A Branch-and-Bound-based solution method for solving vehicle routing problem with fuzzy stochastic demands V P SINGH 1 , KIRTI SHARMA 1 and DEBJANI CHAKRABORTY 2, * 1 Department of Mathematics, VNIT Nagpur, Nagpur, India 2 Department of Mathematics, IIT Kharagpur, Kharagpur, India e-mail: [email protected]; [email protected]MS received 15 March 2021; revised 3 August 2021; accepted 14 August 2021 Abstract. In this paper, a capacitated vehicle routing problem (CVRP) with fuzzy stochastic demands has been presented. Discrete fuzzy random variables have been used to represent the demands of the customers. The objective of CVRP with fuzzy stochastic demands is to obtain a set of routes that originates as well as terminates at the source node and while traversing the route, the demands of all the customers present in the network are satisfied. The task here is to carry out all these operations with minimum cost. CVRP in imprecise and random environment has been considered here, and an a priori route construction technique has been adopted for which Branch and Bound algorithm has been used. The recourse policy used in this work is reactive, i.e. recourse to depot is done only upon the occurrence of the failure. The delivery policy considered here is unsplit delivery. Demands of the customers are the only source of impreciseness and randomness in the problem under con- struction. Parametric graded mean integration representation (PGMIR) method has been used for the comparison purposes, whenever required. A numerical example with four customers has been solved to present the proposed methodology. Keywords. Vehicle routing problem; Branch and Bound algorithm; discrete fuzzy random variable; fuzzy stochastic demands. 1. Introduction Capacitated vehicle routing problem (CVRP) [1] is a very- well-known problem of operations research, which aims at finding a set of routes in a network beginning and ending at the same node (usually called depot node) and fulfilling the demands of every customer present in the network with minimum possible cost. Because of varied applications of the problem in transportation management, logistic ser- vices, pickup and delivery services, communication net- works, etc., it has gained attention of researchers from both academia and industrial backgrounds in recent decades. In CVRP, usually a weighted graph [2] is presented in which the edge weights represent cost (time or distance) required to traverse the particular edge, customers are supposed to be present in the network and the task is to design a set of minimum cost routes satisfying demands of all the cus- tomers in the network. The journey of the travelling salesman (service provider) originates as well as terminates on the depot node. In conventional CVRPs all the param- eters of the problem, namely number of customers, edge weights, carrying capacity of the vehicles and various others, are well known in advance. The very first work regarding vehicle routing problem (VRP) was performed by Dantzig and Ramser [3] in 1959. The problem was then popularly known as truck dispatch- ing problem and was concerned with the delivery of gasoline to gas stations. Various variants of VRP arise because of uncertainty and variability of different param- eters of the network. In general, if all the information about the network is available well in advance, such a VRP is known as dynamic VRP whereas if the system conditions are not known earlier in advance then the corresponding VRP is known as stochastic vehicle routing problem (SVRP) [4]. Different variations of SVRP [5] arise because of dif- ferent attributes of the problem. One of the attributes of the problem is the time at which the demand of the customers becomes known [6]. The first extreme case is when the demands of all the customers become known before exe- cuting the route and this gives rise to the classical version of CVRP [7]. Another extreme case is when the demands of the customers are known only when the vehicle arrives at the particular customer [7]. In between these two extreme cases, there lies a whole spectrum of possibilities. Another attribute of the problem is regarding the service policy. There are basically two types of service policies, namely split and unsplit deliveries. In case of split delivery, the *For correspondence Sådhanå (2021)46:195 Ó Indian Academy of Sciences https://doi.org/10.1007/s12046-021-01722-0
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A Branch-and-Bound-based solution method for solving vehiclerouting problem with fuzzy stochastic demands
V P SINGH1, KIRTI SHARMA1 and DEBJANI CHAKRABORTY2,*
1Department of Mathematics, VNIT Nagpur, Nagpur, India2Department of Mathematics, IIT Kharagpur, Kharagpur, India
Definition 6 Fuzzy random variable [28]: If the range of a
random variable extends from the set of real numbers to the
set of fuzzy numbers, then such a random variable is known
as a fuzzy random variable [29].
