Piotr Kwasniak
email: piotr.kwasniak@pw.edu.pl

Emmanuel Clouet

Basal slip of a screw dislocations in hexagonal titanium

Keywords: Titanium, Plasticity, Dislocations, Ab initio, Basal slip

Basal slip of a screw dislocations in hexagonal closed-packed titanium is investigated with ab initio calculations. We show that a basal dissociation is highly unstable and reconfigures to other structures dissociated in a first order pyramidal plane. The obtained mechanism for basal slip corresponds to the migration of the partial dislocations and of the associated stacking fault ribbon in a direction perpendicular to the dissociation plane. Presented results indicate that both basal and pyramidal slip will operate through the Peierls mechanism of double-kink nucleation and will be equally active at high enough temperature.

Plastic deformation of hexagonal titanium depends on complex interplay of temperature, purity and activity of the different slip modes [START_REF] Churchman | The slip modes of titanium and the effect of purity on their occurrence during tensile deformation of single crystals[END_REF][START_REF] Conrad | Effect of interstitial solutes on the strength and ductility of titanium[END_REF][START_REF] Naka | Prismatic slip in the plastic deformation of α-Ti single crystals below 700 K[END_REF][START_REF] Caillard | Thermally activated mechanisms in crystal plasticity[END_REF]. Experiments recognized that prismatic slip of a = 1/3 1210 dislocations is the easiest slip mode with a lattice friction opposing the motion of the screw orientation for temperatures below 550 K while a dislocation with an edge character remain highly mobile [START_REF] Naka | The low-temperature plastic deformation of α titanium and the core structure of a-type screw dislocations[END_REF][START_REF] Biget | Low-temperature plasticity of high-purity α-titanium single crystals[END_REF]. In situ TEM straining experiments [START_REF] Farenc | An in situ study of prismatic glide in α titanium at low temperatures[END_REF] confirmed that deformation is governed at low temperatures by long rectilinear screw dislocations which glide in the {1010} prismatic planes through a jerky motion associated with some crossslip in the {1011} pyramidal planes. This thermally activated mobility of the screw dislocation in pure Ti could be explained by a locking-unlocking mechanism [START_REF] Farenc | An in situ study of prismatic glide in α titanium at low temperatures[END_REF][START_REF] Farenc | A new model for the peak of activation area of α titanium[END_REF]: the screw dislocation, which is sessile in its ground state, needs to reconfigure to a higher energy configuration corresponding to a dissociation in a prismatic plane where the dislocation can easily glide. Various cores have been proposed for this ground state [START_REF] Naka | The low-temperature plastic deformation of α titanium and the core structure of a-type screw dislocations[END_REF][START_REF] Šob | Sessile splittings of screw dislocations in h.c.p. metals[END_REF][START_REF] Šob | Theory of strengthening of alpha titanium by interstitial solutes[END_REF] and recent ab initio calculations [START_REF] Clouet | Dislocation locking versus easy glide in titanium and zirconium[END_REF] have shown that it corresponds to a planar dissociation in the pyramidal plane.

Besides prismatic and pyramidal glide, a dislocations in hcp Ti are also known to glide in the (0001) basal planes [START_REF] Levine | Deformation mechanisms in titanium at low temperatures[END_REF][START_REF] Shechtman | Orientation dependent slip in polycrystalline titanium[END_REF][START_REF] Akhtar | Prismatic slip in zirconium single crystals at elevated temperatures[END_REF]. Recent TEM observations [START_REF] Caillard | Glide and cross-slip of a -dislocations in Zr and Ti[END_REF][START_REF] Barkia | In situ TEM observations of dislocation dynamics in α titanium: Effect of the oxygen content[END_REF] at and above room temperature show straight screw dislocations moving viscously in the basal planes. This deformation mode is, however, part of profuse cross slip and is always linked with prismatic glide, leading to wavy slip traces. Caillard et al. [START_REF] Caillard | Glide and cross-slip of a -dislocations in Zr and Ti[END_REF] have proposed that basal glide of the screw dislocations proceeds through a kinkpair mechanism, where the screw dislocation remains dissociated either in a pyramidal or prismatic plane and a kink-pair nucleates in the neighboring Peierls valley, with the kinks lying in the basal plane. A basal dissociation of the gliding dislocation will therefore not be required to allow for basal slip. Ab initio calculations have actually not managed to stabilize such a basal configuration of the screw dislocation until now [START_REF] Tarrat | Screw dislocation in hcp Ti: DFT dislocation excess energies and metastable core structures[END_REF], probably because the basal I 2 stacking fault associated with such a dissociation has a too high energy in titanium [START_REF] Kwaśniak | Plasticity of hexagonal systems: Split slip modes and inverse Peierls relation in α-Ti[END_REF][START_REF] Kwaśniak | Solid solution strengthening of hexagonal titanium alloys: Restoring forces and stacking faults calculated from first principles[END_REF][START_REF] Yin | Comprehensive first-principles study of stable stacking faults in hcp metals[END_REF][START_REF] Rodney | Ab initio modeling of dislocation core properties in metals and semiconductors[END_REF]. Here, we use ab initio calculations to examine more closely the stability of such a basal dissociation of the a screw dislocation in pure Ti, before determining the transition pathway corresponding to basal slip, taking into account all configurations which have been found stable for the a screw dislocation.

The computational supercell used in this study contains 288 atoms and exhibits m = 9, n = 8 and l = 1 unit cell periodicity along e x = [0001], e y = [ 10 10] and e z = 1/3 [1 210] directions, respec- tively. As a result, the periodicity vectors of the simulation box are u 1 = mc e x , u 2 = na e y and u 3 = la e z , where a and c are the Ti lattice parameters. The calculations of the screw dislocations were performed in accordance to full periodic conditions approach [START_REF] Rodney | Ab initio modeling of dislocation core properties in metals and semiconductors[END_REF][START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF] and the initial structures of two line defects with opposite Burgers vectors were generated using anisotropic linear elasticity thanks to the Babel code [START_REF] Clouet | Babel software[END_REF]. The two dislocations composing the dipole are separated by a vector d = 1/2 ( u 1 + u 2 ), thus leading to a quadrupolar arrangement which minimizes the elastic interaction between the dislocations and their periodic images. The plastic strain created by the dipole is compensated by adding the tilt vectors δ 1 = b/2 = a e z /2 and δ 2 = -δ 1 to u 1 and u 2 , respectively. It should be highlighted that the selected supercell geometry has an odd number of α-Ti unit cell repetition along e x (m = 9) which is different from those used in previous studies of 〈a〉 type screw dislocations in hcp systems [START_REF] Clouet | Dislocation locking versus easy glide in titanium and zirconium[END_REF][START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF][START_REF] Chaari | First-principles study of secondary slip in zirconium[END_REF][START_REF] Kwaśniak | Screw dislocation mediated solution strengthening of substitutional α-Ti alloysfirst principles investigation[END_REF]. This ensures that the plastic strain arising from the edge com-ponents of a basal dissociation exactly compensates for the two dislocations composing the dipole, thus preventing the existence of a back-stress caused by the fixed periodicity vectors and which would acts against such a dissociation.

Ab initio calculations were performed with VASP code [26,27], with projector augmented wave (PAW) method for core-valence electron interaction [28] and Perdew-Burke-Ernzerhof (PBE) [29] generalized gradient functional. The Brillouin zone was sampled in accordance with the Monkhorst-Pack scheme [30] using 1 × 1 × 9 gamma centered k-points grid and a 0.3 eV Methfessel-Paxton electronic occupancy smearing.

The Ti pv pseudopotential with 10 valence electrons (3p 6 3d 2 4s 2 ) was employed, with a 500 eV cutoff energy for plane waves. The atomic structures of the screw dislocations were determined by performing ionic relaxations with a fixed-shape simulation box using a criterion of 3 meV/ Å for Hellmann-Feynman forces convergence. The relative line defects energies are equal to:

2 ∆E = E tot -E π l 2h (1)
where, E tot is the total energy of α-Ti supercells with different positions or configurations of dislocation cores, E πl is the total energy of the simulation box with dislocations in their ground state (low energy pyramidal configuration [START_REF] Clouet | Dislocation locking versus easy glide in titanium and zirconium[END_REF]), and h is the height of the atomic model along dislocation line.

The dislocation initial configuration is dissociated in the basal plane. It is introduced in the simulation cell by applying to all atoms the elastic displacement predicted by anisotropic elasticity for a pair of 1/3 1100 and 1/3 0110 Shockley partials in the same prismatic plane (Fig. 1a), thus creating a stacking fault ribbon between them. Two core positions, which correspond to the center of symmetry of the ground state or of the high energy pyramidal metastable configuration, marked as B 1 and B 2 , respectively, are considered. Width of the core is characterized by the distance λ between the two Shockley partials which is measured in units of the distance k = a √ 3/2 between two Peierls valleys in the [1010] direction. Such prepared basal screw dislocations subjected to ab initio ionic relaxation spontaneously reconfigure to the ground state π l or high energy pyramidal π h geometry depending on their initial position (Fig. 1b), with an energy difference ∆E = 11.9 meV/ Å between these two states in good agreement with previous ab initio studies [START_REF] Clouet | Dislocation locking versus easy glide in titanium and zirconium[END_REF][START_REF] Kwaśniak | Screw dislocation mediated solution strengthening of substitutional α-Ti alloysfirst principles investigation[END_REF][START_REF] Poschmann | Convergence of calculated dislocation core structures in hexagonal close packed titanium[END_REF]. Our investigation therefore conclude to the instability of basal dissociation in pure Ti, which is in line with the ab initio calculations of Tarrat et al. [START_REF] Tarrat | Screw dislocation in hcp Ti: DFT dislocation excess energies and metastable core structures[END_REF] relying on different boundary conditions.

To unveil the energy variation during decay of the basal dissociation, i.e. to estimate the core energy as function of the basal dissociation width, constrained relaxations have also been performed. In these calculations, atoms in the dissociation plane located between the two partial dislocations were not allowed to relax along the [1210] screw direction. These atoms are indicated by light-blue squares in Fig. 1c. Such a constraint allows maintaining the basal stacking fault. Because of hcp lattice symmetry, the dissociation distance λ where this constraint is applied needs to be equal to an odd (respectively even) number of distances k between Peierls valleys for the B 1 (respectively B 2 ) position. Independently from the selected core position and width, the leading and trailing partials are never purely basal but spread onto the pris-matic or first-order pyramidal planes (Fig. 1c). Moreover, the energy of the considered line defects strongly depends on the basal dissociation distance as shown in Fig. 1d. The excess energy ∆E continuously increases with the dissociation distance λ, without showing any minimum at small distances, thus indicating the absence of a metastable configuration dissociated in the basal plane. The energy variation corresponding to a basal dissociation was also calculated according to elasticity theory (Eq. 3 in Ref. [START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF]) using a core radius equal to 1 k, a stacking fault energy γ b = 306 mJ/m 2 and the elastic constants given by our ab initio calculations. As shown in Fig. 1d, linear elasticity, although not fully quantitative, provides a reasonable estimate of ∆E(λ): this energy variation therefore mainly results from the increase of the dissociation width. Elasticity theory predicts a dissociation distance d eq b = 2.6 Å = 1.03 k, a distance which is too small to lead to any real basal dissociation, in agreement with ab initio results.

