Solving capacitated vehicle routing problem with demands as fuzzy random variable

Capacitated vehicle routing problem (CVRP) is a classical combinatorial optimization problem in which a network of customers with specified demands is given. The objective is to find a set of routes which originates as well as terminates at the depot node. These routes are to be traversed in such a way that the demands of all the customers in the network are satisfied and the cost associated with traversal of these routes come out to be a minimum. In real-world situations, the demand of any commodity depends upon various uncontrollable factors, such as season, delivery time, and market conditions. Due to these factors, the demand can always not be told in advance and a precise information about the demand is nearly impossible to achieve. Hence, the demands of the customers always experience impreciseness and randomness in real life. The decisions made by the customers about the demands may also have some scope of hesitation as well. In order to handle such demands of customers in the network, fuzzy random variables and intuitionistic fuzzy random variables are used in this work. The work bridges the gap between the classical version of CVRP and the real-life situation and hence makes it easier for the logistic management companies to determine the routes that should be followed for minimum operational cost and maximum profit. Mathematical models corresponding to CVRP with fuzzy stochastic demands (CVRPFSD) and CVRP with intuitionistic fuzzy stochastic demands (CVRPIFSD) have been presented. A two-stage model has been proposed to find out the solution for the same. To explain the working of the methodology defined in this work, different examples of networks with fuzzy and intuitionistic fuzzy demands have been worked out. The proposed solution approach is also tested on modified fuzzy stochastic versions of some benchmark instances.


