CHAPTER 2

SURVEY OF METHODOLOGIES FOR TSP and VRP

2.1 SOLUTION APPROACHES OF TSP and VRP

Solution approaches of TSP and VRP are detailed here

2.1.1 TSPThe approaches for solving the TSP range from simple

heuristics to algorithms based on the working of the human

mind (Ganesh and Narendran, 2008). Many heuristics have been

proposed to find near-optimal solutions in reasonable time

(Flood, 1956; Croes, 1958; Lin and Kernighan, 1973). Apart

from the special purpose heuristics applicable only to the

TSP, there are meta-heuristics such as Simulated Annealing,

Thershold Accepting, Genetic Algorithm, Tabu Search, and Ant

Colony Optimisation, are useful in a variety of optimisation

problems. Several authors have proposed Branch and Bound

algorithms based on the assignment problem relaxation

(Christofides et al. 1981). Heuristic approaches for TSP mostly

fall into three broad classes:

Route construction procedures Route improvement procedures and Composite procedures (Bodin et al. 1983)Route constructive procedures generate an approximate

optimal route from the distance matrix. They consist of

nearest neighbour procedure (Rosenkrantz et al. 1977), savings

procedure (Clark and Wright, 1964), insertion procedures

(including nearest insertion, cheapest insertion, arbitrary

insertion, quick insertion, greatest angle insertion),

minimal spanning tree approach (Christofides, 1976). Route

improvement procedure attempts to find a better route from a

given initial route, generally using branch exchange

approach. The 2-opt and 3-opt heuristics were introduced by

Lin (1965) while the k-opt procedure was presented by Lin

and Kernighan (1973). Recently for many variants of TSP,

meta-heuristics have been used for improvement of given

initial solutions (Goldberg, 1989). Composite procedures

construct a route using an existing procedure and then

attempt to find a better route using one or more of the

route improvement procedures (Russell, 1995; Baraglia et al.

2000).

2.1.2 VRPDaskin (1985) formulated a conceptual model of logistics

incorporating consumers, producer/shippers, carriers and

governments. Private and public sector problems including

vehicle routing, inventory management, fleet selection and

facility location were identified. Bodin et al. (1983)

presented a comprehensive review of VRP. Bott and Ballou

(1986) discussed various approaches to the vehicle

scheduling and routing problems and identified the

restrictions and extensions that should be incorporated in a

generalised vehicle routing and scheduling methodology.

Laporte and Nobert (1987) presented an extensive survey

which was entirely devoted to exact methods for the VRP and

gave a complete analysis of the state of the art up to the

late eighties. Other surveys on the use of exact algorithms

and heuristic methods were presented by Laporte (1992) and

Fisher (1995). Laporte (1992) surveyed some of the well-

known results related to the VRP. Useful techniques for the

general VRP have been outlined by Golden and Assad (1988),

Ball et al. (1995) and Aarts and Lenstra (1997). Many

references on meta-heuristics have been reported by Laporte

and Osman (1995), Osman and Laporte (1996) and Gendreau et al.

(1998) for applications on routing problems. Laporte et al.

(2000) presented a survey of heuristics divided into two

parts, classical and modern heuristics. The first part

contains well-known schemes such as the savings method, the

sweep algorithm and various two-phase approaches. The second

part is devoted to TS heuristics which has been proved to be

successful. Tarantilis et al. (2005) surveyed the recent

research efforts on meta-heuristic solution methodologies

for the standard version of the VRP.

Literature abounds with the solution approaches for TSP

and VRP. These can be classified as follows (Figure 2.1)

(Ganesh et al. 2007a):

1. Mathematical Modeling

2. Heuristics

3. Meta-heuristic

4. Interactive approaches

5. Hybrid Approaches

2.2 MATHEMATICAL MODELLING

This approach one is inclined to think, is capable providing

exact solutions. Unfortunately, even for a TSP and VRP of

modest size, it is computationally too complex to solve.

Branch and Bound algorithms (Radharamanan and Choi,

1986; Laporte et al. 1987; Ralphs, 2003), Branch and Cut

algorithms (Bard et al. (1998); Lysgaard et al. (2004); Fukasawa

et al. (2006) Dynamic programming (Magnanti, 1981; and Kolen et

al. 1987) and Lagrangean relaxation procedure (Kallehauge et al.

2001; Magnanti, 1981; Stewart and Golden, 1984) are some of

the exact approaches that have been used for solving VRPs.

TSP was also approached with Branch-and-bound by Diderich and

Gengler (1996), Cotta et al. (1995) and Tschöke et al. (1995).

