NETWORK FLOW ORIENTED APPROACHES FOR VEHICLE SHARING RELOCATION PROBLEMS

Managing a one-way vehicle sharing system means periodically moving free access vehicles from excess to deficit stations in order to avoid local shortages. We propose and study here several network flow oriented models and algorithms which deal with a static version of this problem while unifying preemption and non preemption as well as carrier riding cost, vehicle riding time and carrier number minimization. Those network flow models are vehicle driven, which means that they focus on the way vehicles are exchanged between excess and deficit stations. We perform a lower bound and approximation analysis which leads us to the design and test of several heuristics. One of them involves implicit dynamic network handling.


INTRODUCTION
Vehicle Sharing systems [14,20,28,29] are emerging mobility systems which aim at compromising between purely individual mobility and rather rigid public transportation.Such a system is composed of a set of stations, at which free access vehicles are parked.Those vehicles may be bicycles or electric cars.There exists a special station called Depot, in which a set of carriers (trucks, self-platoon convoys, . . . ) are waiting: they periodically exchange vehicles between the stations and eventually provide them with additional vehicles.A trend is to make the system be a one-way system: users are not imposed to give vehicles back at the stations where they picked them up.This feature makes the system more attractive, but also raises the eventuality of unbalanced situations: stations may become overfilled other under-filled, provoking local shortages or making users unable to give their vehicles back.This makes arise two decision problems:

NETWORK FLOW ORIENTED APPROACHES FOR VEHICLE SHARING RELOCATION PROBLEMS
-a strategic level problem [8,9,14,25,29], about the way stations are located and capacitated and about the pricing of the system [29].One must simultaneously maximize some global Access Demand, and minimize costs which involve not only infrastructure costs but also running costs related to periodical vehicle relocation.Though this Vehicle Sharing Station Location (VSSL) problem looks like a standard Facility Location problem [13,19,22,26,30], addressing it is difficult in practice, since estimating the way Access Demand depends on the way stations are located can only be done through rough approximation.
-an operational (or tactical) level problem (see [5,6,7,10,11,18,20,21,23,24,25,27]), about the way vehicles are periodically moved from excess to deficit stations in order rebalance the system (Relocation Process).Performing this process while meeting both economic and quality of service purposes means addressing a Vehicle Sharing Relocation problem (VSR).
This contribution is devoted to the operational level, that means to the VSR Problem, which also appears as a slave sub-problem in any bi-level VSSL formulation.Related VSR models may be: static: at some time during the process, excess and deficit stations are identified, together with excess and deficit amount of vehicles.One must make the carriers move vehicles from excess to deficit stations, while minimizing some operational cost, function of the vehicle riding time, of the number of carriers and of the carrier riding time, while keeping the total duration of the process from exceeding a makespan threshold; dynamic: one knows, for every station x, at which time vehicles are going to be demanded or given back by the users.Then one schedules the carriers in order to meet most demands and avoid any unbalanced situation, while minimizing some operational cost; on line: the context is the same as in the dynamic case, but knowledge about demands is incomplete and uncertain.
Preemption may be allowed: a carrier may load some vehicle at some station and drop it at another station, before some other (or eventually the same) carrier comes, loads it again and brings it until a third station.
In practice, VSR models have to be handled on line [4,18,20,23]: relocation is performed in a continuous way and, at any time, knowledge about customer requests is incomplete and uncertain.Still, as it is usual when it comes to scheduling or routing decisional problems, it is appropriated, in order to better understand the problem and design efficient strategies, to first deal with a static or eventually a dynamic version.VSR literature makes appear several static models (see [3,4,10]) which have been addressed through metaheuristic schemes or through hierarchical decomposition into a routing master model, handled through local search, and a load/unload network flow slave model [1].Most authors impose restrictions on the number of carriers and the components of the cost function [5,6,7,10], most often reduced to the carrier riding time.
Some authors consider time indexed requests [11,27] and address the resulting model through a Benders decomposition scheme.None of them links non preemption and preemption, while, even if not practical from the point of view of a central manager, preemption may be used as a relaxation of non preemption and help into designing algorithms.
More, one may notice that a common feature of the above mentioned static and dynamic models is that they are carrier oriented [24,25], in the sense that they focus on the construction of the recollection tours which are run by the carriers, and consider the routing of the vehicles inside the carriers as a kind of slave object [10].Such an approach may be criticized because of the lack of a backward link between the master carrier tour collection and the vehicle sub-problem: the search for the master carrier tour collection is then performed in a somewhat blind way (genetic algorithms, . . .).
So we adopt here the opposite point of view and consider that performing a relocation process means routing vehicles from excess stations to deficit ones in a way which make them share, as often as possible, related carriers.This leads us to propose models which stress the role played by the vehicle network flow induced by the relocation process, and then derive alternative approaches to carrier driven ones, which we say to be vehicle driven: the vehicle routing strategy becomes the master object, which determines in turn the carrier routes.This allows us to link preemptive and non preemptive VSR models and point out that understanding preemption as a relaxation of non preemption leads us to a common Network Flow framework.
The paper is organized as follows.We first provide (Section 2) a general framework for both preemptive and non preemptive static VSR, which mixes several performance criteria: economic cost of the relocation process (carrier number and carrier riding cost), and quality of service (unavailability of the vehicles during the process).We reformulate resulting models as Network Flow models, making appear preemption as a relaxation of non preemption.We keep on (Section 3) by performing a lower bound analysis of this VSR model.In Section 4, we propose a first heuristic scheme, which considers the way vehicles are distributed from excess stations to deficit ones as the master object of a Min Cost Assignment/Pick up and Delivery hierarchical decomposition scheme, and state an approximation result for what we call the min-cost assignment strategy.In Section 5 and 6, we propose and test heuristics, which deal with aggregated vehicle and carrier flow vectors and turn them into solutions of respectively Non Preemptive and Preemptive VSR.One of those heuristics involves the implicit management of large size dynamic network.

