a branch-and-bound-based solution method for solving
TRANSCRIPT
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
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 � Indian Academy of Sciences
https://doi.org/10.1007/s12046-021-01722-0Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)
demand of the customers can be split among 2 vehicles or
multiple visits of the same vehicle and hence in the case of
split delivery [8], the demand of the customer may exceed
the carrying capacity of the vehicle. In case of unsplit
delivery problem [9], a customer is visited exactly once and
hence the customers with demands greater than the carrying
capacity of the vehicle are not allowed so as to keep the
problem feasible. A bi-objective model for disaster relief
location routing problem with stochastic demands is dis-
cussed in [10]. In this work, a model for risk-averse opti-
mization of disaster relief facility location is presented. The
problems of selection of distribution centers, assignment of
vehicles, route selection of vehicles and delivery opti-
mization are handled simultaneously in this work. A multi-
objective model for location routing problem with uncertain
data after disasters is discussed in [11]. In [11] three
objectives are considered: namely, minimization of total
costs (which include fixed costs and vehicle transportation
costs), maximization of satisfaction rates and maximization
of transport capacities of worst path. The uncertainty in
parameters is handled by taking interval numbers as a
method of representation. Such objectives improve the
working of disaster relief teams.
Some other attributes of the problem are recourse poli-
cies and re-routing policies. Recourse policies are used to
decide when the vehicle should return to the depot for
replenishment. There are two recourse policies: namely
preventive and reactive. In reactive recourse policy [12]
vehicle returns to depot only after the realization of
occurrence of failure, whereas in preventive recourse policy
[13] a preventive trip to depot for replenishment can be
made when the residual capacity of the vehicle falls below
a certain threshold. In case of solution methods, an a prioriroute is constructed in the first stage and that route is
executed in the second stage. One of the important features
is re-routing when the failure of the route occurs. There are
two possibilities in this case. The first possibility is that the
salesman sticks to the original route after route failure and
resumes the delivery from the point where the failure
occurred. The second possibility is to construct a route
again by considering only those customers who are yet not
serviced. Constructing a route after occurrence of failure
may increase the intractability of the problem but it may
help in reducing the cost of the tour.
On the basis of different service policies and the point at
which the demands become known, the problem may be
solved in various number of stages. For example, in case of
unsplit delivery problem where the information about
demands of customers is known well in advance, the
problem can be solved in exactly one stage. In case of
SVRP when demands are not known well in advance there
are many stages, where each stage corresponds to delivery
of goods for some customers and then taking some decision
about whether to go to depot node or go to the next cus-
tomer. In case of unsplit delivery with stochastic demands
there may be at most n stages but in case of split delivery
with stochastic demands, there can be more than n stages
where n is the number of customers. Table 1 presents the
literature survey in a gist.
In this paper, we consider CVRP in an imprecise and
random environment: an environment which is an amal-
gamation of impreciseness and randomness [21–23]. To the
best of author’s knowledge, a combination of impreciseness
and randomness with CVRP has never been dealt with in
the literature. In this work, the demands of the customers
are assumed to be imprecise as well as uncertain. To handle
the randomness (state uncertainty) and impreciseness, the
problem can be generalized to CVRP with fuzzy stochastic
demands (CVRPFSD). CVRPFSD, in this paper, has been
modeled as a two-level stochastic process. While solving
the problem, the first task is to design a minimum cost apriori route. In the second stage, the corresponding path is
executed. This is because the demands are imprecise and
random, which means that the demands are revealed only
upon the arrival of vehicle and such randomness of
parameters may give rise to a situation when the residual
capacity of the vehicle is less than the demand of the
customer. This is the case when failure occurs and upon
occurrence of failure, vehicle returns to the depot, performs
replenishment and then services are resumed. In this work,
unsplit delivery policy is adopted and vehicle re-routes to
depot only upon the occurrence of failure; i.e. the reactive
recourse policy is in action. Here, in this work, re-opti-
mization of routes is not performed after failure and
demands of customers are known only upon the arrival of
vehicle at the customer node; however, on the basis of past
experience, a fuzzy probability function [24] can be esti-
mated for every customer that gives a fuzzy probability of a
customer having some fuzzy demand.
