Hydrogen Fleet Size and Mix
Location Routing Problem with
Partial Recharging
July 15, 2017
By:
Wessel Kamer
2399350
The University of Groningen
Abstract: The increasing popularity of hydrogen as a source of alternative fuel gives rise to complex optimisation problems. The limited fuel-cell capacity makes the use of hydrogen refuelling stations necessary, requiring efficient routing decisions to be competitive with fossil fuel cars. Additionally, the infrastructure needed for the implementation of hydrogen fuel-cell vehicles is not yet in place. To simultaneously optimise the tour planning and location decisions for the hydrogen refuelling stations this thesis introduces the Hydrogen Fleet Size and Mix Location Routing Problem with Partial Recharging. Apart from routing- and location decisions the fleet composition and size need to be chosen and decisions need to be made on the level to which the fuel-cell is recharged. This problem is solved using a using a framework to identify subsets of opened refuelling stations for which the routing problem needs to be solved. The routing problem is then solved using column generation. The results obtained from solving newly created instances shows the importance of considering location decisions and partial recharging in this problem setting.
W. Kamer
Master’s thesis for MSc Econometrics, Operations Research and Actuarial Studies
Supervisor: dr. X. Zhu
W. Kamer CONTENTS
Contents
1 Introduction 3
2 Literature Review 4
2.1 Contributions by this thesis . . . 5
3 Problem description 5 3.1 Notation . . . 6
3.2 Mixed-integer linear programming model . . . 7
3.3 Set Partitioning formulation . . . 8
4 Column Generation Algorithm 9 4.1 Labelling Algorithm . . . 10
4.1.1 Description of the algorithm . . . 11
4.1.2 Label extension . . . 11
4.1.3 Dominance criterion . . . 15
4.2 Solving the set partitioning model . . . 15
5 Computational Experiments 16 5.1 Instances . . . 16 5.2 Results . . . 18 6 Discussion 20 7 Conclusion 21 Appendices 24 A Feasibility check 24 B Labelling for load-dependent energy consumption 24 B.1 Feasibility check for load-dependent energy consumption . . . 27
W. Kamer 1 Introduction
Hydrogen Fleet Size and Mix Location Routing
Problem with Partial Recharging
Wessel Kamer
1
Introduction
In recent years the use of sustainable energy has been a major topic of research. A particular application of sustainable energy which is expected to see a large increase in the near future is the use of hydrogen fuel-cell cars. Montenegro Camacho et.al.(2017)[1] showed that hydrogen fuel-cell cars are a viable option by examining a low-cost supply of biogas-derived hydrogen, while Xu et.al. (2017)[2] showed a roll-out strategy for fuel-cell vehicles and required hydrogen stations, comparing it in cost and emission to other vehicles. The driving force behind the introduction of hydrogen fuel-cell cars is the need to reduce emission, for example Mac Kinnon et.al.(2016)[3] showed intro-duction of fuel-cell electric hydrogen vehicles contributed to improvements in ground-level ozone. Apart from the environmental benefits it is important to address the economic implications of the introduction of hydrogen cars. Concerns such as range and recharging times are very similar for electric vehicles which are already widely in use. However in the case of hydrogen cars the required infrastructure is not yet in place. This plays an important role in the economic considerations of the widespread introduction of hydrogen cars, with the costs of building one hydrogen refuelling station ranging from $1.5 to $5.5 million according to the California Fuel Cell Partnership (2009)[4]. This thesis introduces the Hydrogen Fleet Size and Mix Location Routing Problem with Time Win-dows and Partial Recharging, or Hydrogen Location Routing Problem (H-LRP), which extends the Electric Fleet Size and Mix Vehicle Routing Problem with Time Windows and Recharging Stations introduced by Hiermann et.al.(2016)[5] with location decisions. Furthermore, this thesis relaxes the limiting assumption in the model of Hiermann et.al.(2016)[5] by allowing vehicles to leave while only partially recharged. Considering Partial recharging can potentially significantly improve solution quality if recharging times are high and time windows tight. In addition to this another limiting assumption, namely the energy consumption being a constant function of carrying load, is briefly addressed, though this is not implemented in the algorithm.
W. Kamer 2 Literature Review
2
Literature Review
The research in this thesis combines three streams of literature, Location Routing (LRP), Electric Vehicle Routing (EVRP) and Heterogeneous Fleet Vehicle Routing or Fleet Size and Mix Vehicle Routing (FSMF).
In the area of Location Routing Problems the number of exact solution methods is scarce, the main area of research being the development of heuristics for which a recent survey was presented by Drexl and Schneider (2015)[6]. An exact algorithm was first introduced by Laporte and Nobert(1981)[7], who used a branch-and-bound algorithm to solve the problem of locating a single depot at one of the customer locations. Problems with up to 50 locations and 5 vehicles were solved. Recently, Contardo et.al.(2014)[8] developed an exact algorithm for the capacitated location-routing problem. The algorithm enumerated over possible subsets of depot locations that could lead to an optimal solution, and solved the corresponding Multi-Depot VRPs(MDVRP) using a branch-and-cut-and-price algorithm. The presented approach could improve upon the bounds found in the literature and solve some previously unsolved instances to optimality.
Electric Vehicle Routing Problems are an extension of the classical VRP (for a survey on recent work in the area of VRPs the reader is referred to Braekers et.al.(2016)[9]), first introduced as the Green VRP by Erdo˘gan and Miller Hooks (2012)[10]. They considered a capacitated VRP with Alternative Fuel Vehicles and Fuelling Stations. The problem was solved heuristically using the well-known Clarke and Wright savings heuristic and a Density Based Clustering method. Both methods were used to solve randomly generated instances and a real world case study.