Introduction
CVRP (Toth and Vigo 2002), a very well-known problem in operational research, aims at finding a set of routes beginning and ending at source node in a way such that total cost incurred comes out to be a minimum.In CVRP, we are usu-problem in logistic management, communication networks, urban solid waste collection, vehicle scheduling, etc., the problem has received close attention from the optimization community since then.CVRP is an extended version of the well-known travelling salesman problem (TSP) (Cormen et al. 2009), where the objective is to determine a minimum cost Hamiltonian circuit.Thus, many approaches may it be exact, random, approximate, heuristics based or metaheuristics based, which are used for finding a solution for CVRP inherited from the successful works done for finding the solution of TSP.The book edited by Toth and Vigo (2002) presented an overview of various exact and heuristic methods for solving a CVRP.Exact algorithms used for solving VRP are based on branch-cut-and-price algorithms (Gauvin et al. 2014).An interesting survey comprising of early exact methods for solving CVRP is given by Laporte and Nobert (1987).
VRP is known to be NP hard in nature; thus, all the exact algorithms used for solving the problem are only applicable for small instances only and face time complexity issues for big networks.The inability of the exact approaches to solve medium-and large-scale vehicle routing problem and the difficulty in evaluating objective function in real-life complex problems are two main reasons why heuristic and meta-heuristics are needed for solving such problems.One of the very early works in heuristics was presented by Clarke and Wright (1964), where a savings heuristics was designed to solve the problem.A heuristic method based on nearest neighbour algorithm was defined in Pop et al. (2011).Nearest neighbour algorithm is a greedy algorithm in which the nearest vertex is traversed first when given a set of vertices to be traversed.A tabu search algorithm was presented by Úbeda et al. (2014) for solving a green CVRP.In tabu search method, a local heuristic method is used to explore the solution space beyond optimality.Meta-heuristic methods based on genetic algorithm was presented in Mohammed et al. (2012) and Awad et al. (2018).Genetic algorithm mimics the natural evolution process as defined in theory of evolution by Charles Darwin.Meta-heuristic methods based on ant colony optimization and particle swarm optimization were presented in Dorigo et al. (2006) and Fang et al. (2007), respectively.Such method imitates the behaviour of ants and swarms, and a local optimal solution is obtained.A hybrid genetic algorithm-bacteria foraging optimization algorithm, was defined in Barma et al. (2021).A hybrid particle swarm optimization and simulated annealing method was presented in Fang et al. (2007).Christofides heuristic method was extended by Frieze (1983) to solve a k-person TSP.A modified fuzzy C-means clustering approach was presented in Shalaby et al. (2020) for solving CVRP.The approach is based on cluster first route second method.In Shalaby et al. (2020), the first stage of problem solving corresponds to the formation of clusters, and in second phase, each cluster is solved independently as a TSP.
Recent years have witnessed various variants of CVRP based on the uncertainty and variability of different attributes of the problem.In the classical version of the problem, all the information is available well in advance and such a VRP is known as a dynamic VRP.If one or more parameters of the problem are not known well in advance, then such a problem is known as stochastic VRP (SVRP).The early works in SVRP were presented in Gendreau et al. (1996), Dror et al. (1989) and Laporte and Nobert (1987).The stochastic nature of the problem is generally found in two parameters, namely the demands of the customers and the travel times.Cordeau et al. (2007) studied the most basic and most studied version of SVRP, i.e.CVRP with stochastic demands (CVRPSD).In CVRPSD, the customer demands are not known well in advance and are only revealed when a vehicle arrives at a customer location.Two well-known accepted algorithms, namely branch-cut-and-price (Gauvin et al. 2014) and branch-and-price (Christiansen and Lysgaard 2007) algorithms, are used for solving CVRPSD.In Bernardo and Pannek (2018), a robust solution approach was presented for solving dynamic and stochastic vehicle routing problem.A two-stage stochastic program model with recourse is presented where the objective of first stage is to minimize the a priori routing plan and the second stage dealt with minimizing the recourse cost.A local search-based algorithm with the objective of minimizing the total number of vehicles used and total distance visited was developed in Kaur and Singh (2017) for solving CVRPSD.In Marković et al. (2020), a case study of real-life problem of municipal waste collection in the city of Nis is considered and a solution approach based on heuristic and meta-heuristic methods was presented for solving CVRPSD.In Xia et al. (2021), a hybrid algorithm called DSMO-GA which combines genetic algorithm (GA) and discrete spider monkey optimization (DSMO) algorithm was proposed for solving CVRPSD.
When the customer demands are not known precisely but are given as uncertain quantities, i.e. as fuzzy numbers, then such a VRP is known as VRP with fuzzy demands.In Singh and Sharma (2020), CVRP with fuzzy demands given by interval type 2 triangular fuzzy numbers was presented.They used a modified Clarke and Wright algorithm to solve the problem.In Tordecilla et al. (2021), a hybrid approach combining simulations, meta-heuristics, and fuzzy logic has been used to generate near optimal strategies to large-scale NP hard problems.They used the simheuristic approach for optimizing VRP with fuzzy demands.In Werners and Drawe (2003), CVRP with fuzzy demands was handled by using fuzzy multi-criteria modelling approach.In Kuo et al. (2012), hybrid particle swarm optimization with a genetic algorithm for solving CVRP with fuzzy demands is proposed.In Mirzaei-khafri et al. (2020), a location arc routing problem was solved by using a mixed integer programming model.In Mirzaei-khafri et al. (2020), the demand of each edge belongs to a bounded uncertainty set.The proposed model determines a subset of potential depots to be opened along with their allocated routes.In Dutta et al. (2022), a biobjective green vehicle routing problem with heterogeneous fleet is proposed.Along with the minimization of distance, the greenhouse gases emitted by the vehicles during the operation are also minimized.Multi-criteria optimization and VIKOR method are used to determine the best solution from the Pareto front.
In certain situations, the cost matrix stores the time taken to traverse edges in the network.In many practical problems, the travel time between two locations in a routing problem cannot be told precisely because of traffic congestion or road conditions.Such situations can be handled by using fuzzy numbers for representing travel times.A fuzzy approach for solving VRP with fuzzy travel times was presented in Brito et al. (2010).There also exist situations when the travel times cannot be told well in advance.Such a version of CVRP with stochastic travel times was presented in Oyola (2019).In Oyola (2019), a non-dominated sorting genetic algorithm (NSGA) together with a variable neighbourhood search (VNS) heuristic was proposed.The demands of the customers were assumed to be deterministic.In addition to stochastic travel time, a soft time window is also associated with every customer and there exists a penalty for serving the customer outside their specified time window.In Zulvia et al. (2012), a hybrid ant colony optimization and genetic algorithm are used to solve VRPTW (VRP with time windows) where the demands of the customers, as well as travel time between every pair of customers, are imprecise.In Gupta et al. (2022), a particular case of VRP where triangular fuzzy numbers are used to represent the fuzzy travel times and the deliveries are split into bags was modelled.In Gultom and Napitupulu (2020), an algorithm is based on Clarke and Wright savings heuristic where three criteria of the network, namely the distance of various edges, time required to travel these edges, and road quality, are considered as fuzzy.In Kondratenko et al. (2020), fuzzy and evolutionary algorithms were used for planning the routes of tankers when the demand of fuel at various nodes in the network is presented by using triangular fuzzy numbers.

