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Operations Research Letters 41 (2013) 576–580

Contents lists available at ScienceDirect

Operations Research Letters

 journal homepage: www.elsevier.com/locate/orl

Routing vehicles to minimize fuel consumption

Daya Ram Gaur a, Apurva Mudgal b,∗, Rishi Ranjan Singh b

a Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, Canadab Department of Computer Science and Engineering, Indian Institute of Technology Ropar, Nangal Road, Rupnagar, Punjab- 140001, India

a r t i c l e i n f o

 Article history:

Received 11 January 2013Received in revised form27 July 2013Accepted 27 July 2013Available online 9 August 2013

Keywords:

Approximation algorithmsFuel consumptionCumulative vehicle routing problem

a b s t r a c t

We consider a generalization of thecapacitatedvehicle routing problem known as thecumulative vehicle

routing problem in the literature. Cumulative VRPs are known to be a simple model for fuel consumptionin VRPs. We examine four variants of the problem, and give constant factor approximation algorithms.Our results are based on a well-known heuristic of partitioning the traveling salesman tours and the useof the averaging argument.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Fuel consumption in transportation and logistics is an impor-

tant area of study. Fuel consumption can account for as much as60% of the operating cost of a vehicle according to one study [17].Therefore, a direct reduction in the operating costs can be obtainedby minimizing the fuel consumption of the vehicle. Fuel consump-tion by a vehicle is affected by various factors, important ones be-ing the distance traveled, the weight of the vehicle, vehicle speed,road inclination, aerodynamic drag, traffic congestion, etc. [8]. Animportant factor that determines the fuel consumption is the to-tal weight of the vehicle, i.e., the weight of the empty vehicle plusthe weight of the cargo being carried by the vehicle. In a simplifiedmodel of fuel consumption, the fuel consumed by the vehicle perunit distance is assumed to be proportional to the total weight of the vehicle [16,11,12,19].

In the vehicle routing problem (VRP) introduced by Dantzig

and Ramser [7], a vehicle with finite capacity is located at a depotin a graph   G(V , E )  with positive edge lengths. Each vertex of   G

except the depot has an object of positive weight located at it. Theobjective is to devise a travel schedule for the vehicle such that allthe objects are brought to the depot and the total distance traveledby the vehicle is minimized.

Several authors have studied generalizations of VRPs wherethe objective is to minimize fuel consumed by the vehicle under

∗  Corresponding author. Tel.: +91 1881 242166.E-mail addresses: gaur@cs.uleth.ca (D.R. Gaur), apurva@iitrpr.ac.in (A. Mudgal),

rishirs@iitrpr.ac.in (R.R. Singh).

the simplified model discussed above. For the background anddescription of the vehicle routing problem please refer to theexcellent books by Golden and Assad [9]andbyTothandVigo[18].

1.1. Cumulative VRPs

Kara et al. [11,12] define cumulative VRPs (CumVRP s) where theobjective is to minimize fuel consumption of the vehicle given thatfuel consumed per unit distance is proportional to the total weightof the vehicle. Further, they note that cumulative VRPs generalizeseveral previously studied problems in literature such as theminimum latency problem and the k-travelingrepairman problem.A linear model of fuel consumption and heuristic algorithmsfor the capacitated vehicle routing problem with the objectiveof minimizing fuel consumption are given in   [19]. We nowdescribe the cumulativevehicle routing problemas defined by Kara

et al. [11,12].CumVRP-Input.

1. We are given a complete, undirected graph G(V , E ) with posi-tivelengthsoneachedge e ∈ E . Theedge lengths satisfy triangleinequality and form a metric space. If the graph is not completeor the edges do not satisfy triangle inequality, we take the met-ric closure of graph  G(V , E ). The vertices of graph  G(V , E ) arenumbered from 0 to n. Vertex 0 is called the depot.

2. For each  i  ∈   V  \ {0}, there is an object of positive weight  wi

which is located at vertex i.