Schneider et.al.(2014)[11] extended the problem by adding time windows(EVRP-TW), on which Hiermann et.al.(2016)[5] further extended by including a Heterogeneous Fleet(E-FSMFTW). The problem was solved using a Branch-and-Price (BnP) algorithm for which the pricing problems are solved using a bi-directional labelling algorithm. Additionally, an Adaptive Large Neighbourhood Search (ALNS) heuristic with embedded local search was developed. The methods were tested on the modified Solomon instances defined by Schneider et.al. (2014)[11], using vehicle definitions taken from Lui and Shen (1999)[12]. The ALNS was able to find optimal solutions on smaller instances and solutions within 1% of the best known solution for larger instances. Additionally the BnP algorithm proved the optimality of several of the best known solutions.
For the related minimum cost path problem (MCPP), Arslan et.al.(2015) researched the problem of finding the minimum cost path for Plug-in Hybrid Electric Vehicles in a road network with refuelling and charging stations. A mixed integer quadratically constrained formulation (MIQCP) as well as a dynamic programming heuristic are used to solve the problem and provide insights into the economics of Plug-in Hybrid Electric Vehicles. However, their solution approaches focus on runtime aspects, taking the location of recharging stations as given and ignoring additional constraints arising in a logistic context.
W. Kamer 3 Problem description vehicle mass and gradients of the terrain. The problem was addressed by developing an Adaptive Large Neighborhood Search algorithm with embedded local search. The effect of considering the actual load distribution on the structure and quality of the generated solutions was investigated in numerical studies on newly designed E-VRPTWMF test instances.
The last related stream of research is the Heterogeneous Fleet VRP, an extensive survey on the topic has been presented by Ko¸c et.al.(2016)[14].
Mancini (2016)[15] developed an ALNS heuristic for the Multi Depot Multi Period Vehicle Routing Problem with a Heterogeneous Fleet, it was shown on realistic sized instances that the method was effective, as well as being very versatile in its application. A possible advantage of using such an approach for the problem presented in this thesis is the intuitively straightforward option of including location decisions as perturbations of the solution.
Another rich VRP with similar characteristics to the problem in this thesis is presented by Yao et.al. (2016)[16]. They created a Particle Swarm Optimisation heuristic strengthened by a self-adaptive inertia weight and local search strategy to obtain solutions for a Heterogeneous VRP with both demand- and supply-side routing. The presented model was able to improve on solutions found in practice.
2.1
Contributions by this thesis
In this thesis the E-FSMFTW introduced by Hiermann et.al.(2016)[5] is extended by including location decisions and the option to partially recharge a vehicle. The inclusion of location decisions combines the EVRP with the LRP. The problem is defined using a general MIP model and solved using a column generation algorithm.
A parallel and independent paper by Schiffer and Walther(2017)[17] considered the Electric Vehicle Routing Problem (EVRP) with location decisions and partial recharging. The main difference in model formulation is the use of a homogeneous fleet by Schiffer and Walther(2017)[17], as opposed to the heterogeneous fleet used in this thesis. Furthermore in stead of constructing an exact algo-rithm their paper opts to solve the problem using the MIP formulation strengthened by modified constraints.
The computational experiments in this thesis show the approach developed to solve the problem is able to find optimal solutions on small instances, but for instances with 15 customers the com-putation times were on average very high. However, the results presented in the thesis show the importance of considering both location decisions and partial recharging in further research in this area.
3
Problem description
W. Kamer 3 Problem description single depot after which visits to recharging stations can be used to extend the route beyond the battery capacity of the vehicles. When vehicles visit a recharging station a choice needs to be made on the amount of recharging which gives the opportunity to save time when compared to the usual assumption that vehicles need to be completely recharged before leaving.
3.1
Notation
The mathematical notation and model presented in Section 3.2 are based on the notation found in Hiermann et.al.(2016)[5] though some of the notation has been changed. The parameters and variables used are presented here:
Parameters
C set of all customers.
F set of all recharging stations.
Fj0 duplicates of recharging station j ∈ F . F0 set of all duplicated recharging stations. b, (e) depot node for the start(end) of a route.
V set of all vertices, given by V = {b} ∪ {e} ∪ C ∪ F0. VB start depot nodes, given by VB = {b} ∪ C ∪ F0.
VE end depot nodes, given by VE = {e} ∪ C ∪ F0.
VP set of nodes excluding the depot, given byVP = C ∪ F0.
E the edge set on the vertices in V .
K set of vehicle types.
Yk fuel capacity of vehicle type k.
gk recharging time per fuel unit of vehicle type k.
rk(·) fuel consumption as a function of the current load of vehicle type k,
given by rk(q) = (1 + αkq)ˆrk.
ˆ
rk fuel consumption of an empty vehicle of type k.
αk load-dependence coefficient of vehicle type k.
Qk load capacity of vehicle type k. fk acquisition cost of vehicle type k. [`i, ui] time window at node i.
si service time at node i.
dij distance from node i to j.
tij travel time from node i to j.
Uj cost of opening a station at location j ∈ F .
Variables xk
ij binary decision variable that indicates a vehicle of type k travels from node i to node j.
W. Kamer 3 Problem description wj binary decision variable that indicates the station at location j ∈ F is opened.
τk
i start of service of a vehicle of type k in node i.
qk
i current load of a vehicle of type k in node i.
yk
i current fuel level of a vehicle of type k in node i.
The edge set E is preprocessed to remove any edge between two stations, and edges from customers i and j for which `i+ si + tij > uj or pi + pj > Qk, ∀k ∈ K, or rk(pj)dij > Yk, ∀k ∈ K.
This means a vehicle can’t travel from one recharging station to another and infeasible customer sequences are eliminated.
3.2
Mixed-integer linear programming model
The model for the H-LRP is given by:
W. Kamer 3 Problem description The model uses the big-M method, where M is defined as:
M = max k∈K ( Yk+ rk X i∈C pi ! ¯ d ) , ¯ d = max i∈VB,j∈VE {dij} .
Equation (3.1) is the objective function of the minimisation problem, it consists of the cost of opening stations, the acquisition costs of the vehicles and the total travel distance.
Constraints (3.2) and (3.4) ensure every customer is visited exactly once and a visiting vehicle will also depart. Constraints (3.3) imply a recharging station can only be visited if it is open.