As basal slip in pure Ti does not stem from a basal dissociation, we consider now the ground state of the screw dislocation, where the core is dissociated in a pyramidal plane (configuration π l in Fig. 1b), and examine how such a core can glide in a basal plane. The studied scenario involves basal motion of the screw dislocation from its π l ground state to the next equivalent π l position in the [1010] direction. The minimum energy path determined with the nudged elastic band (NEB) method [START_REF] Sheppard | Optimization methods for finding minimum energy paths[END_REF] using 5 intermediate images corresponds to the reaction coordinate 0 ≤ ζ ≤ 2 in Fig. 2. According to the preliminary results obtained with a force convergence criterion of 30 meV/ Å, direct π l → π l glide goes through an intermediate metastable configuration which corresponds to the high energy π h core (Fig. 2e). The NEB calculation was thus further refined by considering the two half-paths π l → π h and π h → π l with 5 intermediate images and a 20 meV/ Å convergence criterion. The obtained energy barrier is only about +2 meV/ Å higher than the metastable π h core and corresponds to a configuration where the dislocation is spread in both prismatic and pyramidal planes (Fig. 2c).

We then consider the prismatic configuration of the screw dislocation. As previously shown [START_REF] Clouet | Dislocation locking versus easy glide in titanium and zirconium[END_REF], a screw dislocations in Ti can adopt a configuration where the core is fully dissociated in a prismatic plane and where the dislocation center is at the same position as for the ground state (Fig. 2b). This metastable P core has an energy The energy variation ∆E is shown as a function of a reaction coordinate ζ, with fully colored points and opened circles corresponding to (meta-)stable cores and transition configurations obtained by the NEB method, respectively. The corresponding path is sketched on top with the basal slip plane marked by a vertical red line. Some core structures along this path are shown in subfigures (a-e) (see Fig. 1 for a detailed caption of these subfigures). ∆E = 5.6 mev/ Å, still in good agreement with previous ab initio calculations [START_REF] Clouet | Dislocation locking versus easy glide in titanium and zirconium[END_REF][START_REF] Kwaśniak | Screw dislocation mediated solution strengthening of substitutional α-Ti alloysfirst principles investigation[END_REF][START_REF] Poschmann | Convergence of calculated dislocation core structures in hexagonal close packed titanium[END_REF], and the energy barrier leading to this stationary core reconstruction is only slightly higher than the energy of P state (energy path shown for 2 ≤ ζ ≤ 3 in Fig. 2). The minimum energy path for basal glide of this prismatic core geometry (3 ≤ ζ ≤ 5 in Fig. 2) goes through the same configurations as for the π l → π l transition, thus leading to the same energy barrier. Similarly to π l → π l slip, after an initial NEB calculation with 5 intermediate images,where the intermediate π h configuration appeared, the P → P transition has been refined by considering half the path, corresponding thus to the P → π h transition. Besides the intermediate π h configuration, a well defined additional metastable state (Fig. 2d) ap-pears both during P → P basal slip (ζ ∼ 3.7 and 4.3), and during π l → π l basal slip (ζ ∼ 0.7 and 1.3), where the dislocation is partly dissociated in the pyramidal and prismatic plane.

We finally consider basal slip of the metastable π h core (Fig. 3), using 5 intermediate images in the NEB calculation, as this configuration appears in both π l → π l and P → P transition paths. No new path was actually obtained as this π h → π h transition goes through the prismatic configuration and is equivalent to the P → P basal slip.

Summarizing the present study, it was found that basal glide of the 1/3〈11 20〉 screw dislocations in pure Ti is realized without any basal dissociation. Instead, the dislocation remains dissociated in pyramidal and prismatic planes during the transition, with a motion of the partial dislocations and of the stacking fault ribbon in a direction perpendicular to the dissociation plane. This basal slip mechanism is similar to the one already identified in Zr [START_REF] Chaari | First-principles study of secondary slip in zirconium[END_REF]. Because of the high energy barrier associated with such a path, basal slip will proceed through the nucleation of double kinks thanks to thermal activation, in agreement with the Peierls mechanism inferred from TEM observations [START_REF] Caillard | Glide and cross-slip of a -dislocations in Zr and Ti[END_REF]. The same configurations being involved for basal as for prismatic and pyramidal motion, this explains the wavy slip traces experimentally observed when basal slip is active [START_REF] Caillard | Glide and cross-slip of a -dislocations in Zr and Ti[END_REF][START_REF] Barkia | In situ TEM observations of dislocation dynamics in α titanium: Effect of the oxygen content[END_REF]. Finally, as the Peierls barrier of basal and pyramidal [START_REF] Clouet | Dislocation locking versus easy glide in titanium and zirconium[END_REF] glide are close, both basal and pyramidal slip are activated at high enough temperature [START_REF] Caillard | Glide and cross-slip of a -dislocations in Zr and Ti[END_REF][START_REF] Barkia | In situ TEM observations of dislocation dynamics in α titanium: Effect of the oxygen content[END_REF].

Figure 1 :

 1 Figure 1: Core structures corresponding to a basal dissociation of a 1/3 [1210] screw dislocation for two different positions B 1 and B 2 of the dislocation center. The initial configurations are shown in a), the fully relaxed configurations in b) and the configurations obtained with constrained minimization in c). Different dissociation distances λ are studied. The arrows between two atomic columns are proportional to the differential displacement created by the dislocation in the [1 210] direction. Displacement smaller than 0.1 b are not shown. The contour map shows the dislocation density according to the Nye tensor. Ti atoms belonging to particular types of stacking faults or different (1 210) atomic planes are plotted with open or full-colored symbols as presented in top-right corner. Positions of the dislocation centers are marked by green crosses and constrained atoms with a fixed position along the dislocation line are marked by light-blue squares. The energy variation ∆E as a function of the basal dissociation distance λ obtained from ab initio calculations (B 1 and B 2 ) and linear elasticity (LE) are shown in d) with filled and opened symbols respectively for unconstrained and constrained relaxations.

Figure 2 :

 2 Figure2: Basal slip of the a screw dislocation starting from its ground-state pyramidal π l and its prismatic P configurations. The energy variation ∆E is shown as a function of a reaction coordinate ζ, with fully colored points and opened circles corresponding to (meta-)stable cores and transition configurations obtained by the NEB method, respectively. The corresponding path is sketched on top with the basal slip plane marked by a vertical red line. Some core structures along this path are shown in subfigures (a-e) (see Fig.1for a detailed caption of these subfigures).

Figure 3 :

 3 Figure 3: Minimum energy path for basal slip of a screw dislocation, corresponding to the transition between two neighboring π h metastable core (shown in the top-left corner). The labels (b), (c), and (e) refer to the configurations shown in Fig. 2.

Preprint submitted to ElsevierDecember 29, 2018

Acknowledgments -This work supported by the National Science Center under SONATINA 1 project No. 2017/24/C/ST8/00123. Computing resources were provided by the HPC facilities of the PL-GRID and CI-TASK infrastructure.


Diego Cattaruzza

Luce Brotcorne

Frédéric Semet

Bayrem Tounsi

A three-phase matheuristic for the Packaging and Shipping Problem

A three-phase matheuristic for the Packaging and Shipping Problem

Diego Cattaruzza, Luce Brotcorne, Frédéric Semet, Bayrem

1 Introduction E-commerce has been continuously growing in the last years to a primary retail market. In 2016 in France, the threshold of 1 billion of online transactions was overcome for a total of 72 billion Euros 1 . At the same time, new challenges arise in the e-commerce sector. Due to high demand uctuation, the workforce sizing for the logistic chain is a challenging problem: a high number of workers guarantees demand satisfaction to face all situations, but at a high cost. On the other side, reducing the personnel results in delays that lead to customer dissatisfaction and negative impacts on the company image. The company has to develop appropriate strategies leading a sustainable workforce size while guaranteeing a high-level service.

In this paper, we aim to model and to provide a solution method for a problem occurring in the management of workforce for a warehouse of an e-commerce company. This problem was originally motivated by a real-case study introduced to us by one of the leading third-party logistics company. We address here three main issues: i) How the workforce at the warehouse can be determined; ii) What is the daily operational production planning; iii) How the demand peaks can be smoothed and the production maintained ideally constant over the horizon.

Warehouse

Storage zone

To address the last issue and face the high uctuation of the demand, we propose two strategies to smooth the workload, i.e., production postponement and demand reassignment. Roughly speaking, the rst strategy allows delaying production on successive days, while the second strategy allows assigning a given order to delivery service dierent than the one selected by the customer. We detail such policies below.

To provide answers to the rst two issues, we address the Packaging and Shipping Problem (PSP) to optimize the preparation order process. In this problem, we deal with tactical and operational decisions simultaneously. In particular, we take into account the management of the workforce over a multi-day horizon and the determination of the number of workers required for all the shifts of each day. Moreover, we provide the operational planning required to prepare the total demand received during each day of the planning horizon.

From a managerial point of view, the model and the matheuristic we propose can help companies in two dierent ways. First, based on historical data it can help understanding the value of postponement and re-assignment strategies. Companies would have an insight on the gain that would achieve adopting such strategies and comparing this gain with the cost of implementing them. Second, it can help to determine the number employees needed to ensure production in dierent scenarios. Such decision support can be obtained in an oine fashion, based on historical data to hire employees, or in an online fashion based on real-time data to determine the number of temporary workers that should be hired to guarantee production. Finally, the model provides a detailed production planning taking into account postponement and re-assignment strategies.

The process of an order received at a warehouse consists of dierent sequential phases. First, the items that form the order need to be picked up in the storage zone, then they need to be put in standard packages and nally loaded in appropriate trucks. In the case considered here, the trucks transfer packages to carrier hubs where they are sorted according to their nal destination. A a representation of the dierent areas of a warehouse is given in Figure 1.

For simplicity, a volume is associated with each order. As an order can include several items, the volume represents the number of packages required to contain these items. Also, during online ordering, the customer can choose among dierent delivery options, called modes, e.g., standard or express delivery service. Each mode is associated with dierent delivery delays (for example the standard mode delivers within 3-5 business days or the express mode within 24 hours) and prices to be paid by the customers. Notice that the word mode is used to distinguish between dierent transportation services. In other papers, as in Alptekino §lu and Tang [START_REF] Alptekino | A model for analyzing multi-channel distribution systems[END_REF], the modes are the dierent modes of distribution, as in-store, mail-order or internet-based services.

The tackled problem consists in determining the daily operations within the warehouse. Each day is composed by shifts, and a shift by periods. During each shift, a certain number of employees is in charge of the demand preparation. Workers are associated with a productivity, i.e., the number of packages they can prepare during a period. The employees can be permanent or temporary. An employee works for the entire shift and guarantees a productivity level during each period of the shift. Temporary workers can be hired for one shift only, usually when the workload is estimated to be high. On the other side, they cost more than a permanent worker. It is assumed that the productivity of a permanent worker is higher than the productivity of a temporary worker.

The number of workers assigned to each shift is a decision variable of the problem. The preparation plan for each time period needs to be determined, namely, for each order, the number of packages prepared at each time period needs to be set. Note that an order with more than one package can be prepared during dierent, and not necessarily consecutive, periods.