3. An empty vehicle is located at the depot, which at any point of time can carry objects of total weight not exceeding Q .

0167-6377/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.orl.2013.07.007

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The goal is to devise a travel schedule for the vehicle so thatall the objects are brought to the depot and the fuel consumed isminimized. We allow the vehicle to offload cargo at the depot anarbitrary number of times.

Fuel consumption model. We assume that the fuel consumed perunit distance is proportional to the total weight of the vehicle. Lettheweightof the empty vehicle be W 0 andsuppose thecargo being

carried has weight w. Then, fuel consumed by the vehicle to traveldistance l  is given by µ(W 0 + w)l =   µW 0l + µw l, where µ  is aconstant. In the following we take a = µW 0 and b = µ, sothat thefuel consumed per unit distance by a vehicle with cargo weight w

is a + bw.

Feasible solution to CumVRP.  A feasible solution  S  to  CumVRP  con-sists of a set of k ≤ |V  \{0}| directed cycles C 1, C 2, . . . , C k contain-ing the depot such that:

1. Each vertex i ∈ V  \ {0} belongs to exactly one cycle.2. The total weight of objects located at the vertices of each cycle

C  j, 1 ≤ j ≤ k, is at most Q .

Given a feasible solution   S , a travel schedule for the vehiclelocated at the depot can be constructed as follows. For 1

 ≤ j

 ≤ k ,

the vehicle at the depot goes once around cycle  C  j in the preferreddirection of traversal, picks up the objects located at the vertices of C  j, and drops them at the depot.

Cost function. We now compute the fuel consumed by schedule  S .Since the fuel consumption per unit distance varies linearly withthe total vehicle weight, we will compute the fuel consumptiondue to every component weight separately and add them up. Fori ∈   V  \ {0}, let  dS 

i  denote the distance traveled by a vehicle be-tween picking up the object at vertex i and dropping it at the depotaccording to the travel schedule given by  S . In the following, |C  j|denotes the length of cycle C  j.

Let us first compute the fuel consumed by a vehicle whiletraversing cycle C  j, where 1 ≤ j ≤ k. The fuel consumption due tothe weight of the empty vehicle is a

|C  j

|, as the vehicle goes around

cycle C  j once. For each non-depot vertex  i ∈ C  j, the vehicle carriesweight   wi  for a total distance   dS 

i . Therefore, the contribution of weight wi, i ∈ C  j is bwid

S i . The fuel consumed by the vehicle while

traversing cycle C  j:

 f (C  j) = a|C  j| + bi∈C  j

widS i .

The total fuel consumption f (S ) of the travel schedule given bysolution S  is f (S ) =k

 j=1 f (C  j) = ak

 j=1 |C  j|+ bk

 j=1

i∈C  j

widS i .

Since each vertex occurs on exactly one cycle, we get that:

 f (S ) = a

k

 j=1

|C  j| + b ·n

i=1

widS i .

Objective. Ourobjective is to find thesolutionS ∗ with theminimumfuel consumption f (S ∗) among all feasible solutions with less thanor equal to |V  \ {0}| cycles.

1.2. Our contributions

Different variants of cumulative VRPs have been studied in lit-erature. A special case is when there is a single vehicle with infinitecapacity and the travel schedule must consist of a single travelingsalesperson (TSP) tour C  of graph G. Constant factor approximationalgorithms have been given for this case by Blum et al. [5] who callit the positive-linear time-dependent traveling salesman problem

and note the relation of this problem to minimizing fuel consump-tion.

The main contribution of this paper is a factor 4 approximationalgorithm for cumulative VRP when the vehicle has finite capacityand an arbitrary number of depot offloads are allowed. For ourconsiderations, it will be convenient to consider four differentversions of this problem:

1.   EQ-INF-FM : All objects have equal weights, and the vehicle hasinfinite capacity.

2.   UNEQ-INF-FM : The objects have unequal weights, and thevehicle has infinite capacity.

3.   EQ-CAP-FM : All objects have equal weights, and the vehicle hascapacity Q .

4.   UNEQ-CAP-FM : The objects have unequal weights, and thevehicle has capacity Q .

For the above four problems, we will give approximationalgorithms with factors 2.5, 2.5, 3.186, and 4 respectively. Tothe best of our knowledge, these are the first constant factorapproximation algorithms for these problems.