The time windows are handled by constraints (3.5) - (3.7). Service needs to start within the time window of a customer(3.5). Constraints (3.6) ensure service at a node cannot start before the service time si and travel time tij from the previous customer have elapsed. In case the previous node was
a recharging station the recharging time gkz
i is used in stead of the service time, this is given by
constraints (3.7).
Constraints (3.8) ensure the demand of customers have been fulfilled, while constraints (3.9) restrict the load to be non-negative and less than the maximum capacity.
Constraints (3.10) - (3.14) handle the fuel levels of the vehicles, from here on also called energy level. Constraints (3.10) and (3.11) ensure energy is depleted when travelling to a node from the depot or a customer respectively. Separate constraints for the depot are needed in the case of linear load-dependence since there is only a single value of qk
b for all routes of vehicles with type k.
Similarly constraints (3.12) handles the energy levels after a recharging station has been visited. At a recharging station the vehicle leaves with an energy level of yki + zi, after which rk(qik)dij units are
used if node j is visited next. In the remainder of this thesis rk is used in stead of rk(·), indicating the constant energy consumption resulting from setting αk = 0.
Constraints (3.13) ensure the fuel-cells (in the remainder also called batteries) are fully charged when a vehicle sets out from the depot. Lastly, Constraints (3.14) ensure the fuel-cell is recharged to at most its full capacity.
Equation (3.15) defines the decision variables.
3.3
Set Partitioning formulation
For the purpose of developing the algorithm presented in Section 4 the problem is reformulated as a set partitioning problem. From this point onwards the notion of duplicate stations which was needed in the previous formulation is no longer used, since in the setting of the algorithm the route order allows for differentiation between recharging station visits. For this notation Ωk is defined as
the set of all routes for a vehicle of type k which are feasible with respect to constraints (3.2)-(3.15). Furthermore, let ck
r give the cost of route r ∈ Ωk, akir is 1 if route r contains customer i, λkr is a
W. Kamer 4 Column Generation Algorithm is then given by:
minX j∈F Ujwj+ X k∈K X r∈Ωk ckrλkr (3.16) subject to X k∈K X r∈Ωk akirλkr = 1 ∀i ∈ C (3.17) λkr ∈ {0, 1} ∀r ∈ Ωk, ∀k ∈ K (3.18)
To solve the LP relaxation a restricted master problem is used. Ωkis replaced by the set of restricted
routes Ω0k and (3.18) is replaced by λkr ≥ 0, r ∈ Ω0 k.
Generating columns with negative reduced cost to the restricted master problem is done by the labelling algorithm presented in section 4.1, which solves the pricing problems:
min fk+X lim i∈VB,j∈VB,i6=j dijxkij − X lim i∈C,j∈VB,i6=j πixkij, (3.19)
subject to the constraints in the MIP formulation, where πi gives the negative reduced cost of node
i.
The pricing problems, of which one is created for every vehicle type, are solved until no negative reduced cost columns can be generated. The original set partitioning problem is then solved giving the solution.
4
Column Generation Algorithm
To solve the set partitioning model described in Section 3.3 this thesis proposes a column generation algorithm embedded in a Graph Restriction Framework(GRF). The column generation algorithm is used to solve the routing problems arising on the restricted graphs, the framework then uses the routing solution to reduce the number of graphs that need to be solved.
The idea of the GRF is similar to the approach given by Contardo et.al.(2014)[8] to enumerate over possible subsets that could lead to an optimal solution. To describe the GRF a set D containing objects d is defined, where d gives a restricted graph(dG) and the best lower bound currently known
for this sub graph(dl). A graph dG has nodes {b} ∪ {e} ∪ C
S
j∈F,wj=1Fj, and the set D contains
objects for the full-factorial combination of the variables wj. Solving the routing problem for any
restricted graph gives a lower bound on all of its sub graphs, equal to the routing cost plus the cost of the stations included in this graph. Let d be an object for which the routing problem has been solved, with a routing cost of cd, then for any object d0, such the d0G ⊂ dG, the lower bound is given
by the maximum of the current bound and
cd+
X
Fj∈d0G
W. Kamer 4 Column Generation Algorithm
Algorithm 1. Graph Restriction Framework
for d in D if d_l is l o w e r t h a n b e s t s o l u t i o n s o l v e ( d_G ) if i m p r o v e d on b e s t s o l u t i o n 5 s t o r e S o l u t i o n () for d ’ _G sub g r a p h of d_G u p d a t e B o u n d ( d ’ _G )
This means that by solving the most complex graph, i.e. dG = V , a lower bound can be set on all
other graphs, which reduces the number of restricted graphs that need to be solved. Therefore, the set D is sorted in order of graph complexity, i.e. in decreasing order of included number of nodes. In the pseudo-code presented in algorithm 1 the functions solve(), storeSolution() and updateBound() are used. solve() creates a new master problem and solves it using column generation, where the sub problems are solved by the labelling algorithm described in section 4.1. updateBound() sets the lower bound for any the graph given as an argument using the information provided by solving the current graph. In pseudo-code this approach is presented in Algorithm 1.
To solve the routing problem on a given graph dG a restricted master problem needs to be created.
For dG = V the master problem is initialised by, for every customer i ∈ C, including the route for
the cheapest vehicle that can perform the sequence b − i − e. To speed up the algorithm the columns generated for the first graph are used for the initialisation of all sub graphs, where routes are only used if all stations visited on the route are included in the graph dG that is currently being solved.
The column generation algorithm iteratively solves all sub problems until no negative reduced cost columns can be generated. Kallehauge et.al.(2005)[18] give an acceleration tactic for column generation algorithms by returning many negative cost columns to the master problem, in this thesis at most |C| columns are returned, i.e. as many as there are customers. When no more negative cost columns can be generated the solution λ optimally solves the restricted master problem, and as stated by Desrosiers and L¨ubbecke(2005)[19] it also optimally solves the master problem.
4.1
Labelling Algorithm
The labelling algorithm described here uses the same general structure as presented by Feillet et.al.(2004)[20], and extends the label definitions presented by Hiermann et.al.(2016)[5].