Once an order has been processed, its packages need to be loaded in the trucks. Trucks need to be present at the warehouse docks to be loaded. Each truck is associated with one delivery mode, and it contains only packages of orders delivered with that mode. Then, if orders associated with dierent modes are processed at a given period, at least one truck for each mode must be docked.

Since packages related to an order can be prepared during dierent periods, they can be loaded in dierent trucks. Moreover, at each period, the number of packages prepared is limited by the number of workers and the number of docked trucks.

The warehouse, where operations take place, has a limited number of docks that, consequently, limits the number of trucks that are loaded simultaneously. The docks are a critical resource that impacts the overall uidity of picking and shipping operations. We thus dene a truck movement policy that manages spatially and temporally the use of docks. The proposed truck movement policy is detailed in Section 3.

After describing the general structure of the warehouse and the classical steps of order picking and shipping, we introduce two new strategies that aim to enhance the exibility of the process and the eciency of resources use: reassignment and postponement.

Reassignment allows delivering an order, i.e., all associated packages, with another mode than the one selected by the customer. When reassignment is operated, the company pays a penalty that represents either an additional delivery cost (when for example the change is from the standard mode to the express mode) or the dissatisfaction of the customer for a possible late delivery (when the change is from the express mode to the standard). Due to the penalty cost, reassignment takes place when it allows to hire fewer workers or to use fewer trucks.

Postponement comes into play due to the multi-period nature of the problem. Since we need to plan the operations for several successive days, we then introduce the possibility to postpone to next days the process of some orders. As a counterpart, a penalty is paid because a postponement leads to a delay. As for the reassignment case, when the preparation of a given order is postponed, the preparation of all its packages is postponed. Postponement can be combined with reassignment when, for instance, some orders associated with the standard delivery mode are delivered with the express mode of the day after.

The PSP looks for a solution that determines the workforce over a multi-period horizon and operational daily plans which minimize the total cost of shipping and picking operations. Specically, the total cost is computed as the sum of the cost of hiring the workers (permanent and temporary), the cost of the trucks used, the penalty cost for order postponements and reassignments, and the dock utilization.

The contribution of this paper is threefold. First, we dene the PSP and provide a mathematical programming formulation. Second, we consider two strategies, reassignment and postponement, to smooth the high demand peaks that characterize e-commerce demand proles. Third, we present a three-phase matheuristic induced by a natural decomposition of the proposed model to obtain ecient solutions.

The paper is organized as follows. Section 2 reviews the related literature. Section 3 formally introduces the problem and presents a mathematical model for the PSP. Section 4 describes the three-phase algorithm we developed to obtain good solutions for the PSP. Section 5 presents the computational results. Finally conclusions are drawn in Section 6.

Literature review

The Packaging and Shipping Problem described in Section 1 aims to simultaneously determine the number of employees, the reassignment and postponement of demands, and a complete and detailed production planning. In the following, we review some academic papers that share similarities with our work. From a general point of view, the PSP belongs to the category of integrated problems, where dierent level decisions are taken simultaneously, i.e., strategic and tactical decisions, or tactical and operational decisions. It is well known that integrated decisions lead to cost savings (Chandra and Fisher [START_REF] Chandra | Coordination of production and distribution planning[END_REF]) against the increase of the problem complexity. Integration of decisions increases the coordination of the distribution system and, consequently, its eciency. It results in cost reductions (from 3% to 20%, Strack et al. [START_REF] Strack | Comparison of heuristic procedures for an integrated model for production and distribution planning in an environment of shared resources[END_REF]) or storage levels reduction and leading to better use of the resources. Moreover, taking decision sequentially and without taking into account interaction among decisions can lead to approaches that provide infeasible solutions. In the recent past, the scientic community devoted a growing interest to this class of problems.

The PSP integrates tactical decisions (determination of the number of employees) as long as operational decisions (production planning). An example of the tactical-operational integrated problems is the Production Routing Problem (PRP, Absi et al. [START_REF] Absi | A two-phase iterative heuristic approach for the production routing problem[END_REF]), where a lot-sizing problem is combined with a vehicle routing problem. Another example of an integrated problem where lot-sizing decision, as well as production and transportation decisions, are considered can be found in [START_REF] Shu | A productiondelivery lot sizing policy with stochastic delivery time and in consideration of transportation cost[END_REF]. The reader interested in production routing problems is referred to the survey of Adulyasak et al. [START_REF] Adulyasak | The production routing problem: A review of formulations and solution algorithms[END_REF]. The work of Hiassat et al. [START_REF] Hiassat | A genetic algorithm approach for location-inventoryrouting problem with perishable products[END_REF] goes a step further. In they work they deal with the location-inventory-routing problem where perishable products are considered. Here strategic, tactical and operational decision are simultaneously taken.

PSP does not deal with routing issues but coordinates production and vehicle loading (shipment release). Two examples of works coordinating production and shipping are Baptiste et al. [START_REF] Baptiste | Integration of production and shipping planning : A cooperative approach[END_REF] and Lee et al. [START_REF] Lee | A dynamic model for inventory lot-sizing and outbound shipment scheduling at a third-party warehouse[END_REF]. In Baptiste et al. [START_REF] Baptiste | Integration of production and shipping planning : A cooperative approach[END_REF] production lines are organized with batches that are subsequently loaded into trucks and shipped to their nal destinations. The problem consists in determining the production plan in such a way that a set of full trucks is dispatched to dierent destinations. The plan needs to respect logistics constraints (number of production lines, number of docks) and to minimize the delivery costs.

Lee et al. [START_REF] Lee | A dynamic model for inventory lot-sizing and outbound shipment scheduling at a third-party warehouse[END_REF] study the coordination of inbound and outbound ows of a product at a warehouse. The inbound ow guarantees inventory level satisfaction, and it is managed by coor-dinating production at the manufacturer level. From the warehouse, a set of distribution centers is served using full truckload. The problem is modeled as a two-echelon inventory lot-sizing problem with shipment scheduling and looks for a solution that minimizes the transportation costs and inventory cost required for the replenishment of the warehouse and the distribution centers.

The PSP considers dierent delivery services. With this respect, the work of Wang and Lee [START_REF] Wang | Production and transport logistics scheduling with two transport mode choices[END_REF] addresses a variant of the PSP. As in the PSP, they consider the possibility to deliver the products manufactured by the company using dierent delivery services: standard and express shipping modes. Modes have dierent costs and travel times. In Wang and Lee [START_REF] Wang | Production and transport logistics scheduling with two transport mode choices[END_REF] production has to be scheduled in such a way that products can be transported at customer locations before prespecied due dates. Another example of integrated optimization problem that considers dierent transportation modes is presented in Siddiqui et al. [START_REF] Siddiqui | An integrated framework for inventory management and transportation of rened petroleum products: Pipeline or marine?[END_REF]. This work deals with an integrated inventory and transportation problem that occurs when petroleum needs to be moved to reneries. The authors consider the possibility of choosing between pipelines and maritime transportation taking into account environmental risks.

Employees management in the PSP relies only on demand modeling (following the terminology of Ernst et al. [START_REF] Ernst | Sta scheduling and rostering: A review of applications, methods and models[END_REF]). In the PSP, we have to determine the number of employees that are needed during each shift to process the total demand. More complex rostering problems require the determination of a detailed working plan for each employee. Applications can be found in freight handling at cargo terminals (Rong and Grunow [START_REF] Rong | Shift designs for freight handling personnel at air cargo terminals[END_REF]), in demand satisfaction of time-dependent requirements for check-in of individual ights (Stolletz and Zamorano [20]).

Postponement strategies have been widely studied. The results obtained highlight the benet of such policies. We can say that a problem involves a postponement strategy when it allows delaying some operations. In Carbonara and Pellegrino [START_REF] Carbonara | Real options approach to evaluate postponement as supply chain disruptions mitigation strategy[END_REF] postponement consists of delaying activities in the supply chain until customer order information becomes available. Here we decide to postpone a delivery if this is benecial for the company. The Inventory Routing Problem (IRP, Bertazzi et al. [START_REF] Bertazzi | The Vehicle Routing Problem -Last Advances and New Challenges[END_REF]) can be seen as a vehicle routing problem with a postponement policy. If a customer consumes q units of product per day, the delivery company is not obliged to deliver q units per day, but T q units of products over a horizon of T days to avoid stockouts. This allows to postpone services and to deliver larger quantities in fewer visits. Signicant cost savings can then be generated (Pang and Muyldermans [START_REF] Pang | Vehicle routing and the value of postponement[END_REF]). Mahar and Wright [START_REF] Mahar | The value of postponing online fulllment decisions in multichannel retail/e-tail organizations[END_REF] develop a dynamic strategy for online orders fulllment for a multi-channel retail company. Online orders are accumulated before being assigned to a specic site for fulllment. This policy results in cost reductions compared with an assign-as-order-arrive strategy.

The third main characteristic of our problem is the reassignment policy, i.e., assignment of an order to a delivery channel dierent than the one selected by the customer. The Lot-Sizing Problem (LSP) with one-way substitution shares similar characteristics. In this case, the demand can be satised by recovering used items. In case of running out of used items, new items can be supplied instead (see for example Piñeyro and Viera [START_REF] Piñeyro | The economic lot-sizing problem with remanufacturing and oneway substitution[END_REF]). This generates a loss for the company that sells products at a lower price. Opposite substitution (supply recovered items instead of new ones) is not allowed. In our problem, the reassignment strategy can be seen as substituting delivery mode. However, the PSP diers from the LSP, since both substitutions are considered.

Last, problems integrating decisions of the same level have been tackled. For example, when order-to-order production (scheduling) and transportation decisions are coupled, decisions at the operational level are integrated. The reader is refereed to Chen [START_REF] Chen | Integrated production and outbound distribution scheduling: Review and extensions[END_REF] for a recent survey of such models. In this section, we formally dene the PSP and we provide a mix-integer mathematical formulation for the problem. Table 1 contains the notation used in the paper.

H

We consider a planning horizon of H periods indexed in H = {0, . . . , H -1}. Typically, a period is as a day. At each period h ∈ H, D h orders have to be processed (indexed in D h = {0, . . . , D h -1}). Orders revealed on period h need to be prepared in one of the following H periods, indexed in H = {0, . . . , H -1}. When h = 0, h ∈ H, orders are not postponed. There are V available delivery modes, indexed in V = {0, . . . , V -1}.

Each order d ∈ D h of period h is characterized by its

• volume vol hd : the number of packages it is composed;

• delivery mode v hd ;

• time slot at which the order becomes known (release date of the order) r hd ;

• penalty p h v hd v for processing the order at period h + h and assigning it to mode v.

As an example, h = 1 corresponds to a postponement of one day, while v hd = v corresponds to a reassignment. Naturally p 0 v hd v hd = 0. We assume that the penalty for postponing a package from period h to period h + h or changing its mode is identical for all orders.

Each delivery mode v is characterized by its departure slot t v . No truck associated with mode v will be available after t v . All the trucks have the same capacity Q and the same cost c truck . Moreover, N max is the number of available docks at the warehouse, thus at most N max trucks can be simultaneously loaded.