NeuralNetwork Ant

Colony

ParticleSwarm

LagrangeanRelaxation

DynamicProgramming

Tabu

SimulatedAnnealing

GeneticAlgorithm

s

Branch & Bound

MathematicalProgramming

PreferenceBased Approach

Intuitive

Simulation

Graphics

Meta -Heuristic

s

ExactTechniques

Heuristics Interactive TSP & VRP

MemeticAlgorithm

s

Figure 2.1 Solution Methodologies of TSP and VRP

SavingsProcedure

InsertionProcedure

Improvement(2-opt, 3-

Cluster-FirstRoute-Second

Route-FirstCluster-Second

Combination Space-Filling

Heuristic

Cross-EntropyMethod

2.3 HEURISTICS

As in the case of other combinatorial problems, heuristics

procedures are widely used for solving TSP and VRP.

Heuristics limit their exploration of the search space but

aim at producing a good solution in a reasonably short time.

Funke et al. (2005) provided a review of both classical and

modern local search neighborhoods for VRP. Ball (2011) has

made a broad survey on the heuristics based on mathematical

programming models and methods.

Three basic categories of heuristics have been proposed

for solving the VRP (Laporte, 1992)

1) Constructive heuristics

a) Nearest neighbour

b) Savings procedure

2) Two-phase heuristics

a) Cluster-first, route-second procedure

b) Route first, cluster-second procedure

3) Local search improvement heuristics

a) Insertion procedure

b) Improvement procedure

2.3.1 Constructive Heuristics

Constructive heuristics use the data of a problem to

construct a solution step by step. (Tarantilis et al. 2005)

stated that no solution is obtained until the procedure is

complete and a special constructive approach is the greedy

method, where, at each step, the node selected for insertion

in a route is based on the lowest contribution to cost.

2.3.1.1 Nearest Neighbour

It starts with any node at the beginning, finds the closest

to the last-added node at each step and completes the route

with all the nodes included. It requires high computational

time (Rosenkratz et al. 1977).

2.3.1.2 Savings procedure

This procedure builds a solution, which may be infeasible,

by calculating savings generated by a new route

configuration. But the savings procedure may produce sub-

optimal routes (Clark and Wright, 1964; Dror and Trudeau,

1986; Altinkemer and Gavish, 1991; Reimann et al. 2004). A

robust enhancement to the Clarke-Wright CW savings

formulation was proposed by Doyuran and Çatay (2011).

2.3.2 Two-Phase Heuristics

Laporte et al. (2000) indicated that the two-phase heuristics

cluster the nodes into feasible routes and then constructs

actual routes, using feedback loops between these two

stages. Two-phase heuristics do not employ a unified

approach for combining the clustering and routing phases;

each of the proposed methods uses a unique recursive loop

for implementing the two-phase method and obtaining the best

possible results.

2.3.2.1 Cluster-first, route-second procedure

Cluster the nodes, and determine feasible routes for each

cluster -- this is the principle of this approach. It is

difficult to use this procedure when vehicles have different

capacities (Renaud and Boctor, 2002). Some of the methods

are sweep algorithm (Gillet and Miller, 1974) and Fisher and

Jaikumar Algorithm (Fisher and Jaikumar, 1981). Hiquebran et

al. (1993) applied Cluster-First Route-Second Algorithm for

VRP. A recent approach can be seen with Bräysy and Hotokka

(2007).

2.3.2.2 Route first, cluster-second procedure

This starts with a large route, which is often infeasible,

and partitioned it to smaller clusters (Beasley, 1983; Mole

et al. 1983; Hachicha et al. 2000). This approach is not suitable

for small problems.

2.3.3 Local search improvement heuristics

(Tarantilis et al. (2005) stated that the local search

improvement heuristics are iterative search procedures that

start from an initial feasible solution (often the result of

a constructive heuristic), and then progressively improve

the solution by applying a series of local modifications

called moves. At each iteration of a local search heuristic,

the moves applied to the current solution, define a set of

neighbouring solutions in the search space. The simplest

neighbourhood structures for the VRP involve moving, within

each iteration cycle, a single node from its current route

and then the selected node is inserted in the same route or

in another route with sufficient residual capacity. An

important feature of neighbourhood development is the way in

which insertions are performed: one could use random

insertion or insertion at the best position in the target

route; alternatively, one could use more complex insertion

schemes that involve a partial re-optimisation of the target

route (Gendreau et al. 1997). Zeng et al. (2005) also proposed an

assignment-based local search method for VRP.

