applied with multiple modes of transportation including air, water, rail, road and any type of intermodal combinations. The model may also be used to study the effect of speed reduction as speed is recognized to have some major impacts on costs, fuel consumption [START_REF] Corbett | The effectiveness and costs of speed reductions on emissions from international shipping[END_REF][START_REF] Fransoo | The Critical Role of Ocean Container Transport in Global Supply Chain Performance[END_REF] and safety. The model is flexible enough to account for realistic transportation costs and carbon emissions structures as any type of piecewise linear functions may be considered. Second, from a theoretical perspective, we apply the concept of multiobjective optimization in order to identify the existing tradeoffs that a company can face when jointly optimizing transportation mode selection and order quantity decisions. Multiobjective optimization helps the decision maker to build a conviction of what is possible and to use this knowledge to identify the most valuable trade-off. We identify analytically the set of efficient solutions for the problem. Our third contribution consists in providing new insights to decision makers. We show that switching to a greener mode of transportation while continuing to optimize the total logistics costs function may lead to a dominated solution. We prove that the modal shift occurs only under strong carbon emissions reduction requirements. We show that the efficient frontier is non-convex and we analyze some implications. Finally, we analyze the impacts of an increase in truck capacity.

Model description and multiobjective optimization results

Model description

In this article, we consider that several modes of transport are available for inbound transportation. Each mode is characterized by a cost function in the form of a fixed cost (per vehicle) and a variable cost (per product unit). Moreover, a fixed lead time is associated to each mode. The lead time has an effect on the average in-transit inventory level. We also take the capacity of each mode into account. The per shipment transportation cost function is defined as follows (  

x represents the nearest integer x  ):

LQ h Q Q T Q T Q T TC FC VC C          max ) ( , (1)
with: The total inventory holding and transportation cost function per time unit is defined as follows:

Q = order
D P Q D Q T O Q D h Q Q Z C C C C C     ) ( 2 ) ( , (2) with
C C C Z Q Z  ).
Lemma 1 provides a sufficient condition ensuring that the optimal order quantity does not exceed one vehicle load.

Lemma 1. Let * C Q be the order quantity minimizing Expression (2). Assume that D

Q h O c C 2 2 max  , then: max * Q Q C  .
All proofs may be found in Appendix A. In the following, we assume that the retailer has no control over the production scheme of the manufacturer (that certainly supplies several other retailers) as in most practical situations. In this case,

C

O represents the fixed costs for order forms, authorization, receiving, inspection and/or handling of invoice from the supplier [START_REF] Axsäter | Inventory control[END_REF] as transport costs are accounted for separately. These costs are usually small comparing to the costs of requiring a second vehicle for inbound transportation as well as the costs of holding extra inventory. These costs have also been reduced by new technologies such as electronic data interchange or radio frequency identification. Consequently,

we assume that max * Q Q C 
in what follows, i.e., that there is no incentive to order more than a full vehicle.

In this case, the cost function to be considered may be expressed as follows:

D L h T P T O Q D h Q Q Z TC VC C FC C C C ) ( ) ( 2 ) (       , max Q Q  , (3)
We refer to [START_REF] Konur | Carbon constrained integrated inventory control and truckload transportation with heterogeneous freight trucks[END_REF] and references therein for an analysis of the complexity and for providing algorithms to solve the problem in the cases for which the optimal order quantity consists in ordering more than a full vehicle load.

In practice, several transportation tariffs are often available, depending on the shipment quantity. We focus on heavy duty truck transport as an example. order 15 pallets. The tariff leads to 450€ per shipment. On the other hand, to retailer may over-declare its shipment to 21 pallets to take advantage of the all unit discount less than truckload tariff leading to 420€ per shipment. Consequently, the transportation costs function follows a modified all-unit discount structure. This cost structure is commonly used in the literature (see e.g. [START_REF] Chan | Effective Zero-Inventory-Ordering Policies for the Single-Warehouse Multiretailer Problem with Piecewise Linear Cost Structures[END_REF][START_REF] Croxton | Models and Methods for Merge-in-Transit Operations[END_REF][START_REF] Rieksts | Optimal inventory policies with two modes of freight transportation[END_REF]. The per shipment transportation cost as a function of the order quantity (up to the truck capacity) is shown in Figure 1.

We intend to develop a model that enables accounting for realistic transportation costs structures such as the modified all-unit discount cost structure introduced above. To do so, let max min Q Q  be the minimum amount per shipment to make a mode available. By making use of min Q and max Q , any type of piecewise linear transportation cost function may be taken into account by considering each piecewise linear segment of the transportation cost function as a separate mode characterized by a fixed cost (per vehicle) and a variable cost (per product unit). Indeed, the minimum and maximum capacity limits enable making the mode available only for the quantities related to the segment under consideration.

Consequently, we consider 4 different transportation modes with different cost parameters and minimum quantities for the example above.

Q Q Q C  
, the order quantity minimizing Equation 3 is obtained easily

as C Z is convex on ] ; [ max min Q Q
. First, the optimal order quantity without taking the minimum and maximum quantity constraints into account can be calculated. Second, the feasible order quantity which is the closest from this theoretical quantity is selected:

                  C FC C C h D T O Q Q Q ) ( 2 , min ; max max min * .
(4)

Modeling carbon emissions across the supply chain is attracting more and more research (see e.g. [START_REF] Scipioni | Monitoring the carbon footprint of products: a methodological proposal[END_REF][START_REF] Sundarakani | Modeling carbon footprints across the supply chain[END_REF]. In our model, three main sources of carbon emissions may be identified. First, carbon emissions are generated by producing the item. This amount of carbon emissions is independent of the decisions taken by the retailer as we assume that the retailer has no control over the production scheme of the manufacturer. A fixed amount of carbon emissions is thus associated to each item due to production. Second, carbon emissions are generated by inbound transportation. The transportation mode selection as well as the order quantity decisions may affect the amount of carbon emissions generated by inbound transportation. For a given transportation mode, the generated carbon emissions are modeled with a fixed term (due to emissions generated by the vehicle if running empty) and a linear term as a function of the order quantity (due to the extra energy consumption generated by transporting the items). Note that this modeling is commonly used in the transportation literature (see e.g. [START_REF] Pan | The reduction of greenhouse gas emissions from freight transport by pooling supply chains[END_REF] for rail and road transportation) and is in accordance with the available methodologies for estimating carbon emissions from transportation (see e.g., [START_REF] Demir | A review of recent research on green road freight transportation[END_REF][START_REF] Ntm | Environmental data for international cargo transport[END_REF][START_REF] Hickman | Methodology for calculating transport emissions and energy consumption[END_REF]. Third, an amount of carbon emissions is associated with the storage of each product unit per time unit. This amount is mainly due to indirect carbon emissions from energy consumption (mainly electricity) in the warehouse. This amount may become important in case of refrigeration.

The total carbon emissions as a function of the order quantity may be expressed as follows for all min

Q Q  : D T Q D Q Q T h Q D P Q Z VE FE E E E           max 2 ) ( , (5)
with:

E P = purchase emissions per product unit, E h = inventory holding emissions per product unit and time unit, FE T = fixed amount of carbon emissions per shipment, VE T = variable amount of carbon emissions per product unit.

Assume that * E Q is the order quantity that minimizes the carbon emissions function (i.e. * * ) (

E E E Z Q Z  ).
We can observe that

max * Q Q E 
(there is no incentive to order more than the maximum transportation capacity). It follows that:

                 E FE E h D T Q Q Q 2 min ; max max; min * .
(6)

Multiobjective optimization results

We consider costs and carbon emissions as two distinct objective functions that have to be minimized. An alternative a is thus said to be dominated if there exists another alternative b that performs at least as good as a on one objective and that performs better than a on the other objective. Multiobjective optimization consists in identifying all the non-dominated alternatives called efficient solutions. The results presented in this section enable identifying the set of efficient solutions in the situations for which multiple modes of transportation are available.

A single mode of transportation

As a first step, we identify the set of efficient solutions when considering a single mode of transport.

In this case, the only decision variable for the problem is the order quantity and the set of possible values

for Q is   max min ;Q Q A  . Let     A Z : ,   ) ( ); ( ) ( a Z a Z a Z E C 
, for all A a  , with C Z defined by Formula 3 representing the total costs and E Z defined by Formula 5 representing the total carbon emissions.

  

) ( ); ( ) ( Q Z Q Z A Z E C   A Q  is the image of A in the criterion space (evaluation space).
The set of efficient solutions is a subset of A noted E . Its image in the criterion space referred as the efficient frontier is

)] ; max( ); ; [min( * * * * E C E C Q Q Q Q E  . Note that * * E C Q Q  in many practical cases, thus ] ; [ * * E C Q Q E  . Indeed, C FC C h T O  is often lower than E FE h
T as the holding cost includes the opportunity cost of the capital tied up into inventory (meaning that C h is high) and as transportation is recognized as a major source of carbon emissions (meaning that FE T is high). Accordingly, we focus on the situations for which

E FE C FC C h T h T O  
in what follows for the sake of relevance and clarity.

We define now the convexity property for the efficient frontier (which corresponds to a subset of

  
) and we prove that the efficient frontier is convex when considering a single mode of transportation. Let S be a subset of

) (E Z is convex.
Proposition 2 implies that the problem behaves nicely when considering a single mode of transportation. Indeed, all the elements of a convex efficient frontier may be generated by minimizing a weighted sum of objectives. We prove later that the efficient frontier is non-convex when more than one mode of transportation is considered (see Proposition 6).

Two modes of transportation

Consider now that a second mode of transportation (mode 2) is available. In what follows, subscript 1 refers to mode 1 and subscript 2 refers to mode 2. As for mode 1, * 2 C Q and * 2 E Q may be obtained by using Formula 4 and Formula 6 respectively. Moreover, let

) ( * 2 2 * 2 C C C Q Z Z  and ) ( * 2 2 * 2 E E E Q Z Z  .
Proposition 1 and Proposition 2 are also valid for mode 2. Without loss of generality, we assume that

* 2 * 1 C C Z Z 
(mode 1 is less costly than mode 2). Propositions 3 and Lemma 2 restrict the possible number

of intersection between ) ( 1 1 E Z and ) ( 2 2 E Z
. Note that S corresponds to the cardinality of the set S .