Each period h is divided into S shifts , and each shift into T slots. It follows that each period is divided into T = S T slots. Each shift s of period h is characterized by its

• starting slot start hs ;

• ending slot end hs ;

• cost for a permanent employee c per hs ;

• cost for a temporary employee c temp hs ;

• number of packages a permanent employee can prepare prod per hs ;

• number of packages a temporary employee can prepare prod temp hs ;

• maximum number of permanent employees e max hs . Trucks are managed according to the following truck movement policy. Each truck is assigned to one and only one delivery mode, i.e., it will carry only packages assigned to that mode. Trucks can be made available at the docks at any slot. When a truck is fully loaded, it is undocked by the end of the slot. As a consequence, the dock, it has occupied, becomes free for use at the beginning of the next slot. If necessary, more than one truck per mode can be simultaneously docked. If the truck is not fully loaded at the end of a slot, it remains docked for the next slot. Non-full trucks for mode v are undocked in two cases: at slot t v or when no package for mode v will be assigned to the corresponding mode during the following slots of the period.

Over the planning horizon, the PSP aims to determine the number of employees and trucks, an order process planning that consists in identifying the exact slot during which each package of each order is processed, and a truck management planning (i.e., when to dock and undock trucks) in order to minimize the sum of the employees and trucks costs, penalties, and the docks occupation.

Let us now present the mathematical formulation of the PSP. We rst introduce the variables of the model. For each period h ∈ H, for each postponement period h ∈ H, for each d ∈ D h , for each mode v ∈ V and for each shift s ∈ S we have:

• tactical variables: z per hs the number of permanent workers working on shift s of period h; z temp hs the number of temporary workers working on shift s of period h;

• operational variables:

x hv hd equals 1 if the order d of period h is prepared in period h + h and assigned to mode v, 0 otherwise; y vt h equals 1 if the number of empty trucks for mode v during period h at a slot t ≥ t is not null, 0 otherwise; f hvt hd indicates the number of packages of order d prepared in slot t of period h + h assigned to mode v; w vt h is the number of docked trucks for mode v at period h in slot t; u vt h is the number of empty trucks for mode v that are docked at period h in slot t; k vt h is the residual capacity of trucks at period h in slot t for mode v.

The mathematical model for the PSP reads as follows:

(P SP ) min h∈H d∈D h h∈ H v∈V p h v hd v vol
f hvt hd = vol hd ∀h ∈ H, ∀d ∈ D h (2) t∈T f hvt hd ≤ vol hd x hv hd ∀h ∈ H, ∀d ∈ D h , ∀ h ∈ H, ∀v ∈ V (3)
h∈ H v∈V

x hv hd = 1 ∀h ∈ H, ∀d ∈ D h (4)
f 0vt hd = 0 ∀h ∈ H, ∀d ∈ D h , ∀v ∈ V, 0 ≤ t < r hd (5) h∈ H h-h≥0 d∈D h- h f hv0 (h-h)d + k v0 h = Qu v0 h ∀h ∈ H, ∀v ∈ V (6) h∈ H h-h≥0 d∈D h- h f hvt (h-h)d + k vt h = k v(t-1) h + Qu vt h ∀h ∈ H, ∀v ∈ V, 0 < t ≤ t v (7) tv t= t u vt h ≤ N max y v t h ∀h ∈ H, ∀v ∈ V, 0 ≤ t ≤ t v (8)
y v t h ≤ tv t= t u vt h ∀h ∈ H, ∀v ∈ V, 0 ≤ t ≤ t v (9)
u v0 h ≤ w v0 h ∀h ∈ H, ∀v ∈ V (10)
Qu vt h + k v(t-1) h ≤ Qw vt h + Q(1 -y vt h ) ∀h ∈ H, ∀v ∈ V, 0 < t ≤ t v (11)
Qu vt h + k v(t-1) h -k v(tv-1) h ≤ Qw vt h ∀h ∈ H, ∀v ∈ V, 0 < t ≤ t v (12)
v∈V

w vt h ≤ N max ∀h ∈ H, ∀t ∈ T ( 13
) h∈ H h-h≥0 d∈D h-h v∈V f hvt (h-h)d ≤
x hv hd ∈ {0, 1} ∀h ∈ H, ∀d ∈ D h , ∀ h ∈ H, ∀v ∈ V (17)
y vt h ∈ {0, 1} ∀h ∈ H, ∀v ∈ V, ∀t ∈ T (18)
z per hs , z temp hs ∈ N ∀h ∈ H, ∀s ∈ S (19)
f hvt hd ∈ N ∀h ∈ H, ∀d ∈ D h , ∀ h ∈ H, ∀v ∈ V, ∀t ∈ T (20)
w vt h , k vt h , u vt h ∈ N ∀h ∈ H, ∀v ∈ V, ∀t ∈ T (21)
Before analyzing the model, we can underline from variable x and f denitions that orders are processed individually. The model oers a highly accurate tracking of order processing information, and it is possible to retrieve for each order the exact slot of its process. Such accurate tracking is required in e-commerce for the management of the whole delivery route of the order as well as for the management of customer relationship.

The objective function (1) aims to minimize the cost of processing all the orders. This cost is given by the sum of four terms computed over the planning horizon. The rst term is the sum of all penalties due to postponements and reassignments. The second term in the objective function is the total labor cost computed as the sum of all workers costs, while the third is the cost of the used trucks. The fourth term is a measure of the docks occupation, and it is incremented each time a dock is occupied by a truck during one time slot. We express this term in dock-slot as it is the case when we measure an amount of work in man-hour or man-day units. The coecient α converts this term from the dock-slot unit into a cost unit.

Constraints [START_REF] Adulyasak | The production routing problem: A review of formulations and solution algorithms[END_REF] ensure that all the packages that compose an order are prepared. Constraints (3) and Constraints (4) impose that each order is assigned to only one mode and processed entirely during the same period. Constraints (5) forbid to prepare orders before their release date. Constraints (6)( 7) is the packages ow conservation: processed packages are loaded in an already docked truck with some residual capacity or in an empty truck. These constraints are formulated dierently for the rst slot of each period. Note that the truck residual capacities are updated at every slot. Constraints (8) (resp. Constraints ( 9)) force variables y vt h to be one (resp. zero) if (resp. if no) additional trucks for the mode v will be used during the slot t or the slots after t of period h.

Together constraints (10) [START_REF] Gendron | Multicommodity capacitated network design[END_REF] and [START_REF] Hiassat | A genetic algorithm approach for location-inventoryrouting problem with perishable products[END_REF] enable to apply the truck movement policy explained earlier in this section. They update variables w vt h that represent the exact number of docks occupied by the trucks associated to each mode at each slot. At the rst slot of each period, constraints [START_REF] Gendron | Multicommodity capacitated network design[END_REF] [START_REF] Hiassat | A genetic algorithm approach for location-inventoryrouting problem with perishable products[END_REF] have specic formulation [START_REF] Ernst | Sta scheduling and rostering: A review of applications, methods and models[END_REF] that corresponds to the rst slot of each period. These formulations are dierent from the general form because they do not implicate truck residual capacities. In Constraints [START_REF] Ernst | Sta scheduling and rostering: A review of applications, methods and models[END_REF] [START_REF] Gendron | Multicommodity capacitated network design[END_REF], a new truck docked at a given slot (variables u vt h ) naturally implies that a dock is occupied. On the other side, a truck remains on dock only if other trucks associated with the same mode are expected to be used in the upcoming slots (y vt h = 1). Constraints (12) complete the truck movement policy by handling the particular case, not handled by Constraints [START_REF] Ernst | Sta scheduling and rostering: A review of applications, methods and models[END_REF] [START_REF] Gendron | Multicommodity capacitated network design[END_REF], where a truck already docked should remain on dock because a number of packages, inferior to the current residual capacity, is expected in the upcoming slots. Note that it is possible to formulate Constraints (10)( 11) and ( 12) in a more compact way, by expressing variables y vt h in terms of expected upcoming packages instead of expected upcoming trucks. The proposed formulation was preferred since it presents a good separability, in the sense that Constraints (8) and ( 9) involve variables associated with one same period.

Constraints (13) impose a limit on the number of docks available. Constraints [START_REF] Mahar | The value of postponing online fulllment decisions in multichannel retail/e-tail organizations[END_REF] impose that the number of packages to be prepared in each slot should not exceed the production capacity of the workers. Constraints (15) impose a limit on the number of permanent workers. Constraints [START_REF] Piñeyro | The economic lot-sizing problem with remanufacturing and oneway substitution[END_REF] ensure that there are not more temporary workers than permanent workers. Otherwise, we assume that permanent workers should be on duty. Constraints [START_REF] Rong | Shift designs for freight handling personnel at air cargo terminals[END_REF] [START_REF] Strack | Comparison of heuristic procedures for an integrated model for production and distribution planning in an environment of shared resources[END_REF] dene the integrality or binary requirements.

To conclude the section, we give in the following complexity results for the PSP.

Proposition 1. The Packaging and Shipping Problem (PSP) is N P-hard.

Proof. The proof is given in Appendix A.

A three-phase matheuristic

To solve the PSP, we propose an algorithm based on a three-phase matheuristic. The choice for developing a matheuristic is motivated by the observation that the model introduced in Section 3 presents a natural decomposition into subproblems involving types of decisions ranging from tactical ones to operational ones. Our three-phase approach sequentially solves three subproblems of the PSP, in a way that the solution of each sub-problem is the input for the next phase. The solution of the third sub-problem is, in turn, a solution for the PSP. Following the classication of matheuristics proposed by Ball [START_REF] Ball | Heuristics based on mathematical programming[END_REF], our procedure falls into the decomposition approach category: sub-problems are sequentially solved to identify a feasible solution for the original problem.

In our three-phase approach, the rst phase solves a relaxation of the PSP model presented in Section 3. It determines the workers needed to process all the orders. In other words, for all h ∈ H and s ∈ S we x the values of variables z per hs and z temp hs . This phase leads to take the tactical decisions under aggregated operational constraints. The second phase determines the complete orders process planning and sets the reassignments and the postponements. Specically, it determines, for each h ∈ H, d ∈ D h , h ∈ H, v ∈ V, the values of variables x hv hd . This phase focuses at the operational level based on tactical decisions taken at the previous phase of the method. The solution provided by the second phase does not consider docks occupation minimization but provides a feasible solution for PSP. Therefore, the algorithm could be stopped after this phase.

If the algorithm is continued, the output of the second phase is used as input for the last phase which considers the truck movement policy and minimizes the dock occupation. This phase renes operational decisions to optimize the dock occupation. The dierent phases are detailed in the next sections. An outline of the three-phase procedure is given in Algorithm 1.

For each of the three phases, we propose a speed-up technique to decrease computation times. In particular, we compute two valid lower-bounds on the objective function for the models solved in phase I and phase II. Each of these lower bounds is used to dene a stopping criterion. Before starting the third phase of the algorithm, we implement an order aggregation procedure which groups all the orders with the same characteristics. Indeed, the reassignments and the postponements are determined in phase II, and the aggregation procedure does not reduce the set of feasible solutions as it will be further explained in Section 4.4.3.