The paper is structured in the following way. In Sect. 2,
some basic concepts and definitions of fuzzy set theory
have been discussed. In Sect. 3, some properties related to
the travelling salesman problem (TSP) have been re-ex-
amined in the context of SVRP. In Sect. 4, a mathematical
model for solving CVRP in an imprecise and random
environment has been presented. Flowchart of the method
is provided in Sect. 5. Algorithm of the model is presented
in Sect. 6. In Sect. 7, a numerical example with 4 cus-
tomers having fuzzy stochastic demands represented with
the help of fuzzy random variable has been discussed. In
Sect. 8, an analysis of various methods is presented and the
work has been compared to several other methods. The last
section consists of the concluding remarks.
2. Preliminary concepts and definitions
In this section, various concepts about fuzzy set theory and
the Branch and Bound algorithm for finding minimum
weighted Hamiltonian tour have been reviewed. For
numerical examples, readers may see ‘‘Appendix I’’.
195 Page 2 of 17 Sådhanå (2021) 46:195
Definition 1 Fuzzy number: A fuzzy set [25] ~A on R is
said to be a fuzzy number if the following three properties
are satisfied:
1. Fuzzy set ~A must be normal, i.e. 9 x such that
sup l ~AðxÞ ¼ 1, where sup stands for supremum.
2. The support of fuzzy set, i.e. set of all the elements with
nonzero degree of membership, must be bounded.
3. a level set, i.e. set of all the elements with membership
degree greater than a, must be a closed interval for a 2[0,1].
Definition 2 A triangular fuzzy number: The membership
function of a generalized triangular fuzzy number [26]
denoted by ~A ¼ ða; b; cÞ is given by Eq. (1) and is shown byfigure 1.
l ~AðxÞ ¼
0 if x\ax� a
b� aif a� x� b
c� x
c� bif b� x� c
0 if x[ c
8>>>>><>>>>>:
ð1Þ
Definition 3 Graded mean integration representation
(GMIR) [27]: Let L�1 and R�1 be the inverse functions of
L(left) and R(right), respectively; then the GMIR of a
generalized triangular fuzzy number ~A is given by Eq. (2):
Gð ~AÞ ¼R 10h L�1ðhÞþR�1ðhÞ
2
� �dhR 1
0hdh
: ð2Þ
Thus, the GMIR [27] of ~A is given by Eq. (3):
Gð ~AÞ ¼ aþ 4bþ c
6ð3Þ
Definition 4 Symmetric triangular fuzzy number: A tri-
angular fuzzy number is said to be symmetric iff its left and
right spreads are equal, i.e. if ~A ¼ ða; b; cÞ is a triangular
fuzzy number then it is said to be symmetric if and only if
ðb� aÞ ¼ ðc� bÞNote A symmetric triangular fuzzy number is denoted by
ðm; a; aÞ where m is the constant value at which the grade of
membership is 1 and a is the spread on left as well as on
right side.
Definition 5 Arithmetic operations on symmetric trian-
gular fuzzy number: Let ~A ¼ ðm1; a; aÞ and ~B ¼ ðm2; b; bÞbe two symmetric TFNs, then
1. Addition
ðm1; a; aÞ � ðm2; b; bÞ ¼ ðm1 þ m2; aþ b; aþ bÞ2. Symmetric image
�ðm; a; aÞ ¼ ð�m; a; aÞ3. Subtraction
Table 1. Literature survey.