2.3.3.1 Insertion procedure

This procedure builds a solution by determining the least

expensive insertion of a node into a route. It may end up

with sub-routes. Some of the examples are nearest

insertion, cheapest insertion, farthest insertion, quick

insertion and the convex hull insertion algorithms (Chung

and Norback, 1991; Gendreau et al. 1992; Foisy and Potvin,

1993). The push-forward insertion heuristic (Jee 2000,

Ghoseiri and Ghannadpour 2009) and virtual vehicle heuristic

(Kilby et al. 1997) are two useful heuristics for search

initialisation in solving difficult VRPs.

2.3.3.2 Improvement procedure

Given a route, the algorithm examines all the routes that

are neighbouring to it and tries to find a short route.

Starting with a small initial route, chosen arbitrarily or

by some other method, if there is no neighbouring route

which is shorter than the original one, the process stops.

This modifies the routes step by step and maintains

feasibility of the solution. It requires long computational

time (Psaraftis, 1983; Sule et al. 1991; Mak and Morton, 1993;

Bianchi, 2005). Some of the examples are 2-opt method, 3-opt

method (Lin, 1965, Alfa et al, 1991), Lin-Kernighan algorithm

(Lin and Kernighan, 1973, Papadimitriou (1992) and Helsgaun

(2000) and Or-opt (Or, 1976; Taillard et al. 1997). Kytöjoki et

al, (2007) presented an efficient variable neighborhood

search heuristic and Goel and Gruhn (2008) proposed

iterative improvement approaches based on the idea of

changing the neighbourhood structure during the search for

near optimal solutions for VRP.

2.4 META-HEURISTICS

Gendreau and Potvin (2005) highlighted that the emergence of

metaheuristics for solving difficult combinatorial

optimisation problems is one of the most notable

achievements of the last two decades. Notable among them are

the emergence of TSP and VRP.

The most promising and effective solution methods for

the TSP and VRP are meta-heuristics (Gendreau et al. 2002),

which are general-purpose mechanisms for solving hard

optimisation problems. In meta-heuristics, the emphasis is

on performing a deep exploration of the most promising

regions of the solution space. These methods typically

combine sophisticated neighbourhood search rules, memory

structures, and recombination of solutions. The quality of

solutions produced by them is usually much higher than those

obtained by classical heuristics. Nevertheless the increased

computing time is the only problem. The procedures are

usually context-dependent and require finely tuned

parameters for effective search. Each meta-heuristic has one

or more adjustable parameters. This permits flexibility, but

for any application to a specific class of problems,

requires careful calibration on a set of numerical instances

as well as testing on an independent set of instances

(Ganesh and Narendran, 2008). Meta-heuristics are classified

as memory-less and memory-based, according to the use of

previously exploited areas of the solution space (Blum and

Roli, 2003). Gendreau and Potvin (2005) provided an account

of the most recent developments in the field metaheuristics

and identified some common issues and trends with respect to

VRP. nevertheless

2.4.1 Memory-less meta-heuristics

The two most important memory-less meta-heuristics are

Simulated Annealing (SA) (Kirkpatrick et al. 1983) and

Threshold Accepting (TA) (Dueck and Scheuer, 1990).

2.4.1.1 Simulated annealing (SA)

SA is inspired from the physical annealing process emanating

in statistical mechanics. It is a local search meta-

heuristic, in the sense that it conducts local search while

guiding the overall exploration process intelligently,

offering the possibility of accepting, in a controlled

manner, solutions that do not descend along the path of

search. This feature allows SA to escape from a low quality

local optimum (Malairajan et al., 2009). Osman (1993) proposed

an SA algorithm whose neighbourhood structure uses λ-

interchanges, in which exchanges of up to λ nodes between

two routes take place. Lin et al, (2006) applied SA for

capacitated VRP. Janakiram et al, (1996) and Pepper et al,

(2002) used annealing based heuristics to solve TSP.

2.4.1.2 Threshold accepting (TA)

TA is a modification of the SA. Specifically, it leaves out

the stochastic element in accepting worse solutions by

introducing a deterministic threshold. During the

optimisation process, the threshold level is gradually

lowered like the temperature in SA (Tarantilis et al. (2005).

Tarantilis et al. (2002a) presented a variant of a TA

algorithm, called Backtracking Adaptive Threshold Accepting

(BATA), in which the neighbourhood structure is defined in a

similar way to Osman’s (1993). Tarantilis et al. (2002b)

developed another variant of TA, called List-Based Threshold

Accepting (LBTA). LBTA expands the standard TA algorithm by

introducing a list of threshold values. Both BATA and LBTA

have been used in numerous real-life distribution operations

involving transportation of goods and materials (Tarantilis

and Kiranoudis, 2001b; Tarantilis and Kiranoudis, 2002a;

Tarantilis and Kiranoudis, 2002b; Tarantilis and Kiranoudis,

2002c; Tarantilis and Kiranoudis, 2002e).