Definition 7 Expectation of a fuzzy random variable: The
concept of expectation of random variable can be easily
extended to the expectation of a fuzzy random variable. If ~X
is a discrete fuzzy random variable and Pð ~X ¼ ~xiÞ ¼ ~pi,i=1, 2, 3, . . .,n then the expectation of given fuzzy random
variable [30] is given by Eq. (4):
E ~X ¼Xni¼1
~xi � ~pi: ð4Þ
2.1 Branch and Bound algorithm
Branch and Bound algorithm [31] was proposed by Ailsa
Land and Alison Doig in 1960. Branch and Bound design
technique is used for solving mathematical optimization
problems as well as combinatorial optimization problems
[2]. In Branch and Bound algorithm, the set of all candidate
solutions is systematically enumerated using a state space
search. In state space search, a rooted tree represents the set
of all possible solutions. In this algorithm the branches of
the tree are explored, which represent the subsets of the
solution set. With every node in the state space tree a bound
is attached, which is an estimate on the bound that will be
achieved if the tree below that node is explored. So, while
exploring the tree using Branch and Bound technique, only
those branches are explored that provide a better estimate
of bounds.
The major challenge in the technique is to compute a
bound on the best possible solution. In the case of TSP [2]
the lower bound on the tour can be obtained by adding the
cost of two minimum weighted edges corresponding to
every vertex and then dividing that by 2, i.e.
cost ¼Pu2V
sum of two minimum weighted edges wrt u
2
2666
3777:ð5Þ
An example discussing the solution procedure of finding
minimum weighted Hamiltonian tour using the Branch and
Bound algorithm is discussed in ‘‘Appendix I’’.
3. Properties of SVRP
There are various techniques to solve VRP with constraints
that are supposed to produce optimal results. Re-opti-
mization of routes after occurrence of failure, an optimal
restocking policy and an optimal service policy are a few of
them. Solving an SVRP includes solving a TSP in the first
stage. There are also various properties that a solution of
TSP should follow if it is obtained using optimal tech-
niques. Properties such as Bellman’s principle of optimal-
ity, optimal cost being independent of the direction of
traversal and the non-intersection property of optimal cir-
cuit are a few of them. In this section, we re-examine a few
of these properties in the context of SVRP.
Property 1 The re-optimization of tour after occurrenceof failure on the designed a priori route does not alwaysreduce the cost of operation.
Solution of SVRP consists of two stages. In the first
stage, an a priori route is designed without considering the
demands of the customers and in the second stage that route
is executed. While executing the route, it may happen that
the demand of a customer at hand may exceed the residual
capacity of the vehicle. In such cases, a route failure is said
to occur and vehicle is supposed to return to the depot node.
After returning to depot, the travelling salesperson has two
options: to resume the service as per the route designed in
stage 1 or to reconstruct a route for servicing the remaining
customers. Re-constructing/re-optimizing a route leads to
an increase in time complexity of the algorithm as well as
an increase in time for decision making. In some cases, re-
optimization of routes may lead to a least-cost tour but the
chances of landing up at a non-optimal tour can also not be
ignored. To see how re-optimization affects the cost of tour
in SVRP, with the help of an example, readers may see
‘‘Appendix II’’.
Property 2 The Bellman principle of optimality does notalways hold in the case of SVRP.
The Bellman principle of optimality states that an opti-
mal policy possesses the property that whatever the initial
states and initial decisions are, the decision that will follow
must create an optimal policy from the state resulting from
the first decision. In the case of SVRP, the initial state is the
network in which the customers are present and initial
decision is opting the a priori route obtained by solving the
corresponding TSP. According to the Bellman principle of
optimality, the decision of opting a priori route obtained
using any optimal method should give us the least-cost tour.
For more information, readers may see ‘‘Appendix III’’.
Property 3 The cost of the tour is independent of direc-tion of traversal.
For SVRP, the cost of the route designed using Branch
and Bound algorithm is not independent of the direction of
195 Page 4 of 17 Sådhanå (2021) 46:195
the traversal of the route. This is illustrated with the help of
figure 13 in ‘‘Appendix III’’.