Proposition 3. Let ) ( 1 1 E Z and ) ( 2 2 E Z
be the efficient frontiers for transportation mode 1 and transportation mode 2 respectively, then:

2 ) ( ) ( 2 2 1 1   E Z E Z . Lemma 2. Let ) ( 1 1 E Z and ) ( 2 2 E Z
be the efficient frontiers for transportation mode 1 and transportation mode 2 respectively, then: . The results we present in Proposition 3 and Lemma 2 enable providing additional insights compared to a classical single objective analysis.

If * 2 * 1 E E Z Z  , then 1 ) ( ) ( 2 2 1 1   E Z E Z , Else   2 ; 0 ) ( ) ( 2 2 1 1   E Z E Z .
If * 2 * 1 E E Z Z 
, mode 2 is more expensive but greener. In this case, there exists at most one intersection between the efficient frontiers of the two modes considered separately. In this case, mode 1 will be preferred from a pure cost perspective, and mode two will be preferred if the requirements to green the operations reach a given threshold. This result implies that adjusting the order quantity is the most efficient way of greening operations until a given threshold. This also implies that switching to a greener mode of transportation while continuing to optimize the total logistic cost function may lead to a dominated solution (see Figures 2a and2c). In this case, the same level of carbon emissions may be obtained with a lower cost by only increasing the order quantity. This result proves that poor decisions may be taken when ignoring the strong interrelationship between inventory control and transportation mode selection. )

( 1 1 E Z ) ( 2 2 E Z ) ( 1 1 E Z ) ( 2 2 E Z ) ( 1 1 E Z ) ( 2 2 E Z ) ( 1 1 E Z ) ( 2 2 E Z ) ( 1 1 E Z ) ( 2 2 E Z ) ( 1 1 E Z ) ( 2 2 E Z ) ; (   e c ) ; ( 1 1   e c ) ; ( 2 2   e c
2g ) ( 1 1 E Z ) ( 2 2 E Z Cost Cost Cost Cost Cost Cost If * 2 * 1 C C Z Z  and * 2 * 1 E E Z Z 
, we may expect mode 2 to be out of interest as this is possible to obtain a cheaper solution as well as a greener solution with mode 1. However, our results proves that some solutions obtained with mode 2 may still be efficient in case

2 ) ( ) ( 2 2 1 1   E Z E Z
(see Figure 2f). Mode 2 may indeed be considered in some situations as a better compromise between costs and carbon emissions.

The efficient frontier of the problem with two modes of transportation may be identified by applying Lemma 2 and by acknowledging that

* 2 * 1 C C Z Z  and * 2 * 1 E E Z Z  then: If 1 ) ( ) ( 2 2 1 1   E Z E Z , then the intersection point is noted     e c ;
and:

            e e E Z e c c c E Z e c E Z 2 2 2 2 2 1 1 1 1 1 ) ( ) ; ( ) ( ) ; ( ) ( . Else 0 ) ( ) ( 2 2 1 1   E Z E Z : If ) ( ) ( * 1 1 E Z Q Z E  , then:   * 1 2 2 2 2 2 1 1 ) ( ) ; ( ) ( ) ( E Z e E Z e c E Z E Z     , Else,   ) ( ) ( ) ; ( ) ( 2 2 * 2 1 1 1 1 1 E Z Z c E Z e c E Z C     .
Proposition 5. Let

) (E Z be the efficient frontier for the problem with two modes of transportation with

* 2 * 1 C C Z Z  and * 2 * 1 E E Z Z  then: If 0 ) ( ) ( 2 2 1 1   E Z E Z then: If ) ( * 2 2 C Q Z is efficient then:   * 2 1 1 1 1 1 ) ( ) ; ( ) ( C Z c E Z e c E Z    ) ( 2 2 E Z    * 2 1 1 1 1 1 ) ( ) ; ( E Z e E Z e c    Else: ) ( ) ( 1 1 E Z E Z  . Else 2 ) ( ) ( 2 2 1 1   E Z E Z . Let   1 1 ;   e c and   2 2 ;   e c
be the two intersection points and assume that

1 2    e e
, then:

  1 1 1 1 1 1 ) ( ) ; ( ) (     e e E Z e c E Z   2 2 1 2 2 2 2 ) ( ) ; (       e e e E Z e c   1 2 1 1 1 1 ) ( ) ; ( e e E Z e c    
Propositions 4 and 5 enable identifying analytically the efficient frontier when two modes of transport are considered. We additionally provide a sufficient condition ensuring that the efficient frontier of the problem with two modes of transportation is non-convex in Proposition 6.

Proposition 6: Let

) (E Z be the set of efficient solutions for the problem with two modes of transportation, then:

If ) ( ) ( 1 1 E Z E Z  then ) (E Z is non-convex.
Proposition 6 proves that the efficient frontier is non-convex in most of practical situations. The fact that the efficient frontier is non-convex as soon a more than one mode of transportation is available implies that some efficient solutions are non-supported ( see e.g. [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF] for more details about nonsupported solutions). These solutions, even if they might be of interest for the decision maker, would not be generated by single objective approaches such as the weighted sum method (pricing emissions is an example of a weighted sum). The multiobjective approach developed in this article enables identifying such non-supported solutions and thus provides a holistic view of the efficient frontier. We provide a discussion on the implications of using a weighted sum method for the problem in Section 3.4.

More than two modes of transportation

The study of the global problem with 2  n modes of transportation may be conducted as follows. First, the set of efficient solutions for mode k denoted as k E (with corresponding efficient frontier denoted as

) ( k k E Z
) may be identified by using

; 1 [ n j  (such that k j  ). ) ( , E Z j k
is the corresponding efficient frontier (by extension, we also consider

) ( ) ( , k k k k E Z E Z  ). Let define  n j j k k k E Z E Z 1 , ) ( ) (    . Note that  ) ( k k E Z
 n k k k E Z E Z 1 ) ( ) (    .
This results enables identifying all the efficient solutions in the case with multiple transportation modes.

We provide an example of application in the next section.

Example and insights

We present here an application that illustrates the type of outcome generated by the multiobjective optimization analysis. We decide to focus on a single application as we provide analytical results that enable understanding the behavior of the model for any set of parameters. The application is followed by an illustration of the main insights highlighted in Section 3.2. The application is based on real data in order to ensure the practical relevance of insights.

Example of the French retailer

We consider a French retailer who orders bottles of wine from an external supplier. The bottles are delivered on pallets and the retailer has to order an integer amount of pallets. Therefore, we consider that one product unit equals to one pallet. We assume that the assumptions of the EOQ model are fulfilled. We exclude purchase costs and purchase emissions from the analysis as they do not affect the decisions we consider. The data relative to the problem may be found in Table 1. . Figure 3 displays the efficient frontier as well as its convex hull. The x-axis corresponds to the costs and the y-axis corresponds to the carbon emissions.

Q max Q FC T FE T C FC C h D T O ) ( 2  E FE h D T 2 * C Q * E Q * C Z * E Z
Figure 1 shows that ) (E Z is convex as stated in Proposition 2. We also include in Figure 3 some dominated solutions obtained when ordering more than a full truckload (dashed line). We can observe that ordering more than a full truckload is not effective as this implies an increase in cost (as stated in Lemma 1) as well as an increase in carbon emissions (as emissions cannot be decreased by ordering more than a full vehicle according to Expression (5)). € per freight car is considered.

In opposition to truck transportation, several types of cargo (from several retailers) may be included into the same train. In this case, the carbon emissions associated with the train when running empty may be split between the different users. A fixed amount of carbon emissions per freight car is then derived from the average utilization rate of the train. Moreover, a variable amount of carbon emissions is associated to each pallet. By using average values provided by the NTM methodology, we obtain that 333  there is no incentive to order more than one full freight car. Note that the results of Section 3.1 and 3.2.1 may be applied to train transportation. The results are summarized in Table 3.

min Q max Q FC T FE T C FC C h D T O ) ( 2  E FE h D T 2 * C Q * E Q * C Z * E Z
E Z Z c E Z e c E Z    
. The results are illustrated in Figure 4. We can also notice that the efficient frontier . Table 4 provides

Q

is set to 14 pallets as a better tariff (with the same level of carbon emissions) may be obtained with mode 2 when 14  Q

. Figure 5 illustrates the results obtained in Table 4 in the criterion space. min

Q max Q FC T FE T C FC C h D T O ) ( 2  E FE h D T 2 * C Q * E Q * C Z * E Z

Insights from multiobjective optimization

The new multiobjective optimization results applied to the example of the French retailer enables to exemplify the insights highlighted in Section 3.2.2.

First, assume that the retailer currently orders 10  Q pallets (i.e. that the retailer minimizes its total logistics costs). Figure 5 shows that a carbon emissions reduction of 57% can be achieved by increasing the order quantity up to 33 pallets without switching to a greener mode of transportation. This feature illustrates that increasing the order quantity (i.e., increasing the load factor for transportation) is very efficient for reducing the supply chain emissions. Moreover, the required financial effort first remains reasonable when decreasing carbon emissions. For instance, choosing truck transportation with 7 . 16  Q enables a 34% reduction in carbon emissions for a 6% costs increase. On the opposite, the financial effort will increase as Q is getting closer to the order quantity that minimizes the amount of carbon emissions for truck transportation. This feature is commonly highlighted in the literature on inventory control with carbon emissions concerns [START_REF] Bouchery | Including sustainability criteria into inventory models[END_REF][START_REF] Chen | The carbon-constrained EOQ[END_REF] due to the relative insensitivity of the costs to a variation in the order quantity. We prove here that the results remain valid when explicitly including transportation features into the economic order quantity model. This implies that switching to a greener mode of transportation is efficient only in case of a strong carbon emissions reduction target. For the example proposed, shifting from truck to train becomes interesting if the carbon emissions reduction target is a least 50% or for a carbon price greater than 1670€/ton CO2 (this result is obtained by identifying the common tangent between truck and train transportation). In other situations, adjusting the order quantity while continuing to use truck transportation would be more efficient to green the supply chain.