Bräysy et al, (2003) pioneered TA for VRP with Time

Windows (VRPTW). Tarantilis et al, (2004) presented a

backtracking adaptive TA algorithm, for solving the

heterogeneous fixed fleet VRP. Nikolakopoulos and Sarimveis

(2007) proposed TA method, enhanced with intense local

search, while the candidate solutions are produced through

an insertion heuristic scheme. Liu (2007) developed the

hybrid scatter search by incorporating the nearest neighbor

rule, TA and edge recombination crossover into a scatter

search conceptual framework to solve the probabilistic TSP.

2.4.2 Memory-based meta-heuristics

Memory based meta-heuristics exploit the previously examined

area of the solution space through one or multiple lists of

solutions kept within a limited memory. The term memory was

used explicitly by the Tabu Search (TS) algorithms (Glover,

1989) and the Adaptive Memory Based Algorithms (AMBA)

(Rochat and Taillard, 1995). However, a number of other

meta-heuristics such as Genetic Algorithm (Holland, 1975),

Ant Colony Optimisation (Dorigo, 1992), Particle Swarm

Optimisation (Kennedy and Eberhart 1995) and Memetic

Algorithm (Moscato 1989), use mechanics and structures that

can be considered as memories. The literature pertaining to

routing problems of each major group of memory based meta-

heuristics is summarised here.

2.4.2.1 Tabu Search (TS)

TS is a local search meta-heuristic. TS explores the solution

space by moving at each iteration, from a current solution

to the best solution in its neighbourhood or in a subset of

neighbourhood for computational efficiency (Tarantilis et al.

(2005).

Osman (1993) presented one of the earliest successful

implementations of TS for solving VRP. Taillard (1993)

developed one of the most effective TS algorithms for

solving VRP. Gendreau et al. (1994) proposed a sophisticated TS

algorithm, called Tabu-route. Zachariasen and Dam (1995)

proposed TS for the Geometric TSP.

The TS in Rochat and Taillard (1995) exploits a

neighborhood based on the exchange of customers between

routes. Carlton (1995) described a reactive TS that

dynamically adjusts its parameter values based on the

current search status. Potvin et al. (1996) described a

standard TS heuristic based on 2-opt and Or-opt exchanges.

Xu and Kelly (1996) presented an approach for defining the

neighbour structure of a TS algorithm. Rego and Roucairol

(1995) and Rego (2001) developed a TS that defines its

neighbourhood structure by employing node-ejection chains.

Toth and Vigo (2003) developed a TS algorithm, called

Granular TS, based on the idea of excluding non-promising

areas of the search space. Cordeau et al. (2002) proposed a TS

algorithm, called Unified TS Algorithm (UTSA) which shares

some common features with Taburoute. Bräysy and Gendreau

(2002) applied TS for the VRPTW. Chiang and Russell (1997),

Cordeau et al. (2001), De Backer et al. (2000), Rochat and

Taillard (1995) and Taillard et al. (1997) have also made

levelheaded contributions to VRP using TS. Paraskevopoulos et

al, (2008) solved the heterogeneous fleet VRPTW using a two-

phase solution framework based upon hybridised TS, within a

new Reactive Variable Neighborhood Search metaheuristic

algorithm. Potvin and Naud (2011) proposed a tabu search

heuristic with a neighbourhood structure based on ejection

chains to solve VRP.

2.4.2.2 Adaptive Memory-Based Algorithms (AMBA)

The first AMBA was presented by Rochat and Taillard (1995).

The adaptive memory rationale constitutes one of the most

powerful tools for automatic diversification and

intensification of the search process (Tarantilis et al.

(2005).

Rochat and Taillard (1995) introduced the concept of

adaptive memory for the VRP, according to which a set of

high quality VRP solutions is stored in a pool that is

dynamically updated during the search process. Tarantilis

and Kiranoudis (2002d) developed an AMBA called BoneRoute.

Tarantilis (2005) presented a modified BoneRoute, called

SEPAS, generating an initial population of diversified

solutions in a systematic way. Tarantilis et al (2005) studied

Capacited VRP with AMBA. Tarantilis and Kiranoudis (2007)

applied a flexible AMBA for real-life transportation

operations. Derigs and Reuter (2009) presented results on an

implementation of the attribute-based hill climber heuristic

to the open VRP which is a parameter-free variant of the

tabu search principle. Zachariadis et al., (2010) introduced

an Adaptive Memory for the algorithmic framework for the VRP

with Simultaneous Delivery and Pickup (VRPSDP) which collects

and combines promising solution features to generate high-

quality solutions.