4. CVRP in an imprecise and random environment
The deterministic version of the VRP generally deals with
the distribution of a particular commodity from a depot
node to a set of customer nodes. The depot node is assumed
to have sufficient amount of commodity and the demands of
the customers are known precisely well in advance. The
task here is to design the routes such that the demand of
every customer is fulfilled, no customer is visited more than
once and the cost incurred in performing such operation is
minimum.
However, different variations of VRP may arise in real-
life problem because of components like randomness and
impreciseness. Because of wide range of applications of
VRP in real-life problems, mathematical model of the
problem in an imprecise and random environment should
also be discussed and worked out. Factors like randomness
may creep into the environment because of the random
nature of parameters like edge weights, customers’ demand,
number of customers in the network, structure of the net-
work and many more. Impreciseness may creep into the
environment because of factors like demands of the cus-
tomers, edge weights, etc. In this work, the nature of the
network has been assumed to be known precisely and well
in advance and the factors like impreciseness and ran-
domness creep in only because of the demands of the
customers. In this work, the demands of the customers are
neither known in advance nor known precisely; however,
an estimate of the imprecise demands of the customers can
be obtained on the basis of past experiences of serving these
customers. Thus, with respect to every customer, there
exists a fuzzy random variable that entails the fuzzy
probability of the fuzzy demands of the customers. In this
work, we consider only the demands of the customers as a
reason for impreciseness and randomness.
4.1 Assumptions of the model
Before discussing the mathematical model of CVRP in
mixed environment, some basic assumptions regarding the
model have been presented that will be used throughout the
paper. The very first assumption is regarding the network
and it states that the network under consideration is sym-
metric and follows triangle inequality, i.e. if the cost of
traversal from i to j is denoted by cij then cij ¼ cji and
cij � cik þ ckj. In this work, we have assumed that edge
weight represents the cost of traversal of edges and is given
in deterministic form. A set of assumptions is also made for
the demand of customers present in the network. The
demands of the customers in this work are known to be the
source of impreciseness as well as randomness. The
demands of the customers are not known well in advance
and are specified by the customers only when the delivery is
made. It is assumed that the demands, as told by customers,
are imprecise in nature. The demands of the customers in
this work are given by fuzzy random variables where a
fuzzy probability is associated with a fuzzy demand. The
demands of the customers are assumed to be independent of
each other and the demand of every customer is less than
the capacity of the vehicle, so as to make the problem
feasible in unsplit delivery conditions.
The journey of the fleet is assumed to originate and
terminate at the source vertex only. The service policy in
this work is assumed to be unsplit delivery, i.e. a customer
is serviced only once. An a priori route construction
technique is used in this work and route is constructed only
once. We assume that while executing the a priori routeobtained in stage 1, there is a possibility of more than one
failure. A reactive recourse policy has been adopted in this
work, i.e. a return trip to depot node for the replenishment
of goods is performed only upon the occurrence of failure.
After replenishment, the route formed earlier is re-executed
from the point of occurrence of failure and no new route is
constructed. The demand at the depot node is assumed to be
0 units and the fleet of vehicles present at depot is assumed
to be homogeneous, i.e all vehicles have identical operating
costs and they have the same carrying capacity.
4.2 Applications of CVRP in mixed environment
Real world applications of the SVRP include among others
the planning of cash distribution to various branches of a
bank or ATMs in a city [4]; in this case, the amount of the
cash to be delivered to various branches is a random vari-
able; the randomness in cash collection occurs due to the
unpredictability of demands and impreciseness occurs
because of lack of exact knowledge of next day’s require-
ments. Other examples include the delivery of essential
commodity (milk, oil) where daily customer consumption
is random in nature. Sometimes, the amount of these
commodities to be delivered is also not known precisely.
Such situations arise when the commodity is not measured
in units, rather it is weighed; e.g. ‘‘ approximately n ton-
nes’’ of goods is to be delivered at a particular node. In the
absence of a weighing device, weighing of good is per-
formed on the basis of human intelligence and past expe-
riences. Such conditions give rise to imprecise and random
demands of customers. The nature of network depends on
the entities stored in the adjacency matrix.