This decomposition approach is based on the distinction of the dierent decisions regarding their nature. In the rst phase, the tactical decisions, namely, the workers needed for production, are determined. The second and the third phases focus on the operational decisions. First, we determine a complete and feasible planning, then we optimize the production planning again to minimize the quay occupancy. Sections 4.14.3 present the three phases of the algorithm. Section 4.4 presents the speed-up techniques. for all Feasible solutions found do

v∈V ( h∈ H h-h≥0 d∈D h- h f hvt (h-h)d + k vt h ) ≤ QN max + QV ∀h ∈ H, 0 ≤ t ≤ t v (23) (14)( 20
)
is a valid relaxation for model (PSP).

In this model, we change the objective function of (PSP I) compared to [START_REF] Absi | A two-phase iterative heuristic approach for the production routing problem[END_REF], and we substitute Constraints (6)(13) by Constraints (23).

Proof. By summing Constraints ( 6) over all modes in V, we obtain (using Constraints [START_REF] Ernst | Sta scheduling and rostering: A review of applications, methods and models[END_REF]):

v∈V ( h∈ H h-h≥0 d∈D h- h f hvt (h-h)d ) ≤ Q v∈V (w vt h + (1 -y vt h )) - v∈V k vt h
From Constraint (13) it follows:

v∈V

( h∈ H h-h≥0 d∈D h- h f hvt (h-h)d + k vt h ) ≤ QN max + Q( v∈V (1 -y vt h )) - v∈V k vt h
From Constraint [START_REF] Strack | Comparison of heuristic procedures for an integrated model for production and distribution planning in an environment of shared resources[END_REF] on the variables, it follows that the term v∈V k vt h is positive, then

v∈V ( h∈ H h-h≥0 d∈D h- h f hvt (h-h)d + k vt h ) ≤ QN max + Q( v∈V (1 -y vt h ))
Finally, the term v∈V (1 -y vt h ) ∈ {0, . . . , V } equals V when all the variables y vt h equal 0, i.e., when the process is ended. Then, we obtain:

v∈V ( h∈ H h-h≥0 d∈D h- h f hvt (h-h)d + k vt h ) ≤ QN max + QV
(PSP I) is solved with a commercial solver, and the solution is used to determine the number of workers assigned to each shift over the planning horizon.

4.2

Phase II -Reassignment and postponement

Based on the decisions obtained in phase I, the second phase of the algorithm determines the assignment of each order to a period and to a delivery mode (variables x hv hd ). The productivity capacity during each shift is known from phase I, i.e., the values of variables z per hs and z temp hs are now xed. Moreover, we do not minimize the platform occupation, i.e., the term Note that the solution obtained after phase II is a feasible solution for the (PSP) model: by construction, it satises Constraints (2) [START_REF] Strack | Comparison of heuristic procedures for an integrated model for production and distribution planning in an environment of shared resources[END_REF]. This solution can be used as an initial feasible solution in the last phase.

(2), (5)(13) (18), (

The solution provided by phase III is the nal solution obtained for the PSP.

Speed-up techniques

To speed-up the algorithm, we have developed two valid lower-bounds on the objective function values of models (PSP I) and (PSP II). The lower-bounds are given by the solution of two specic arc-ows problems. Since the computation of these lower-bounds follows the same lines, we detail only the computation of the lower bound for (PSP II).

Lower-bound for (PSP II)

We rst recall that in phase II, the objective function is given by the sum of the penalties due to the reassignments and the postponements plus the cost of used trucks. The lower-bound is obtained by solving the following relaxation of (PSP II).

(RPSP II) min h∈H v∈V h∈H v∈V

p h vv ξ hv hv + c truck h∈H v∈V ζ v h (29) D hv + h∈ H h-h≥0 v∈V ξ hv hv -ξ v h - h∈ H v∈V ξ hv hv = 0 ∀h ∈ H, ∀v ∈ V (30) ξ v h ≤ Qζ h v ∀h ∈ H, ∀v ∈ V (31) ξ hv hv ∈ N, ∀h, h ∈ H, ∀v, v ∈ V (32) ξ v h , ζ h v ∈ N, ∀h ∈ H, ∀v ∈ V (33)
where

• ξ hv hv represents the total volume of orders for period h and mode v treated on period h by mode v

• ξ v h represents the total volume of orders for period h and mode v that is not postponed or reassigned

• ζ v h represents the total number of required vehicles for mode v in period h and D hv is the total number of packages which should be prepared on period h and delivered by mode v.

Proposition 3. Model (RPSP II) is a relaxation of model (PSP II).

Proof. From Equations ( 4 vol hd x hv hd as the total demand of day h assigned to mode v that is delivered on period h by mode v. We then have

v∈V h∈ H v∈V ξ h h vv = d∈D h vol hd = v∈V d∈D h v hd =v vol hd = v∈V D h v = D h ∀h ∈ H (34)
where D h v and D h are respectively the total demand of day h originally associated with mode v and the total demand of day h. Note that since, for each h ∈ H, for each d ∈ D h , for each v ∈ V and for each h ∈ H there exists only one variable x hv hd equal to one, we can write Equations (34) as:

h∈ H v∈V ξ h h vv = d∈D h v hd =v vol hd = D h v ∀h ∈ H, ∀v ∈ V D h v - h∈ H v∈V ξ h h vv = 0 ∀h ∈ H, ∀v ∈ V (35)
From Equations ( 6), summing on t ∈ T we obtain:

t∈T h∈ H h-h≥0 d∈D h- h f hvt (h-h)d + t∈T k vt h = t∈T t>0 k v(t-1) h + Q t∈T u vt h ∀h ∈ H, ∀v ∈ V t∈T h∈ H h-h≥0 d∈D h- h f hvt (h-h)d + k vT h = Q t∈T u vt h ∀h ∈ H, ∀v ∈ V t∈T h∈ H h-h≥0 d∈D h- h f hvt (h-h)d ≤ Q t∈T u vt h ∀h ∈ H, ∀v ∈ V Let us now dene ξ h v = t∈T h∈ H h-h≥0 d∈D h- h f hvt (h-h)d and ζ h v = t∈T u vt h .
Then we obtain:

ξ h v ≤ Qζ h v ∀h ∈ H, ∀v ∈ V (36)
that are Constraints (31). ξ h v represents the total volume of packages that must be prepared in day h and delivered by mode v after postponing and re-aecting operations. ζ h v represents the number of vehicles needed to transport the ξ h v packages. From the denition of ξ h v we have:

ξ h v = t∈T h∈ H h-h≥0 d∈D h- h f hvt (h-h)d , ∀h ∈ H, ∀v ∈ V ξ h v = h∈ H h-h≥0 v∈V d∈D h- h v=v hd t∈T f hvt (h-h)d , ∀h ∈ H, ∀v ∈ V ξ h v = h∈ H h-h≥0 v∈V ξ hh vv , ∀h ∈ H, ∀v ∈ V h∈ H h-h≥0 v∈V ξ hh vv -ξ h v = 0, ∀h ∈ H, ∀v ∈ V (37)
where we have dened

ξ hh vv = d∈D h- h v=v hd t∈T f hvt (h-h)d
, that represents all packages originally assigned to day h -h and mode v that are prepared on day h (i.e., are postponed by h) and mode v.

Summing Equations ( 35) and (37) we obtain:

D h v + h∈ H h-h≥0 v∈V ξ hh vv -ξ h v - h∈ H v∈V ξ h h vv = 0 ∀h ∈ H, ∀v ∈ V (38)
that are Constraints (30). All the other constraints in the model (PSP II) are relaxed.

From Proposition 3 it follows that the value of the optimal solution of (RPSP II) is a lowerbound for (PSP II).

The model (RPSP II) is a special case of the multi-commodity capacitated network design problem where only one commodity has to be routed on the network, and capacities have to be respected or installed to satisfy the demand (see, for example, Gendron et al. [START_REF] Gendron | Multicommodity capacitated network design[END_REF]). In particular, (RPSP II) is equivalent to

(AF-RPSP II) min v∈V v∈V h∈H h∈H p h vv ξ hv hv + c truck v∈V h∈H ζ v h (39) Aξ = b (40) 0 ≤ ξ ≤ c(ζ) (41) ζ, ξ ∈ N (42)
Model (AF-RPSP II) denes an arc-ow problem on an oriented graph G = (N , A) where N = {s, t} ∪ N h v , and N h v contains a node n h v for each pair (v, h), v ∈ V, h ∈ H and

A = {(s, i)|i ∈ N h v } ∪ {(i, t)|i ∈ N h v } ∪ {(i, j)|i, j ∈ N h v , i = j}
A is the adjacency matrix of graph G. Vectors b and c are as follows:

b i =      -v∈V h∈H D h v if i = s v∈V h∈H D h v if i = t 0 otherwise
and,

c a =      D h v if a = (s, n h v ) Qζ h v if a = (n h v , t) v∈V h∈H D h v if a = (n h v , n h 1 v 1 ), i = s, t
The model (RPSP II) is solved with a commercial solver, and the optimal solution value gives a lower-bound for phase II. With respect to our testbed, the size of instances remains small, and optimal solutions are obtained almost instantly. Each time a feasible solution for model (PSP II) is identied and its value is equal to the lower-bound, the solution of model (PSP II) is stopped.

Lower-bound for phase I

The model (RPSP II) determines orders that require postponement or reassignment to minimize the number of vehicles, and it computes the resulting penalties. A valid lower-bound for phase I is obtained accordingly: an estimation of postponed orders over the horizon is computed to minimize the number of required workers. The problem can be formulated as an arc-ow problem similar to (AF-RPSP II). Due to similarities shared between both constructions we omit the details here.

Order aggregation for phase III

Phase II determines the quantities of orders assigned to each mode in each period. Based on these decisions, phase III looks for a packages loading plan, in other words, the quantities loaded at each slot and the required trucks movements, which minimize the docks occupation. To speed up the solution of phase III model, we aggregate orders. We group orders that

• have the same release date,

• have to be processed on the same period h ∈ H,

• have to be delivered using the same mode v ∈ V as a single order whose volume is the sum of individual order volumes. Orders in D h postponed by h > 0 are included in the volume of the single order created for period h + h associated with a release date equal to zero. Since these orders were available h periods before, they are available at the beginning of period h + h.

This aggregation can be performed since no postponement or reassignment is allowed during this phase. Postponement or reassignment must be done on the total volume of an order. Aggregation is not possible when postponement or reassignment is admissible since the packages associated with each order have to be known. This information is lost in case of aggregation.

Formally, for each h ∈ H, for each v ∈ V and for each t ∈ T we dene a unique order D t hv with a volume vol D t hv dened as follows:

vol D t hv =      d∈D h r hd =t vol hd x 0v hd + d∈D h- h h-h≥0, h>0 vol (h-h)d x hv (h-h)d if t = 0 d∈D h r hd =t vol hd x 0v hd if t > 0 ( 43
)
When t = 0, the volume of D t hv corresponds to the sum of volumes of all orders released exactly at t = 0 on the period h and processed on the same period, plus the volume of all the orders released during period h-h and postponed by h periods. When an order is postponed to a given period h, it is known at the beginning of period h. When t > 0, the volume D t hv corresponds to the sum of volumes of all orders released exactly at t > 0 on the same period h. Moreover, D t hv is characterized by its mode v D t hv = v, and its release date r D t hv = t. Let Dh denote the set of all aggregated orders. These orders are the input of the model solved by the commercial solver in phase III. Let us indicate with f hvt hd the variables corresponding to orders in Dh . A solution of phase III determines an operational planning for orders d ∈ Dh . The solution of the problem in terms of variables f hvt hd can easily be obtained by applying a greedy algorithm using the values of variables f hvt hd .