2.4.2.3 Genetic Algorithm (GA)

GAs are population-based algorithms that simulate the

evolutionary process of species that reproduce. GA causes

the evolution of a population of individuals encoded as

chromosomes by creating new generations of offspring through

an iterative process that continues until some convergence

criteria are met. At the end of this process, it is expected

that an initial population of randomly generated chromosomes

will improve and be replaced by better off-springs. The best

chromosome obtained by this process is then decoded to

obtain the solution (Holland, 1975).

Literature on the application of GA to VRP is limited.

Baker and Ayechew (2003) presented a GA for solving the VRP.

The representation of this algorithm was influenced by the

non-binary representation of Chu and Beasley (1997) for the

Generalised Assignment Problem. Xiong et al, (1998) modeled

rolling batch planning as VRPTW and used GA and heuristics

to solve the problem. Prins (2004) developed an effective GA

in which a VRP solution is represented as a sequence in

which a vehicle must visit all clients, assuming the same

vehicle performs all the trips by turn. Hanshar and Ombuki-

Berman (2007) applied GA for the Dynamic VRP. Salhi and

Petch (2007) proposed a hybrid GA, which used a new non-

binary chromosome representation and which is enhanced by a

domain specific data structure, appropriate genetic

operators and a scheme for chromosome evaluation for the VRP

with multiple trips. Prins (2008) proposed a simple and

effective GA for the VRP. Ho et al, (2008) proposed two hybrid

GAs for the multi depot VRP. In the first one the initial

solutions are generated randomly and in the second one the

Clarke and Wright saving method and the nearest neighbour

heuristic are incorporated into hybrid GA for the

initialisation procedure and found that the performance of

the second one is superior in terms of the total delivery

time.

Literature on the application of GA to TSP is in a good

number. GA was explored by Ulder et al, (1991) and Potvin

(1996) for TSP; Cotta et al, (1995) Hybridising with Branch

and Bound Techniques for TSP; Kureichik et al, (1997) used GA

with new features to TSP against Premature Convergence;

Nagata and Kobayashi (1997) used a high-power GA for TSP.

Schmitt and Amini (1998) developed GA and evaluated by

solving 5000 TSPs; Ochi et al, (1998) presented a new hybrid

metaheuristic which uses Parallel GA and Scatter Search

coupled with a decomposition-into-petals procedure for

solving a class of Vehicle Routing and Scheduling Problems.

While addressing TSP Tesfaldet and Hermosilla (1999)

used a Lamarckian GA; Larranaga et al, (1999) and Andal

Jayalakshmi et al, (2001) used a Hybrid GA. Hwang (2005)

proposed GA for VRP and TSP as well; Pankratz (2005) proposed

a Grouping GA for solving the pickup and delivery problem

with time windows which features a group-oriented genetic

encoding in which each gene represents a group of requests

instead of a single request. Snyder and Daskin (2006)

presented an effective heuristic which combines GA with a

local tour improvement heuristic applied to generalised TSP

and termed as random-key GA. Marinakis et al, (2007)

formulated a new bi-level formulation for VRP and tested

with a solution method using GA and also tested it with TSP.

Snyder and Daskin (2006) proposed a random-key GA for

the generalised TSP. Samanlioglu et al, (2007) arrived at

approximate and sometimes optimal solutions to the symmetric

TSP using a hybrid approach that combines a Random-Key GA

with a local search procedure. Bae et al, (2007) developed an

integrated VRP model using heuristic method and the improved

GA of which operators and initial population are improved.

This was tested for TSP Problems too. Marinakis et al, (2007)

proposed a bi-level GA for both VRP and TSP. Ganesh and

Narendran (2007) proposed a multi-phase constructive

heuristic that clusters nodes based on proximity, orients

them along a route using shrink-wrap algorithm and allots

vehicles using generalised assignment procedure and employed

GA for an intensive final search.

2.4.2.4 Ant Colony Optimisation (ACO)

ACO is one more memory-based meta-heuristic, which simulates

the ant’s ability in determining the shortest path between

food and the nest (Tarantilis et al. 2005).