Novelty and contribution
Higher cost of precise information retrieval and random nature of demands of customers give rise to CVRPFSD.In CVRPFSD, the demands of the customers are neither presented precisely and are also not given well in advance.The decisions about the demands also face hesitation because of fluctuating demand-supply chain network.This work considers such fuzzy stochastic nature of customers' demands, and such nature of customer demands is handled by using fuzzy random variables.In this work, the basic version of CVRPSD has been extended and the current version deals with CVRPFSD and CVRPIFSD.In this paper, the edge weights denote the travel cost.The customers' demands are only revealed upon the arrival of the vehicle, and even then the demands of the customers are not told precisely.Various uncontrollable factors like delivery hours, market conditions, demand-supply chain fluctuations contribute to this imprecise and random nature of the customers' demands.In this work, two cases have been considered.The first case corresponds to the situation when the demands of customers are represented by using fuzzy random variables, whereas in the second case, the demands of the customers are represented by using intuitionistic fuzzy random variables.Various methods have been developed in the literature for solving CVRP with fuzzy demands and CVRPSD separately, but a combination of randomness and impreciseness has never been dealt in the literature.To the best of author's knowledge, such an amalgamation of randomness and impreciseness will bridge the gap between conventional and real-life problems.
In this work, CVRPFSD (CVRPIFSD) has been modelled as a two-level stochastic process.While solving the problem, the first task is to design a minimum cost path which traverse each node exactly once.Here, we used branch-andbound algorithm for finding a minimum cost Hamiltonian circuit.The branch-and-bound algorithm can only be used for symmetric networks of small sizes, and this comes up as a major limitation of this work.After designing, the execution of these routes takes place.Since the demands of the customers are not known in advance and are imprecise in nature, it may happen that upon reaching a customer, the salesman will realize his/her inability to serve the customer.This is the case when route failure occurs, and in such cases, the salesman is required to return to the depot node, replenish their vehicle, and then re-continue the planned route from the point of recent failure.In this work, the customer is served only in one visit of one vehicle, i.e. un-split delivery service policy is followed.The recourse policy followed in this work is reactive in nature, i.e. a vehicle will only return to depot when a route failure will occur.The results obtained by this method can only be used when the demands of the customers are represented by discrete fuzzy (intuitionistic fuzzy) random variables, where the demand of the customers and their respective probability are given by triangular (triangular intuitionistic) fuzzy numbers.The results of this work can be extended for scheduling problems faced by service agents since the service time windows are always stochastic and imprecise.The model can also be further extended to include the randomness and impreciseness of the network, and such networks can be represented by using neutrosophic graph theory (Kandasamy et al. 2015).The work can also be extended when the demands of the customers present in the network are expressed by using linguistic environment using dual-connection numbers (Irvanizam et al. 2020).The use of dual-connection numbers for representing demands will help the customers to formulate the membership degree under certain and uncertain situations and thus help the customers in making feasible and rational judgements.