In addition to the resources Rc, Rq and Rt the resource Rb is introduced, furthermore the recourse
Ry is divided into minimum energy depletion Ry and maximum energy depletion Ry¯. The complete
forward label is defined as LF = {node, (R
c, Rq, Rt, Rb, Ry, Ry¯), prev, ν, reach}. Rcgives the cost of
the path, the load after visit to current node is Rq, earliest start of service at the current node Rt.
The new resource introduced to incorporate the recharge decision is the buffer time Rb resulting
W. Kamer 4 Column Generation Algorithm maximum depletion is needed since it is not possible to express the relation between Rt, Rb and a
single energy depletion recourse Ry. prev is a pointer to the previous label on the path. ν is the
set of visited nodes on a path and reach gives the nodes that can be reached with respect to the time windows and the capacity constraint.
In the labelling algorithm the service time si and demand pi of recharging stations and the depot
are used since this is more convenient for the notation, they are set to 0. As mentioned before no duplicates of the stations are used, instead the current node is subtracted from the reach only if it is a customer.
4.1.1 Description of the algorithm
The following notation is used to describe the labelling algorithm.
• E: List of nodes to be treated. • Λi: List of labels on node i. • Fj: List of new labels on node j.
• Extend(LF, i): Function that returns the label resulting form extension of LF to i. • Dominance(Λi): Procedure that removes dominated labels from a set of label Λi.
The algorithm described by Feillet et.al.(2004)[20] is reformulated to make it a bit more intuitive. In pseudo-code it is given in Algorithm 2. The order of operations is changed such that a label is selected which is extended to all its successors rather than selecting a successor to which all labels are extended. This is done purely for the presentation since it has no benefits in terms of coding efficiency. To reduce computation times a label is only extended if, after subtracting the negative reduced cost of the nodes in its reach from the current cost resource Rc plus the cost of travelling
to the depot is negative, since this is the best possible value of Rc when reaching the depot.
4.1.2 Label extension
To create a new label the earliest possible arrival time and lowest possible energy depletion need to be known. Since the recourse Rt gives the earliest possible start of service time it does not contain
the charging time needed to reach customer i. Therefore, this charging time needs to be added to the travel time. The additional charging time is given by
gk(Ry¯(LF) + dnode(LF)irk− Yk)+.
Of this additional charging time only the amount in excess of the buffer time will propagate through the route, therefore the earliest possible arrival time is given by
ˆ
τi(LF) = Rt(LF) + snode(LF)+ tnode(LF)i+ gk(Ry¯(LF) + dnode(LF)irk− Yk)+− Rb(LF)
+
W. Kamer 4 Column Generation Algorithm
Algorithm 2. Labelling algorithm
I n i t i a l i s e LF( b ) = new L a b e l ( b ) Λb = {LF( b ) } for i in V_E 5 Λi = ∅ E = { b } r e p e a t c h o o s e i in E 10 for j in V_E , i 6= j F_j = ∅ for LF in Λ i for j in r e a c h (LF) if e x t e n s i o n LF to j is f e a s i b l e 15 F_j = F_j ∪ E x t e n d (LF, j ) for j in V Λj = D o m i n a n c e (Λj ∪ F_j ) if Λj c h a n g e d E = E ∪ j 20 E = E \{ E_0 } u n t i l E = ∅
Similarly for the lowest possible energy depletion the time windows in the route have not been incorporated. Since the resource Ry assumes the vehicle is charged for as long as possible it may
not be possible to reach node i even if Ry(LF) + dnode(LF)irk ≤ Yk. A term needs to be added for
the recharging that is sacrificed to reach node i within its time window. The foregone recharging depends on the difference in arrival time between a maximally- and a minimally charged vehicle. The difference in departure time from the recharging station between the two vehicles is
gk(Ry¯(LF) − Ry(LF)).
The buffer time Rb should be subtracted, since this is time that a minimally charged vehicle loses
by waiting at customers. Therefore the difference in arrival time is given by:
gk(Ry¯(LF) − Ry(LF)) − Rb(LF)
+ ,
so if the waiting time exceeds the charging time both vehicles arrive at the same time. The foregone charging time is then:
W. Kamer 4 Column Generation Algorithm which means the lowest possible energy depletion is
ˆ yi(LF) = Ry(LF) + dnode(LF)irk+ Rt(LF) + snode(LF)+ tnode(LF)i− ui gk + Ry¯(LF) − Ry(LF) − Rb(LF) gk +!+ . (4.3)
Given the information provided by calculating ˆτi(LF) and ˆyi(LF) the feasibility of the extension of
LF to node i can be expressed. For an extension to be feasible equations (4.4) - (4.8) need to hold. Firstly, the feasibility of the next extension is given by:
Rq(LF) + pi ≤ Qk, (4.4)
ˆ
τi(LF) ≤ ui, (4.5)
ˆ
yi(LF) ≤ Yk. (4.6)
Equation (4.4) ensures feasibility of the extension for the maximum capacity of the vehicle. Equation (4.5) is the feasibility of an extension with respect to the time-window.
In addition to conditions (4.4)-(4.6), a vehicle should be able to return to the depot within the time window. The following equation takes into account the time needed to reach the depot as well as the recharging time needed.
max{Rt(LF)+snode(LF)+tnode(LF)i+
(R¯y(LF) + (dnode(LF)i+ die)rk− Yk)+ gk − Rb(L F) + , `i} + si+ tie ≤ ue. (4.7)
If the next node is a recharging station no time is added, because this has already been covered by the recharging need. For the feasibility of the extension it is irrelevant when the recharging occurs. Lastly, the minimum energy depletion cannot exceed the energy needed to visit the next recharging station or return to the depot. This only needs to be checked for a visit to a customers, since the recharge needed to reach to depot after visiting a station visit can be added at any station on the route. Therefore if an extension to a recharging station is feasible with respect to condition (4.7) it is possible to recharge the required amount to reach the depot from a station node after extension of the current label. For extension of the label to a customer node the following needs to hold.