The approximation factors are established by a novel applica-tion of the iterated tour partitioning technique of Haimovich andRinnooy Kan [10] and the travel schedule for the vehicle is com-puted using a variation of dynamic programming on a travelingsalesperson tour [4,15].

2. Previous work

We now discuss the algorithms for the capacitated minimumdistance vehicle routing problem (CVRP), where the objective isto compute a schedule with minimum total travel distance. Weassume that there is a single vehicle with capacity  Q  at the depot.Let wi be the weight of object at vertex i and let di be the shortestdistance between vertex   i   and the depot. The following lowerbound on thelength of anytravel schedule wasgiven by Haimovichand Rinnooy Kan [10].

 Theorem 1   ([10]).   Let C ∗  denote an optimal traveling salespersontour ofthe graph G(V , E ). Then, the total distance traveled by a vehicleto bring all objects to the depot is at least:

max

|C ∗|, 2

ni=1

widi

Q 

.

Let   C   be a traveling salesperson tour of graph   G(V , E ). Asalesperson tour that visits a subset of vertices in  V  is referred toas a subtour. At times when it is clear from the context we will usetour to refer to a subtour. Without loss of generality we will labelthe depot 0, and number the vertices 1, 2, . . . , n in the order inwhich they are visited by a vehicle following tour C . The length of tour C  will be denoted by |C |.

A   cluster  [i, j]   is a contiguous set {i, i +  1, . . . , j}   of verticeson tour   C . A   cluster partition P 

  = [1, i1, i2, . . . , ik

−1, n

], k

  ≥2,  and 1   <   i1   <   i2   < · · ·   <   ik−1 ≤   n is a decomposition of tourC  into k clusters [1, i1 − 1], [i1, i2 − 1], . . . , [ik−1, n]. If the wholetour is a single cluster, the cluster partition is denoted by [1, n].

In the following, we assume that the maximum total weight of objects in any cluster of a cluster partition is at most the vehiclecapacity Q .

Given a cluster partition  P   = [1, i1, i2, . . . , ik−1, n]  of  C , weconstruct k  subtours C 1, C 2, . . . , C k  such that tour  C  j  correspondsto the  jth cluster. A vehicle following subtour  C  j  starts from thedepot, visits all vertices in the jth cluster in increasing order, andreturns back to the depot. The  length l(P ) of the cluster partitionP   is defined as the sum of the lengths of all the   k   subtoursC 1, C 2, . . . , C k, i.e., l(P ) = |C 1| + |C 2| + · · · + |C k|.

The next theorem due to [10,2] gives an upper bound on the

length of the optimal partition for the case when all objects haveequal weights.

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578   D.R. Gaur et al. / Operations Research Letters 41 (2013) 576–580

 Theorem 2   ([10,2]).   Suppose all vertices have objects with equal

weights. Let C be a TSP tour of G and let Q ′   be the maximum

number of objects a vehicle can carry at any point of time. Then, there

exists a cluster partition P of C of total length at most  2n

i=1 di

Q ′   +1 −   1

Q ′

|C |.

A partition   P  satisfying the conditions of  Theorem 2  can becomputed in O(n) time [10]. Further, if  C  is an α-optimal travelingsalesperson tour, the lower bound in Theorem 1 shows that the

schedule corresponding to P  is of length at most 1 + α

1 −   1

Q ′

times the optimal schedule [10,2].

The following theorem due to [1] gives an upper bound on thelength of the optimal partition when objects have unequal weightsless than or equal to the vehicle capacity  Q .