The example of the French retailer also illustrates that switching to a greener mode of transportation while continuing to optimize the total logistic cost function may lead to a dominated solution (we refer to Figure 5). Without performing the multiobjective optimization analysis provided in this article, the retailer may decide to switch to rail for inbound transportation while continuing to minimize the total cost function in order to decrease the carbon emissions of the supply chain. This solution leads to 1 350 € and 441 kg CO2 per month and may be perceived as appropriate as this leads to a 40% decrease in carbon emissions for a 13% increase in costs, when compared to minimizing costs with truck transportation. However, Figure 5 clearly shows that the solution consisting in switching to rail while continuing to minimize the cost function is a dominated solution. The same decrease in carbon emissions may be achieved with a cost reduced by more than 5% by continuing to use truck for inbound transportation and by choosing 19  Q (leading to 1 277 € and 440 kg CO2 per month). This example clearly shows that switching to a greener mode of transportation while continuing to optimize the cost function may not be the best option to green the supply chain. This highlights the necessity of taking an integrated inventory control and transportation mode selection perspective when intending to green the supply chain.

Cost model with a carbon price

In this section, we aim at comparing our results to the ones obtained if a price is associated to carbon emissions. We indeed argued that the weighted sum method was not appropriate to generate all the efficient solutions if the efficient frontier is non-convex, and we concluded that carbon pricing might not be a proper way of generating efficient solutions for the problem we consider as we proved that the efficient frontier is not convex in many cases.

We consider that a price is associated to the company's carbon emissions. This price can be imposed to the company in the case of a carbon tax. However, this price can also come from an internal evaluation from the company, by considering the cost of the energy used or the cost obtained with an environmental accounting analysis. This price per amount of carbon emissions is noted

    ; 0 
. We perform the analysis by considering two transportation modes. In this context, there exists a value

    ; 0 C L that
allows deciding which mode is the most interesting as we show in Proposition 8.

Proposition 8. Assume that a price  is associated to the company's carbon emissions and assume that the company can decide between modes 1 and 2 with

* 2 * 1 C C Z Z  and * 2 * 1 E E Z Z 
. Then there exists a value C L such that:

-if C L   , then )) ( ) ( min(( )) ( ) ( min( 2 2 1 1 Q E Q Z Q E Q Z      , -if C L   , then )) ( ) ( min(( )) ( ) ( min( 2 2 1 1 Q E Q Z Q E Q Z      .
The value of C L is unique and may be found by identifying a value 1

 such that )) ( ) ( min(( )) ( ) ( min( 2 1 2 1 1 1 Q E Q Z Q E Q Z      (i.e. C L  1 
) and by applying the bisection method on the interval   1 ; 0  (at each iteration, we select the interval

  1 ;  i i   such that )) ( ) ( min(( )) ( ) ( min( 2 2 1 1 Q E Q Z Q E Q Z i i      and )) ( ) ( min(( )) ( ) ( min( 2 1 2 1 1 1 Q E Q Z Q E Q Z i i       
). We apply the method to the example proposed in Figure 4 and we illustrate the results in Figure 7. The common tangent as well as the efficient solutions not identified by the weighted sum approached (the nonsupported efficient solutions) are displayed (dashed lines) in addition to the efficient solutions obtained with the weighted sum method.

Figure 7: The efficient frontier obtained with the weighted sum method

Figure 7 clearly highlights that the weighted sum approach is not appropriate for identifying all the efficient solutions of the problem. This could provide misleading impression to the decision maker. For instance, the decision maker could conclude that the maximum reduction in carbon emissions achievable by efficiently using truck transport is 14% (from 735 kg CO2 per month to 634 kg CO2 per month), instead of 33% (from 735 kg CO2 per month to 495 kg CO2 per month), if we include the non-supported efficient solutions. In addition, the emissions may be reduced by 20% (from 735 kg CO2 per month to 590 kg CO2 per month), for an increase in cost of 6% (from 1192€ per month to 1258€ per month) if we include the non-supported In opposite, the cost has to be increased by 14% (from 1192€ per month to 1361€ per month) to reduce the emissions by at least 20% if we ignore the non-supported solutions.

Moreover, a marginal change in carbon price around C L would have a huge effect on the amount of carbon emissions generated. Assume that  raises from

  C L to   C L with  being arbitrarily small.
Then, the optimal level of carbon emissions for the example presented above would drop from 634 kg CO2 per month to 395 kg CO2 per month (i.e., 38% reduction). This implies that a small change in carbon price may have huge impact on carbon emissions.

The case with

C L   is also very interesting. In this case, both transportation modes give the same overall result (operational costs + carbon tax). However, the costs and the carbon emissions are different for both options. In the example highlighted above, when 542 . 0   , using truck would lead to 1231+0.542*634=1575€ per month while using train would lead to 1361+0.542*395=1575€ per month.

Consequently, the company may decide among the two options. This operational flexibility implies that the total amount of carbon emissions is hardly controllable by setting a carbon price as the amount of carbon emissions for a given carbon price is not necessarily unique and may widely differ. We provide additional insights on the impact of vehicle capacity in the next section.

Impact of vehicle capacity

Vehicle capacity is often limited by regulation. For instance, the maximum length of a truck in France is limited to 18.25 meters (33 euro pallets) while the maximum weight eligible for truck transportation is 26 tons of goods. Logistics providers often argue that increasing vehicle capacities could strongly impact the performance of deliveries both in terms of costs and carbon emissions. In this section, we intend to analyze the impact of increasing trucks capacity in France, based on our results and on the example we provide in Section 3.3. Most of the existing studies on this topic only focus on transportation and only compare the performances of existing trucks with the performances of longer trucks. Our results enable a more extensive study by including the effects of the variation in inventory and by also comparing to a shift to train.

We focus on the impact of allowing European Modular System (EMS) in France. EMS consists of combining existing loading units to form a longer truck. These longer trucks are currently tested in many European countries with good results in terms of carbon emissions. An EMS truck can be up to 25.25 meters long and carries up to 52 pallets. We investigate the impact of allowing for EMS trucks in France by using the example presented in Section 3.3. We assume that the trucks can now be loaded with 52 pallets and that the full truckload tariff does not change (i.e.,  FC T 600€ per truck). We also take the optimistic assumption that the emissions due to empty running are similar to a conventional heavy duty truck, i.e., 324  FE T kg CO2. Finally, we assume that the variable emissions from transportation remain unchanged, i.e., 69 .

3

 VE T
kg CO2 per pallet. The results we obtain in this case are shown in Figure 8.

Figure 8 highlights that the increase in truck capacity does not affect the results. Indeed, the new solutions available are dominated by train transportation. This may be explained as follows by comparing the results obtained when 52 pallets are transported with an EMS truck with the results obtained when 36 pallets are transported by rail. EMS trucks perform better than rail transport both in terms of costs and emissions for the example we propose. Indeed, transportation costs (respectively emissions) equal 247€ per month (respectively 200 kg CO2 per month) with a full EMS truck as compared to 316€ per month (respectively 211 kg CO2 per month) for a full freight car. However, inventory holding costs equal to 1 950€ per month if the order quantity is 52 pallets, instead of 1 350€ per month if the order quantity is 36 pallets. This explains why extending the truck capacity is not efficient from a cost perspective for the example we consider. Similarly, inventory holding emissions raise from 48 kg CO2 per month to 69 kg CO2 per month when increasing the order quantity from 36 to 52 pallets. This increase is larger than the difference in transportation emissions. This example enables us to derive the following insights. First, we can notice that when focusing solely on transportation, increasing the truck capacity in Europe may be viewed as very effective. Indeed, if we increase truck capacity from 33 to 52 pallets in the example we consider, transportation costs decrease from 380€ per month to 247€ per month (35% decrease in cost) and transportation emissions decrease from 270 kg CO2 per month to 200 kg CO2 per month (26% decrease in emissions). These results also highlight that truck can compete with train in terms of carbon emissions if truck capacity is appropriately extended.

Second, we have highlighted that the increase in inventory while loading 52 pallets in an EMS truck was detrimental both in terms of costs and emissions for the example studied. We conclude that using EMS trucks can be competitive only for a limited set of products (with very low inventory holding cost and emissions) except if several loads can be combined into a single shipment.

Conclusions

This article applies multiobjective optimization to show how the efficient tradeoffs between different types of objectives may be obtained in the context of sustainable operations. The article focuses on joint inventory and transportation model with cost and carbon emissions objectives. We prove that switching to a greener mode of transportation while continuing to optimize the total logistics costs function may lead to a dominated solution. The proposed results give additional flexibility to supply chain managers who are likely to be focused on low-carbon transportation projects implementation without taking a total supply chain costs and carbon emissions perspective. We also prove that the modal shift is interesting only for strong carbon emissions reduction requirements. In the provided example base on industrial data, the modal shift is interesting for a 50% carbon emissions reduction target or for a carbon price greater than 1670€/ton CO2. On other situations, adjusting the order quantity while continuing to use truck transportation would be more efficient to green the supply chain. We also highlight the drawbacks of using carbon pricing to account for carbon emissions and we investigate the impact of an increase in vehicle capacity.

Several research directions can be considered. First, other models related to sustainable operations could be revisited by using multiobjective optimization. Multiobjective optimization could indeed be efficiently used as a first step of analysis in order to provide an effective decision making support tool to the decision maker by enabling a graphical representation of the existing tradeoffs. The decision maker may thus easily build a conviction of what is possible and use this knowledge to identify the most valuable trade-off. Moreover, this process of simultaneously optimizing several conflicting objectives adequately reflects the concept of sustainable operations as the different dimensions of sustainability are a good illustration of multiple objectives.