W. Kamer 4 Column Generation Algorithm A more detailed derivation is given in Appendix A.
The extension of a label to a new node is handled by so-called resource extension functions. The functions used to create the new label LF
new resulting from the extension of LF to node i are defined
as follows: node(LFnew) = i (4.9) Rc(LFnew) = Rc(LF) + ˆcnode(LF)i (4.10) Rq(LFnew) = Rq(LF) + pi (4.11) Rt(LFnew) = max ˆτi(LF), `i (4.12) Rb(LFnew) = min{(Rb(LF) − gk(R¯y(LF) + dnode(LF)irk− Yk)+)++ (`i− ˆτi(LF))+,
gk(Ry¯(LFnew) − Ry(LFnew))} (4.13)
Ry¯(LFnew) = min{R¯y(LF) + dnode(LF)irk, Yk} (4.14)
Ry(LFnew) = ( ˆ yi(LF) if i ∈ C 0 if i ∈ F (4.15) ν(LFnew) = ν(LF) ∪ {i} (4.16)
reach(LFnew) = reach(LF)/ (
unreachables(i) ∪ i if i ∈ C
unreachables(i) if i ∈ F (4.17)
Equations (4.9) - (4.11) set the node, costs and load.
Equation (4.12) sets the start of service time. It takes extra time for recharging into account by adding the time needed to recharge until the node can be exactly reached. Rb(LF) is subtracted
because this waiting time can be used to absorb the extra charging time.
Equation (4.14) gives the maximum battery depletion, therefore it assumes a vehicle is only recharged to the level needed to reach the current node. Equation (4.15) gives the minimum battery depletion, an explanation of the terms used has been given earlier.
The buffer time is calculated in equation (4.13), if a vehicle arrives before the start of the time window this time is added to the current buffer, after subtracting any recharging time that is used. The buffer cannot exceed the time needed to charge the difference between Ry¯(LF) and Ry(LF),
since the charging time is the only thing that affects the arrival time, combining these conditions with the energy extension functions implies recharging will not violate the upper bound of any time-windows earlier in the route.
Equation (4.16) updates the visits, and equation (4.17) updates the reach.
re-W. Kamer 4 Column Generation Algorithm sources can be used to tackle to increased difficulties resulting from the linearly dependent energy consumption shows the versatility of the chosen set of resources.
4.1.3 Dominance criterion
The function Dominance(Λi) used in Algorithm 2 checks the new nodes in the label set against
those that were already present. If a new label is dominated by one of the labels with a lower cost resource currently in the set it is removed, otherwise it is checked whether the new label dominates any of labels with a higher cost resource. A forward label LF1 dominates LF2 (LF1 LF
2) if
node(LF1) = node(LF1), Rc(LF1) ≤ Rc(LF2), Rq(LF1) ≤ Rq(LF2), Rt(LF1) ≤ Rt(LF2),
Rb(LF1) ≥ Rb(L2F), Ry(LF1) ≤ Ry(L2F), Ry¯(LF1) ≤ Ry¯(LF2), ν(LF1) ⊇ ν(LF2) (4.18)
with one of these inequalities being strict.
Proof. This extends the proof of the dominance criterion given by Feillet et.al.(2004)[20].
If Rt(LF1) ≤ Rt(LF2) holds the arrival time at the current node is lower for LF1 than for LF2, then
Ry(LF1) ≤ Ry(LF2) and R¯y(LF1) ≤ Ry¯(LF2) imply that the recharging needed to reach any next node
is lower for LF1 than for LF2. Additionally Rb(LF1) ≥ Rb(LF2) implies that more of the additional time
used for recharging can be absorbed by waiting times at previous stops. therefore the arrival-time at the next node will also be lower for LF
1 than for LF2.
With respect to the energy constraint Rt(LF1) ≤ Rt(LF2) also needs to be taken into account. Since
the current time is lower the vehicle corresponding to LF
1 will have to forego on at most as much
recharging time as LF
2 to be able to arrive before the upper bound of any time-window. Combining
this with the conditions for the current energy level implies the energy level after an extension will still satisfy the dominance criteria.
Lastly the buffer resource is considered. Since Ry¯(LF1) ≤ R¯y(LF2) the reduction of buffer time is
lower for LF
1 than it is for LF2, and since Rt(LF1) ≤ Rt(LF2) the new buffer time due to early arrival
at a customer is greater. Therefore this dominance criteria also still holds after an extension. The rest of the proof follows from Feillet et.al.(2004)[20].
4.2
Solving the set partitioning model
After the set of all feasible non-dominated routes is created the original set partitioning formulation is solved using the CPLEX Optimization Studio V12.7.1. This gives an optimal selection of routes ω for which the following model, solved for every route r ∈ ω, gives possible recharge actions (in this model k gives the vehicle used for route r, and n is the last position in the route)
minX
j∈r
W. Kamer 5 Computational Experiments subject to
`i ≤ τi ≤ ui ∀i ∈ r (4.20)
τi+ ti,i+1+ si ≤ τi+1 ∀i ∈ r/{n ∪ F } (4.21)
τi+ ti,i+1+ gkzi ≤ τi+1 ∀i ∈ r ∩ F (4.22)
qi− pi+1≥ qi+1≥ 0 ∀i ∈ r (4.23)
q0 ≤ Qk (4.24)
yi− rk(qi)di,i+1 ≥ yi+1 ≥ 0 ∀i ∈ r/{n ∪ F } (4.25)
yi− rk(qi)di,i+1+ zi ≥ yi+1 ≥ 0 ∀i ∈ r ∩ F (4.26)
y0 = Yk (4.27)
0 ≤ zi ≤ Yk− yi ∀i ∈ r ∩ F (4.28)
zi = 0 ∀i ∈ r/F (4.29)
This model is a reformulation of the MIP presented in Section 3.2. In stead of referring directly to a node, i refers to the node at position i in the route. The decision variable zi is defined for every
position in the route, but constraints (4.29) restrict customers and the depot to a zero recharge.