Name of authors Type of environment Concerned parameter
Laporte [7] Deterministic Minimizing cost of the operation
Bertsimas [4] Stochastic Minimize cost when demands are uncertain
Salavati-Khoshghalb et al [14] Stochastic SVRP with active recourse policy
Salavati-Khoshghalb et al [15] Stochastic SVRP with preventive recourse policy
Gendreau et al [16] Stochastic SVRP with active recourse policy
Laporte et al [17] Stochastic VRP with stochastic travel times
Singh and Sharma [18] Fuzzy VRP with fuzzy demands
Zulvia et al [19] Fuzzy VRP with fuzzy travel time and fuzzy demand
Kuo et al [20] Fuzzy VRP with fuzzy demands
Gaur et al [8], Gupta et al [9] Stochastic SVRP with split delivery
Gaur et al [8], Gupta et al [9] Stochastic SVRP with unsplit delivery
Figure 1. A generalized triangular fuzzy number ~A ¼ ða; b; cÞ:
Sådhanå (2021) 46:195 Page 3 of 17 195
ðm1; a; aÞ � ðm2; b; bÞ ¼ ðm1 � m2; aþ b; aþ bÞ4. Multiplication
ðm1; a; aÞ � ðm2; b; bÞ ¼ ðm1m2;m2aþ m1b;m2aþ m1bÞ
for ~A and ~B positive.
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:
total effective failure cost ¼Xni¼1
EFCi
¼ ð�27:2; 30; 87:2Þexpected cost ¼ cost obtained in stage 1 þ
total effective failure cost
¼ ð�0:2; 57; 114:2Þ
8. Discussion and analysis
In the presented work, the first stage of the solution finding
corresponds to the very famous problem of finding the
travelling salesman path in which the objective is to visit
every vertex in the network exactly once such that the cost
of operation is minimum [31]. There are several methods
Figure 5. Path by Branch and Bound.
Table 5. Comparison of various methods.
Name of the method Path Expected cost Time complexity
Brute force approach [2] 0-1-2-3-4-0
0-4-3-2-1-0
(- 0.2, 57, 114.2)
(- 42.8080, 57.256, 171.88)
O(n!)
Christofides algorithm [38] 0-4-3-1-2-0 (- 53.568, 64.132, 181.832) O(n4)Clark and Wright algorithm [33] 0-1-2-3-4-0 (- 0.2, 57, 114.2) O(n2 log n)Bellman–Held–Karp algorithm [39] 0-1-2-3-4-0
0-4-3-2-1-0
(- 0.2, 57, 114.2)
(- 42.8080, 57.256, 171.88)O(n22n)
Nearest neighbour algorithm [32] 0-2-1-4-3-0 (- 47, 63, 173) O(n2)Genetic algorithm [34] 0-1-3-4-2-0 (- 13.19, 61.06, 135.31) O(n2 log n)Proposed method 0-1-2-3-4-0 (- 0.2, 57, 114.2) O(n22n)
195 Page 10 of 17 Sådhanå (2021) 46:195
present in the literature for solving TSP and in this work,
we have used Branch and Bound algorithm because this is
an optimal solution method for solving combinatorial
optimization problems. Though the time complexity of the
algorithm is Oðn22nÞ, which is exponential, no algorithm
better than Branch and Bound algorithm has been discov-
ered, as of now. The major advantage of using Branch and
Bound algorithm is that we can control the quality of the
solution to be expected, even if it is yet not found. The
exploration of all feasible paths in the network is done only
in the worst case scenarios. In other classical versions of
VRP, randomness and impreciseness have never been dealt
together. In this work, the demands of the customers are
imprecise as well as random in nature and probability
theory and fuzzy set theory are used to handle randomness
and impreciseness of the model, respectively. GMIR
method has been used for comparison purposes in this
work, whenever required.
The first stage of solution finding corresponds to finding
a minimum weighted Hamiltonian circuit starting and ter-
minating at depot node and the literature comprises several
exact methods like Dynamic Programming, Brute force
approach, Branch and Bound algorithms and several more.
Apart from exact methods there are several algorithms that
although do not give an optimal solution, yet give a good
solution in lesser time using some intelligent heuristics.