In this work, we consider VRP where the network is
deterministic and precise and the demands of the customers
are imprecise as well as random. Such a situation is rep-
resented by figure 2. The a priori route that is constructed
in stage 1 of the problem using various algorithms for
finding minimum cost Hamiltonian circuit is represented by
figure 2a; figure 2b refers to the second stage when the
Sådhanå (2021) 46:195 Page 5 of 17 195
route obtained in stage 1 is executed and failure occurs.
Upon occurrence of failure, re-routing to depot is done and
then services are resumed after replenishment. The objec-
tive of the problem is to find a minimum cost tour such that
demands of all the customers are satisfied without violating
any constraint of the problem.
4.3 Mathematical model
A VRP with fuzzy stochastic demands is represented by a
complete weighted graph G ¼ ðV ;EÞ where V is the set of
vertices in the network and E is the set of edges joining
these vertices. The set of vertices include a depot node and
a finite number of customer nodes. The depot node is
assumed to have ample stock of the commodity, which is to
be delivered to the customers. A homogeneous fleet of
vehicles is also present at the depot node. The remaining
nodes (customer nodes) are the nodes where customers with
fuzzy stochastic demands of a commodity are present and
wait for the commodity to be delivered. ~Di is the fuzzy
random variable representing the fuzzy demands of the
customer located at node i. The edge weights represent the
cost of traversal of a particular edge. Table 2 comprises the
symbols used in the mathematical model and their
descriptions.
A route is defined as a path of the form r ¼ði1; i2; . . .; ijrjÞ where i1 ¼ ijrj ¼ depot node with ik 2customer nodes for k 2 f2; 3; . . .; jrj � 1g. We define
~TDð~lik ; ~r2ikÞ ¼Pk
l¼1~Dil to be the fuzzy random variable
indicating total actual cumulative demand at ik for
k 2 f2; 3; . . .; jrjg. Since the demands of the customers are
independent, we have ~lik ¼Pk
l¼1 Eð ~DilÞ and
~r2ik ¼Pk
l¼1 Varð ~DilÞ. Given a route r ¼ ði1; i2; . . .; ijrjÞ, wedenote EFCikð~lik ; ~r2ikÞ as the expected failure cost at cus-
tomer ik.So, we write
EFC ~lik ;~r2ik
� �¼ 2c0ik
X1u¼1
P ~TD ~lik�1; ~r2ik�1
� �� uQ
� �n
�P ~TD ~lik ;~r2ik
� �� uQ
� �oð6Þ
where P(E) denotes the probability of occurrence of event
E. Pð ~TDð~lik�1; ~r2ik�1
Þ� uQÞ � Pð ~TDð~lik ; ~r2ikÞ� uQÞ whereQ, the capacity of the vehicle, can therefore be interpreted
as the probability of having the uth failure at ik customer
with the condition that failure has yet not occurred on any
previously visited customer along the route.
Thus, an a priori model for solving VRP with fuzzy
stochastic demands is given as follows:
minimizeX
cijxij þX
EFCihþ1ð~lihþ1
; ~r2ihþ1Þ
subject to
Xnj¼2
xij ¼2m ð7Þ
Xi\k
xik þXk\j
xkj ¼2 ð8ÞXi\k
xik þXk\j
xkj ¼2 ð9Þ
Xvi;vj2S
xij � j S j �P
vi2S G E½Di�ð ÞQ
ð10Þ
(a) A priori route.
(b) The final route.
Figure 2. Real-life application of SVRP.
195 Page 6 of 17 Sådhanå (2021) 46:195
S � V � fv0g; 2� j S j � n� 2 ð11Þ
xij ¼f0; 1g j ¼ 2; . . .; n ð12Þ
x0j ¼f0; 1; 2g 8f0; jg 2 E ð13Þ
x ¼xij an integer array ð14ÞIn this mathematical model presented here, constraint rep-
resented by Eq. (7) ensures that exactly m vehicles start
their journey from depot node and end their journey at
depot node. Constraint represented by Eq. (9) ensures that
every vertex is traversed exactly once. Constraint repre-
sented by Eqs. (10) and (11) eliminates the infeasible routes
with excessive commodity demand. First stage of solution
finding is finding an a priori solution that deals with finding
the minimum cost Hamiltonian circuit and the first com-
ponent of objective function deals with the corresponding
problem statement. The second component of objective
function finds out the effective failure cost incurred in
executing the path obtained in first stage of solution-finding
approach.