5
Computational Experiments
This section presents the computational results of the exact algorithm described in the previous section. The algorithm is implemented in Java using CPLEX to solve the (restricted) master problem, and run on a Windows computer with a 2 GHz AMD A6-5200 and 8 GB of RAM. First the instances created based on the well-known Solomon instances are described, followed by insights obtained form the solutions.
5.1
Instances
W. Kamer 5 Computational Experiments Table 2: Optimal solution selection of instances ‘Obj’, and optimal solution when not allowing partial recharging ‘p’, not considering location decisions ‘l’ and not considering both ‘pl’ for instances with 15 customers. The computation times in seconds are given. ‘Mix’ gives the vehicle mix that was used, ‘g’ gives the fraction of sub graphs that needed to be solved and ‘Open’ the number of stations that were opened.
Type Name Obj l p pl Time(sec) Mix g Open
W. Kamer 5 Computational Experiments time. To make sure recharging stations have to be used the battery capacity is defined as follows
Yk= (k − 1)β1 |K| − 1 +
(|K| − k)β2
|K| − 1 (5.1)
where β1 is 2.1 times the distance from the depot to the furthest customer, and β2 is two times the
average distance from the customers to the recharging stations or 2/3 of β1, whichever is smallest.
5.2
Results
A selection of results for individual instances is shown in Table 2. The same results for instances with 5 and 10 customers are given in Appendix C. The column ‘Mix’ shows the fleet composition, using the notation of Repoussis and Tarantilis(2010)[21]. Vehicle types are represented by letters, where ‘A’ is the smallest vehicle, the number of vehicles of a used is given by the superscript of the corresponding letter. As expected the solutions use larger vehicles when the vehicle acquisition prices go down. Since the price of vehicles also becomes lower compared to the cost of opening a station there is also a small decrease in the number of stations used. In terms of vehicle mix the most interesting results are found in the ‘BC’ instances. Rather than use larger vehicles to be able to service multiple customers on a route many small vehicles are used which use recharging actions to be able to complete their routes.
It can also be seen that the computation times are very high, the average over all instances with 15 customers being 853.83 seconds. This is nearly as high as the highest computation time recorded by Hiermann et.al.(2016)[5] which was 953.47 sec. Given this increase in computation times it needs to be shown that taking the location decisions and partial recharging into account improves solution quality.
Table 3 shows the percentage increase in cost when disregarding the station costs and the addi-tional routing opportunities afforded by partial recharging. On average the cost increases 0.61% when disregarding location decisions, or in other words stations are opened based on the routing information provided by solving the routing problem on the complete graph. However this cost statistic depends heavily on the relative cost of opening a station. More informative is the cost increase as a percentage of the average cost of opening a station, which is 33.37%. It can be seen that for instances within the same category this percentage increases when moving from vehicle definitions A to C, meaning that the increase in objective value depends on the relative cost of the stations compared to the vehicles. It is of course also reasonable this will increase if the price relative to the routing costs increases, however this has not been tested in this thesis.
By restricting the resource Ry¯ to be equal to Ry and adding the recharging time corresponding to a
full recharge to Rt after a station visit, the recharging behaviour is reverted to full recharging. The
W. Kamer 5 Computational Experiments Table 3: Average and maximum percentage increase in cost compared to the best solution when not allowing partial recharging ‘p’, not considering location decisions ‘l’ and both ‘pl’ for instances with 15 customers. ‘s’ give the increase in cost as a percentage of the average station cost. ‘g’ gives the fraction of sub graphs that needed to be solved.
Type Name l p pl s l max p max pl max g
A R115 0.14 1.14 1.22 27.45 0.52 5.39 5.39 48.96 IC115 0.06 5.87 5.92 8.62 0.23 32.04 32.04 70.83 BC115 0.23 0.87 1.07 24.79 0.45 4.79 4.79 50.0 RC115 0.42 2.66 3.07 64.96 0.74 8.11 8.8 42.19 B R115 0.18 1.38 1.59 16.32 1.39 6.67 6.67 53.13 IC115 0.19 3.01 3.37 7.95 0.77 9.33 9.33 75.0 BC115 0.86 8.59 9.69 27.42 2.31 20.86 21.18 56.25 RC115 1.12 3.02 4.21 66.93 1.92 12.71 13.27 45.31 C R115 0.27 1.42 1.69 20.09 1.37 6.15 6.15 50.0 IC115 0.49 2.24 2.83 14.49 0.97 6.37 6.37 62.5 BC115 1.68 6.55 8.36 36.49 3.55 15.4 18.35 50.0 RC115 1.72 0.53 2.54 84.88 3.38 3.95 5.65 40.63 Avg 0.61 3.1 3.8 33.37 1.47 10.98 11.5 53.73
a single customers. To create longer routes in this setting multiple station visits are needed, which gives extra opportunities to benefit from partial recharging of the fuel-cell.
When combining both restrictions the extra routing options provided by opening all stations cancel out the inefficiency introduced by full recharging action for the maximum cost increase, however for average cost increase there is no clear relation. For all instance types and vehicle definitions the cost increase is close to the sum of the increase in cost for the individual restrictions.
The last statistic presented in Table 3 is the percentage of the total number of sub graphs that have been solved to obtain the solution. This is highest for the ‘IC’ instances which only have 2 stations, for all other instances it is around 50% with the average over all instances being 53.73%. Compared to the same statistics for instances with 5 and 10 customers, which are 42.8% and 54.04% there is no drop-off in the efficiency of the Graph Restriction Framework when increasing the number of customers. Table 4 shows characteristics of the solutions that were computed. First the energy recharged on station visits is compared to a full recharge in the same situation. The clustering of customers has no apparent effect on the charging behaviour, however the recharging as a percentage of a full recharge goes down with when moving from vehicle definitions A to C. Since larger vehicles can be used in these instances the same recharge action constitutes a lower percentage of battery capacity.