Applications of CVRPFSD and CVRPIFSD
Real-world applications of the CVRPFSD and CVRPIFSD include the distribution of perishable items from stockholders to retailers, who further sell it to the consumers.For this work, the retailers play the role of customers and the stockholders play the role of travelling salesman.Since perishable items decay after a period of time which usually depend upon the uncontrollable weather conditions.So, the demands of perishable items are always random in nature.Factors like the fluctuations of market supply and demand of any perishable item make it difficult for the retailers to precisely determine the demand.In such case, the demands are told randomly and imprecisely.Such real-life situations give rise to CVRPFSD.Psychologically, the experience of retailers also makes them hesitant because an unreasonable stock of such items would only either lead to decay and hence the loss (if more stock is bought), in terms of money, or it will lead to unsatisfied consumers (if less stock is bought), which will cost their market reputation.So, the decision of the retailer has some scope of hesitancy in addition to the randomness and impreciseness while entailing their demand.Such real-life situations give rise to CVRPIFSD.Another applications include distribution of cash to different automatic teller machines (ATMs) in the city.Other examples include the delivery of essential commodity (milk, oil) where daily customer consumption is random in nature but can be predicted with the use of discrete random variable.
This paper is structured as follows: Sect. 2 deals with the basic concepts and definitions of fuzzy set theory and intuitionistic fuzzy set theory.Branch-and-bound algorithm for solving TSP is also presented.In Sect.3, two mathematical models for solving CVRP in an imprecise and random environment when the demands of the customers are represented by using triangular fuzzy numbers and triangular intuitionistic fuzzy number have been presented.Section 4 deals with a procedure based on branch-and-bound algorithm to solve CVRPFSD and CVRPIFSD.This section comprises of a twolevel algorithm and a two-level flow chart which presents the working of the two-level stochastic process defined in this paper.In Sect.5, several numerical examples with customers having fuzzy stochastic demands and intuitionistic fuzzy stochastic demands have been presented.Section 5 also comprises of the comparison of the results obtained by the other methods and the method proposed in this work and thus explains the supremacy of the method proposed.The proposed algorithm is also tested on the modified version of some benchmark instances.Section 6 comprises of the concluding remarks.

Preliminaries: concepts and definitions
Definition 1 Fuzzy number: A fuzzy set (Zimmermann 2011) Ã on R is said to be a fuzzy number, if the following three properties are satisfied: 1. Fuzzy set Ã must be normal, i.e. ∃ x such that sup μ Ã(x ) = 1. 2. The support of fuzzy set, i.e. set of all the elements with nonzero degree of membership, must be bounded.3. α level set, i.e. set of all the elements with membership degree greater than α, must be a closed interval for α ∈ [0,1].
Definition 2 Triangular fuzzy number: A generalized triangular fuzzy number is written as Ã = (a, b, c), and the membership function of triangular fuzzy number (a, b, c) is given by Eq. ( 1).
Thus, the graded mean integration representation (Chiao 2016) of a triangular fuzzy number Ã = (a, b, c) is given by Eq. ( 5).
The accuracy function (Singh and Yadav 2018) of ÃI is denoted by f ( ÃI ) and is defined by Eq. ( 6).
123 Fig. 2 A triangular intuitionistic fuzzy number Definition 9 Fuzzy random variable: a fuzzy random variable (Puri et al. 1993) is a random variable whose value is not real, but a fuzzy number.
Definition 10 Expectation of discrete fuzzy random variable: if X is a discrete fuzzy random variable, in such a way that P( X = xi ) = pi , i=1, 2, 3, . .., then the fuzzy expectation is given by