VSR (Vehicle Sharing Relocation Problem) Instances:
We consider here a set X of stations, one of them being a specific station Depot.Any station x is provided with a coefficient v(x), which tells us that v(x) vehicles are in excess at station x: if v(x) is strictly negative, then carriers need to bring -v(x) vehicles to station x (x is then said to be a deficit station); if v(x) is strictly positive, then x is an excess station and carriers have to remove v(x) vehicles from x; if v(x) = 0 then x is said to be neutral.Carriers are initially located at Depot and they all have a same capacity CAP.We suppose that x∈X v(x) = 0, which means that some stations may be used to bring additional vehicles to the system, or, conversely, to remove some of them.We also suppose that Depot is neutral.Any station x is provided with a capacity C(x).DIST denotes the X.X time matrix: DIST x,y is the time required for a carrier to go from station x to station y.T-Max is the maximal makespan of the relocation process: the total time for this process cannot exceed T-Max.By the same way, COST denotes the X.X carrier cost matrix: COST x,y is the integrated cost (energy, human resource, . . . ) induced by a move of a carrier from station x to station y, when this move is performed in DIST

Non Preemptive VSR Model
A VSR tour is a finite sequence Rout e = {x 0 = Depot, x 1 , . . ., x n( ) = Depot } of stations, which is called a route, given together with a loading strategy, that means with 2 sequences of coefficients whose meaning is: a carrier which follows the route Route loads, at time T i , L i vehicles at station x i (unloads in case L i < 0).The length of Route in the COST sense is given by L-COST( Route ) = j COST x j,x j +1 .The length of Route in the DIST sense is given by L-DIST( Route ) = j DIST x j,x j+1 .The cost L-E-COST( ) of is given by: L-E-COST( ) = L-COST( Route ) + Idle-Cost.(Tn( ) − T 0 ).For any i, we denote by L * i = j =0..i L j the load of the carrier when it leaves station x i .This VSR tour is Non Preemptive VSR feasible if: ( E 1 ) Explanation: (E1): A carrier needs at least DIST xi,xi+1 time units to go from x i to x i+1 ; (E2, E3): Current carrier load L * i cannot exceed the capacity CAP, and this load is null at the end of the tour; (E4): loading (unloading) operations are respectively restricted to excess (deficit) stations, which means that we impose a given vehicle to be moved from an origin station to a destination station by exactly one carrier (Non Preemption).

Remark 2 (About MIP Models and Complexity).
Modeling VSR through a MIP (Mixed Integer Linear Program) is possible, but inefficient.The reason is that there is no a priori bound about the number of times a given station is going to be visited by a same carrier.As for complexity, in the case when K = 1 (α very large), v(x) values are equal to 1 or −1, CAP = 1 and δ = 0, our problem is equivalent to the Travelling Salesman Problem set on a bipartite graph (the excess stations on one side and the deficit ones on the other side), which is NP-Hard.Non Preemptive VSR also contains the Uncapacitated Swapping Problem, which is also NP-Hard (see [1]).

Route
Let us suppose now that we are provided with a collection * Route = { Rout e (1), . . ., Route (K )} of K carrier routes, all with length ≤ T-Max.Following [10], we define a network H ( Route ) as follows (see Fig. 1): • Nodes of H ( * Route ) are: • copies of the nodes x(k) j of Route (1), . . ., Route (K ) considered as being all distinct; • a source s and a sink p; • nodes Exc(x), x ∈ X , excess nodes; • nodes Def (x), x ∈ X , deficita nodes.
• Arcs e of H ( Route ) and related costs C e are: • Route arcs e = (x(k) j , x(k) j +1 ) of the routes Route (k), with cost C e = DIST x(k) j,x(k) j +1 ; • Excess arcs e = (Exc(x), x(k) j ), x ∈ X , x excess, such that the image in X of x(k) j is x, with C e = 0; • Deficit arcs e = (y(k) j , Def(y)), y deficit, such that y(k) j is y, with C e = 0; • Input arcs e = (s, Exc(x)), x excess, and output arcs e = (Def (y), p), y deficit, with C e = 0.This construction yields, as in [10]: Lemma 0. Any optimal solution (if it exists) of Load-NP-VSR provides us with an optimal loading strategy related to the route collection * Route .
Proof.Any loading strategy related to the tour collection * may be turned into a feasible solution of Load-VSR whose cost is exactly the vehicle riding time: k , j (DIST(x(k) j , x(k) j +1 ).L * j ).Conversely, any flow vector Z which is a feasible solution of Load-NP-VSR can be interpreted as a loading strategy.
It comes that Non Preemptive VSR may be reformulated:

Preemptive VSR Model
In case preemption is allowed, then we say that the VSR tour is preemptive VSR feasible if (E1, E2, E3) are true.Besides, for any collection * = ( (k), k = 1..K ≤ K-Max) of such non-preemptive feasible tours, we set, for any time value t , and any station x: Clearly, H ( , x, t ) denotes the number of vehicles which are really located in station x at time t after all loading/unloading transactions have been performed.Then we say that the collection E5) holds and if, for any time value t and any station x: 0 ≤ H ( , x, t ) ≤ C(x).
( E 6 ) (E6) expresses the fact that, at any time t , the number of vehicles currently located at x is non negative and cannot exceed the capacity of the station x.Then we may set: Preemptive VSR Model: {Compute a preemptive VSR feasible tour collection * = ( (k), k = 1 . . .K ) such that (E1, E2, E3, E5 and E6 hold) and which minimizes the following global cost:

Remark 3. The use of preemption leads to the introduction of synchronization mechanisms.
A carrier k which arrives at some station x may wait for another vehicle k before leaving x.So L-E-COST cannot any more be replaced by L-COST in above Global-Cost.The role of the Extended Cost Hypothesis is that there is no difference, from the Global-Cost point of view, between moving from some station x until some station y according to a maximal speed strategy and next waiting some time t at y, and moving from x to y at a reduced speed in order to arrive in y with a delay t .Also, the vehicle riding time k j (DIST x(k) j,x(k) j +1 .L * j ) quantity expresses the time vehicles spend into the carriers: in case some carrier k arrives to some station x at time T and leaves it at time T +t , vehicles unloaded at time t and loaded again at time T +t (provided C(x) is large enough) are not involves in this vehicle riding time since they are available for users between T and T + t .Remark 4. Taken together, above Non Preemptive and Preemptive VSR models extend [5,7,10,11,14,25], since they unify preemption and non preemption, and mix carrier numbers, vehicle riding time and carrier riding cost into a same criterion.Still, in case of non feasibility, we do not take into account, as in [10,11], the eventual deviation between the wanted balanced state and the true state of the system at the end of the process.

A Network Flow Framework
Let us recall that a flow vector defined on a network G = (N , A), with node set N and arc set A, is a rational (or integral) valued A-indexed vector g such that, for any node z, the following flow conservation law holds: e such that origin(e) = z g e = e such that destination(e) = z g e Let us now suppose that all values DIST x,y are integral (it is always possible to do it).Then we derive from the VSR instance ( • Input arcs (s, (x, 0)) and ((x, T-Max), p), with null vehicle and carrier costs; • idle arcs ((x, t ), (x, t + 1)) Out , with null vehicle and carrier costs; • carrier-idle arcs ((x, t ), (x, t + 1)) In with unit vehicle costs and carrier costs equal to β.Idle-Cost if x = Depot and 0 else; • active arcs ((x, t ), (y, t +DIST x,y ), with vehicle costs equal to δ.DIST x,y and carrier cost equal to β.COST x,y ; • backward arc ( p, s) with null vehicle costs and carrier costs equal to α.
Then, we may set on this network the following multi-commodity flow model:

Network-Flow-VSR Model: {Compute non negative integral flow vectors F and f , respectively carrier and vehicle flow vectors, such that:
• For any idle arc e = ((x, t ), (x, t + 1)) Out , f e ≤ C(x) and F e = 0; (E7) • For any carrier-idle arc e = ((x, t ), (x, t + 1) I n , f e ≤ C AP.F e ; ( E 8 ) • For any active arc e = ((x, t ), (y, t + DIST x,y ), f e ≤ C AP.F e ; ( E 9 ) Proof.We first notice that, if a feasible preemptive VSR tour collection * = ( (k), k = 1..K ≤ K-Max) is given, then the Extended Cost Hypothesis implies that inequalities (E1) may be supposed to be tight in case x i = x i+1 .It comes that may be turned into a feasible Network-Flow-VSR solution F, f , with same cost, by setting: - -For any carrier-idle arc e = ((x, t ), (x, t + 1)) In : • F e = number of carriers k located in x between t and t + 1 according to the tours (k); • f e = the sum of all quantities L * (k) j , taken for all carriers k as above and j such that x(k -For any idle arc e = ((x, t ), (x, t + 1)) Out : F e = 0 and f e = H ( , x, t ); -For any active arc e = ((x, t ), (y, t + DIST x,y ), x = y: • F e = number of carriers k such that (k) involves a move from x to y at time t ; • f e = sum of all L * (k) j , for all k as above and j such that x(k -For any arc e = (s, (x, 0))(e = (Depot, T-Max), p), F e and f e are defined according to (E10) and (E11).
Conversely, if (F, f ) is some Network-Flow-VSR feasible solution, then we know that F may be decomposed as a sum of {0, 1}-valued flow vectors Let us now try to extend Lemma 0 to this framework.In order to do it, we consider a carrier flow vector F and denote by X (F) the node subset of X T-Max which contains s, p, all nodes (x, 0) and (x, T-Max), together with all nodes (x, t ) which are origin or extremity of some arc e = ((x, t ), (y, t + DIST x,y ), x = y, such that F e = 0. We provide X (F) with an arc set E(F) which contains arcs (s, (x, 0)), (x, T-Max), p), x ∈ X , as well as: -related active arcs e = ((x, t ), (y, t + DIST x,y ), x = y; extended idle arcs ((x, t ), (x, t )) In and ((x, t ), (x, t )) Out , with t, t such that no (x, t ) exists in X (F) such that t < t < t ; those arcs are provided with vehicle-cost values respectively equal to (t − t ) and 0; Since values F e defined on idle arcs e = ((x, t ), (x, t + 1)) In can be turned in a natural way into values F e defined on extended idle arcs ((x, t ), (x, t )) In , F may be viewed as a flow vector on the network (X (F), E(F)).Then we set: Load-P-VSR: {Compute, on the network (X (F), E(F)) a non negative flow vector f , such that: • For any active arc e = ((x, t ), (y, t + DIST (x,y) , x = y and any extended-idle arc e = ((x, t ), (x, t )) In we have f e ≤ CAP.F e ; • For any station x, f (s,(x,0)) = Sup(v(x), 0) and f ((x,T-Max), p) = Sup(−v(x), 0); • f minimizes the linear cost e∈E f e .Vehicle-Cost e .}This allows us to state the following extension of Lemma 0 (proof left to the reader): Lemma 1. F being given, solving Load-P-VSR provides us with an optimal loading strategy.