Bullnheimer et al. (1998) presented an application of Ant

System (AS) for solving the VRP. Reimann et al. (2002)

developed an AS algorithm based on the transformation of the

simultaneous route construction mechanism proposed by Clarke

and Wright (1964) into a rank-based AS. Reimann et al. (2004)

proposed an AS approach, called D-Ants, built on the

algorithm developed by them earlier (Reimann et al. 2002).

Bell and McMullen (2004) applied ACO to a set of VRPs.

ACO was employed by Mazzeo and Loiseau (2004) for the

capacitated VRP, Manfrin (2004) and Reimann et al, (2004) for

VRP and Lin and Cai (2006) for VRP and mail delivery

problems. Montemanni et al, (2005) proposed a dynamic VRP,

based on the Ant Colony System paradigm. Li and Tian (2006)

presented an ant colony system hybridised with local search

for solving the Open VRP. Reimann and Ulrich (2006) compared

the backhauling strategies in VRP using ACO. Rizzoli et al,

(2007) proposed ACO for realworld VRP. Lee et al, (2008)

proposed an enhanced ACO for capacitated VRP.

Fuellerer et al (2009) approached VRP with an ACO

algorithm which combines two different heuristic measures

(with respect to loading and routing) within one pheromone

matrix. Yu et al. (2009) proposed an improved ACO, which

possesses a new strategy to update the increased pheromone,

called ant-weight strategy, and a mutation operation, to

solve VRP. Gajpal and Abad (2009) used a multi-ant colony

system to solve VRPB.

Dorigo and Gambardella (1997) proposed an ACO for the

TSP. Binachi et al. (2002) proposed an ACO approach to the

Probabilistic TSP. Stüzle and Dorigo (1999) and Hung et al,

(2007) proposed the ACO for TSP. Li and Gong (2003) proposed

a Dynamic ACO for TSP. Branke and Guntsch (2004) showed that

ACO works well even when only an approximate evaluation

function is used, which speeds up the algorithm, leaving

more time for the actual construction and applied it for the

Probabilistic TSP. Qingbao and Lingling (2007) analysed of

the convergence of ant colony with the TSP. Liu (2005)

applied Rank-based ACO applied to dynamic TSP. Yang et al

(2008) focused on the generalised TSP with ACO and to avoid

locking into local minima, a mutation process and a 2-opt

local searching technique are also introduced. Donati et al

(2008) proposed a multi ant colony for VRP. Çatay (2010)

proposed an ACO employing a new saving-based visibility

function and pheromone updating procedure. Yu et al., (2011)

presented an improved ACO with coarse-grain parallel

strategy, ant-weight strategy and mutation operation for the

multi-depot vehicle routing problem.

2.4.2.5 Particle Swarm Optimisation (PSO)

PSO is also one of the population based stochastic

optimization techniques which is inspired by the social

behavior of bird flocking or fish schooling. This technique

searches a space by adjusting the trajectories of individual

vectors, called “particles”, conceptualized as moving points

in multidimensional space. The individual particles are

drawn stochastically on the basis of the positions of their

own previous best performance and the best previous

performance of their neighbours. The application of this

technique to routing problems emerged recently.

Wang et al. (2006) proposed a novel real number encoding method

of PSO for an Open VRP. In addition they also applied

several heurist methods into the post-optimisation

procedure, such as Nearest Insertion algorithm, GENI

algorithm, and 2-Opt, after decoding. Teodorović (2008)

presented a classification and analysis of the Swarm

intelligence systems for the Transportation problems. The

techniques include ACO, PSO, Bee colony optimisation and

stochastic diffusion search. Belmecheri et al., (2010)

proposed a PSO to solve the VRP with Heterogeneous fleet,

Mixed Backhauls, and time windows.

Ai and Kachitvichyanukul (2009) presented a PSO

algorithm for solving a VRPSDP. The formulation is a

generalisation of three existing VRPSDP formulations. Li et al,

(2008) proposed PSO for the Electronic design automation

modeled as TSP. Shi et al. (2007) presented a PSO based

algorithm for the TSP. An uncertain searching strategy and a

crossover elimination technique were used to accelerate the

convergence speed. They also proposed another PSO-based

algorithm applied to solve the generalised TSP by employing

the generalised chromosome. Two local search techniques were

also used to speed up the convergence. Onwubolu and Clerc

(2004) solved the optimal path problem for automated

drilling operations by a new heuristic approach using PSO.

Wang et al, (2003) also proposed PSO for the TSP. Contribution

of Sofge et al. (2002), Secrest (2001) and Secrest and Lamont

(2001) in solving TSP with PSO are also significant.