Branch-and-bound algorithm
The branch-and-bound algorithm (Cormen et al. 2009) is used to find exact solution for a wide variety of combinatorial optimization problems like TSP, knapsack problem, job scheduling problem, and many more.The algorithm works by systematically enumerating all the candidates solutions and then discarding some useless candidates by using the estimated bound on the quantity being optimized.In branch-and-bound algorithm, a tree search strategy is used to enumerate all possible solutions of a given optimization problem and only those branches of tree are explored which satisfies a bound.Figure 3 shows a branch-and-bound tree search strategy corresponding to a network with 4 nodes.To avoid unnecessary exploration of entire tree, the algorithm calculates a value (known as bound) corresponding to each node.The bound of any node represents the cost of best possible solution that can be obtained if a path through that node is traversed.If a better solution cannot belong to the sub tree rooted at a considered node, then the sub-tree need not be further explored.Otherwise, the process of exploration continues.For minimization problem, a lower bound corresponding to every node is calculated and the branches with lesser lower bounds are explored for better solution.For max-imization problem, an upper bound corresponding to every node is calculated and the branches with higher upper bounds are explored first since that will provide better solution.

Mathematical model
In this section, mathematical models for solving a CVRP with random and imprecise demands have been formulated.The mathematical models depicted in Sects.3.1 and 3.2 represent the situation when the demands of the customers are given by discrete fuzzy random variables and impreciseness of customer demands is handled by using triangular fuzzy numbers and TIFNs, respectively.In this work, some assumptions regarding the structure of the network, demands of the customers in the network, the recourse policy, and the service policy used have been made.The structure of the network is assumed to be symmetric, i.e. c i j = c ji , which means that the cost of traversal from node i to node j is same as the cost of traversal from node j to node i.The demands of the customers are assumed to be independent and positive.The demand of every individual customer is assumed to be less than the carrying capacity of the vehicle used to maintain the feasibility of the network.In this work, the factors like randomness and impreciseness have been used to represent the demands of the customer.The random(stochastic) nature of the customer demands means that the demands of the customers are only known when the vehicle arrives and this stochastic nature of demands leads to the failure of the route.The failure of route means that when a salesman arrives at a particular customer, they realize that they do not have enough capacity of goods to serve the customer.In this work, it has been assumed that the route failure may occur any number of times.Upon the failure of the route, the salesman is supposed to return to the depot, i.e. a reactive recourse policy is used.After returning to the Fig. 3 A tree-based search strategy for TSP in a network with 4 nodes depot, the vehicles are supposed to replenish themselves and continue the service again.In this work, it has been assumed that each customer is visited exactly once and is served in that visit only, i.e. the delivery policy used is un-split in nature.The journey of the travelling salesman is always assumed to originate and terminate at depot node only.The fleet used for serving the customers is assumed to be homogeneous, i.e. all the vehicles have the same carrying capacity, and they all operate at the identical cost.
We will now formulate the mathematical model for CVRP with fuzzy stochastic demands.Let G = (N = C∪{s, s }, A) be a graph.The notation C is used for the set of customers, and s and s represent the source node and its copy, respectively.c i j represents the cost of travelling from node i to node j.Let D i and D I i be the fuzzy random variable and intuitionistic fuzzy random variable representing the fuzzy and intuitionistic fuzzy demands of the customer at node i, respectively.Demand of source node is assumed to be 0 units.
A route is defined as a path of the form P = ( p 1 , p 2 , . . ., p |P| ) where p 1 = s and denote the total actual fuzzy cumulative demand and the total actual intuitionistic fuzzy cumulative demand at the customer p h for h ∈ {2, 3, 4, . . .|P| − 1}.
Given a route P = ( p 1 , p 2 , . . ., p |P| ), let E FC( μ p h , σ 2 p h ) and E FC I ( μ I p h , σ 2 I p h ) denote the expected fuzzy and expected intuitionistic fuzzy failure cost at customer p h , respectively.It can be written as the fuzzy probability of having the uth failure at p h customer when the demands are given by discrete fuzzy random variables with the condition that failure has yet not occurred on any other node which has been previously visited on the route.
the intuitionistic fuzzy probability of having the uth failure at p h customer under the same conditions as explained for the case when demands are given by fuzzy random variables.