 Theorem 3   ([1]).   Suppose the objects at vertices have arbitraryinteger weights less than or equal to Q . Let C be a TSP tour of G and

let Q be the vehicle capacity. Then, there exists a cluster partition P of 

C of total length at most  4n

i=1 widi

Q   +

1 −   2

Q 

|C |.

A partition   P  satisfying the conditions of  Theorem 3  can becomputed in   O(n2)   time   [4,15], and if we have an   α-optimaltraveling salesman tour, the schedule corresponding to partition P 

has length at most 2 + α

1 −   2

Q 

times the optimal schedule [1].

Note that if   w′i s   are rationals then   Theorem 3   can be applied

by scaling all   w′i s   to integers. The bound in  Theorem 3  is then

4n

i=1 widi

Q   +

1 −   2

sQ 

|C | for some scaling factor s.

3. Our results

Let   C   be a traveling salesperson tour of graph   G. Let   P    =[1, i1, i2, . . . , ik−1, n]  be a cluster partition of tour   C . Let   C 1, C 2,. . . , C k be the subtours corresponding to partition P . A vehicle can

traverse each subtour   C  j   in either a clockwise or anticlockwisedirection, and the fuel consumption for these two directions maybe different. We define the fuel consumption f (C  j) of subtour C  j asthe fuel consumption for the optimum direction of traversal. Thefuel consumption of partitionP  isdefinedas f (P ) = f (C 1)+ f (C 2)+·· · + f (C k).

Next we give a lower bound on the cost of the fuel consumptionwhich is a straightforward and important extension of the boundin [10].

 Theorem 4.  Let C ∗ denote an optimal traveling salesperson tour, andlet Q be the capacity of the vehicle. Let  w i be the weight of the object 

at vertex i and let di be the shortest distance between vertex i and the

depot. Then, the minimum fuel consumed by the vehicle to bring all

objects to the depot is at least:

a · max

|C ∗|, 2

ni=1

widi

Q 

+ b

  ni=1

widi

.

Proof.   Let   C ∗1 , C ∗2 , . . . , C ∗l   be a set of tours which achieve theoptimal fuel consumption for a vehicle of capacity   Q . Let  d∗

i   bethe distance traveled by a vehicle following the optimal set of tours between picking up the object with weight  wi   at vertex  i

and dropping it at the depot. Then, the optimal fuel consumption

is a ·

l

 j=1 |C ∗ j |

+ b ·

n

i=1 wid∗i .

We note two facts to complete the proof. First, l

i= j |C ∗ j |   isthe distance traveled by a vehicle of capacity  Q   in graph G(V , E )

to bring all objects to the depot. By  Theorem 1,   this is at least

max|C ∗|, 2

ni=1 wi di

Q 

. Secondly, for all  i  ∈   V  \ {0}, d∗

i   ≥   di , a

vehicle has to travel at least the shortest distance from vertex i tothe depot to bring object  i  to the depot. Therefore, the optimumfuel consumption is at least:

a · max|C ∗|, 2

n

i=1

widi

Q 

+ b   n

i=1

widi

.  

Lemma 1.   Let C be a tour of length L which visits a subset S  ⊂  V of 

vertices of total weight W . Then, a vehicle can visit all vertices in set 

S with fuel consumption at most aL + bWL/2.

Proof.  Without loss of generality suppose the tour visits verticesnumbered 1, 2, . . . , n′.Wenotethatthefuelconsumptioncanvarybased on whether we take theclockwise or anticlockwise directionaround C . Let d1

i   and d2i  be the distances of vertex i from the depot

along tour C  in the clockwise and anticlockwise directions. Clearly,d1

i

 + d2

i

 = L

 ∀i.

The fuel consumed by a vehicle following  C  in the clockwisedirection is   a|C | +   b

n′i=1 wid

1i

, and in the anticlockwise

direction is   a|C | +   b

n′i=1 wid

2i

. The sum of the two costs is

2a|C |+ bn′

i=1(wi · (d1i + d2

i )) which is equal to2aL + bWL. Hence,the minimum of the two tours will have cost at most half of thisquantity.  