2.4.2.6 Memetic Algorithm

Memetic Algorithm was introduced in the late 80s to denote a

family of meta-heuristics that has central theme as

hybridisation of different algorithmic approaches for a

given problem. Special emphasis was given to the use of a

population-based approach in which a set of cooperating and

competing agents were engaged in periods of individual

improvement of the solutions while they sporadically

interact (Moscato and Cotta 2003).

Prins and Bouchenoua (2005) tried Memetic Algorithms

for solving the VRP, Capacitated VRP and General Routing

Problems with Nodes, Edges and Arcs. Contribution of Lacomme

et al. (2004) and Belenguer et al. (2006) is also notable in

investigating vehicle routing with memetic algorithms. Lima

et al. (2004) described a Memetic Algorithm for the

Heterogeneous Fleet VRP.

Tavakkoli-Moghaddam et al. (2006) proposed a memetic

algorithm which uses different local search algorithms. To

make use of the power of memetic algorithm, inter and intra-

route node exchanges were also used as a part of their

evolutionary algorithm. Fallahi et al. (2008) proposed a

memetic algorithm with a post-optimisation phase based on

path relinking and TS method. Créput and Koukam (2009)

studied the hybridisation of the self-organising map in an

evolutionary algorithm to solve the Euclidean TSP which is

considered to be memetic neural network algorithm.

Moscato and Norman (1992), Moscato and Tinetti (1992),

Buriol et al, (2004) and Gutin et al, (2008) approached TSP with

a memetic approach. Krasnogor and Smith (2000) introduced a

Memetic Algorithm with Self-Adaptive Local Search. Merz and

Freisleben (2001) proposed Memetic Algorithms for the TSP.

Merz (2002) compared several memetic algorithms,

incorporating local search methods.

Aarts and Verhoeven (1997) provided some test results

using Memetic Algorithms for the TSP with 2-opt (Lin, 1965)

and variable depth neighborhoods (Lin and Kernighan, 1973)

as local search techniques. Larranaga et al, (1999) studied

various representations and operators used in GA for solving

TSP. They presented crossover and mutation operators to

tackle the TSP with GA having different representations such

as binary representation, path representation, adjacency

representation, ordinal representation and matrix

representation. Liu et al. (2006) proposed an effective PSO

based Memetic Algorithm for the TSP. In this, a novel

encoding scheme was developed and an effective local search

based on SA with adaptive meta-Lamarckian learning strategy

was proposed and incorporated into PSO. Buriol et al. (2004)

proposed the algorithm for the Asymmetric TSP. Duan and Yu

(2007) addressed TSP with Hybrid ACO using Memetic

Algorithms. Créput and Koukam (2008) presented an extension

of the self- organising map by embedding it into the memetic

algorithm for VRP.

2.5 INTERACTIVE APPROACHES

These are simple approaches that can be tailored to suit a

particular application. It can be based on intuition,

simulation, preference or some type of graphics to aid the

decision maker in a ‘what if’ mode. This can be called as

quick and dirty procedure (Doll, 1980; Cullen et al. 1981; Hill,

1988; Potvin and Rousseau, 1994; Nussbaum et al, 1997; Hwang,

1999; Du, et al. 2007).

Hurrion (1980) described a visual interactive method of

improving solutions for the TSP. Wu and Liou (1993) devised a

mechanism to include the elastic ring in the Potts neural

encoding approach to solve the TSP. Baker and Carreto (2003)

described a graphical-user-interface and a heuristic based

on a greedy randomised adaptive search procedure which was

developed to work in combination to tackle the basic VRP.

2.6 HYBRID APPROACHES

Analysts have also attempted hybrid approaches, combining

two or more of those suggested in the preceding paragraphs.

Some of these are reported to have a high potential to

provide good solutions at low computational time (Laporte et

al. 2000).

Given the nature of the combinatorial optimisation

problem, a lot of approaches have been made for variants of

both VRP and TSP and have been discussed all the way through

the review of the variants. Nevertheless, to be precise a

few of the recent works are due to Liu et al. (2006), who

combined PSO with Memetic Algorithms and proposed it for TSP.

They also developed a novel encoding scheme and an effective

local search based on SA with adaptive meta-Lamarckian

learning strategy and incorporated into PSO. Tam and Ma

(2004) combined Guided local search, TS and SA and applied to

solve VRPTW. Liu et al. (2006) proposed an effective PSO based

Memetic Algorithm for the TSP. Fang et al, (2007) proposed a

hybrid algorithm which integrates PSO with simulating SA to

solve the TSP. Duan and Yu (2007) proposed a hybrid ACO

Using Memetic Algorithms for TSP. Tiejun et al, (2008)

proposed a hybrid new method named multi-agent approach

based on GA and ACO to solve the TSP. Perboli et al, (2008)

presented a hybrid algorithm based on GA and TS for solving

the Capacitated VRP.