Mathematical model for CVRPFSD
123 A binary decision variable whose value is 1 when route P is traversed and 0 otherwise α i p A binary decision variable whose entry is 1 when node i is traversed in route P and 1 when node i is traversed in route P

R
Set of all the routes Lower bound on the number of vehicles when demands are given by Fuzzy (intuitionistic fuzzy) random variable subject to

Mathematical model for CVRPIFSD
In the mathematical models proposed above, α i p and λ p are binary decision variables.The value of λ p is 1 if route p is chosen and 0 otherwise.α i p is also a binary decision variable which takes the value 1 if the node i is traversed while traversing the route p. R is the set of all feasible routes originating and terminating at the source node.The terms ϑ F (n) and ϑ I F (n) are lower bounds on the number of vehicles required when the service is performed by a fleet of vehicles or the number of times the vehicle may be required to return to source node because of failure in fulfilling the fuzzy and intuitionistic fuzzy demands of the customers, respectively, if only one vehicle is used.These can be easily computed by the formulas given in Eqs.(17 and (18), respectively.
where Q is the capacity of the vehicle.The objectives of minimization of operation cost are represented by Eqs. ( 9) and ( 13) for CVRPFSD and CVRPIFSD, respectively.The condition of serving each customer exactly once is represented by Eqs. ( 10) and ( 14).The use of minimum number of vehicles needed for successful execution 14:

15:
Effective failure cost = 2c 0r i Effective failure Prob(C r i )

Methodology
In this work, a two-level stochastic process has been defined for solving CVRPFSD and CVRPIFSD.The first level corresponds to the determination of an a priori route and the cost associated with that route.Various algorithms used for solving a TSP can be used for this purpose.In this work, we have used branch-and-bound algorithm for this purpose because the method ensures an optimal solution.The second level corresponds to the execution of the a priori route found in the first level and serving the customers.While executing the a priori route, a route failure may occur and in the second stage, an effective fuzzy failure cost corresponding to every customer in the network is calculated.The sum of deterministic cost obtained in stage 1 and effective failure cost obtained in stage 2 gives the cost of operation.Table 1 comprises of the various symbols used in the algorithm and their descriptions.

Algorithm
The stage 1 of methodology is represented by Algorithm 1.The stage 2 of methodology for fuzzy and intuitionistic fuzzy demands is represented by Algorithms 2 and 3, respectively.
Algorithm 3 An algorithm for calculating the Total Intuitionistic fuzzy effective failure cost Input: 1. Intuitionistic Fuzzy random variable D I i representing demands of every customer.2. r = (d, r 1 , r 2 , . . ., r n , d).

Capacity, Q of the vehicle.
Output: The total effective failure cost 14:

Numerical examples for CVRPFSD
Example 2 To illustrate the working of the mathematical model that has been explained in Sect.3.1, let us assume a network with 7 customers waiting for the goods to be delivered.The depot node is denoted by Node-0, and the customers are waiting at the remaining nodes.The information about the cost of traversal between various nodes is given by C 1 , where The demand at the depot node is considered to be 0 units, and the fuzzy stochastic demands for the customers in the network are presented in Table 2.The carrying capacity of the vehicle is assumed to be 30 units.
Example 3 We now assume a network with 5 customers.The depot node is denoted by Node-0.The demand at the depot node is considered to be 0 units.The information about the cost of traversal between various nodes is given by C 2 , where The fuzzy stochastic demands for the customers in the network are presented in Table 4.The carrying capacity of the vehicle is assumed to be 40 units.
Table 5 comprises of the comparison of the solutions obtained by various conventional methods defined in the literature for solving a TSP for Example 3. Figure 7 represents the comparison of total fuzzy cost obtained by using various methods.
Example 4 We now assume a network with 4 customers.The depot node is denoted by Node-0, and the customers are waiting at the remaining nodes.The information about the cost  .15, 0.20, 0.25) of traversal between various nodes is given by C 3 , where The demand at the depot node is considered to be 0 units, and the fuzzy stochastic demands for the customers in the network are presented in Table 6.Let the carrying capacity of the vehicle be assumed to be 30 units.Table 7 comprises of the comparison of the solutions obtained by various conventional methods defined in the literature for solving a TSP for Example 4. Figure 8 represents the comparison of total fuzzy cost obtained by using various methods.