W. Kamer 6 Discussion Table 4: Average, minimum and maximum recharging as a percentage of the battery depletion (Yk − yk
i), and number of stations visits per route ‘rs’, stations on the first ‘1’ and last ‘n − 1’
position of a route and the expected number of station visits on any given position in the route ‘i’, for instances with 15 customers
Type Name avg min max rs 1 n − 1 i
A R115 73.4 33.81 97.35 1.25 0.33 0.12 0.29 IC115 59.23 30.68 85.69 1.66 0.18 0.54 0.31 BC115 72.71 28.39 97.61 2.19 0.5 0.74 0.42 RC115 65.01 31.25 89.15 1.5 0.74 0.48 0.22 B R115 66.74 29.21 94.12 1.03 0.29 0.23 0.24 IC115 52.05 31.96 66.7 1.28 0.1 0.43 0.18 BC115 63.9 37.45 89.66 1.42 0.19 0.4 0.18 RC115 60.58 39.91 82.35 1.22 0.64 0.44 0.16 C R115 64.71 29.44 96.16 1.06 0.32 0.23 0.23 IC115 43.48 32.68 50.42 1.03 0.0 0.42 0.12 BC115 65.32 53.32 77.32 0.67 0.19 0.06 0.1 RC115 51.64 34.92 68.53 0.98 0.56 0.38 0.12 Avg 62.67 33.8 85.17 1.24 0.35 0.34 0.22
on the way back to the depot. If a recharging station is visited in the middle of a route a vehicle leaves a cluster before visiting the cluster again or moving to a different cluster, since the instances solved in this thesis only included 2 clusters this kind of routing behaviour is not useful.
6
Discussion
This thesis extends the E-FSMFTW with the option to partially recharge vehicles on a visit to a recharging station. Due to the fact that the resulting label definitions are significantly less tight the labelling algorithm used to solve the routing sub problems is less efficient. In addition the location decisions are added, which is known to greatly increase run times of exact solution methods. However the results presented in this paper show that due to the economic impact of these decisions it is important to incorporate them in the model for this problem. A direction for future work implied by the high computation times and economic impact is the development of heuristic solvers, which is not straightforward since recharging and routing decisions need to be optimised simultaneously. Therefore checking feasibility of moves performed by local search operators which could usually be computed in O(1) are no of O(n).
W. Kamer 7 Conclusion
7
Conclusion
As an extension of the Electric Fleet Size and Mix Vehicle Routing Problem with Time Windows and Recharging Stations introduced by Hiermann et.al.(2016)[5] this thesis examines the impact of location decisions for the placement of recharging stations and the ability to partially recharge a vehicle when it visits one of these stations. The resulting problem, called the Hydrogen Fleet Size and Mix Location Routing Problem with Pratial Recharging, adds additional complexity to the routing sub problems to be able to deal with the recharging decisions. On top of that the overarching structure of the problem is more complex due to the location decisions.
In this thesis an exact solution methodology based on column generation within a Graph Restriction Framework are used to obtain optimal solutions for instances with at most 15 customers and 3 possible station locations.
The results show that although computation times for the exact solution methodology are high there are economic gains to be had from including these extensions in the model. Furthermore, some characteristics of optimal solutions are shown, which might be useful when constructing heuristics for the problem presented in this paper, which as mentioned in Section 6 is an important area for further research.
Acknowledgements
I would like to thank dr. Evrim Ursavas for her valuable input during the research process and the feedback on the first draft of this thesis.
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W. Kamer
Appendices
A
Feasibility check
This section gives the mathematical derivation of Equation (4.8). Since the feasibility check for the battery level only needs te be performed when a customer is visited ˆyh(LFnew) can be rewritten
using the definitions for ˆyi(LF) and ˆτi(LF) and the resource extension functions. Equation (A.1)
uses the definition of ˆyi(LF) when extending label LFnew to a node h ∈ F ∪ e, the remaining steps
are straightforward algebraic operations.
ˆ yh(LFnew) = ˆyi(LF) + dihrk+ Rt(LFnew) + si+ tih− uh gk + Ry¯(LFnew) − Ry(LFnew) − Rb(LFnew) gk +!+ (A.1) = ˆyi(LF) + dihrk+ max{ˆτi(LF), `i} + si+ tih− uh gk + R¯y(LFnew) − Ry(LFnew) − min n Rb(LF) + (`i− ˆτi(LF))+, gk(Ry¯(LFnew) − Ry(LFnew)) o gk + + (A.2) = ˆyi(LF) + dihrk+ max{ˆτi(LF), `i} + si+ tih− uh gk + Ry¯(LFnew) − Ry(LFnew) − Rb(LF) + (`i− ˆτi(LF))+ gk ++ (A.3) = ˆyi(LF) + dihrk+ max{ˆτi(LF), `i} + si+ tih− uh gk + min{Ry¯(LF) + dnode(LF)irk, Yk} − ˆyi(LF) − Rb(LF) + (`i− ˆτi(LF))+ gk ++ (A.4)
B
Labelling for load-dependent energy consumption
The introduction and Section 3 refer to the relaxation of the assumption of constant energy de-pendence. Due to time constraints the labelling for load-dependent energy consumption was not implemented in the algorithm, but the redefinition of the labels described in Section 4.1 is shown in this section of the appendix.
W. Kamer B Labelling for load-dependent energy consumption Similar to the forward label extensions ˆτj and ˆyj in this case denoting the latest possible departure
time and again lowest possible energy depletion. The definition of the latest departure is similarly derived to its forward counterpart and is given by
ˆ
τj(LB) = Rt(LB) − sj − tjnode(LB)− gk(Ry¯(LB) + djnode(LB)rk(Rq(LB)) − Yk)+− Rb(LB)
+ . (B.1)
For the lowest possible energy depletion the definition is only similar if a customer is visited on the next extension: ˆ yj∈C(LB) = Ry(LB) + djnode(LB)rk(Rq(LB))+ `j− (Rt(LB) − sj− tjnode(LB)) gk + Ry¯(LF) − Ry(LF) − Rb(LF) gk +!+ . (B.2)
The main difference from the forward labelling lies in the backward extension to a station node, because it may not be possible to charge the vehicle completely and still perform the rest of the route. The energy level given the recharge activity in the remainder of the route is Ry(LB) +
djnode(LB)rk(Rq(LB)). Given the departure time ˆτj(LB) the extra time needed in the rest of the
route to attain this level is
gk(Ry¯(LB) − Ry(LB)) − Rb(LB)
+ ,
so the time available to recharge the vehicle at the current station will be
ˆ τj(LB) − gk(Ry¯(LB) − Ry(LB)) − Rb(LB) + − `j + ,
multiplying by gk, adding the current level and rewriting gives
ˆ yj∈F(LB) = Ry(LB) + djnode(LB)rk(Rq(LB))− ˆ τj(LB) − `j gk − Ry¯(LB) − Ry(LB) − Rb(LB) gk +!+!+ . (B.3)
Label extensions are feasible if (B.4)-(B.8) hold.