VSR LOWER BOUNDS
We propose here 2 classes of easy to compute vehicle driven lower bounds for the VSR Problem: the first one relies on Min-Cost Assignment models which separately bound the active carrier number, the carrier riding cost and the vehicle riding time.The second one directly derives from the previous Network-Flow-VSR model.

Min-Cost Assignment Based Lower Bounds
Let us consider the following ILP models: VMCA Vehicle-Min-Cost-Assign: {Compute integral vector Q = (Q x,y , x excess, y deficit) ≥ 0, such that: • For any excess station x, y deficit station Q x,y = v(x) • For any deficit station y, x excess station Q x,y = −v(y) • Minimize x,y DIST x,y .Q x,y } LB-VMCA denotes the related optimal value, which may be computed while relaxing the integrality constraint on the vector Q.In any case (preemption or not), LB-VMCA provides us with a lower bound of the vehicle riding time: k j (DIST x(k) j,x(k) j +1 .L * j ).
CMCA Carrier-Min-Cost-Assign: {Compute integral vector R = (R x,y , x, y stations) ≥ 0, such that: • For any neutral station x = Depot, y R x,y = 0 = y R y,x • For any excess station x, CAP.y R x,y = CAP.y R y,x ≥ v(x) • For any deficit station y, CAP.x R x,y = CAP.x R y,x ≥ −v(y) UCMCA Unit-Carrier-Min-Cost-Assign: {Compute rational vector R = (R x,y , xstations) ≥ 0, such that: • For any neutral station x = Depot, y R x,y = 0 = y R y,x • For any x, y, both deficit or both excess, R x,y = 0 • For any y deficit, R Depot,y = 0 and for any x excess, R x,Depot = 0 • For any excess station x, y deficit or Depot R y,x = y deficit R x,y = v(x) • For any deficit station y, x excess or Depot R y,x = x excess R x,y = −v(y) • y excess R Depot,y = y R y deficit,Depot = 1

Projected Flow Lower Bound
We derive from the dynamic network G T-Max = (X T-Max , E T-Max ) of Section 2.4 a projected network G Proj = (X Proj , E Proj ) as follows (see Fig. 3): • X Proj = X ∪ {s, p} where nodes s and p are additional nodes source and sink; • The restriction of G Proj to X is a complete network: any arc e = (x, y) is provided with a carrier cost CC e = β.COST x,y + (α/T-Max).DIST x,y and with a vehicle cost CV e = δ.DIST x,y .
• There is an arc (s, x) from s to any excess station x, with null carrier and vehicle costs; • There is an arc (y, p) from any deficit station y to p, with null carrier and vehicle costs; • There is a backward arc ( p, s), with null carrier and vehicle costs.Then we set:

Projected-VSR-Flow Model: {Compute on the network G Proj two integral flow vectors H and h such that:
• For any arc e = ((x, y), x, y = s, p, h e ≤ CAP.H e (E18)

• For any excess (or neutral) station x, h (s,x) = v(x) and for any deficit station y, h
• y H Depot,y = y H y,Depot ≥ 1 (E20) • Remark 6.The Projected-VSR-Flow model does not solve our VSR problem, even according to its preemptive version.For instance one may consider a station set X = {Depot, A, B, C}, a carrier flow H related to the route (Depot, A, B, C, A, Depot) followed by 1 carrier with capacity 1, and a vehicle flow h which routes 1 flow unit from excess station C to deficit station B. Then the carrier cannot deliver its load in B before picking it up in C. Remark 7. LB-Proj-Flow value provides us with a better lower bound than the LB-MCA lower bound of Theorem 3. Still, Projected-VSR-Flow is a complex NP-Hard model, whose rational relaxation yields a poor lower bound as soon as CAP is large.The Lagrangean relaxation of the coupling constraint (E18) yields a Lagrangean value Sup λ∈ (

•
= {λ such that the restriction of the graph G Proj to X does not contain any negative But, because of the total unimodularity of flow constraint matrices, this value is the same as the value obtained by performing Lagrangean relaxation of (E18) on the rational relaxation of Projected-VSR-Flow.That means that the above Lagrangean value does not improve the standard relaxation of the integrality constraint.

A VEHICLE MIN-COST ASSIGNMENT BASED HEURISTIC FOR NON PREEMPTIVE VSR
We focus here on Non Preemptive VSR, and derive from the LB-MCA lower bound a decomposition of this problem into a Master Vehicle-Min-Cost Assignment problem and a Slave Pick-up&Delivery (PDP) Problem.