Numerical examples for CVRPIFSD
Example 5 To illustrate the working of the mathematical model that has been explained in Sect.3.2, let us assume a network with 5 customers waiting for the goods to be delivered.The depot node is denoted by Node-0, and the customers are waiting at the remaining nodes.The information about the cost of traversal between various nodes is given by C 4 , where The demand at the depot node is considered to be 0 units, and the intuitionistic fuzzy stochastic demands for the customers in the network are presented in Table 8.The carrying capacity of the vehicle is assumed to be 30 units.9 represents the comparison of total intuitionistic fuzzy cost obtained by using various methods.
Example 6 We now assume a network with 4 customers.The depot node is denoted by Node-0, and the customers are waiting at the remaining nodes.The information about the cost of traversal between various nodes is given by C 5 , where The demand at the depot node is considered to be 0 units, and the intuitionistic fuzzy stochastic demands for the customers in the network are presented in Table 10.Let the carrying capacity of the vehicle be 30 units.
Table 11 comprises of the comparison of the solutions obtained by various conventional methods for the example described in Example 6. Figure 10 represents the comparison of total intuitionistic fuzzy cost obtained by using various methods.
Example 7 Assume a network with 4 customers waiting for the goods to be delivered.The depot node is denoted by Node-0, and the customers are waiting at the remaining nodes.The information about the cost of traversal between various nodes is given by C 6 , where The demand at the depot node is considered to be 0 units, and the intuitionistic fuzzy stochastic demands for the customers in the network are presented in Table 12.Let the carrying capacity of the vehicle be 25 units.
Table 13 comprises of the comparison of the solutions obtained by various conventional methods for Example 7. Figure 11 represents the comparison of total intuitionistic fuzzy cost obtained by using various methods.

Benchmark instances
The benchmark instances for CVRP given by Set P defined by Augerat (1995) (7 instances), Set A defined by Augerat (1995) (10 instances) and Set E defined by Christofides and Eilon (1969) (3 instances) are modified for the imprecise and random environment.The cost of traversal of edges in the network is taken as the distance which is calculated by using Euclidean distance norm.The coordinates of the vertices are taken as the same given in the benchmark instances.
The carrying capacity of the vehicle is considered same as that of benchmark instances.Two realizations for the fuzzy demand of the customers in the network are taken, where one realization is obtained by fuzzifying the demand as given in the benchmark instance and another realization is generated randomly in neighbourhood of first demand.The probability distribution function for customers' demand is also generated randomly.Table 14 comprises of the results of some benchmark instances obtained by using the proposed methodology.− 52,296.38, 393.42, 53,078.25) (− 51,826.83, 862.96, 53,547.79)A-n39-K5 542.02 (− 76,493.56, 535.45, 77,559.47) (− 75,951.54, 1077.47, 78,101.49)A-n39-K6 548.24 (− 60,740.21, 519.55, 61,777.35) (− 60,191.97, 1067.79, 62,325.58)− 12,569.77, 659.49, 13,888.753)*Operational costs about these modified instances are not available in repositories cise in nature which aligns well with the modified inputs provided to the benchmark instances.