Rq(LB) + pj ≤ Qk, (B.4)
ˆ
τj(LB) ≥ `i, (B.5)
ˆ
W. Kamer B Labelling for load-dependent energy consumption Again stronger feasibility checks are included to ensure a vehicle could have started within the time window of the depot and has enough energy for the entire route if the next extension is the depot or a recharging station. max ( Rt(LB) − sj − tjnode(LB)− (Ry¯(LB) + (djnode(LB)+ dbj)rk(Rq(LB)) − Yk)+ gk − Rb(L B) + , ui ) − tbj ≥ `b (B.7)
The feasibility of the energy constraint again only needs to be checked if the current label is extended to a customer node. The constraint is given by
∃h ∈ F ∪ e such that ˆyh(LFnew) ≤ Y k ˆ yh(LFnew) = ˆyi(LF) + dihrk+ max{ˆτi(LF), `i} + si + tih− uh gk + min{Ry¯(LF) + dnode(LF)irk, Yk} − ˆyi(LF) − Rb(LF) + (`i− ˆτi(LF))+ gk ++ ≤ Yk. (B.8) The derivation is given in Section B.1. The new Resource extension functions for an extension of label LB to node j are given below.
node(LBnew) = j (B.9) Rc(LBnew) = Rc(LB) + ˆcjnode(LB) (B.10) Rq(LBnew) = Rq(LB) + pj (B.11) Rt(LBnew) = min ˆτj(LB), uj (B.12) Rb(LBnew) = min{(Rb(LB) − gk(Ry¯(LB) + djnode(LB)rk(Rq(LB)) − Yk)+)++ (ˆτj(LB) − ui)+,
gk(Ry¯(LBnew) − Ry(LBnew))} (B.13)
R¯y(LBnew) = min{Ry¯(LB) + djnode(LB)rk(Rq(LB)), Yk} (B.14)
Ry(LBnew) = ˆyj(LB) (B.15)
ν(LFnew) = ν(LF) ∪ {j} (B.16)
Reach(LFnew) = Reach(LF)/ (
unreachables(j) ∪ j if j ∈ C
W. Kamer B Labelling for load-dependent energy consumption
B.1
Feasibility check for load-dependent energy consumption
In this section the derivation of energy feasibility condition (B.8) is derived similar to the derivation in Appendix A. ˆ yh∈F(LBnew) = (B.18) ˆ yj(LB) + dhjrk(Rq(LBnew)) − ˆ τh(LBnew) − `h gk − Ry¯(LBnew) − Ry(LBnew) − Rb(LBnew) gk +!+!+ (B.19) = yˆj(LB) + dhjrk(Rq(LBnew))− Rt(LBnew) − thj −
gk(Ry¯(LBnew) + dhjrk(Rq(LBnew)) − Yk)+− Rb(LBnew)
+ − `h gk − Ry¯(LBnew) − Ry(LBnew) − Rb(LBnew) gk +!+!+ (B.20) = yˆj(LB) + dhjrk(Rq(LBnew))− Rt(LBnew) − thj −
gk(Ry¯(LBnew) + dhjrk(Rq(LBnew)) − Yk)+− Rb(LBnew)
W. Kamer C Further Results = yˆj(LB) + dhjrk(Rq(LB) + pj) − min{ˆτj(LB), uj} − thj − `h gk − min{R¯y(LB) + djnode(LB)rk(Rq(LB)), Yk} + dhjrk(Rq(LB) + pj) − Yk + − (min{Ry¯(LB) + djnode(LB)rk(Rq(LB)), Yk} − ˆyj(LB)) !+!+!+ (B.24)
For the step from (B.22) to (B.23) part of the equation can be written as
f (x, y, z) = (x − min{y, z})+− (y − z)+, (B.25) with x = Ry¯(LBnew) + dhjrk(Rq(LBnew)) − Y k+ , (B.26) y = R¯y(LBnew) − Ry(LBnew), (B.27) z = Rb(L B) + (ˆτ j(LB) − ui)+ gk . (B.28)
The 6 cases are given by
x < y < z, f (x, y, z) = 0, x < z < y, f (x, y, z) = z − y < 0, y < x < z, f (x, y, z) = x − y > 0, z < x < y, f (x, y, z) = x − z − (y − z) = x − y < 0, y < z < x, f (x, y, z) = x − y > 0, z < y < x, f (x, y, z) = x − z − (y − z) = x − y > 0.
Since the expression is enclosed in a positive part operator in (B.22) this gives
(f (x, y, z))+ = (x − y)+. (B.29)
C
Further Results
W. Kamer C Further Results
Table 5: Definitions of the columns are the same is in Table 3, table for 10 customers
Type Name Obj l p pl Time(sec) Mix g Open
W. Kamer C Further Results
Table 6: Definitions of the columns are the same is in Table 3, table for 5 customers
Type Name Obj l p pl Time(sec) Mix g Open
W. Kamer C Further Results
Table 7: Gives the same results as Table 3 for instances with 10 and 5 customers
Type Name l p pl s l max p max pl max g
W. Kamer C Further Results
Table 8: Gives the same results as Table 4 for instances with 10 and 5 customers
Type Name avg min max rs 1 n − 1 i