Second, other transportation mode selection and inventory optimization models could be studied. For instance, considering both stochastic demand and stochastic transportation lead times may enable enriching the analysis. In this case, safety stock optimization decisions has to be considered in addition to transportation mode selection and order quantity optimization decisions. Note that [START_REF] Konur | Economic and environmental comparison of grouping strategies in coordinated multi-item inventory systems[END_REF] paved the way for such type of study. More complex supply chain structures could also be considered by revisiting the multi-echelon lot sizing models. Considering a carbon-sensitive demand could also be of great interest as customers' pressure is nowadays considered as a main driver of environmental improvements for companies.

Third, other dimensions of sustainability could also be considered in order to take a broader sustainable operations perspective. For instance, very interesting analyses may be developed by taking the impact of working conditions into account such as the level of training into operations management models. Other people related dimensions of sustainable operations such as the risk of accident resulting in injuries or deaths may also be included into the models. This research direction may however be viewed as challenging as there is still a lack of consensus how to assess the social dimension of sustainable operations.

Appendix A

Proof of Lemma 1:

According to Expression (2), Q D Q T Q Z Q Z C C C ) ( ) ( ' ) (   with D P Q h Q D O Q Z C C C C    2 ) ( ' . C Z ' is convex in Q and its minimum is obtained for C C h D O Q 2 ' *  . Let D Q h O C C 2 2 max  , then max * ' Q Q 
and we obtain that for all

max Q Q  , ) ( ' ) ( ' max Q Z Q Z C C  . In addition, for all max Q Q  , max max ) ( ) ( Q D Q T Q D Q T C C  . We conclude that for all max Q Q  , ) ( ) ( max Q Z Q Z C C  thus max * Q Q 
. This concludes the proof.

Proof of Proposition 1:

Both ) (Q Z C and ) (Q Z E are convex on   max min ;Q Q . If * * E C Q Q  , * C Q E 
as *

C

Q is the optimal order quantity for both costs and carbon emissions. Assume that

* * E C Q Q  , then ) (Q Z C is strictly increasing on ] , [ * * E C Q Q , ) (Q Z E is strictly decreasing on ] , [ * * E C Q Q , and both ) (Q Z C and ) (Q Z E are strictly increasing on ] , [ max * Q Q E and strictly decreasing on ] , [ * min C Q Q . Any solution ] , [ * * E C Q Q Q
is then dominated. By using the same argumentation as for

* * C E Q Q  , it follows that: )] , max( ); , [min( * * * * E C E C Q Q Q Q E 
. This concludes the proof.

Proof of Proposition 2:

Let  

; ; E Q Q Q  such that:     ) ( ) ( ) ( ); ( ) ( ) ( , 3 3 2 2 1 1 3 3 2 1 1 Q Z Q Z Q Z Q Z Q Z Q Z e c E E E C C C            with 1 3 1    i i  . As ) (Q Z C and ) (Q Z E are convex on E , we obtain that ) ( 3 3 2 2 1 1 Q Q Q Z c C       and ) ( 3 3 2 2 1 1 Q Q Q Z e E       . Moreover, the equalities hold only if   ) ( , E Z e c  . We conclude that ) ( ))) ( ( ( E Z E Z Conv Eff  as E Q Q Q    3 3 2 2 1 1   
. This concludes the proof.

Proof of Proposition 3:

Let   e c, be an element of ) ( ) ( 2 2 1 1 E Z E Z  . Then there exists   2 1 2 1 ; E E Q Q   such that ) ( ) ( 2 2 1 1 Q Z Q Z c C C   and ) ( ) ( 2 2 1 1 Q Z Q Z e E E  
. These conditions are equivalent to:

     ) ( ) ( ) ( ) ( 2 2 1 1 2 2 1 1 Q Z Q Z Q Z Q Z E E C C                       1 2 1 1 2 2 2 1 1 2 1 2 1 2 2 2 1 ) ( 2 ) ( ) ( 2 VE VE FE FE E VC VC TC FC C FC C C T T Q T Q T Q Q D h T T L L h Q T O Q T O Q Q D h                             1 2 1 1 2 2 1 2 1 2 1 1 2 2 ) ( VE VE E E C VC VC TC FC C FC C E T T Q T Q T h T T L L h Q T O Q T O h C Q B Q A    1 2 , with: E C FE C FC C E h h T h T O h A 2 2 ) (    , E C FC C E FE C h h T O h T h B ) ( 1 1    , E VE VE C VC VC TC h T T h T T L L h C 1 2 2 1 2 1 ) (       . As E FE C FC C h T h T O 1 1   and E FE C FC C h T h T O 2 2   , we obtain that 0  A and 0  B . If 0  C , 2 1 Q A B Q  
. By substituting, we obtain that:

1 2 2 1 2 2 2 2 ) ( 2 VE VE FE FE E T T Q T B A Q T Q Q A B D h       1 2 2 1 2 2 2 2 ) ( ) 1 ( 2 ) ( FE FE VE VE E T B A T Q T T Q A B D h Q P        . Then ) ( 2 Q P
is a second degree polynomial that has at most two roots, thus 2

) ( ) ( 2 2 1 1   E Z E Z . If 0  C , then:     0 2 2 1 A CQ BQ Q C A Q  2 ,     0 1 1 2 B CQ AQ Q C B Q  1 . If 0  C , then 0  C B and 0 ) ( ) ( 2 2 1 1   Z E Z . Assume now that 0  C . By taking A CQ BQ Q   2 2 1
, we obtain that:

1 2 2 2 1 2 2 2 2 2 ) ( ) ( 2 VE VE FE FE E T T BQ A CQ T Q T Q A CQ BQ D h                 2 2 2 3 2 2 1 . . . ) ( Q Q Q Q P , with: 0 2    D C h E  , C T T B C T B A D h VE VE FE E ) ( ) ( 2 2 1 2 1       , C T B AC T T T A FE FE VE VE 2 1 1 2 2 ) (      , B T A AT FE FE 1 2 2    . If 0   then 0 ) 0 ( 1  P and ) ( 2 1 Q P has at least non-positive root as ) ( 2 1 Q P tends to  as 2 Q tends to   ( 0   ). Therefore, ) ( 2 1 Q P has at most two strictly positive roots, thus 2 ) ( ) ( 2 2 1 1   E Z E Z .
Assume now that 0   :

We start proving that 0

0      .   0           0 1 2 B AT T A FE FE 0 1 2   B AT T FE FE . If 0 2 1   VE VE T T then we obtain that 0   . Else, 0 ) ( 2 1 2 1     C VC VC TC h T T L L h as 0  C
. In addition, we can see that

E FE h T B 1  .   E FE h T B 1       C h T T B C T T T E VE VE FE VE VE 1 2 1 1 2 0 ) ( 2 1 2 1 1 1 2         C VC VC TC E FE VE VE h T T L L h h B C T T T and we obtain that 0   .         2 2 2 2 2 1 2 3 ) ( Q Q Q Q P . Let  be the discriminant of 2 2 1 ) ( Q Q P   , then   12 4 2    . If 0   , then 2 2 1 ) ( Q Q P  
has at most one root. In this case, the sign of

) ( ) ( 2 2 1 1   E Z E Z . If 0   , the two roots of 2 2 1 ) ( Q Q P   are   6 2 1     q and   6 2 2     q .   0   2   thus 0 6 2 2       q
. As the sign of

) ( ) ( 2 2 1 1   E Z E Z
. This concludes the proof.

Proof of Lemma 2:

Recall that

* 2 * 1 C C Z Z  . If * 2 * 1 E E Z Z 
, then the number of intersections between . This proves that

 n k k k E Z E Z 1 ) ( ) (   
. This concludes the proof.

Proof of Proposition 8:

Assume that a price  is associated to the company's carbon emissions and assume that the company can decide between modes 1 and 2 with

Q E Q Z Q E Q Z      If C L   , then )) ( ) ( min(( )) ( ) ( min( 2 2 1 1 Q E Q Z Q E Q Z     
. This concludes the proof.

  transportation cost per product unit, FC T = fixed transportation costs per vehicle, TC h = in-transit inventory holding costs per product unit and time unit, L = transportation lead time.

  : D = demand per time unit, C h = inventory holding costs per product unit and time unit,

Figure

  Figure 1: Transportation cost

  identifying the set of efficient solutions in the situations for which a single mode is available. This one can be expressed as a function of * set of efficient solutions when considering a single mode of transportation, then:

  Figure 2 is an illustration of the possible outcomes when two modes are considered. Figures 2a to 2d

Figure 2 :

 2 Figure 2: The seven situations when considering two modes of transportation

  Figure 2b

Figure

  Figure 2g

  frontier for the problem with two modes of transportation with

  may be an empty set. The efficient frontier of the global problem may be identified by

Figure 3 :

 3 Figure 3: The efficient frontier with a single mode of transportation

  pallet. Due to both costs and carbon emissions structure,

Figure 4 :

 4 Figure 4: Truck and train transportation efficient frontiers

Figure 5 :

 5 Figure 5: Efficient frontiers for the different modes

Figure 8 :

 8 Figure 8: Impact of vehicle capacity

  two strictly positive roots and we can conclude that 2

  The truck capacity is 

								max Q	33 euro pallets. Assume
	that the logistics provider offers three different tariffs. The first one is a less than truckload tariff with
	T		30€ per pallet and	T			0€ per truck. A second less than truckload tariff is available if the shipment
	VC				FC
	size is at least 21 pallets. In this case,	T		20€ per pallet and	T		0€ per truck. Finally, a full truckload
								VC	FC
	tariff is available with	T		600€ per truck and	T		0€ per pallet. Assume that the retailer decides to
					FC			VC

Table 1 :

 1 Application's dataAssume first that the French retailer decides to use heavy duty trucks for inbound transportation. The At first, we assume that the truck transportation cost is linear in the order quantity with  0€ per truck (i.e., following a less than truckload tariff) and we consider that the minimum capacity for getting this tariff consists in ordering The Network for Transport and Environment methodology[START_REF] Ntm | Environmental data for international cargo transport[END_REF] is used to evaluate the carbon emissions related to transportation. This methodology developed by a Swedish not-for-profit organization provides estimates for emissions generated by different modes of transportation in Europe. The NTM methodology may be applied at the aggregate level for less than truckload transport by computing a per product amount of carbon emissions by considering a given average load factor. However, Velázquez-Martínez et al. (2013) have shown that such aggregate approach may lead to substantial errors as the effects of lot sizing decisions on the load factor are disregarded. The NTM methodology also enables a more detailed estimation of carbon emissions by considering the fixed emissions generated by the truck when running empty and the per pallet emissions due to the extra energy consumption. The detailed model provided by the NTM methodology is used in our application as the load factor depends on the order

		time unit	1	month
		quantity unit	1	pallet
		unit weight	500	kg
		inbound distance	500	km
		D	20	pallets/month
		hC	75	euro/pallet.month
		hTC	50	euro/pallet.month
		OC	100	euro/order
		hE	2.65	kg CO2/pallet.month
	truck capacity is	

max

Q

33 euro pallets and the transportation leadtime is  L 0.017 month (i.e., 0.5 day).