Nearest neighbour algorithm [32] and Clark and Wright
algorithm [18, 33] are two such heuristic-based algorithms
used for solving TSP. Several other methods based on
meta-heuristics are Genetic algorithm [34], Tabu Search
methods [35], Ant colony optimization [36], Simulated
annealing method [37], Particle swarm optimization [37]
and many more. Table 5 comprises comparison of method
presented in this work with various other methods. The
comparison has been done on the network given in the
numerical example presented earlier.
The algorithms like Bellman–Held–Karp algorithm and
Brute force approach give optimal solutions. The time
complexity of Brute force approach is O(n!), factorial innature, which cannot be used for practical situations when
the number of nodes in the network exceeds even 10. The
complexity of Held–Karp algorithm is the same as that of
Branch and Bound algorithm but the memory requirement
of Held–Karp algorithm is more, and moreover the quality
of the solution cannot be controlled. Other methods like
Clark and Wright, nearest neighbour and Christofides
algorithm are based on heuristics and thus do not guarantee
optimality of the solution. Methods such as genetic algo-
rithm mimic the natural process of evolution but the solu-
tion obtained by such algorithms may get stuck in local
optima and hence do not always give optimal solutions. In
the proposed method the algorithm can be practically
applicable for approximately 70 customers in the network,
still providing an optimal solution with lesser memory
requirements. So, the criterion of optimality as well as
memory requirements has been taken care of in the pro-
posed method.
9. Conclusion
In this work, a mathematical model of VRP with fuzzy
stochastic demands and the algorithm to solve such a
problem has been presented. In practical life, while deliv-
ering a certain commodity to a set of customers in a net-
work, sometimes the demands of the customers are neither
known precisely nor in advance. Such practical life situa-
tions give rise to VRP with fuzzy stochastic demands. In
mathematical modeling of CVRP with fuzzy stochastic
demands, the objective is to find a minimum weighted
Hamiltonian circuit starting as well as terminating on depot
node in such a way that the demands of all the customers
present in the network are fulfilled when the route is exe-
cuted. In this work, the demands of the customers are
imprecise as well as stochastic in nature. A probability
mass function of demands with respect to every customer is
estimated on the basis of past experiences and the demands
of the customers and their respective probability are rep-
resented using symmetric triangular fuzzy number.
The approach used in this work is a priori in nature, i.e.
the route construction is done first and while doing so the
demands of the customers are not kept in mind. Here, we
used Branch and Bound algorithm for route construction.
The recourse policy used here is reactive in nature, i.e. the
return trip to depot node is performed only upon the
occurrence of failure of the route. A numerical example has
also been solved using the proposed approach. These results
may be useful to find the minimum weighted tour for any
commodity delivery problem when the demands of the
customers are imprecise as well as random in nature and the
network under consideration is deterministic in nature. This
amalgamation of uncertainty and randomness will help us
to cover more realistic situations while modeling VRP.
Appendix I. Appendix A
Example 1 If ~A ¼ ð2; 6; 8Þ is a triangular fuzzy number,
then its GMIR representation is given as
Gðð2; 6; 8ÞÞ ¼ 2þ 24þ 8
6¼ 5:66
Example 2 (2, 5, 8) is a symmetric TFN. The graph of the
membership function of a symmetric triangular fuzzy
number is given by figure 6:
Example 3 Let ~A ¼ ð6; 2; 2Þ and ~B ¼ ð4; 3; 3Þ be two
symmetric triangular fuzzy numbers; then
Sådhanå (2021) 46:195 Page 11 of 17 195
~A� ~B ¼ ð6; 2; 2Þ � ð4; 3; 3Þ ¼ ð10; 5; 5Þ~A� ~B ¼ ð6; 2; 2Þ � ð4; 3; 3Þ ¼ ð2; 5; 5Þ~A� ~B ¼ ð6; 2; 2Þ � ð4; 3; 3Þ ¼ ð24; 26; 26Þ� ~A ¼ �ð6; 2; 2Þ ¼ ð�6; 2; 2Þ
Example 4 Suppose we are given a network as that of
figure 7; the task is to find a minimum cost tour that starts at
0, traverses every vertex and at last returns to 0. This
problem is the same as finding the minimum cost Hamil-
tonian circuit or finding the minimum cost travelling
salesman tour. There are several exact as well as heuristic
methods in the literature that can be used to solve this
problem. The exact methods at one hand include the
methods like Bellman–Held–Karp algorithm and Branch
and Bound technique whereas the approximation methods
include the use of heuristics like nearest neighbour, inser-
tion, sweep and many others. In this example, we will look
for optimal solution of the problem stated earlier using
Branch and Bound technique.