Computational results

This section discusses the eciency of the three-phase procedure we developed for the PSP.

First we describe the instances we created from data provided by an industrial partner (Section 5.1). The results on these instances are reported in Section 5.2.1. Sensitivity analyses of the three-phase algorithm with respect to slight modication of instances and with respect to dierent penalty proles are reported, respectively in Sections 5.2.2 and 5.2.3. In Section 5.2.4 we assess the performance of the lower-bounds introduced in Sections 4.4.1 and 4.4.2. Finally, in Section 5.2.5 we compare the three-phase algorithm with the solution of the (PSP) using a commercial solver.

Instance generation

Since the PSP is a new problem, we have to generate a set of instances to test the algorithm described in Section 4. The instances are based on real data provided by a logistics company operating in the e-commerce sector.

Each working day is identied by a "prole", i.e. a number of orders processed during that day. We dene three proles, named low, normal, high, respectively characterized by 1000, 3000, 5000 orders.

A list of common data is shared among the dierent instances. Specically, we consider a three-day horizon, H = {0, 1, 2}, whereas the order process can be postponed by one day, i.e., H = {0, 1}. Orders are received only during days 0 and 1. The third day is only used if the whole demand cannot be prepared during days 0 and 1. Each day consists of two shifts including eight time slots each.

Two delivery modes are available, the express mode and the standard mode. Trucks associated with the express mode leave the warehouse earlier than the other trucks, which are scheduled to leave at the end of the last shift. As an example, the express mode departure time is slot 12. This means packages can be loaded into vehicles until slot 11. Trucks have a capacity of 1300 packages and a xed cost of 650 Euros for both modes.

For permanent workers, the productivity is set to 40 packages per time slot, and the cost to 185 Euros. Temporary workers produce up to 30 packages per time slot and cost 210 Euros. Temporary workers are hired for at least one shift. We limit the number of permanent workers to 15 for each shift of each day.

The number of available docks is set to 10. The penalties for a postponement or a reassignment are as follows:

p hv hd =          0 if h = 0 and v = v hd , 1 if h = 0 and v = v hd or h = 1 and v = v hd , 2 if h = 1 and v = v hd , ∞ otherwise. ( 44
)
We consider nine types of instances associated with all possible prole combinations for day 0 and day 1 chosen among low, normal and high proles. For each type of instance, ve instances are generated randomly xing the values of the release date, the number of packages that constitute an order as well as their delivery mode. Order volumes are uniformly drawn among values {1, 2, 3} and release dates are drawn uniformly among the slots of the day. Modes are initially assigned to orders according to a uniform distribution.

Discussion

The algorithm was implemented in C++ with Visual Studio environment. The models presented in Sections 4.14.3 were solved with Cplex 12.6. All tests were performed on an Intel R Core TM i7-4600U CPU 2.10 GHz. We allowed a maximum computation time of 30 minutes for each phase of the algorithm. Moreover, the resolution in phases I and II is stopped if the optimality gap with respect to the lower bound provided by Cplex 12.6 or with respect to the lower bounds computed as explained in Sections 4.4.2 and4.4.1 is less than 2%. For all computational experiments the value of α in the objective function ( 1) is set to 1.

Results on the basic instances

First, we ran our three-phase algorithm on one instance of each type described in Section 5.1. Detailed results are reported in Table 2. Column Instance reports the name of the instance type as a couple corresponding to the proles of day 0 and 1. For each instance, we report results in four lines: the rst three lines correspond to each of the algorithm phases. The fourth line gives the total cost and the CPU time values.

Column Phase indicates the considered phase of the algorithm. Column Cost reports the cost of the objective function for each phase as well as the total solution cost. Note that the objective function at phase I takes into account the penalties that occur when setting the workforce. Then the sum of workers cost is only included in the value of the nal solution. The penalties are computed again in phase II and then contribute eectively to the nal solution cost. Columns workers Per and workers Temp indicate respectively the number of permanent and temporary workers. Note that these columns are empty for phase II and phase III, as they are determined during phase I. Column Pen reports the value of the sum of all postponement and reassignment penalties. Column Truck indicates the number of trucks needed to deliver the orders. Phase III does not modify the values in these columns, and the corresponding slots are left blank. Column Dock-slots reports the total number of time slots during which docks are occupied. The solution of the phase I model does not provide the values of Truck and Dock-slots columns (variables u vt h and w vt h are not present in model PSP I) while in phase II this term is not minimized. As a consequence the value is reported only for phase III, when it is optimized. Since we provide to the solver specic lower bounds in phase I and II, the reported gap is calculated using either the value of the linear relaxation or the corresponding lower-bound (see Section 4.4). The gap is reported only when it is strictly positive. When the slot is empty, an optimal solution (for that phase) is identied. Finally, column Time provides the CPU time in seconds.

For one type of instances and each of the three phases, a (local) optimal solution is obtained. For other eight types, phase II fails to reach the optimal solution within 30 minutes of CPU time. However, the optimality gap is less than 2% for six of them. Phase II reveals to be the bottleneck of the procedure.

For ve types of instances, phase I suggests hiring temporary workers even if the total availability of permanent workers for the rst two days has not been used. We recall that 15 permanent workers are available for each shift, for a total of 60 workers for day-0 and day-1. The penalty scheme considered guides the optimization through solutions that favor temporary workers hiring, rather than order postponement.

For seven types of instances, phase II can reduce the number of trucks that was rst determined in phase I. This is possibly due to the reassignment strategy. Note that the increase in penalty costs is always lower than the savings due to unused trucks. This result highlights the potential benets of incorporating postponement and reassignment into the process planning. Finally, phase III always reduces the number of dock-slots which is crucial for handling high activity peaks.

Algorithm behavior analysis on Normal-Low type instances

We ran our algorithm on ve dierent instances of the type Normal-Low. Since e-commerce enterprises often experience the same sequence of day proles, but with dierent orders quantities, we selected the Normal-Low sequence to analyze the sensitivity of the algorithm. Table 3 presents detailed results on the ve runs. Column headings correspond to those reported in Table 2. It can be seen that the results for dierent instances are equivalent. It is worthwhile to note that the solution time for the three phases does not vary signicantly among runs. We can conclude that our solution algorithm is not deeply impacted by the structure of the instance solved.

We made the same analysis for the other instance types solving each time ve instances, and we ended up each time with the same conclusions. Thus we omit to report detailed results on these instances. In this section we compare the results obtained according to dierent penalty schemes for the postponement and reassignment policies. In Table 4 we report results obtained when penalty values in Equation ( 44) are divided by 10. The algorithm is run on one instance of each type.

There are two main dierences with the results reported in Table 2. The rst is related to the number of temporary workers while the second one concerns the optimality gap. For the High-High instance, the obtained solution postpones the orders processed to the third day (day 2) with a consequent use of 3 permanent workers. Note that 3 permanent workers guarantee a production (40 packages per slot * 8 slots * 3 equals to 960) equal to 4 temporary workers (30*8*4), but cost 555 instead of 840, leaving room for a large postponement that is favored by the low penalization scheme considered.

Moreover larger optimality gaps are obtained in phase II. An explanation could be the following. Let us consider two orders d 1 and d 2 for the same day, with the same volume and the same release date. Let us suppose to have in hand the complete planning. Exchanging production of d 1 with d 2 would provide an equivalent planning. This leads to equivalent solutions which the solver needs to consider to prove optimality. When penalty is low, this symmetry is also present for the postponement and reassignment policies, making computation even harder.

Table 5 reports results when the penalty scheme proposed in Equation ( 44) is modied to move orders from the standard to the express delivery mode for free (even if associated with postponement). On one hand, earlier deliveries increase the company's image. On the other hand, a postponement coupled with a change to a faster mode leads to on-time deliveries. Similar observations as those formulated for Table 4 can be drawn. Low-cost reassignments and postponements make disadvantageous to hire temporary workers and increase solution symmetry. The latter leads to signicant optimality gaps that are reported in the table.

Lower-bound eectiveness

In Table 6 we report the deviations of the lower-bounds dened in Sections 4.4.14.4.2 on the instances considered in Table 2. Columns Instance and Phase are self-explanatory. Column Cplex gap reports the gap value between upper-and lower-bounds provided by Cplex 12.6 when the solution of the corresponding phase is stopped. Column LB gap indicates the gap value of the lower-bound computed solving the related arc-ow problem. When the gap is null, the cell is left empty.

An empty cell (a zero gap value) or a value lower than 2% in column LB gap associated with a value grater than 2% in column Cplex gap certies the eectiveness of the lower-bound used to stop the corresponding model solution. It can be seen that the lower-bound for the phase I (LB1) allows for an earlier stop of the computation 3 times, while the lower-bound for phase II (LB2) does it in 8 cases. When the optimal values are not reached, LB2 provides a better optimality gap compared to the one given by Cplex gap on all instances except the Low-Low type ones.

Comparison with a commercial solver

Last we report on the comparison between our algorithm and the commercial solver Cplex 12.6. The result on the complexity of the PSP suggests that only small size instances can be solved to optimality.

In Table 7 we compare the performances of the three-phase method against Cplex 12.6 on the same instances as those used to obtain results reported in Table 2. In Table 7 Columns Cost report the value of the solution obtained. Finally, columns Gap report the gap between both solutions. Negative gaps correspond to better solutions obtained by the threephase algorithm. A time limit of 1 hour of computation is given to Cplex 12.6. Note that the three-phase algorithm never runs for more than 1808 seconds. On the other hand, we allow 30 minutes for each phase of our algorithm even if only phase-II could use the whole allowed amount of time. Thus, giving to Cplex 12.6 1 hour of computation time allows for a fair comparison.

We can notice from Table 7 that when one of the days has a Low prole, Cplex 12.6 is competitive with respect to the quality of the solution obtained. For the Low-Low instance, it can even nd the optimal solution. On the other hand, the three-phase heuristic systematically provides better results on instances that consider days characterized by Normal and High productivity.

To assess the eciency of our procedure we run the three-phase procedure on 5 other instances for each combination of days with Normal and High proles. Table 8 reports the results obtained. The three-phase heuristic always provides a better solution and is always quicker. In two cases, Cplex 12.6 cannot even nd a solution after one hour of computational time, while the threephase heuristic provides one in a little longer than half an hour.

It can then be stated that Cplex 12.6 can be competitive as long as the instance is easy. On the other hand, when solving instances characterized by Normal or High activities, the threephase algorithm becomes necessary to obtain good quality results in reasonable computational times. To better evaluate the performance of the three-phase heuristic, we compare the results obtained with optimal solutions provided by the commercial solver Cplex 12.6. Optimal solutions can systematically be obtained (within a reasonable amount of CPU time) for Low-Low type instances. We thus ran our three-phase heuristic and Cplex 12.6 on 5 instances with a Low-Low prole. Results are reported on Table 9.

Three-phase

It can be noticed that the three-phase heuristic can always identify near-optimal solutions with a maximal optimality gap of 0.55%. Moreover, the heuristic procedure allows solutions to be obtained in computational times that are almost two orders of magnitude lower than those of the commercial solver.