FC T quantity decision. Indeed, we assume that the retailer does not allow the logistics provider for including other type of cargo for inbound transportation and consequently has a direct control on the load factor.

This situation is classical for retail distribution and this explains why the minimum shipment size is 10 pallets. According to the NTM methodology, the fixed carbon emissions are equal to 324  min

Table 3 :

 3 Parameters for train transportationFrom Table3 and Table 2, we can notice that *

	Train transportation	1		36	449			333				17.1	71.0	17.1	36.0	1350.29 258.86
																				Z		Z	and	Z	*		Z	*	thus Proposition 4 can be
																				Ctruck	Ctrain	Etruck	Etrain
	applied. By solving the system of equations leading to an intersection between	Z	( truck E truck	)	and
	Z	( train E train	)	, we obtain that	Z	truck	(	truck E	)		Z	train	(	train E	)		Ø. Moreover, we can notice that
	Z	( * Etruck truck Q	)		 1914 . 77	;	 92 . 313	is dominated by	Z	( * Etrain train Q	)		 1721 . 79	;	 86 . 258	. Then, by
	applying Proposition 4, we obtain that	(		)	 (	1	;	1	)	truck	(	truck	)	1	* Ctrain		train	(	train	)

Table 4 :

 4 Synthesis of the different modesThe results of Section 3.2.3 are applied to identify the set of efficient solution for the global problem.

	mode 1	10	14	0	324	7.3	70.0			10.0	14.0	1191.67 555.25
	mode 2	14	21	420	324	16.7	70.0			16.7	21.0	1265.67 410.24
	mode 3	21	30	0	324	7.3	70.0			21.0	30.0	1299.40 329.58
	mode 4	1	36	449	333	17.1	71.0			17.1	36.0	1350.29 258.86
	mode 5	30	33	600	324	19.3	70.0			30.0	33.0	1608.33 313.92
	With 5 different modes, 24 pairwise comparisons are theoretically required, but we can notice that the
	analysis is totally disjointed for mode 1 and mode 2 and that	Z	( 5 5 E	)				by comparing to mode 4. The
	results of Section 3.2.3 boil down to apply the results of Section 3.2.2 to mode 1 and mode 2 and to mode
	3 and mode 4 separately. The final results are presented in Figure 6.

Acknowledgements

The authors would like to thank the editor as well as two anonymous referees for their valuable comments and suggestions.


Susumu Ariki
email: ariki@kurims.kyoto-u.ac.jp

Nicolas Jacon
email: njacon@univ-fcomte.fr

DIPPER-JAMES-MURPHY'S CONJECTURE FOR HECKE ALGEBRAS OF TYPE B n

Keywords:

We prove a conjecture by Dipper, James and Murphy that a bipartition is restricted if and only if it is Kleshchev. Hence the restricted bipartitions naturally label the crystal graphs of level two irreducible integrable Uv( sle)-modules and the simple modules of Hecke algebras of type Bn.

Dedicated to Toshiaki Shoji and Ken-ichi Shinoda on their 60 th birthdays

Introduction

Let F be a field, q and Q invertible elements of F . The Hecke algebra of type B n is the F -algebra defined by generators T 0 , . . . , T n-1 and relations (T 0 -Q)(T 0 + 1) = 0, (T iq)(T i + 1) = 0 (1 i < n)

(T 0 T 1 ) 2 = (T 1 T 0 ) 2 , T i T i+1 T i = T i+1 T i T i+1 (1 i < n -1)
T i T j = T j T i (j ≥ i + 2).
We denote it by H n (Q, q), or H n for short. The representation theory of H n in the semisimple case was studied by Hoefsmit, which had applications in determining generic degrees and Lusztig's a-values. Motivated by the modular representation theory of U n (q) and Sp 2n (q) in the non-defining characteristic case, Dipper, James and Murphy began the study of the modular case more than a decade ago. The first task was to obtain classification of simple modules. For this, they constructed Specht modules which are indexed by the set of bipartitions [START_REF] Dipper | Hecke algebras of type Bn at roots of unity[END_REF]. The work shows in particular that Hecke algebras of type B n are cellular algebras in the sense of Graham and Lehrer. 1 Then they conjectured that the simple modules were labeled by (Q, e)-restricted bipartitions. Their philosophy to classify the simple H n -modules resembles the highest weight theory in Lie theory: let g be a semisimple Lie algebra. It has a commutative Lie subalgebra h, the Cartan subalgebra. One dimensional h-modules are called weights (by abuse of notion). When a g-module admits a simultaneous generalized eigenspace decomposition with respect to h, the decomposition is called the (generalized) weight space decomposition. Let Λ be a weight. Suppose that a g-module M has the property that (i) Λ appears in the weight space decomposition of M , (ii) If N is a proper g-submodule of M then Λ does not appear in the weight space decomposition of N .

Then the standard argument shows that M has a unique nonzero irreducible quotient. In fact, Verma modules enjoy the property and their irreducible quotients give a complete set of simple objects in the BGG category. Now we turn to the Hecke algebra H n . Define the Jucys-Murphy elements t 1 , . . . , t n by t 1 = T 0 and t i+1 = q -1 T i t i T i , for 1 i n -1. They generate a commutative subalgebra A n of the Hecke algebra H n , and A n plays the role of the Cartan subalgebra: one dimensional A n -modules are called weights and the generalized simultaneous eigenspace decomposition of an H n -module is called the weight space decomposition. Any weight is uniquely determined by the values at t 1 , . . . , t n of the weight, and the sequence of these values in this order is called the residue sequence. Let λ = (λ (1) , λ (2) ) be a bipartition (see §2.1) and let t be a standard bitableau of shape λ (see Def. 2.9). Then, t defines a weight whose values at t i are given by c i q b i -a i where a i and b i are the row number and the column number of the node of t labelled by i respectively, c i = -Q if the node is in λ (1) and c i = 1 if the node is in λ (2) . By the theory of seminormal representations in the semisimple case and the modular reduction, a weight appears in some H n -module if and only if it is obtained from a bitableau this way.

Suppose that there is a weight obtained from a bitableau t of shape λ such that it does not appear in S µ when µ ⊳ λ. If such a bitableau exists, we say that λ is (Q, e)-restricted. This is a clever generalization of the notion of e-restrictedness. Recall that a partition λ = (λ 0 , λ 1 , . . . ) is called e-restricted if λ i+1λ i < e, for all i ≥ 0. Recall also that we have the similar Specht module theory for Hecke algebras of type A. Using Jucys-Murphy elements of the Hecke algebra of type A, we can define weights as well. Then, a partition is e-restricted if and only if there is a weight obtained from a tableau of shape λ such that it does not appear in S µ when µ ⊳ λ.

Recall from [START_REF] Dipper | Hecke algebras of type Bn at roots of unity[END_REF] that

[S λ ] = [D λ ] + µ⊳λ d λµ [D µ ],
where the summation is over µ such that D µ = 0, d λµ are decomposition numbers, and µ⊳λ d λµ [D µ ] is represented by the radical of the bilinear form on S λ . As D µ is a surjective image of S µ , it implies that the weight does not appear in the radical, while it appears in S λ . Therefore, D λ = 0 if λ is (Q, e)-restricted. Unlike the case of the BGG category, we may have D λ = 0 and it is important to know when it occurs. When -Q is not a power of q, a bipartition λ = (λ (1) , λ (2) ) is (Q, e)-restricted if and only if both λ (1) and λ (2) are e-restricted. Thus we know when a bipartition is (Q, e)-restricted. Further, [START_REF] Dipper | Representations of Hecke algebras of type Bn[END_REF]Thm 4.18] implies that D λ = 0 if and only if λ is (Q, e)-restricted, that is, simple H n -modules are labelled by (Q, e)restricted bipartitions. Now we suppose that -Q is a power of q. More precisely, we suppose that (a) q is a primitive e th root of unity with e 2, (b) -Q = q m , for some 0 m < e.

in the rest of the paper. We call (Q, e)-restricted bipartitions restricted bipartitions. They conjectured in this case that D λ = 0 if and only if λ is restricted, and it has been known as the Dipper-James-Murphy conjecture for Hecke algebras of type B n . Later, connection with the theory of canonical bases in deformed Fock spaces in the sense of Hayashi and Misra-Miwa was discovered by Lascoux-Leclerc-Thibon [START_REF] Lasoux | Hecke algebras at roots of unity and cystal bases of quantum affine algebras[END_REF] and its proof in the framework of cyclotomic Hecke algebras [START_REF] Ariki | On the decomposition numbers of the Hecke algebra of G(m, 1, n)[END_REF] allowed the first author and Mathas [3] [5] to label simple H nmodules by the n th layer of the crystal graph of the level two irreducible integrable g(A

e-1 )-module L v (Λ 0 + Λ m ). In the theory, the crystal graph is realized as a subcrystal of the crystal of bipartitions, and the nodes of the crystal graph are called Kleshchev bipartitions. More precise definition is given in the next section and λ = (λ (1) , λ (2) ) is Kleshchev if and only if λ (2) ⊗λ (1) belongs to the subcrystal B(Λ 0 +Λ m ) of B(Λ 0 )⊗B(Λ m ), where the crystals B(Λ 0 ) and B(Λ m ) are realized on the set of e-restricted partitions. Now, D λ = 0 if and only if λ is Kleshchev by [START_REF] Ariki | On the classification of simple modules for cyclotomic Hecke algebras of type G(m, 1, n) and Kleshchev multipartitions[END_REF]. Hence, we obtained the classification of simple H n -modules, or more precisely description of the set {λ | D λ = 0}, through a different approach and the Dipper-James-Murphy conjecture in the modern language is the statement that the Kleshchev bipartitions are precisely the restricted bipartitions.