MCA/PDP Decomposition
Let us recall that a Pick-up&Delivery instance (see [2,12,17]) is defined by: • a set J of requests j = (o( j ), d( j ), λ( j )), where o( j ), d( j ) and λ( j ) are respectively the origin, the destination and the load of j ; N denotes the set of all nodes o( j ), d( j ), j ∈ J , augmented with a Depot node and considered as pairwise distinct; these requests have to be served by trucks, initially located in Depot and all with capacity CH; • 2 distance matrices D and CS, indexed on the set N .N and a threshold D-Max; A collection ρ of truck routes ρ(m), m = 1 . . .M defined on the set N is a feasible PDP solution if: • every request j is serviced by some truck m : m first loads λ( j ) at o( j ) and unloads it into d( j ); • the load of a truck never exceeds capacity CH; • the D-length of ρ(m) of any truck route λ(m), m = 1..M, never exceeds D-Max.
It is an optimal PDP solution if it is feasible and minimizes a quantity: where D-Ride( j ) is the D-length which is run by load λ( j ) inside a truck.A Load-Split PDP instance is defined the same way, but every loads λ( j ) may be split into a sum λ( j , of several sub-loads, which are separately handled. Though Load-Split PDP is NP-Hard, it may be in practice efficiently handled through a GRASP-VNS (Greedy Randomized Adaptative Search + Variable Neighborhood Search) process based upon Insert/Remove operators: -Insert operator: Inserting request j = (o( j ), d( j ), λ( j )) into some truck route ρ(m) means: • computing 2 insertion nodes x and y in ρ(m), and some sub-load λ ≤ λ( j ); • inserting o( j )(d( j )) between x(y) and its successor in ρ(m); • adding λ to the current load of ρ(m) between x and y, and updating λ( j ); -Remove operator: Delete o( j ) and d( j ) from ρ(m) and update the load of m and the λ( j ) value accordingly.
Then related GRASP-VNS scheme comes as follows:

PDP GRASP-VNS Algorithm
Randomized Initialization: While all requests have not been inserted do Randomly pick up some non inserted request j ; Compute (in a heuristic way) truck parameter m, together with insertion parameters x, y ∈ ρ(m), and λ ≤ λ( j ) in such a way that related insertion is feasible and such that (bi-criteria choice): • the induced increase of PDP-COST(ρ) is the smallest possible; • λ is the largest possible; Local Search Loop: Identify a set J 0 ⊆ J of poorly inserted requests; Remove J 0 from J and reinsert it according to the same process as in the initialization; Update the current best solution ρ * = (ρ(m), m = 1..M); Update Stop.
Let us now come back to our Non Preemptive VSR instance, and suppose that, for some instance (X, v, C, CAP, T-Max, DIST, COST), we know, for every pair (x, y), x excess, y deficit station, which quantity Q x,y has to move from x to y.Then, we only need to solve the Load-Split PDP instance defined by: • Requests j are all 3-uples (o( j ) = x, d( j ) = y, λ( j ) = Q x,y ), taken for all pairs x, y such that Q x,y = 0; One may conjecture that it is possible to impose assignment vector Q to be an optimal solution, for some cost vector U = (U x,y , x Excess, y Deficit) ≥ 0, of the following VMCA(U ) (Vehicle Min-Cost Assignment) model:

VMCA(U):
{Compute integral vector Q = (Q x,y , x excess, y deficit stations) ≥ 0, such that: • For any excess station x, y deficit station Q x,y = v(x); For any deficit station y, x excess station Q x,y = −v(y); • Minimize x,y U x,y .Q x,y } vector Q, a request set Req(U ) = {r = (x, y, Q x,y ) such that Q x,y = 0} and a Non Preemptive VSR solution * , whose global cost Global-Cost( * ) may be distributed among requests (x, y, Q x,y ) in a natural way: • The carrier cost α + β.L-COST( (k)) related to a given carrier k is shared between the requests which are served by this carrier, proportionally to the value L-COST ( (k) x,y ).Q x,y , where (k) x,y is the sub-route which is induced by the restriction (k) x,y of (k) between x and y (in case Q x,y is split into sub-loads, we deal separately with those sub-loads); It comes that Global-Cost( * ) may be written Global-Cost( * ) = r∈Req(U ) Partial-Cost (r, * ), where Partial-Cost(r, * ) is the part of Global-Cost( * ) which is charged this way to request r.Then, for every request r = (x, y, Q x,y = 0) we set V x,y = Partial-Cost(r, * )/Q x,y and update U as follows: • When U = U 0 , U values may be very different from V values.So we compute the mean value τ of the ratio V x,y /U x,y , x, y such that Q x,y = 0, and replace every value U 0 x,y by = τ.U 0 x,y .

An Approximation Result
A natural question comes about the quality of the Shortest Cost/Distance strategy.Since, in most cases, the COST and DIST matrices are strongly correlated, we consider here the case when those matrices are the same, and when Global-Cost only involves the carrier riding cost.
In such a case, we may state:

focus on carrier riding cost minimization) and if T-Max = +∞, then the Shortest Cost/Distance strategy induces an approximation ratio of (1+CAP). This is the best possible ratio.
Proof.We first notice that we may, since T-Max = +∞, deal with only one carrier.Let us first prove the first part of the result, that means that there is no approximation ratio better than (1+CAP).In order to do so, we build the following Non Preemptive VSR instance: .CAP} where N is a large number; function v is equal to 1 for o n,c (excess) stations and to −1 for d n,c (deficit) stations; -DIST = COST represents the shortest path distance induced on the set X by the following arc set • E 1 = {(Depot, o 0,1 ), (d N-1,1 , Depot)}, both arcs with length equal to 1/2; .CAP −1}, all arcs with small length ε; ), n = 0..N-2}, all arcs with length 1; .CAP} addition being performed modulo N , all arcs with length 1-α, where α is a small number.
One easily checks that an optimal tour for the carrier is the tour {Depot, o 0,1 , .
) (proof left to the reader: if it were not the case, then one could remove related arcs of E 4 ).So, as soon as the carrier has been loading in station o n,c , it moves to station d n−1,c and delivers its load.A consequence is that at any time during the process, the current loads of the carrier does not exceeds 1 and that the optimal PDP solution comes as a sequence {Depot, o In order to prove the first part of the result, that means that (1 + CAP) provides us with an approximation ratio, we first notice that splitting any station x into v(x) copies, all with v value equal to 1 or −1 and to distance 0 to each other does not modify the problem.Then we consider some feasible Preemptive VSR tour γ = {Depot, x 0 , x 1 , . . ., x n(γ ) = Depot}.Clearly, we may suppose that no station is involved more than once in γ .Then we may state: Lemma 2. There cannot exist any sequence (discrete circular interval) J = {x i , x i+1 . . ., x i+t }, addition being taken modulo n(γ ), such that x∈J v(x) ≤ CAP − 1.
Proof.If such a sequence exists then the load of the carrier just before reaching x i is at least equal to CAP+1.
Lemma 3.There exists some one-to-one involutive correspondence u = u γ from X into itself such that: -If x is an excess station then u γ (x) is a deficit station and conversely; -If one runs along γ from some deficit station x, then it visits no more that CAP−1 stations other than (eventually) Depot, x and u γ (x) before reaching u γ (x).We denote by γ (x, u) the related sub-path of γ .
Pesquisa Operacional, Vol.37(3), 2017 By the same way there exists a one-to-one involutive correspondence w = w γ from X into itself such that: -If x is an excess station then w γ (x) is a deficit station and conversely; -If one runs along γ from some excess station x, then it visits no more that CAP−1 stations other than (eventually) Depot, x and w γ (x) before reaching w γ (x).We denote by γ (x, w) the related sub-path of γ .
Proof.For any node x = x i of γ , we set J x = {x i , x i+1 , . . ., x i+CAP }, addition being taken modulo n(γ ).Then, we build a bipartite graph (U, V , E) by setting: -U = {deficit stations of γ }; V = {excess stations of γ }; -E = {(x i , x j ) such that one visits no more than CAP−1 non trivial stations when running from x i to x j along γ }.
The first part of Lemma 3 (existence of u = w γ ) means that this bipartite graph admits a perfect matching.If it is not true, then Koenig-Hall Theorem tells us that there exists U * ⊆ U such that Card({v ∈ V which are the extremity of an edge (u, v), u ∈ U * }) ≤ Card(U * ) − 1.One may choose U * in such a way that the intersection graph defined by the discrete circular intervals J x , x ∈ U * is connected.But then we see that the discrete interval J = ∪ x∈U * J x is such that x∈J v(x) ← CAP, and thus that it contradicts former Lemma 2. We proceed the same way in order to get the existence of w = w γ .Lemma 4. A same transition x i → x i+1 (i + 1 being computed modulo n) of γ = {Depot, x 0 , x 1 , . . ., x n = Depot}, cannot appear more than CAP times in the path collection {γ (x, u), γ (x, w), j = 0, . . ., n − 1} of Lemma 3.
Proof.If the transition x i → x i+1 is involved into γ (x, u) then x is a deficit station and is one of the CAP stations which are located before in γ .If it is involved into γ (x, w) then x is an excess station and is one of the CAP stations which are located before in γ .We conclude.
We may now finish with the proof of Theorem 6.Let us suppose that tour γ is an optimal solution of Non Preemptive VSR and that we are provided with a min-cost assignment Q, which, with any excess station x, associates some deficit station z Q (x) in a one-to-one way and which is such that x excess DIST x,Q(x) is the smallest possible.Then for any excess station x, we may derive a circuit γ (x) as follows: Start from x, then go to the deficit node z Q (x), next go to u γ (x) of Lemma 3 while following γ and keep on this way.Two circuits γ (x) and γ (y) are either identical or disjoint, and induce a partition of X − {Depot} into a collection {γ 1 , . . ., γ P } of circuits, with related representative stations x p labeled in such a way that they come according to this order in the tour γ .
-(I1): Handling of the VSR-Flow model: We do it here through the use of a MIP library, while imposing a threshold on the computing time, as soon as the number of stations exceeds 30.
-(I2): Derive a route collection γ * Route = {γ Route (1), . . ., γ Route (P)} from H and h: Flow vector H defines a collection of arcs (x, y), each of them taken H (x,y) times, in such a way that for any node x, there exists as many arcs which enter into x as arcs which come out x. So, every connected component X j , j = 1 . . .s, of the resulting graph gives rise to some Eulerian route γ j .Then we build γ * Route by starting from Depot, reaching some closest X j into some node x j , running γ j until being back to x j and keeping on with another connected component X j .Every time the length L-DIST of current route γ Route ( p) is on the edge to exceed the T-Max threshold, we close it and start γ Route ( p + 1).
As a matter of fact, since there exists several ways to perform this route construction process, we do it while simulating related loading/unloading transactions and trying to maximize them:

NUMERICAL EXPERIMENTS
Purpose: Our purpose here is to: -get a comparative evaluation of the lower bounds of Section 3; -get a comparative evaluation of the 3 heuristic scheme described in Section 4, 5 and 6.
-test the influence of scaling coefficients α, β, δ and the impact of preemption.
Technical context: Algorithms were implemented in C, on PC AMD Opteron 2.1GHz, while using gcc 4.1 compiler.We used the CPLEX12 library for the handling of linear models.