The first stage solution is obtained without considering
the demands of the customers. However, in the presence of
stochastic demands of the customer, a route obtained earlier
may fail because the observed demands of a customer en-
route may exceed the residual capacity of the vehicle and in
such cases the vehicle is bound to return to the depot, refill
and then resume the delivery. Such a scenario of failure of
route compels the vehicle to perform recourse actions and
these actions, in turn, increase the cost of operation. So the
total cost of operation is given by the sum of deterministic
cost, which is obtained in first stage of solution finding, and
expected cost of recourse actions, which is calculated in
second stage and is denoted by the second component of
objective function.
5. Flowchart of the method
The flowchart of the method discussed is given by figure 3
6. Algorithm of the proposed model
The algorithm of the proposed model has been divided into
two parts. In the first part, the a priori route that the trav-
elling salesman should take and the cost of that route are
determined. This part of the methodology is presented
using Algorithm 1. In the second part of the algorithm, the
route obtained in part 1 is traversed and effective failure
cost corresponding to every vertex are determined. This
part of the methodology is shown by Algorithm 2. Table 3
comprises the symbols and their descriptions used in
Algorithms 1 and 2.
Table 2. Description of symbols used in the mathematical model.
Symbol Description of the symbol
P(E) Probability of an event Ecij Cost of traversal of edge ij
V The set of vertices in the network
E The set of edges in the network~Di Fuzzy random variable denoting demands of customer at node i
r = ði1; i2; . . .; ijrjÞ A route starting and ending at i1 and ijrj, respectively~lik Fuzzy expected demand of customer ijkj in the route r
~r2ik Fuzzy variance of demand at customer ijkj in the route r
~TD ~lik ; ~r2ik
� �Fuzzy cumulative demand at customer ijkj in the route r
Q Capacity of the vehicle
c0ik Cost of traversal from node 0 to node ijkj in the route r
Gð ~AÞ GMIR representation of ~A
EFC ~lik ; ~r2ik
� �Effective failure cost at customer ijkj of the route r
xij A binary integer array whose entry is 0 when edge ij is not traversed and 1
when edge ij is traversed once
x0j A binary integer array whose entry is 0 when edge 0j is not traversed, 1when edge 0j is traversed once and 2 when edge 0j is traversed twice
m The number of vehicles at depot~A The fuzzy number ~Al ~AðxÞ Membership function of ~Asupl ~AðxÞ Supremum of membership function
Sådhanå (2021) 46:195 Page 7 of 17 195
7. Numerical example
To illustrate the working of the method proposed, let us
consider a network in which there are four customers and a
single depot. Suppose that the customers present in the
network have stochastic as well as imprecise demands, i.e.
the demands of the customers are not defined crisply and
are revealed only upon the arrival of the distributor at the
customers’ end. The edge weights of the network repre-
sented by figure 4 store the cost required to traverse the
corresponding edge. The demands of the customers are not
revealed in advance but the probability distribution (mass)
function of demands of all the customers present in the
network can be estimated easily. For the customers and the
network under consideration, the fuzzy probability distri-
bution of the demands of customers is given by Table 4.
The depot node is denoted by node 0 and customers wait at
nodes 1–4. In the graph, the travel costs are assumed to be
symmetric. The demand at the depot node is considered to
be 0 units and the carrying capacity of vehicle is assumed to
be 60 units.
Before starting to find out the formal solution, we first
calculate the expected demand at each node.
E½ ~D1� ¼X2i¼1
~D1i ~p1i
¼ ~D11 ~p11 þ ~D12 ~p12
¼ ð18:50; 22:75; 27Þ
E½ ~D2� ¼X2i¼1
~D2i ~p2i
¼ ~D21 ~p21 þ ~D22 ~p22
¼ ð24:5; 33; 41:5Þ
E½ ~D3� ¼X2i¼1
~D3i ~p3i
¼ ~D31 ~p31 þ ~D32 ~p32
¼ ð30:5; 41; 51:5Þ
E½ ~D4� ¼X2i¼1
~D4i ~p4i
¼ ~D41 ~p41 þ ~D42 ~p42
¼ ð35:5; 53; 70:5ÞThe total actual cumulative demand will be the sum of all
expected demands, i.e.