The aim of this paper is to prove the Dipper-James-Murphy conjecture. Recall that Lascoux, Leclerc and Thibon considered Hecke algebras of type A and they showed that if λ is a e-restricted partition then we can find a 1 , . . . , a p and i 1 , . . . , i p such that we may write

f (a 1 ) i 1 ...f (ap) ip ∅ = λ + ν⊲λ c ν,λ (v)ν
in the deformed Fock space, where c ν,λ (v) are Laurent polynomials. This follows from the ladder decomposition of a partition. Then LLT algorithm proves that Kleshchev partitions are precisely e-restricted partitions. The second author [START_REF] Jacon | On the parametrization of the simple modules for Ariki-Koike algebras at roots of unity[END_REF] proved the similar formula for FLOTW multipartitions in the Jimbo-Misra-Miwa-Okado higher level Fock space using certain a-values instead of the dominance order. Recall that Geck and Rouquier gave another method to label simple H n -modules by bipartitions. The result shows that the parametrizing set of simple H n -modules in the Geck-Rouquier theory, which is called the canonical basic set, is precisely the set of the FLOTW bipartitions. Our strategy to prove the conjecture is to give the analogous formula for Kleshchev bipartitions. To establish the formula, a non-recursive characterization of Kleshchev bipartitions given by the first author, Kreiman and Tsuchioka [START_REF] Ariki | On the tensor product of two basic representations of Uv( sle)[END_REF] plays a key role.

The paper is organized as follows. In the first section, we briefly recall the definition of Kleshchev bipartitions. We also recall the main result of [START_REF] Ariki | On the tensor product of two basic representations of Uv( sle)[END_REF]. In the second section, we use this result to give an analogue for bipartitions of the ladder decomposition. Finally, the last section gives a proof for the conjecture.

Acknowledgments : This paper was written when the second author visited RIMS in Kyoto. He would like to thank RIMS for the hospitality.

Preliminaries

In this section, we recall the definition of Kleshchev bipartitions together with the main result of [START_REF] Ariki | On the tensor product of two basic representations of Uv( sle)[END_REF] which gives a non-recursive characterization of these bipartitions. We fix m as in the inroduction. Namely, the parameter Q of the Hecke algebra is Q = -q m with 0 m < e.

First definitions. Recall that a partition λ is a sequence of weakly decreasing nonnegative integers (λ

0 , λ 1 , • • • ) such that |λ| = i≥0 λ i is finite. If λ i = 0 for i ≥ r then we write λ = (λ 0 , • • • , λ r-1 )
. A bipartition λ is an ordered pair of partitions (λ (1) , λ (2) ). |λ| = |λ (1) | + |λ (2) | is called the rank of λ. The empty bipartition (∅, ∅) is the only bipartition of rank zero. The diagram of λ is the set

{(a, b, c) | 1 c 2, 0 b λ (c) a -1} ⊆ Z 3 ≥0 .
We often identify a bipartition with its diagram. The nodes of λ are the elements of the diagram. Let γ = (a, b, c) be a node of λ. Then the residue of γ is defined by

res(γ) = b -a + m (mod e) if c = 1, b -a (mod e) if c = 2.
By assigning residues to the nodes of a bipartition, we view a bipartition as a colored diagram with colors in Z/eZ.

  2 3 0 1 2 , 0 1 2 3 3 0 2  
If γ is a node with residue i, we say that γ is an i-node. Let λ and µ be two bipartitions such that µ = λ ⊔ {γ}. Then, we denote µ/λ = γ and if res(γ) = i, we say that γ is a removable i-node of µ. We also say that γ is an addable i-node of λ by abuse of notion. 2 Let i ∈ Z/eZ. We choose a total order on the set of removable and addable i-nodes of a bipartition. Let γ = (a, b, c) and γ ′ = (a ′ , b ′ , c ′ ) be removable 2 An addable i-node of λ is not a node of λ.

or addable i-nodes of a bipartition. We say that γ is above γ ′ if either c = 1 and c ′ = 2, or c = c ′ and a < a ′ . 3Let F be the vector space over Q such that the basis is given by the set of all bipartitions. We color the nodes of bipartitions as above. We call it the (level two) Fock space. We may equip it with ŝl e -module structure in which the action of the Chevalley generators is given by

e i λ = ν:res(λ/ν)=i ν, f i λ = ν:res(ν/λ)=i ν.
Using the total order on the set of removable and addable i-nodes given above, we deform the ŝl e -module structure to U v ( ŝl e )-module structure on the deformed Fock space F ⊗ Q Q(v), which is the tensor product of two level one deformed Fock spaces. We refer to [START_REF] Ariki | Representations of quantum algebras and combinatorics of Young tableaux[END_REF] for the details.

2.2. Kleshchev bipartitions. Recall that the crystal basis of the deformed Fock space is given by the basis vectors of the deformed Fock space. Hence it defines a crystal structure on the set of bipartitions. We call it the crystal of bipartitions. As is explained in [START_REF] Ariki | Representations of quantum algebras and combinatorics of Young tableaux[END_REF], the map (λ (1) , λ (2) ) → λ (2) ⊗ λ (1) identifies the crystal of bipartitions with the tensor product of the crystal of partitions of highest weight Λ 0 and that of highest weight Λ m . As is already mentioned in the introduction, Kleshchev bipartitions are those bipartitions which belongs to the same connected component as the empty bipartition in the crystal of bipartitions. Equivalently, Kleshchev bipartitions are those bipartitions which may be obtained from the empty bipartition by applying the Kashiwara operators successively. Rephrasing it in combinatorial terms, we have a recursive definition of Kleshchev bipartitions as follows.

Let λ be a bipartition and let γ be an i-node of λ, we say that γ is a normal i-node of λ if, whenever η is an addable i-node of λ below γ, there are more removable i-nodes between η and γ than addable i-nodes between η and γ. If γ is the highest normal i-node of λ, we say that γ is a good i-node of λ. When γ is a good i-node, we denote λ \ {γ} by ẽi λ. Definition 2.2. A bipartition λ is Kleshchev if either λ = (∅, ∅) or there exists i ∈ Z/eZ and a good i-node γ of λ such that λ \ {γ} is Kleshchev.

Note that the definition depends on m. The reader can prove easily using induction on n = |λ (1) | + |λ (2) | that if λ = (λ (1) , λ (2) ) is Kleshchev then both λ (1) and λ (2) are e-restricted. By general property of crystal bases, the following is clear. Lemma 2.3. Suppose that λ is a Kleshchev bipartition, γ a good i-node of λ, for some i. Then ẽi λ = λ \ {γ} is Kleshchev.

In [START_REF] Ariki | On the tensor product of two basic representations of Uv( sle)[END_REF], the first author, Kreiman and Tsuchioka have given a different characterization of Kleshchev bipartitions.

Let λ be a partition. Then the set of beta numbers of charge h, where we only use h = 0 or h = m in the paper, is by definition the set J h of decreasing integers

j 0 > j 1 > • • • > j k > • • • defined by j k = λ k + h -k, for k 0.
The charge h also defines a coloring of nodes: res(γ) = ba + h (mod e) where a and b are the row number and the column number of a node γ, respectively.

An addable i-node of λ corresponds to x ∈ J h such that x + eZ = i and x + 1 ∈ J h . We call x an addable i-node of J h . Similarly, a removable i-node of λ corresponds to x ∈ J h such that x + eZ = i + 1 and x -1 ∈ J h . We call x a removable i-node of J h .

We define the abacus display of J h in the usual way. The i th runner of the abacus is {x ∈ Z | x + eZ = i}, for i ∈ Z/eZ. Definition 2.4. Let λ be an e-restricted partition, J m (λ) the corresponding set of beta numbers of charge m. We write J m for J m (λ) and define

U (J m ) = {x ∈ J m | x -e / ∈ J m }.
If λ is an e-core then we define up m (λ) = λ. Otherwise let p = max U (J m ) and define

V (J m ) = {x > p | x = p (mod e), x -e ∈ J m , x / ∈ J m }.
Note that V (J m ) is nonempty since λ is e-restricted. Let q = min V (J m ).

Then we define up(J m ) = (J m \ {p}) ⊔ {q} and we denote the corresponding partition by up m (λ).

In [START_REF] Ariki | On the tensor product of two basic representations of Uv( sle)[END_REF], it is shown that up m (λ) is again e-restricted and we reach an e-core after applying up m finitely many times. Definition 2.5. Let λ be an e-restricted partition. Apply up m repeatedly until we reach an e-core. We denote the resulting e-core by roof m (λ). Definition 2.6. Let λ be an e-restricted partition, J 0 (λ) the corresponding set of beta numbers of charge 0. We write J 0 for J 0 (λ) and define

U (J 0 ) = {x ∈ J 0 | x -e / ∈ J 0 }.
If λ is an e-core then we define down 0 (λ) = λ. Otherwise let p ′ = min U (J 0 ) and define

W (J 0 ) = {x > p ′ -e | x ∈ J 0 , x + e / ∈ J 0 } ∪ {p ′ }.
It is clear that W (J 0 ) is nonempty. Let q ′ = min W (J 0 ). Then we define down(J 0 ) = (J 0 \ {q ′ }) ⊔ {p ′ -e} and we denote the corresponding partition by down 0 (λ).

In [START_REF] Ariki | On the tensor product of two basic representations of Uv( sle)[END_REF], it is shown that down 0 (λ) is again e-restricted and we reach an e-core after applying down 0 finitely many times. Definition 2.7. Let λ be an e-restricted partition. Apply down 0 repeatedly until we reach an e-core. We denote the resulting e-core by base 0 (λ).