While finding the tour, a bound is to be calculated
associated with the root node using the formula Eq. (5) and
we call it as lower bound. While calculating the lower
bound associated with the root node, we calculate the sum
of the two minimum weighted edges corresponding to
every vertex in the network and divide it by 2. The
calculation of lower bound with respect to root node can be
shown as follows:
lower bound ¼10þ 15þ 10þ 25þ 30þ 15þ 20þ 25
2
� �¼ 75
While traversing down the tree, if we want to find the lower
bound for the node corresponding to the branch 0-3, then
the weight of the edge 0-3 is to be included (even if it is not
in the two minimum weighted edges corresponding to the
vertex 0). So, the lower bound corresponding to the edge 0-
3 can be calculated in the following manner:
lower bound ¼10þ 20þ 10þ 25þ 30þ 15þ 20þ 25
2
� �¼ 78
In the same way, a lower bound corresponding to every
possible set of solutions can be calculated. The nodes at
depth 1 correspond to the subset of solutions where only
one particular node is traversed after 0. According to the
state space tree, the lowest cost tours, that can be obtained
by traversing 1, 2 and 3 immediately after 0 are 75, 75 and
78 units, respectively. While taking the decision to traverse
only one node, the node that corresponds to minimum cost
is traversed in case of minimization problem. At this
juncture, either the node 1 or 2 can be traversed because of
the same lower bounds. The process continues until all the
nodes are traversed. A state space search tree corresponding
to Branch and Bound algorithm used for solving travelling
salesman problem for the network given in figure 7 is given
by figure 8; the minimum cost path according to the algo-
rithm is 0-1-3-2-0 (0-2-3-1-0) and the cost of this path is 80
units.
As we move down the tree, the lower bound can be
calculated in an almost similar way but keeping in mind to
add the weight of the edge that is to be traversed in that
particular branch. For example, while calculating the lower
bound estimate for branch corresponding to 0-2, the weight
of the edge 0-2 should be added even if 0-2 is not among
the two minimum weighted edges. A node is explored only
when the bound it provides is the best. Example 4 presents
the working of the algorithm in the deterministic
environment.
Figure 6. A symmetric TFN.
Figure 7. A network of four nodes.
195 Page 12 of 17 Sådhanå (2021) 46:195
Appendix II. Property 1
Consider the case of servicing 4 customers, namely cus-
tomer 1, 2, 3 and 4 located at (0, 4), (3, 3), (5, 0) and (0, 7),
respectively, and let location of depot be at the origin. The
vehicle capacity is assumed to be 10 units and service
policy adopted here is unsplit delivery. Upon using Branch
and Bound algorithm to find the least-cost a priori route, itis observed that 0-1-2-4-3-0 is the least-cost tour with the
cost of 25.762 units. Figure 9 presents the network of
customers and the a priori route designed using Branch and
Bound algorithm.
Let the demands of customers observed at customer 1, 2,
3 and 4 are 7, 4, 1 and 2 units, respectively, upon reaching
the corresponding customer. Then, while executing the
route 0-1-2-4-3-0, a failure occurs when the vehicle reaches
customer 2. So, the vehicle returns to depot. After return-
ing, the salesman can either opt to re-continue with the old
route designed (and follow the route 0-2-4-3-0) or may opt
for re-optimization of tour using the unserviced customers.
Re-optimization of routes gives the new route that should
be followed with remaining unserviced customers to be
0-3-2-4-0. So the total route after re-optimization becomes
0-1-2-0-3-2-4-0 and the cost of operation after re-opti-
mization comes out to be 32.004, whereas the final route
that is followed without using the re-optimization technique
is 0-1-2-0-2-4-3-0 and the cost of this route is 34.246 units.