Conclusions

In this paper, we introduced the Packaging and Shipping Problem (PSP) arising in e-commerce logistics. It consists in determining number of employees required to process a set of orders in a multi-day horizon setting. In addition, an operational planning has to be produced as well as a loading of the packages into trucks for deliveries that can be performed with dierent modes. We considered two strategies in order to obtain overall solutions with a lower cost: reassignment and postponement. The rst strategy consists in changing the delivery mode chosen by the customer to another available to decrease operational costs. The second strategy consists in processing the orders in a subsequent day rather than the day of arrival. These strategies generate penalties, but they can lead to hire fewer employees or to use fewer trucks and, as a result, to savings for the company.

We proposed a mathematical model for the PSP and proved that the PSP is NP-hard. It is then unlikely the PSP can be eciently solved to optimality within a reasonable time regardless the size of the instances (unless P = N P). We then proposed a three-phase matheuristic approach that allows us to deal with large real-life instances. Our approach exploits the structure of the PSP by sequentially solving three sub-problems to construct a feasible solution. We rst take the tactical decisions, xing the workforce for each day, and consequently, we determine the operational planning. Moreover, our approach is enhanced with speed up techniques based on lower-bounds for the subproblems.

We created a set of instances for the PSP based on data provided by our industrial partner. Instances with up to 5000 orders per day are then solved by the three-phase procedure that we proposed. Results show the eciency of the method which can provide high-quality solutions in a reasonable amount of time and performs signicantly better than the commercial solver whenever sequences of days with normal or high production activities are considered.

Future work could consider the stochastic nature of the problem. In this paper, we consider all order information to be deterministic. In real life, total demand is only forecast for the following days and, consequently, is subject to variations. Since we have interaction between decisions of consecutive periods, future demand uncertainty should be taken into account in the decision-making process. We plan to apply a rolling horizon based procedure that corresponds well to the dynamics of data acquisition and decision-making in e-fulllment and to investigate appropriate stochastic optimization techniques.

Acknowledgement

This work was supported by the French National Research Agency (ANR-Agence Nationale de la Recherche) and was part of project RESPET -Ecient network managing for the door-to-door transportation services (gestion de RésEaux de Services Porte à porte Ecace pour le Transport de marchandises). Thanks are also due to the referees for their valuable comments. Proof. We prove the N P-hardness of (PSP) by reduction from the knapsack problem (KP). Given a knapsack with volume B and a set N of N items, indexed from 1 to N , each with a volume b i and a value c i , the KP consists in selecting a subset N of N under the budget constraint which imposes that the total volume is less than or equal to B, and that i∈ N c i is maximized. It can be formulated as follows:

(KP) max x i ∈ {0, 1}, ∀i ∈ N (47)

where binary variable x i is equal to 1 if the i-th item has been selected, and zero otherwise. The objective function (45) is to maximize the value of the selected items. Constraint (46) is the budget constraint, while Constraints (47) impose variables to be binary.

For each instance of the KP we construct the following PSP instance. For each item i ∈ N , we construct an order d, such that d ∈ D 0 (i.e., it is an order associated with the rst period of the horizon), vol 0d = b i , r 0d = 0, and is assigned to a unique mode v. The horizon is made of two periods, i.e., H = {0, 1}. Orders received the period 0, can be processed during period 1, i.e., H = {0, 1}. Each period is covered by a unique shift indexed with zero, i.e., S = {0}. Only one slot is associated with the shift, T = {0}. No order is received during period 1. For sake of simplicity, in this section we omit the index related to the period and the shift as well as the mode index.

At most one permanent worker is available for each shift, namely, e max 0 = e max 1 = 1 with a null cost. The productivity is set to i∈N vol i for the permanent worker working during period 0 (this worker can process all orders arrived in period 0), and B for the permanent worker of period 1. On the other side, temporary workers have a null productivity and their cost is xed to a strict positive constant, i.e., 1. By construction, reassignment is not possible (only one mode is available). Postponing order d to period 1 generates a penalty

p h d = cd = -c d vol d if h = 1, 0 if h = 0. (48)
The cost of a truck is set to zero, i.e., c truck = 0. truck capacity is set to i∈N vol i : a truck can contain all the orders received in period 0. It is supposed that only one dock is available, N max is set to 1. Other time related parameters like the shift starting period are trivially xed. This transformation of a KP instance into a PSP instance is polynomial in time and takes O(|N |) operations.

For the obtained instance, the model (PSP) is reduced to (49)(67). Note that variables related to the truck management are not present in the objective function. Then, we can suppose that variables w, u and y xed to 1, which leads to Constraints (54)(58). It is trivial to see that problems dened by models (PSP-KP) and (KP) have the same optimal solution, and this concludes the proof.

min

Figure 1 :

 1 Figure 1: Warehouse sketch

  the number of workers required during each shift of the planning horizon. To achieve this aim, we solve a relaxation of the model (PSP) presented in Section 3. The relaxation does not consider the truck management issues, i.e., Constraints (6)[START_REF] Lee | A dynamic model for inventory lot-sizing and outbound shipment scheduling at a third-party warehouse[END_REF] are not taken into account. The relaxation of the model (PSP) is based on the following proposition. Algorithm 1 Three-phase algorithm 1: Phase I 2: Compute a lower-bound for model PSP I: LBI (Section 4.4

APPENDICESA

  Complexity of the PSP Proposition 1. The Packaging and Shipping Problem (PSP) is N P-hard.

Table 1 : Notation 3 Problem denition, notation and model

 1

		Set indexing periods in the planning horizon
	H	Set indexing postponement periods
	D h	Set indexing orders to be processed during period h
	V	Set indexing the delivery modes
	H	Periods in the planning horizon
	H	Postponement periods in the planning horizon
	D h	Number of orders to be processed during period h
	d	A specic order
	V	Number of delivery modes
	vol hd	Number of packages that compose order d of period h
	v hd	Delivery mode of order d of period h
	r hd	Release date of order d of period h
	p h v hd v	Penalty due to postponement to h and to re-assignment to v order d of period h
	t v	Departure slot of vehicles associated to mode v
	Q	Truck capacity
	c truck	Truck cost
	N max	Number of docks
	S	Number of shifts per period
	T	Number of slots per period
	T	Number of slots per shift
	start hs	Starting slot for shift s at period h
	end hs c per hs c temp hs	Ending slot for shift s at period h Cost of a permanent employee working at shift s of period h Cost of a temporary employee hired at shift s of period h
	prod per hs	Number of packages that can be prepared by a permanent employee during shift s of period h
	prod temp hs	Number of packages that can be prepared by a temporary employee during shift s of period h
	e max hs	Maximum number of permanent employees at shift s of period h

  hd x hv hd +

			(c per hs z per hs + c temp hs z temp hs )+
		h∈H s∈S
	+ c truck	u vt h + α	w vt h	(1)
	h∈H v∈V t∈T	h∈H v∈V t∈T
	s.t.

h∈ H v∈V t∈T

  Fix the number of workers based on the solution of model PSP I 14: Phase II 15: Compute a lower-bound for model PSP II: LBII (Section 4.4.1) 16: while Time limit not reached do

	Proposition 2. The following model :
		(PSP I) min	p h v hd v vol hd x hv hd +	(c per hs z per hs + c temp hs z temp hs )	(22)
		h∈H d∈D h h∈ H v∈V	h∈H s∈S
		s.t. (2)(5)
	6:	if Solution optimality proved or solution cost equal to LBI then
	7:	Go to Step 13
	8:	end if
	9:	end for
	10:	end
	11: end while
	12: end
	13: 17:	Solve model PSP II (Section 4.2)
	18:	for all Feasible solutions found do
	19:	if Solution optimality proved or solution cost equal to LBII then
	20:	Go to Step 26
	21:	end if
	22:	end for
	23:	end
	24: end while
	25: end
	26: Fix reassignments and postponements based on solution of model PSP II
	27: Phase III
	28: Aggregate orders (Section 4.4.3)
	29: while Time limit not reached do
	30:	Solve model PSP III (Section 4.3)
	31: end while
	32: end
	33: Disaggregate orders

  In the last phase, the platform occupancy is optimized, i.e., we minimize the number of slots during which vehicles are present at the docks. The workers to hire, i.e., the values of variables z per hs and z temp hs and the possible reassignments or postponements of orders, i.e., the values of variables x hv hd , are xed and are parameters of the model (indicated with χ hv hd ). The mathematical model solved in this phase is the following:

	4.3	Phase III -Dock management
		(PSP III) min	w vt h	(26)
		h∈H v∈V t∈T
		s.t.
			f hvt (h-h)d ≤ prod per hs ζ per hs + prod temp hs ζ temp
		h∈ h-h≥0 H	d∈D h-h v∈V

t∈T f hvt hd ≤ vol hd χ hv hd ∀h ∈ H, ∀d ∈ D h , ∀ h ∈ H, ∀v ∈ V

(27)

hs ∀h ∈ H, ∀s ∈ S, ∀t = start hs , . . . , end hs

  ), multiplying both terms by vol hd , we obtain:

		vol hd x hv hd =	vol hd =	vol hd ∀h ∈ H
	d∈D h h∈ H v∈V	d∈D h	v∈V d∈D h v hd =v
		vol hd x hv hd =	vol hd =
	v∈V d∈D h v hd =v	h∈ H v∈V	d∈D h	v∈V d∈D h v hd =v

h∈ H v∈V vol hd x hv hd = vol hd ∀h ∈ H, ∀d ∈ D h and summing up on the demands d ∈ D h we obtain:

vol hd ∀h ∈ H and, for v ∈ V, h ∈ H let us dene ξ h h vv = d∈D h v hd =v

Table 2 :

 2 Computational results on the basic instances

	Instance	Phase	Cost	Workers Per Temp	Pen Truck Dock-slots Gap Time
		I	2405 + 75 13	0	75	4	0.00%	11
	Low-Low	II	2677			77	4	1.95%	13
		III	35				35	0.00%	1
			5117						25
		I	4810 + 75 26	0	75	8	0.00%	4
	Low-Normal	II	4665			115	7	1.63% 163
		III	37				37	0.00%	3
			9512						170
		I	7290+75	36	3	75	11	0.43%	18
	Low-High	II	6775			275	10	1.73%	80
		III	51				51	0.00%	4
			14116						102
		I	4810	26	0	0	8	0.00%	14
	Normal-Low	II	4961			411	7	0.02% 177
		III	38				38	0.00%	7
			9809						198
		I	7030	38	0	0	12	0.00%	19
	Normal-Normal	II	6994			494	10	0.09% 1074
		III	42				42	0.00%	8
			14066						1101
		I	9485	49	2	0	14	0.58%	16
	Normal-High	II	8766			316	13	0.01% 481
		III	41				41	0.00% 150
			18292						647
		I	7290 + 2	36	3	2	10	1.40%	31
	High-Low	II	6539			39	10	0.60% 148
		III	44				44	0.00%	14
			13837						193
		I	9485+1	49	2	1	14	0.15%	33
	High-Normal	II	8807			356	13	0.71% 1261
		III	47				47	0.00%	15
			18339						1307
		I	12150	60	5	0	17	1.80%	34
	High-High	II	10560			160	16	0.00% 1758
		III	44				44	0.00%	16
			22754						1808