Finally, let λ be an e-restricted partition, J max 0 the set of beta numbers of charge 0 for base 0 (λ). Define M i (λ), for i ∈ Z/eZ, by

M i (λ) = max{x ∈ J max 0 | x + eZ = i}.
We write M i (λ) in decreasing order

M i 1 (λ) > M i 2 (λ) > • • • > M ie (λ).
Then J max 0 ∪ {M i k (λ) + e} 1 k m is the set of beta numbers of charge m, for some partition. We denote the partition by τ m (base 0 (λ)). Now, the characterization of Kleshchev bipartitions is as follows.

Theorem 2.8 ([4]

). Let λ = (λ (1) , λ (2) ) be a bipartition such that both λ (1) and λ (2) are e-restricted. Then λ is Kleshchev if and only if roof m (λ (1) ) ⊆ τ m (base 0 (λ (2) )).

2.3. The Dipper-James-Murphy conjecture. We recall the dominance order for bipartition. Let λ = (λ (1) , λ (2) ) and µ = (µ (1) , µ (2) ) be bipartitions.

In this paper, we write µ λ if

j k=1 λ (1) k j k=1 µ (1)
k and |λ (1)

| + j k=1 λ (2) k |µ (1) | + j k=1 µ (2)
k , for all j 0. Definition 2.9. Let λ be a bipartition of rank n. A standard bitableau of shape λ is a sequence of bipartitions

∅ = λ[0] ⊆ λ[1] ⊆ • • • ⊆ λ[n] = λ
such that the rank of λ[k] is k, for 0 k n. Let t be a standard bitableau of shape λ. Then the residue sequence of t is the sequence

(res(γ[1]), . . . , res(γ[n])) ∈ (Z/eZ) n where γ[k] = λ[k]/λ[k -1], for 1 k n.
A standard bitableau may be viewed as filling of the nodes of λ with numbers 1, . . . , n: we write k in the node γ[k], for 1 k n. Definition 2.10. A bipartition λ is (-q m , e)-restricted, or restricted for short, if there exists a standard bitableau t of shape λ such that the residue sequence of any standard bitableau of shape ν ⊳ λ does not coincide with the residue sequence of t. Conjecture 2.11 ([7, Conj. 8.13]). A bipartition λ is Kleshchev if and only if it is restricted.

Properties of Kleshchev bipartitions

The aim of this section is to prove some combinatorial results concerning Kleshchev bipartitions.

3.1. Admissible sequence. Definition 3.1. Let i ∈ Z/eZ. We say that a sequence of removable inodes R 1 , ...., R s (where s ≥ 1) of a bipartition λ is an admissible sequence of i-nodes for λ if

• R 1 , ...., R s are the lowest s removable i-nodes of λ and every addable i-nodes is above all of these nodes, and • if there is a removable i-node R above R 1 , ...., R s , there must exist an addable i-node below R.

The following lemma shows the existence of an admissible sequence of inodes, for some i, for a Kleshchev bipartition: choose i as in the lemma and read addable and removable i-nodes in the total order of nodes. Suppose that λ has at least one addable i-node and let η be the lowest addable i-node. Then removable i-nodes below η form an admissible sequence of i-nodes. If λ does not have an addable i-node, all removable i-nodes of λ form an admissible sequence of i-nodes. Lemma 3.2. Let λ = (λ (1) , λ (2) ) be a nonempty Kleshchev bipartition. Then there exists i ∈ Z/eZ and a removable i-node γ such that if η is an addable i-node of λ then η is above γ.

Proof. Recall that both λ (1) and λ (2) are e-restricted. There are two cases to consider.

• Assume that λ (2) is not the empty partition. Let λ

(2) j be the last row. Define γ = (j, λ

(2) j -1, 2) and i = res(γ). Since λ (2) is e-restricted, the residue of the addable node (j + 1, 0, 2) is not i. Hence all the addable i-node of λ are above γ.

• Assume that λ (2) is the empty partition. Let λ (1) j be the last row.

Define γ = (j, λ

(1) j -1, 1) and i = res(γ). Since λ (1) is e-restricted, the residue of the addable node (j + 1, 0, 1) is not i. We show that the residue of the addable node (0, 0, 2) is not i. Suppose to the contrary that the residue is i. As λ is Kleshchev, we may delete good nodes successively to obtain the empty bipartition. Hence γ must be deleted at some point in the process. However, it can never be a good node since we always have an addable i-node (0, 0, 2) just below it and there is no removable i-node between them, so we have a contradiction. Hence the claim follows.

Lemma 3.3. Let λ = (λ (1) , λ (2) ) be a nonempty Kleshchev bipartition and R 1 , ...., R s an admissible sequence of i-nodes for λ. Define µ = (µ (1) , µ (2) ) by λ = µ ⊔ {R 1 , ...., R s }. Then µ is Kleshchev.

at least one addable i-node. Note that it forces λ (1) = µ (1) and R 1 , ...., R s are nodes of λ (2) . Define t ′ ≥ 0 by 2) )}. We prove that µ is Kleshchev by induction on t ′ . Note that if λ is Kleshchev then so is (λ (1) , down 0 (λ (2) )) since roof m (λ (1) ) ⊆ τ m (base 0 (λ (2) )) = τ m (base 0 (down 0 (λ (2) ))).

t ′ = min{k | base 0 (λ (2) ) = down k 0 (λ (
If t ′ = 0 then λ (2) is an e-core and it has removable i-nodes R 1 , ...., R s . Hence, λ (2) has no addable i-node and we are reduced to the previous case. Thus we suppose that t ′ > 0 and that the lemma holds for (λ (1) , down(λ (2) )). Let J = J 0 (λ (2) ) and

r = min{x ∈ J | x + eZ = i + 1, x -1 / ∈ J} -1.
Note that r is on the i th runner. Then there exists N ≥ 1 such that r, r + e, . . . , r + (N -1)e ∈ J and r + N e ∈ J.

Let p ′ = min U (J). Then p ′ ≤ r + N e. Suppose that p ′ is not on the i th runner or the (i + 1) th runner. If a node which is not on one of these two runners moves to p ′e by the down operation, the admissible sequence R 1 , ...., R s is an admissible sequence of i-nodes for (λ (1) , down 0 (λ (2) )) and (λ (1) , down 0 (λ (2) )) = (µ (1) , down 0 (µ (2) )) ⊔ {R 1 , ...., R s }.

Thus, by the induction hypothesis, (µ (1) , down 0 (µ (2) )) is Kleshchev. Hence, roof m (µ (1) ) ⊆ τ m (base 0 (down 0 (µ (2) ))) = τ m (base 0 (µ (2) )) implies that µ is Kleshchev.

If a node in one of the two runners moves to p ′e, then there exists 0 ≤ k ≤ N -1 such that r + ke + 1 ∈ J, r + (k + 1)e + 1 ∈ J and r + ke + 1 moves to p ′e. Suppose that k < N -1. Then, r + ke ∈ J 0 (µ (2) ) and r + (k + 1)e ∈ J 0 (µ (2) ). Hence, r + ke moves to p ′e to obtain down 0 (µ (2) ). As r + ke + 1 corresponds to one of R 1 , ...., R s , say R k , R 1 , . . . , Rk , . . . , R s is an admissible sequence of i-nodes for (λ (1) , down 0 (λ (2) )) and (λ (1) , down 0 (λ (2) )) = (µ (1) , down 0 (µ (2) )) ⊔ {R 1 , . . . , Rk , . . . , R s }.

Hence, (µ (1) , down 0 (µ (2) )) is Kleshchev by the induction hypothesis, and µ is Kleshchev as before. Next suppose that k = N -1. As r + (N -1)e + 1 moves to p ′e, we have r + (N -1)e + 1 < p ′ < r + N e, r + N e + 1 / ∈ J 0 and r + N e is an addable i-node. Let K be the set of beta numbers of charge 0 of µ (2) . For x ∈ Z, we denote J ≤x := J ∩ Z x and K ≤x := K ∩ Z x . We claim that base(J ≤r+N e ) = base(K ≤r+N e ).

Let p ′ = y 0 < y 1 < • • • < y l < r + N e be the nodes in J which are greater than or equal to p ′ and smaller than r + N e. We show that base(J ≤y j ) = s i base(K ≤y j ) ⊇ base(K ≤y j ), for 0 ≤ j ≤ l, where s i means swap of the i th and (i + 1) th runners. J ≤p ′ and K ≤p ′ are s i -cores in the sense of [START_REF] Ariki | On the tensor product of two basic representations of Uv( sle)[END_REF], and direct computation shows the formula for j = 0. Now we use base(J ≤y j+1 ) = base({y j+1 } ∪ base(J ≤y j )) and base(K ≤y j+1 ) = base({y j+1 } ∪ base(K ≤y j )) 4 and continue the similar computation and comparison of base(J ≤y j ) and base(K ≤y j ). At the end of the inductive step, we get base(J <r+N e ) = s i base(K <r+N e ) ⊇ base(K <r+N e ). Now, one more direct computation shows base({r + N e} ∪ base(J <r+N e )) = base({r + N e} ∪ base(K <r+N e )), and we have the claim. Therefore, base 0 (λ (2) ) = base 0 (µ (2) ) and we have roof m (λ (1) ) ⊆ τ m (base 0 (λ (2) )) = τ m (base 0 (µ (2) )).

Recalling that λ (1) = µ (1) , we have that µ is Kleshchev.

Suppose that p ′ is on one of the two runners. As p ′ ≤ r + N e, we have either p ′ = r + N e or p ′ = r + ke + 1, for some 0 ≤ k ≤ N -1. If the latter occurs, down 0 (λ (2) ) is obtained by moving a node outside the two runners to p ′ -e or moving p ′ to p ′ -e, and down 0 (µ (2) ) is obtained from µ (2) by moving the same node outside the two runners to p ′ -1-e or moving p ′ -1 to p ′ -1-e, respectively. Thus, down 0 (µ (2) ) is obtained from down 0 (λ (2) ) by removing the nodes of an admissible sequence of i-nodes for (λ (1) , down 0 (λ (2) )). Hence (µ (1) , down 0 (µ (2) )) is Kleshchev by the induction hypothesis, and it follows that µ is Kleshchev. If p ′ = r + N e and r + N e + 1 ∈ J 0 then the same is true, and if p ′ = r + N e and r + N e + 1 ∈ J 0 then µ (2) = down N 0 (λ (2) ) and we have roof m (λ (1) ) ⊆ τ m (base 0 (λ (2) )) = τ m (base 0 (µ (2) )). Hence, µ = (λ (1) , µ (2) ) is Kleshchev.