Conclusion
In this work, two mathematical models for CVRPFSD and CVRPIFSD have been presented.A two-stage stochastic process for solving such problems has been presented.The first stage of the problem solving corresponds to an a priori route construction.In this work, branch-and-bound algorithm has been used for this purpose.The demands of the customers are dealt in the second stage where the rote obtained in first stage is traversed.The demands of the customers present in the network are random and imprecise in nature and hence are given by discrete fuzzy random variables.An effective failure cost corresponding to every customer is calculated in the second stage.In this work, each customer is traversed exactly once and the vehicle returns to depot only when a route failure occurs, i.e. when the vehicle reaches at a customer and realizes that the customer cannot be satisfied any more.The work can be used by logistic management companies to schedule the delivery of the seasonal/perishable items.Since the demands of seasonal/perishable items are stochastic in nature, various uncontrollable factors cause the impreciseness of customers' demands.This work shows new dimension of estimating the route cost in a mixed environment when information about the customer demands is based on the past experiences and includes factor of impreciseness and hesitation as well.A model for calculating the operation cost when a logical customer also includes the factor of hesitation while representing his/her demand is also provided in this work.Thus, the methods presented in this work deal with the modelling of randomness, impreciseness and hesitance, which a logical decision maker often encounters in real-world problems.The work can further be extended for service provider companies whose objective is to provide services to the customers present at various nodes according to their specified time windows.The work can also be further extended to include randomness and impreciseness of arc lengths, and such case will be helpful when the cost matrix represents the time taken to cover various edges in the network.

Definition 3 Definition 4
Symmetric triangular fuzzy number: A triangular fuzzy number is said to be symmetric if its left and right spreads are equal.If Ã = (a, b, c) is a triangular fuzzy number, then it is said to be symmetric if and only if (b − a) = (c − b).Example 1 (2, 4, 6) is a symmetric triangular fuzzy number.The graph of the membership function of a symmetric triangular fuzzy number is given in Fig. 1.Arithmetic operations on symmetric triangular fuzzy number: let Ã = (a 1 , b 1 , c 1 ) and B = (a 2 , b 2 , c 2 ) be two symmetric triangular fuzzy numbers, then 1. Addition:

Fig. 1
Fig. 1 A symmetric triangular fuzzy number

Definition 6
Figure 2 represents the membership and non-membership functions of TIFN defined above.Arithmetic operations on TIFNs Atanassov (2016): Let ÃI = (a, b, c; a , b, c

5 )Definition 8
Accuracy function: let ÃI = (a, b, c; a , b, c ) be a triangular intuitionistic fuzzy number.The score function for membership function and non-membership function is denoted by S(μ ÃI ) and S(ϑ ÃI ), respectively, where Figures 4 and 5 represent the flow chart for methodology of stage 1 and stage 2, respectively.

Fig. 6 Table 5 Name
Fig. 6 Total cost by using various methods for CVRPFSD presented in Example 2

Fig. 7 Table 7 NameFig. 8
Fig. 7 Total cost by using various methods for CVRPFSD presented in Example 3

Fig. 9 Fig. 10 Table 9 Name
Fig. 9 Total cost by using various methods for CVRPIFSD presented in 5

Table 1
Description of symbols used in mathematical model Cost of traversal of edge i j P = ( p 1 , p 2 , . . ., p |P| ) A route starting and ending at p 1 and p |P| , respectively

Table 2
Fuzzy stochastic demands for Example 2

Table 4
Fuzzy stochastic demands for Example 3

Table 3
Comparison of various methods for CVRPFSD presented in Example 2

Table 6
Fuzzy stochastic demands for Example 4

Table 9
comprises of the comparison of the solutions obtained by various conventional methods for Example 5.Figure

Table 11
Comparison of various methods for CVRPIFSD presented in Example 6

Table 14
Results for benchmark instances modified fuzzy stochastic demands

Table 15
Comparison of various methods on modified benchmark datasets