 Theorem 5   (UNEQ-INF-FM ). Let  β >  0 be a positive rational num-

ber, C be a traveling salesperson tour, and assume that the vehi-

cle has infinite capacity. Then, there exists a cluster partition P   =[1, i1, i2, . . . , ik−1, n] of C with total fuel consumption at most:

1 +

2

β

b   n

i=1widi

+ 1 +

β

2

a|C |.

Proof.   Since β is a rational numberand a, b,andtheobjectweightswi  are finite-precision numbers on a computer, we can choosepositive integers  t , m such that (i)   βa

b  is equal to   t 

m, and (ii)  mwi

is a positive integer for all object weights  wi, 1 ≤ i ≤ n.We construct a new graph  G ′(V ′, E ′) from  G(V , E ) as follows.

We keep the depot as it is, and replace each vertex  i, 1 ≤   i ≤   n,with a set V i = {vi

1, vi2, . . . , vi

mwi} of  mwi  vertices of weight 1/m

each. Any two vertices in the same set V i  are joined by an edge of cost 0, whereas a vertex in set V i is joined toa vertex inset V  j, j ̸   = i

with an edge of length equal to the distance between vertex i andvertex j  in graph G. The distance of the depot from a vertex in set

V  j, 1 ≤ i ≤ n, istakenas d j, the shortest distance from the depot tovertex j in graph G. Clearly, the edge costs in graph G′ form a metricspace.

We now construct a traveling salesperson tour  C ′  of graph G ′

from the given tour  C  as follows. We replace each vertex i on tourC  with a path P i  =   (vi

1, vi2, . . . , vi

mwi) on the  mwi  vertices in  V i.

Since the cost of an edge between any two consecutive verticesin path P i  is zero, path P i  has length 0. The depot will be replacedby a path P 0  consisting of a single vertex. For each  i, 0 ≤   i ≤   n,the last vertex of path  P i  is connected to the first vertex of pathP (i+1)mod(n+1) by the corresponding edge in graph G′. Clearly, C ′ is atour on the N  =   m · n

i=1 wi

vertices in graph G′ and has total

length equal to |C |.Label the depot in graph   G′  as 0 and number the remaining

vertices 1, 2, . . . , N  according to the order in which they occur ontour   C ′. Let  d ′i  denote the shortest distance of vertex   i  from the

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3. Output the subtours  C ∗1 , C ∗2 , . . . , C ∗k  corresponding to partitionP ∗ as the solution.

We now show that there exists a partition  P  of  C  with f (P ) ≤4 · OPT , where OPT  is the optimum fuel consumption. Since  P ∗ isthe optimal partition of  C , we get that f (P ∗) ≤ f (P ) ≤ 4 · OPT .

Applying Theorem 6 on tour C  with β = 2/3, we get a partitionP  of tour C  with fuel consumption at most

4 · b ·

  ni=1

widi

+ (4/3)a|C | + 4a

ni=1

widi

Q .

Since |C | ≤ 1.5|C ∗|, the fuel consumption of  P  is at most:

4 ·

b ·

  ni=1

widi

+ a · max

|C ∗|, 2

ni=1

widi

Q 

.

By Theorem 4, this quantity is less than or equal to 4 · OPT .  

Applying Theorem6 onan α-approximatetraveling salesperson

tour gives usa factormax

1 +  2β , 2 + α

1 +

  β

2

approximationfor UNEQ-CAP-FM .Theminimumfactorisachievedatavalue β > 0

which makes 1 +   2β

 equal to 2 + α

1 +   β

2

. This gives us a factor

 f (α) =  1 +   4α√ 4α2+24α+4−(2α+2)

approximation for UNEQ-CAP-FM .