3.2.

Optimal sequence of a Kleshchev bipartition. Let λ = (λ (1) , λ (2) ) be a Kleshchev bipartition. By the previous lemma, we may define by induction a sequence of Kleshchev bipartitions

λ =: λ[1], λ[2], ..., λ[p], λ[p + 1] = ∅
and a sequence of residues i 1 , ..., i 1 a 1 times , ..., i p , ..., i p ap times such that, for 1 ≤ j ≤ p, λ[j] = λ[j + 1] ⊔ {R j 1 , ...., R j a j } and R j 1 , ...., R j a j is an admissible sequence of i j -nodes for λ[j].

We call i 1 , ..., i 1 a 1 times , ..., i p , ..., i p ap times an optimal sequence of λ.

Example 3.4. Keeping example 2.1, it is easy to see that ((3, 2), (4, 2, 1)) is a Kleshchev bipartition and an optimal sequence is given by 2, 2, 0, 3, 3, 2, 1, 1, 0, 0, 3, 2.

4. Proof of the conjecture 4.1. The result. We are now ready to prove the conjecture. As is explained in the introduction, it is enough to prove that Kleshchev bipartitions are restricted bipartitions. To do this, we introduce certain reverse lexicographic order on the set of bipartitions. Definition 4.1. We write λ ≺ ν if either

• there exists j ≥ 0 such that λ

k = ν (2)
k , for k > j, and λ

(2)

j < ν (2)
j , or • there exists j ≥ 0 such that λ (2) = ν (2) , λ

k = ν (1) (1)
k , for k > j, and λ

(1)

j < ν (1) j . It is clear that if ν ⊳ λ then λ ≺ ν.
Recall that the deformed Fock space is given a specific U v ( ŝl e )-module structure which is suitable for the Dipper-James-Murphy's Specht module theory. Proposition 4.2. Let λ = (λ (1) , λ (2) ) be Kleshchev and let i 1 , ..., i 1 a 1 times , ..., i p , ..., i p ap times be an optimal sequence of λ. Then we have

f (a 1 ) i 1 ...f (ap) ip ∅ = λ + ν≺λ c ν,λ (v)ν, for some Laurent polynomials c ν,λ (v) ∈ Z ≥0 [v, v -1 ], in the deformed Fock space.
Proof. First note that the coefficient of λ is one because each admissible sequence of i j -nodes is a sequence of normal i j -nodes. Now the proposition is proved by induction on p. Let R 1 1 , ...., R 1 a 1 be the admissible sequence of i 1 -nodes for λ and let λ ′ be the Kleshchev bipartition such that λ = λ ′ ⊔ {R 1 1 , ...., R 1 a 1 }. By the induction hypothesis, we have

f (a 2 ) i 2 ...f (ap) ip ∅ = λ ′ + ν ′ ≺λ ′ c ν,λ ′ (v)ν ′ . Let ν = λ be a bipartition such that it appears in f (a 1 ) i 1 ...f (ap) ip ∅ with nonzero coefficient. Then there exist removable i 1 -nodes R ′ 1 1 ,...,R ′ 1 a 1 of ν and a bi- partition ν ′ λ ′ such that ν = ν ′ ⊔ {R ′ 1 1 , ...., R ′ 1 a 1 }. As R 1 1 , ...., R 1 a 1 are the lowest a 1 i 1 -nodes of λ, ν ′ = λ ′ implies ν λ. Hence we may assume ν ′ ≺λ ′ . Suppose that we have λ ≺ ν. If ν ′ (2) = λ ′ (2) then choose t such that ν ′ (2) t < λ ′ (2) t and ν ′ (2) j = λ ′ (2)
j , for j > t. Then we can show (i) ν

(2)

j = λ (2)
j , for j > t, (ii) ν

(2)

t+1 < ν (2) t = ν ′ (2) t + 1 = λ ′ (2) t = λ (2)
t , (iii) at least one of the nodes R 1 1 , ...., R 1 a 1 is above (t, λ

t -1, 2). The condition (ii) implies that res(t, λ

t -1, 2) = res(t, ν ′ (2) t , 2) = i 1 . Thus (t, λ

t -1, 2) is an i 1 -node of λ (2) . Then (iii) implies that it is not a removable node of λ (2) (otherwise it has to be removed to obtain λ ′ ). This implies that λ t+1 and (i) is contradicted. If ν ′ (2) = λ ′ (2) then choose t such that ν ′ (1) t < λ ′ (1) t and ν ′ (1) j = λ ′ (1) j , for j > t. Then we argue as above. Proof. Observe that ν appears in f in • • • f i 1 ∅ if and only if there exists a standard bitableau of shape ν such that its residue sequence is (i 1 , . . . , i n ). Let λ be Kleshchev. Then Proposition 4.2 shows that there is a standard bitableau t such that if the residue sequence of t appears as the residue sequence of a standard bitableau of shape ν then ν λ. Suppose that the residue sequence of t is the residue sequence of a standard bitableau of shape ν ⊳ λ. As ν ⊳ λ implies λ ≺ ν, we cannot have ν λ, a contradiction. Hence λ is restricted. 4.2. Remark. We conclude the paper with a remark. Remark 4.4. There is a systematic way to produce realizations of the crystal B(Λ 0 + Λ m ) on a set of bipartitions, for each choice of log q (-Q). The bipartitions are called Uglov bipartitions. Recent results of Geck [START_REF] Geck | Hecke algebras of finite type are cellular[END_REF] and Geck and the second author [START_REF] Geck | Canonical basic sets in type Bn[END_REF] show that Uglov bipartitions naturally label simple H n -modules, and Rouquier's theory of the BGG category of rational Cherednik algebras as quasihereditary covers of Hecke algebras naturally explains the existence of various Specht module theories which depends on log q (-Q).

We conjecture that Uglov bipartitions satisfy an analogue of Proposition 4.2 except that the dominance order is replaced by an appropriate a-value in the sense of [START_REF] Geck | Canonical basic sets in type Bn[END_REF]Prop 2.1]. As is mentioned in the introduction, it is known that this conjecture is true in the case where Uglov bipartitions are FLOTW bipartitions [START_REF] Jacon | On the parametrization of the simple modules for Ariki-Koike algebras at roots of unity[END_REF]Prop. 4.6].

Example 2 . 1 .

 21 Put e = 4, m = 2 and λ = ((3, 2), (4, 2, 1)). Then the colored diagram associated with λ is as follows.

Corollary 4 . 3 .

 43 The Dipper-James-Murphy conjecture is true.

We now know that there are more than one Specht module theory, and different Specht module theories prefer different total orders on the set of i-nodes of a bipartition. Our choice of the total order is the one prefered by Dipper-James-Murphy's Specht module theory.

See [4, Prop 7.8].

Proof. Recall that λ (1) and λ (2) are both e-restricted. We claim that µ (1) and µ (2) are both e-restricted. We only prove that µ (1) is e-restricted as the proof for µ (2) is the same. Suppose that µ (1) is not e-restricted. Since λ (1) is e-restricted, it occurs only when there exists j such that λ (1)

j+1 -1 and res(j + 1, λ

j+1 -1, 1) = i. But then res(j, λ

(1)

j -1 by definition of µ.

First case. First we consider the case when either λ (2) = ∅ or λ (2) = ∅ and λ (2) has no addable i-node. If λ (1) has no addable i-node then the admissible sequence R 1 , ...., R s is given by all the removable i-nodes of λ, and thus all the normal i-nodes of λ. Hence µ = ẽs i λ, which implies that µ is Kleshchev. Therefore, we may and do assume that λ (1) has at least one addable i-node.

Define t ≥ 0 by ) )}. We prove that µ is Kleshchev by induction on t. Note that if λ is Kleshchev then so is (up m (λ (1) ), λ (2) ) since roof m (up m (λ (1) )) = roof m (λ (1) ) ⊆ τ m (base 0 (λ (2) )).

Suppose that t = 0. Then λ (1) is an e-core. As λ (1) has an addable i-node, λ (1) has no removable i-node. Thus all the removable i-nodes of λ are nodes of λ (2) . As λ (2) has no addable i-node, the admissible sequence R 1 , ...., R s is given by all the normal i-nodes of λ. Hence, µ = ẽs i λ and µ is Kleshchev. Suppose that t > 0 and that the lemma holds for (up m (λ (1) ), λ (2) ). Recall that we have assumed that λ (1) has an addable i-node. Let r be the minimal addable i-node of J m := J m (λ (1) ). The corresponding addable i-node of λ (1) , say γ, is the lowest addable i-node of λ and the admissible sequence R 1 , ...., R s is given by all the removable i-nodes of λ that is below γ. If there is no removable i-node greater than r then all the removable i-nodes of λ are normal and by deleting them, we obtain that µ is Kleshchev. Hence, we assume that there is a removable i-node greater than r. As r + 1 ∈ J m , this implies that there is x ∈ U (J m ) on the (i + 1) th runner such that x > r + 1. Let p = max U (J m ). Then x ∈ U (J m ) implies that p ≥ x > r + 1. As p moves to q > p, it implies that R 1 , ...., R s is an admissible sequence of i-nodes for (up m (λ (1) ), λ (2) ) and that (up m (λ (1) ), λ (2) ) = (up m (µ (1) ), µ (2) ) ⊔ {R 1 , ...., R s }. Now, (up m (µ (1) ), µ (2) ) is Kleshchev by the induction hypothesis. Hence, roof m (µ (1) ) = roof m (up m (µ (1) )) ⊆ τ m (base 0 (µ (2) )) and µ is Kleshchev as desired.

Second case : Now, we consider the case when λ (2) = ∅ and λ (2) has