Instances:
No standardized benchmarks exist for generic VSR.So we built instances as follows: -Station set X is randomly generated as a set of n + 1 points x 0 , x 1 , . . ., x n , inside the -Each station but Depot = x 0 is assigned a random v(x) value chosen between −10 and 10, in such a way that the sum of demands over all stations equal to 0; That means that we allow here few neutral stations.
-CAP is randomly chosen between 10 and 20; -T-Max is randomly chosen between = 30 and 100.

Testing the Impact of Scaling Coefficients α, β, δ
On a given instance (X, v, CAP, T − Max, DIST, COST = DIST), we fix α = 10, make vary β, δ with β + δ = 1, and compute solutions through the Shortest Cost/Distance Strategy.We obtain Comment: We see that Carrier riding cost and vehicle riding time behave like antagonistic criteria.

Comparing the Lower Bounds of Section 3
For several groups of 5 instances each related to a given size n, we compute the mean value of: -LB-Proj-Flow: as defined in Theorem 5; CPU-LB-Proj denotes the related computing times in seconds.
-LB-MCA as defined in Theorem 3; CPU-LB-MCA denotes the related computing times in seconds.
When n is larger than 35, computing times for LB-Proj-Flow are too high.Still, we observe that CPLEX12 converges fast on the LB-Proj-Flow model, and so that imposing a threshold on CPU time is not likely to deteriorate the LB-Proj-Flow value in a significant way, even if it keeps us from mathematically proving that we get a lower bound this way.We get the following results (Symbol * means that we imposed a threshold of 1000 s on the running time for the LB-Proj-Flow model): Comment: Experiments confirm both the better quality and the higher computing cost of the LB-Proj-Flow Lower bound.

Testing the Heuristics of Section 4, 5, 6 and the impact of Preemption
We compute, for the same groups of 5 instances as above, the average of the following Global-Cost values: • SD(50): obtained through GRASP-VSR-MCA, with N = 1 and R = 50.
• VF: obtained through the Projected-Vehicle-Flow heuristic ⇒ CPU-VF is the related CPU time.
• FRP: obtained through application of the Flow-Reconstruction-P-VSR algorithm ⇒ CPU-FRP is the related CPU time.
• LB denotes here the LB-Proj-Flow lower bound of the previous experiment.
We get (the computing time which were necessary in order to deal with the LB-Proj-Flow model re not taken into account in CPU-VF and CPU-FRP): Comment: The improvement margin induced by the local search loop of the VSR-MCA algorithm is not very high, especially when the focus is on the vehicle riding time.A consequence is that performing random diversification through the use of the replication parameter R is most often more efficient.Both require small computational times.The Projected-Vehicle Flow algorithm provides similar results.At the end, the Flow-Reconstruction-P-VSR algorithm produces preemptive solutions whose Global-Cost value is always better that values obtained for the Non Preemptive case.This should be amplified in case we allow more neutral stations.We may extrapolate that, on our instances, lower bound LB-Proj-Flow probably misses the optimal value of Non Preemptive VSR problem by about 10%, and is close to optimal value of Preemptive VSR.Finally, we must notice that a limitation for both Flow-Reconstruction-P-VSR and Projected-Vehicle-Flow is that they rely on the resolution of Projected-VSR-Flow instances defined on an almost complete oriented graph, whose computing costs increase fast with the number n of stations.
y } LB-CMCA denotes the related optimal value, which may be computed in polynomial time through a simple Min Cost Flow algorithm.LB-CMCA is a lower bound for the carrier riding cost k L-E-COST( (k)).If LB-Time-CMCA is the value of the CMCA model obtained by replacing the COST matrix by the DIST matrix, then LB-Time-CMCA/T-Max is a lower bound for the carrier number K .

Figure 3 -
Figure 3 -A network G Proj derived from 3 excess stations and 5 deficit stations.