195 Page 8 of 17 Sådhanå (2021) 46:195
total cumulative demand ¼X4i¼1
E½ ~Di�
¼ ð109; 149:75; 190:5ÞDefuzzifying this triangular fuzzy number using the GMIR
method given by Eq. (3) gives
GX4i¼1
E½ ~Di� !
¼ Gð109; 149:75; 190:5Þ
¼ 149:75
In this example, the capacity of one vehicle is assumed to
be 60 units and since there are chances of failures, it is
better to approximate the number of times the vehicle
should return to the source node and re-continue its service.
The number of times the vehicle should return to the source
node can be obtained by dividing the total actual cumula-
tive demand by the capacity of the vehicle:
#ðnÞ ¼ 149:75
60 2:5
Thus, in order to accomplish the demand of the customers,
minimum three vehicles are required or a single vehicle is
required to make 3 trips.
The solution finding procedure can be divided into two
stages where stage 1 deals with finding an a priori sequenceof edges, which should be travelled so that every vertex is
visited exactly once by the end of the tour traversal and cost
incurred in performing this operation comes out to be a
minimum and stage 2 corresponds to executing the route
obtained in stage 1 and fulfilling the demands of customers
when that customer is visited. While executing the route, it
may happen that the demand of the customer exceeds the
residual capacity of the vehicle and in such situation the
salesperson is bound to return to the depot, refill the vehicle
and resume the services from the customer at whom the
failure occurred. With the occurrence of failure, the total
Figure 3. Flowchart of the model.
Table 3. Description of symbols used in Algorithms 1 and 2.
Symbol Description of the symbol
cost[][] The cost matrix.
curr-
bound
The lower bound of the root node
curr-cost Cost of the path found so far
l The current level in the search space tree
curr-path The path visited till now
visited[] A binary array whose ith entry is 1 if ith node
is visited and 0 if the node is not visited
cost Cost of optimal TSP tour
curr-res Cost of the solution found so far
Fpath The minimum cost tour
EFC[i] The effective failure cost at customer i
TEFC The total effective failure cost
Expcost Expected cost of the minimum length tour
Figure 4. Network.
Table 4. Demands of the customers.
Node Demand Probability
1 (18, 20, 22) (0.40, 0.45, 0.50)
1 (23, 25, 27) (0.50, 0.55, 0.60)
2 (28, 30, 32) (0.3, 0.4, 0.5)
2 (33, 35, 37) (0.5, 0.6, 0.7)
3 (38, 40, 42) (0.7, 0.8, 0.9)
3 (43, 45, 47) (0.1, 0.2, 0.3)
4 (48, 50, 52) (0.2, 0.4, 0.6)
4 (53, 55, 57) (0.5, 0.6, 0.7)
Sådhanå (2021) 46:195 Page 9 of 17 195
cost of the operation increases and this can be estimated by
calculating the effective failure cost at every vertex.
The solution for stage 1 of the problem can be obtained
using any algorithm that is used for solving TSP. In this
work, we use the Branch and Bound algorithm for finding
the path that the salesman should follow because of the
guarantee of the optimality of the method and lesser
memory requirements and lesser complications while
solving. While finding the solution for stage 1, the demands
of the customers are not considered. A schematic diagram
representing the route to be taken is given in figure 5 and
the route that should be taken is 0-1-2-3-4-0 and the cost of
traversal of this route is 27.0 units.
In the second stage of the solution, the route obtained in
first stage is traversed and on the occurrence of failure a trip
to depot node is made to refill the vehicle and then the
service continues. In such a case, effective failure cost gets
associated with every vertex that represents the cost of a
route if a failure occurs at that specific node assuming that
the failure has not yet occurred on any other previous
nodes. The formula to calculate the effective failure cost at
a node in the route is given by Eq. (6).
Then the sum of effective failure costs corresponding to
every vertex gives the total effective failure cost and the
sum of total effective failure cost and cost obtained in stage
1 corresponds to the expected cost of the operation:
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