In general, given an  α-approximate tour, approximation factorsof 1 +  α, 1 +  α, 1 +   4α√ 

4α2+16α−2α, and 1 +   4α√ 

4α2+24α+4−(2α+2)

can be achieved for   EQ-INF-FM ,   UNEQ-INF-FM ,   EQ-CAP-FM , andUNEQ-CAP-FM  respectively. Further, if we have access to a 1 + ϵ

approximate traveling salesperson tour, the approximation factorscan be improved to 2 +   ϵ, 2 +   ϵ, 2.618 +   ϵ, and 3.414 +   ϵ

respectively, and this indeed is the case for the Euclidean andplanar versions of our problem where PTASes exist to approximate

the length of the traveling salesperson tour [14,3,13].

3.1. Computing the optimal partition

Given a TSP tour C , we can compute the minimum length parti-tion of  C  in  O(n2) time using dynamic programming [4,15]. In thissection, we give an O(n2) time algorithm to compute a partition of tour C  with minimum fuel consumption based on the same ideas.

As before, number the vertices 1, 2, . . . , n   according to theorder in which they occur on tour  C . For a cluster [i, j], we define

 A(i, j)  as the fuel cost of traversing the cluster in the clockwisedirection (i.e., in increasing order of labels). Similarly, define B(i, j)

as the fuel cost of traversing the same cluster in the anticlockwisedirection. Define the fuel consumption   F (i, j)  of cluster [i, j]  as

min ( A(i, j), B(i, j)).For a vertex i, let di  denote the shortest distance from  i  to thedepot. Let d1

i   and d2i   denote distance traveled by a vehicle between

vertex  i  and depot while traversing tour  C  in the clockwise andanticlockwise direction respectively. For  i ∈   V  \ {0}, define W i =n

l=i wl. Given tour C , thearrays di, d1i , d2

i ,and W i can be computedin O(n) time.

 A(i, j)’s satisfy the following recurrence: A(i, j + 1) = A(i, j) +a ·   ∆ +   b ·   ((W i −   W  j +   w j) ·   ∆ +   w j+1d j+1)   where   ∆   =d1

 j  − d1 j+1 +  d j+1 −  d j  is the difference in the length of the tours

that define the fuel consumption  A(i, j + 1) and  A(i, j). We take

 A(i, j) = ∞ if  j

l=i wl   >   Q . Thus, given  A(i, j), one can compute A(i, j + 1)  in constant time. As a base case, we have that for alli ∈ V  \ {0}, A(i, i) = 2adi + bwidi. For a given i, we loop on j from

i + 1 to n, and use the above recurrence to compute  A(i, j) from A(i, j −  1). Thus, we can compute all the   A(i, j)’s in  O(n2)  time.

The algorithm for computing  B(i, j)’s is symmetric. Finally, given A(i, j) and B(i, j), we can compute all the F (i, j)’s in O(n2) time, asrequired.

We now proceed to compute the optimal partition. For   i   ∈V  \{0}, let Q (i) denote the minimum fuel consumed by a partitionof vertices {1, 2, . . . , i}. We define  Q (0)  =   0.  Q (i)’s satisfy thefollowing recurrence: Q (i) = min1≤ j≤i (Q ( j − 1) + F ( j, i)) .

We can compute Q (1), Q (2) , . . . , Q (n) in this order, spendingO(n)  time for computing each   Q (i). Thus, we can compute thefuel cost Q (n) of the optimal partition in O(n2) time. The partitioncorresponding to Q (n) can be computed by storing for each  Q (i)

the index  J ∗i  , 1  ≤   J ∗i   ≤   i, which achieves the minimum in theabove recurrence.

4. Conclusion

Our algorithms can be compared with the algorithm of Blumetal.[5] for cumulative VRP where a single offload is allowed atthedepot.The approximabilityof cumulative VRPwhen thenumber of offloads is given as part of input is an open question.

 Acknowledgments

The authors would like to thank the referees for commentsthat helped with the presentation of some of the material. DG wasfunded in part by the DST-SERC grant SR/S3/EECE/0100/2010 andthe work was done while the author was at IIT Ropar. AM was par-tially supported by the ISIRD grant (Ref. No. ISIRD/10-11/AM/19)from IIT Ropar.

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