Figure 6 -
Figure 6 -Pareto frontier carrier riding cost versus vehicle riding time.
x,y time units.Both matrices DIST and COST satisfy the Triangle Inequality and are such that COST x,x = DIST x,x = 0 for any station x.Idle-Cost denotes the waiting cost induced for a carrier when it remains at any station x = Depot during one time unit.We suppose (Extended Cost Hypothesis) that if a carrier moves from x to y at a reduced speed in time t ≥ DIST x,y , then the induced extended cost E-COST x,y,t is equal to COST x,y + Idle-Cost.(t− DIST x,y ).All this defines a VSR instance (X , v, C, CAP, T-Max, DIST, COST).
. Route .By the same way, any neutral stations but the Depot station may be removed from the input of the Non Preemptive VSR model.
define in a canonical way routes (k) Route , together with date sequences (k) Time .Then, for any arc e = ((x, t ), (y, t + DIST x,y ) and e = ((x, t ), (x, t +1)) In , we decompose f e as a sum of non negative values f e (k), with values no more than CAP.This allows us to deduce loading sequences (k) Load = {L(k) 0 , L 1 , . .., L(k) n( k)) }, k = 1 . ..K ,according to a basic j = 0, . . ., n( (k) indexed iterative process.We easily check that the resulting tour collection (k), k = 1 . . .K is preemptive VSR feasible, with a global cost Global-Cost( * ) exactly equal to Cost T-Max (F, f ).
It is pure routine to check that any Non Preemptive VSR feasible solution * gives rise to a feasible solution (F, f ) of NP-Network-Flow-VSR with the same cost value.Conversely, monotony constraint (E17) forbids any carrier k from unloading (loading) at some excess or neutral (deficit or neutral) station x, enabling us to turn flow vector f (k) into a loading strategy for the tour (k) induced by {0, 1}-valued flow vector F(k).
Minimize e CC e .H e + e CV e .hAnyfeasiblesolution F, f of Network-Flow-VSR can be turned into a feasible solution H, h of Projected-VSR-Flow.The cost Cost T-Max (F, f ) = e∈ET-Max F e .Carrier-Cost e + e∈ET-Max f e .Vehicle-Cost e may be decomposed into:Cost T-Max (F, f ) = α.F ( p,s) + e∈ET-Max,e =( p,s) F e .Carrier-Cost e + e∈ET-Max f e .Vehicle-Cost e .Through projection, the two last terms of this sum give rise to the quantity arcs e β.COST e .H e + arcs e CV e .he .The first component corresponds to α.K , where K = F ( p,s) is the carrier number.But we know that this carrier number is at least equal to ( e=(x,y)∈E-Proj H x,y .DIST x,y )/T-Max.We deduce the first part of our statement.As for the second part, we get it by noticing that any solution H, h of the Projected-VSR-Flow program give rise in a natural way to a feasible solution R of the CMCA program and a feasible solution Q of the VMCA program, and by keeping on with the above decomposition of the quantity arcs e CC e .H e + arcs e CV e .he .
e .}Wedenote by LB-Proj-Flow the related optimal value of this program.Then we state: Theorem 5. LB-Proj-Flow is a (Preemptive or Not) VSR lower bound, such that LB-Proj-Flow ≥ LB-MCA.Proof.
. ., o 0,CAP , d 0,CAP , . . ., d 0,1 , o 1,1,... , o 1,CAP , . . ., Depot}, with length L-DIST = 2n + 2n.(CAP-1)ε.For every n = 0, . . ., N-1, this tour makes the carrier load all the excess vehicles located in excess stations o n,c , c = 1 . . .CAP, and next bring them to deficit stations d n,c , c = CAP . . . 1, before moving to node o n+1,1 .On another side, the vector Q deriving from the Shortest Cost/Distance strategy is provided by E 4 .One checks that a related optimal PDP meets every request related to an arc (o n,c , None among previous cases 1 and 2 holds;If x-cour is an excess station, then move back along γ Route (P) until x-cour is a deficit station; Close current route γ Route (P) by coming back from x-cour Route-Reconstruction aims at building γ Route in such a way it maximizes Profit and minimizes both Penalty and P. Tree search would be too costly.Instead, we randomize Route-Reconstruction and launch it several times, before keeping the best collection γ Route ever obtained.We deal now with the preemptive version of VSR, and involve the Dynamic Network Framework of Section 2.4, according to the following algorithmic scheme: Compute largest paths, according to DIST, respectively from Depot 1 to any nodex of G * h , and from any node y of to Depot 2 ; Denote by L-DIST x and L-DIST* y the resulting DIST-length values; In case x = s, set L-DIST s = 0 and do as if any arc (s, x) where provided with null DIST value; Do the same thing with p and L-DIST* p ; Out accordingly.We also connect any node (Depot, t ) obtained this way to any existing node (x, u) such that u = t + DIST Depot,x and any existing node (x, u) to node (Depot, t ) such that t = u + DIST x,Depot .We may consider the resulting network G * f as a sub-network of G T-Max , and its arcs e as provided with carrier costs CC e as in the VSR-Flow-Model.Then we compute on G * f a flow vector F, which satisfies (E7, E8, E9, E10) and which minimizes e∈ET-Max F e .Carrier-Cost e .Step 6 always yields a feasible solution, since no non null vehicle flow value f (x,t ),(y,u) , x = y, is involved with t < DIST Depot,x or (T-Max − u) < DIST y,Depot. . .
Move to y 0 as I the first case with H x-cour, y0 unchanged and Penalty ← Penalty + COST x,y0 ; 3th case:Flow-Reconstruction-P-VSR Algorithm:1th step: Compute an optimal solution (H, h) of the VSR-Flow model; 2th step: Denote by G h the network induced by non null h x,y values; Because of the optimality of (H, h), G h does not contain any circuit; Add 2 nodes Depot 1 and Depot 2 to G h and:• Connect Depot 1 to any minimal node (which admits no predecessor but s) x = s of G h ;• Connect any maximal node (which admits no successor but p) y = p of G h to Depot 2 ;• Provide related arcs with DIST values in a natural way;Denote by G * h the resulting network; flow vector h may be considered as defined on G * h ; 3th step: 6th step: Derive a feasible solution (F, f ) of the Network-Flow-VSR model.Let us now describe into more details the contents of steps 4, 5 and 6.Pesquisa Operacional, Vol.37(3), 2017and ((Depot, t i ), (Depot, t i+1 ))
• SD: obtained through Shortest Cost/Distance Strategy initialization of VSR-MCA ⇒ CPU-SD is the related CPU time (s).