So, in such cases, re-optimization of routes is a better task
to do since there is a significant reduction in cost. The
presented scenario is presented in figure 10.
However, let demands observed at customers 1, 2, 3 and
4 be 8, 4, 3 and 5 units, respectively, upon reaching the
corresponding customer. Then, while executing the route
0-1-2-4-3-0, a failure occurs when vehicle reaches customer
2. If re-optimization of route is not done, then the second
failure occurs at the last customer. In case of absence of re-
Figure 8. A state space search tree for figure 7.
Figure 9. The network of four customers and corresponding apriori route.
Sådhanå (2021) 46:195 Page 13 of 17 195
optimization the route opted will be 0-1-2-0-2-4-3-0-3-0
and the cost of this tour will be 44.246 units, whereas if re-
optimization is allowed then first failure occurs at customer
2 and re-optimization gives the new tour that should be
followed with remaining customers to be 0-3-2-4-0; while
traversing this route the failure occurs at the last customer.
So, the total route after re-optimization becomes 0-1-2-0-3-
2-4-0-4-0 and the cost of this route comes out to be 46.004
units. So, in such a case re-optimization of routes is not a
better option to adopt since the cost of path is not reduced;
moreover the time complexity of algorithm also increases.
The scenario where re-optimization of route after occur-
rence of failure is not a better option is presented in
figure 11.
(a) Without re-optimization. (b) After re-optimization.
Figure 10. Effects of re-optimization of routes after failure.
(a) Without re-optimization. (b) After re-optimization.
Figure 11. Effects of re-optimization of routes after failure.
195 Page 14 of 17 Sådhanå (2021) 46:195
Appendix III. Property 2
Consider the case of servicing 4 customers as shown in
figure 12; customers 1, 2, 3 and 4 are present at (0, 4), (- 4,
2), (- 1, - 4) and (3, - 2), respectively, and let the depot
be at the origin node (0,0). Let the carrying capacity of the
vehicle be 15 units. The tour obtained using Branch and
Bound algorithm is 0-1-2-3-4-0. If demands of customer 1,
2, 3 and 4 are observed to be 2, 4, 5 and 2 units, respec-
tively, upon arriving at the respective customer, then the
cost of operation comes out to be 23.257 units while opting
the tour 0-1-2-3-4-0 and this is the least-cost tour. If the
demands of the customer 1, 2, 3 and 4 are observed to be
12, 8, 9 and 6 units, respectively, then the cost of operation
incurred by opting the route 0-1-2-3-4-0 is 47.657 units,
which is the optimal a priori route obtained using Branch
and Bound algorithm, whereas if the tour 0-1-4-3-2-0 is
traversed then the cost of operation comes out to be 41.145
units, which is of course lesser than the cost obtained using
optimal a priori route. So, we can say that Bellman’s
principle of optimality does not hold in the case of SVRP.
Figure 12. A network and its corresponding a priori route.
(a) Execution of original route. (b) Execution of route in opposite direction.
Figure 13. Effects of direction of traversal of route.
Sådhanå (2021) 46:195 Page 15 of 17 195
Appendix III. Property 3
Consider a network of 3 customers in which the coordinates
of depot node and customers 1, 2 and 3 are given by (0, 0),
(- 4, 0), (0, 6) and (0, - 7), respectively. Suppose there is
only one vehicle of capacity 10 units and the distance
between the points is calculated using Euclidean distance
norm. Then the a priori route designed using Branch and
Bound algorithm is 0-3-1-2-0. Suppose, while executing the
route, the demands of the customers 1, 2 and 3 are realized
as 8, 5 and 3 units, respectively. Thus, the cost incurred
while executing the route 0-3-1-2-0 is 48.27 units and the
cost incurred while executing the route 0-2-1-3-0 (the same
route in opposite direction) is 50.27 units. Figure 13 illus-
trates this property of SVRP.
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