Table 3 :

 3 Algorithm performance on 5 instances with a Normal-Low demand prole 5.2.3 Analysis of penalty schemes

Table 4 :

 4 columns Cplex report the results obtained by the Cplex 12.6, while columns Three-phase report the results obtained by the proposed algorithm. Columns CPU report the computational time in seconds. Computational results with reduced penalties

	Instance	Phase	Cost	Workers Per Temp	Pen Truck Dock	Gap	Time
		I	2405 + 9.2	13	0	9.2	4	0.07%	5
	Low-Low	II	2641				4	1.58%	45
		III	36				36		1
			5082						51
		I	4810+25	26	0	25	10	0.36%	9
	Low-Normal	II	5224.1			24.1	8	6.07% 1800
		III	52				52		4
			10086.1						1813
		I	7030 +72.5	38	0	72.5	12	0.92%	7
	Low-High	II	7235.1			85.1	11	6.09% 1800
		III	67				67		3
			14332.1						1810
		I	4810.0	26	0	0	8		19
	Normal-Low	II	4679.3			129.3	7	1.92%	154
		III	40				40		8
			9529.3						181
		I	7030+30.4	38	0	30.4	12	0.03%	22
	Normal-Normal	II	6605.6			105.6	10	0.73% 1016
		III	39				39		10
			13674.6						1048
		I	9435 +49.1	51	0	49.1	16	0.52%	32
	Normal-High	II	9849.5			99.5	15	16.00% 1800
		III	56				56		11
			19340.5						1843
		I	7030 +48.9	38	0	48.9	11	0.24%	53
	High-Low	II	6574.8			74.8	10	1.15%	643
		III	44				44		21
			13648.8						717
		I	9435+48.1	51	0	48.1	14	0.51%	42
	High-Normal	II	9164.0			64.0	14	8.07% 1800
		III	45				45		38
			18644.0						1880
		I	11865+124.3 63	0	124.3	18	1.94%	100
	High-High	II	11255.7			205.7	17	8.06% 1800
		III	47				47		55
			23167.7						1955

Table 5 :

 5 Computational results when the delivery service is modied free of charge.

Table 8 :

 8 Comparison with Cplex 12.6 on instances not including a low demand day

Table 9 :

 9 Comparison with optimal solutions on Low-Low instances 5.2.6 Quality of the solution on Low-Low instances

			Cpu Cost Cpu Cost	Cpu
		5124	13	5099 1121 0.49% -98.84%
		5121	15	5094 702 0.53% -97.86%
	Low-Low	5104	15	5079 547 0.49% -97.26%
		5093	14	5066 1094 0.53% -98.72%
		5103	13	5075 523 0.55% -97.51%

  Moreover, since periods are constituted by only one period, variables k h become useless and Constraints (53) can be replaced by Due to construction, Constraints (67) are trivially satised (a truck can contain the full orders received in period 0). Finally, due to construction, we are sure that in the optimal solution only one x worker works each period (z per Constraints (60)(61) are trivially satised. Then the model reduces to 1} ∀d ∈ D 0 , ∀ h ∈ H (73)f h d ∈ N ∀d ∈ D 0 , ∀ h ∈ H(74)Replacing p h d with the expression given in Equation (48), the objective function (68) is (perfect relation between variables x and f is due to the fact that periods are made by only one time slot). Then the model is equivalent to (PSP-KP) max

	(z temp 0	= z temp 1	= 0). min	0 d∈D 0 h∈ = z per 1 d∈D 0 H p h d vol d x h = 1), while no temporary workers will be hired c d x 1 d (84) d (68) s.t. h∈ H x h d = 1 ∀d ∈ D 0 (85)
				s.t.	h∈ h∈	d = vol d ∀d ∈ D 0 d∈D 0 vol d x 1 d ≤ B H f h H x h d = 1 ∀d ∈ D 0 x h d ∈ {0, 1} ∀d ∈ D 0 , ∀ h ∈ H	(69) (86) (70) (87)
				f h d ≤ vol d x h d	∀d ∈ D 0 , ∀ h ∈ H	(71)
		d∈D 0 h∈ h∈ H f h d = vol d ∀d ∈ D 0 H p h d vol d x h d + (c per h z per h + c temp h f h d ≤ prod per ∀h ∈ H z temp h h h∈ H d∈D 0 h-h=0 h∈H h∈ H x h d = 1 ∀d ∈ D 0 f h d ≤ vol d x h d ∀d ∈ D 0 , ∀ h ∈ H f h x h d ∈ {0, d∈D 0 h∈ s.t. H p h d vol d x h d = d∈D 0 (p 0 d vol d x 0 d + p 1 d vol d x 1 d ) = d∈D 0 -c d vol d vol d x 1 ) d =	d∈D 0	-c d x 1 d	(72) (49) (50) (51) (52)
	h∈ h-h=0 H and Constraints (72) decompose into d∈D 0
							f 0 d ≤ prod per 0 ≤	vol d	(75)
				d∈D 0	d∈D 0
			Qu h ≤ Qw h ∀h ∈ H d∈D 0 f 1 d ≤ prod per 1 ≤ B	(57) (76)
	w h ≤ N max ∀h ∈ H Constraint (75) is always satised and can be removed. The model becomes	(58)
			h∈ h-h=0 H	d∈D 0 min	d∈D 0 f h d ≤ prod per h z per h + prod temp h -c d x 1 d	z temp h	∀h ∈ H	(59) (77)
			z per h ≤ 1 ∀h ∈ H z temp h ≤ z per h ∀h ∈ H s.t. h∈ H f h d = vol d ∀d ∈ D 0	(60) (61) (78)
			x h d ∈ {0, 1} ∀d ∈ D 0 , ∀ h ∈ H y h ∈ {0, 1} ∀h ∈ H h∈ d = 1 ∀d ∈ D 0 H x h z per h , z temp ∈ N ∀h ∈ H f h d ≤ vol d x h ∀d ∈ D 0 , ∀ h ∈ H d h f h f 1 d ≤ B d ∈ N ∀d ∈ D 0 , ∀ h ∈ H w h , k h , u h ∈ N ∀h ∈ H d∈D 0 x h d ∈ {0, 1} ∀d ∈ D 0 , ∀ h ∈ H	(62) (79) (63) (64) (80) (65) (81) (66) (82)
					f h d ∈ N ∀d ∈ D 0 , ∀ h ∈ H	(83)
							f h d ≤ Q ∀h ∈ H	(67)
				h∈ h-h=0 H	d∈D 0

d + k h = Qu h ∀h ∈ H (53) u h ≤ N max y h ∀h ∈ H

(54)

y h ≤ u h ∀h ∈ H (55) Qu h ≤ Qw h + Q(1 -y h ) ∀h ∈ H

(56)

Note that Constraints (80) are never strict, and inequalities can be changed to

f h d = vol d x h d ∀d ∈ D 0 , ∀ h ∈ H

http://www.fevad.com/bilan-2016-e-commerce-france-cap-70-milliards-a-ete-franchi/


Julia Eberlen

Annique Smeding

Jean-Charles Quinton

AM I COLD?

Overall objective: Can we learn a stereotype about ourselves if... 1. the stereotype is generic (compared to specific, [1]) 2. the stereotype is negative 3. self/group belonging is experimentally induced 4. there is no prior knowledge about the groups

BACKGROUND

RESULTS

Self-Other Categorization:

• Minimal group paradigm successful • Effect reversed after (negative) stereotype learning • similar results for counterbalanced condition

Stereotype -(In-)Group Association

• Self-positivity bias before learning • Reversed after learning • Control: no change • similar results for counterbalanced condition

CONCLUSION & CHALLENGES

• Self-positivity bias trumps learning of negative ingroup-stereotype • Interpretation Framework: Situated Social Cognition [START_REF] Smith | Socially situated cognition: Cognition in its social context[END_REF] • Control Version Self -Warm?

• Implicit Assessment?

• Stronger Self -Avatar association manipulation?

PILOT & SIMULATION

Pilot: is the minimal group paradigm [START_REF] Tajfel | Experiments in intergroup discrimination[END_REF] sufficient to induce self-positivity bias?

• Avatar assignment: Big5 items [START_REF] Plaisant | Validation par analyse factorielle du big five inventory français (BFI-fr)[END_REF] bogus feedback

• Self-other categorization • Warm-cold categorization • N =

Figure 1 : 2 :

 12 Figure 1: Self-Other Classification Figure 2: Warm-Cold Classification

  40 • Largely unsuccessful Simulation: Based on successful selfavatar links • Successful = 60/80 items classified as self • Identify smallest effect of interest • Feasibility of full design • Power METHOD • N = 80 • Order: 1. Avatar assignment 2. Self-other categorization 3. Warm-cold categorization 4. Stereotype learning 5. Self-other categorization 6. Warm-cold categorization • Stereotypes: pretested warm and cold adjectives • Groups: Family A & B Fribbles, recoloured, (tarrlab.org) • Control: Animals (POPORO) Group A Group B


Alisée Bruno

Annique Smeding

Jean-Charles Quinton

Benoit Dompnier

Measuring achievement goals: A methodological contribution based on mouse-tracking data

come

Developing and sustaining students' motivation to learn represents an essential educational challenge for teachers, parents, and policy-makers alike, because this motivation should foster learning and achievement. [START_REF] Hulleman | A Meta-Analytic Review of Achievement Goal Measures: Different Labels for the Same Constructs or Different Constructs With Similar Labels[END_REF] Although it has been assumed that the motivation to learnor mastery (approach) goal endorsementand to perform better than othersor performance (approach) goal endorsement -predicts learning and achievement, findings are still inconsistent in the literature. One explanation may be related to response biases, and more specifically social desirability bias.

The aim of this research was to develop an implicit measure of mastery and performance goals by recording computer mouse movements. Mouse-tracking allows measuring participants' hand movements, which reflect underlying cognitive processes, including neural competition involved in decision making dynamics. [START_REF] Freeman | MouseTracker : software for studying real-time mental processing using a computer mouse-tracking method[END_REF] In this study, Area Under the Curve (AUC) was used to measure implicit achievement goals. Convergent validity between the implicit measure and achievement goals' explicit, self-reported equivalent measures was tested. 2) Implicit measure (Imp).

3) Self report measure (Exp). 2x2 scale of achievement goals Elliot & Murayama (2008)

Results

Discussion

  Participants. Study 1 : 142 psychology students (Mage = 20.3, SD = 3.4) Study 2 : 133 psychology students (Mage = 20.9, SD = 2.78) 1) Pre-test: 10 words for performance goals (COMPETITION) and 10 for mastery goals (COMPREHENSION), χ² (p < .05).

Figure 1 .

 1 Figure 1. Sample item

Table 1 .

 1 Values above the diagonal refer to the Study 1 / values below the diagonal refer to Study 2. * indicate significant correlations. p <.05 *. p <. 001 **.

	Appro
	ach

Measures partially converge Method studies 1 & 2 Introduction Self report measure Implicit indicators appear to partially converge with explicit measure but not systematically in the 2 studies for each goal MT-based indicators are promising for achievement-goal research, but more empirical work is needed to consolidate current findings

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My goal is to learn as much as possible in my studies.