Repoussis (2010) proposed a hybrid evolution strategy

for the open VRP which manipulates a population of ‘μ’

individuals using a (μ+λ)-ES. At each generation, a new

intermediate population of λ offspring is produced via

mutation, using arcs extracted from parent individuals. The

selection and combination of arcs is dictated by a vector of

strategy parameters. A multi-parent recombination operator

enables the self-adaptation of the mutation rates based on

the frequency of appearance of each arc and the diversity of

the population. Each new offspring is further improved via a

memory-based trajectory local search algorithm, while an

elitist scheme guides the selection of survivors.

2.7 APPLICATIONS OF TSP and VRP

The following are some examples of the multitude of TSP and

VRP applications in manufacturing and service sectors:

Routing of automated guided vehicles which are considered

as one of the most appropriate modes for material handling

in contemporary flexibly automated production environments

(Reveliotis, 2000).

Minimisation of the distribution costs in a multi-facility

production system (Dhaenens-Flipo, 2000).

Determination of vehicle routes for material delivery

within the premises of a plant operating under a Just-In-

Time philosophy (Vaidyanathan et al, 1999).

Sequencing of the operations in single or multi-feeder

printed circuit board manufacturing unit (Altinkemer et al.

2000).

Rolling batch planning (Xiong et al.1998).

A few real-life examples (Ganesh et al. 2007a), which are

variants of the classical VRP, are listed below:

Upper limit on the time of delivery - e.g., Milk and

Newspaper

Barred time windows - e.g., Urban solid waste removal

Conflicting/competing time windows - e.g., Mobile catering

Combined routing and scheduling - e.g., Mobile hospital,

Mobile court

Independent multiple depots

Interdependent multiple depots

A diverse application of TSP and VRP with different set of

variants is listed in Table 2.1 and some of the applications

are explained in detail.

2.7.1 Multiple Interdependent Depot VRP (MIDVRP)

In organisations with more than one depot, it is often the

case that each depot is sovereign, with its own fleet of

vehicles and its own geographical customer area to serve. In

such cases, the organisation would simply face a number of

similar single-depot VRPs. In other cases depot operations

are interdependent and vehicles leaving one depot may, after

delivering to customers, end up at another depot (Marinakis

and Migdalas 2007). These problems are called multiple

interdependent-depot-VRP (MIDVRP). Some of the examples are:

Garbage collection in a crowded city, special buses arranged

by the government during functions, etc.

2.7.2 Multiple Commodities VRP (MCVRP)

In some cases, the vehicles are partitioned so that

different commodities are stored in segregated compartments.

Each customer may specify the required quantities of various

commodities. This characterises a multi-commodity-VRP

(MCVRP).

2.7.3 Vehicle Scheduling Problem (VSP)

Vehicle scheduling problems can be thought of as routing

problems with additional constraints imposed by time periods

during which various activities may be carried out (Bodin et

al. 1983).

Some of the constraints which make VSP are:

1. The length of the time that a vehicle may be in

operation before it must return to the depot for

service or refueling.

2. The fact that certain tasks can only be carried out by

certain vehicle types.

3. The presence of a number of depots where vehicle may be

housed.

Example: Grocery store distribution.

2.7.4 School Bus Routing and Scheduling Problem (SBRSP)

In the SBRSP there are a number of schools, wherein each is

assigned a set of bus stops, with a given number of students

assigned to each stop, and time windows for the delivery and

pick-up of the students (Bodin et al. 1983). The problem is to

minimise the number of buses used and total transportation

costs while serving all the students and satisfying all the

time windows.

2.7.5 Routing and Scheduling with Full Loads and Time Window

(RSFLTW)

In the problem of RSFLTW, a set of demands is specified for

a set of origin-destination pairs like transportation

problem. Each demand is a full load that must be loaded onto

a vehicle at an origin and unloaded at a destination. These

stops must satisfy pre specified time window constraints and

the aim is to design routes and schedules for the fleet of

vehicle. (Bodin and Golden 1981). The objective can be

minimising the number of vehicles and the total distance

travelled.

2.7.6 Newspaper Distribution VRP

Newspaper distribution problem of a newspaper company is a

complex one. A newspaper company will have printing centres

where newspapers are printed and distributed to local

distribution centres everyday. Newspapers are delivered to

subscribers from the local distribution centres. Newspapers

should be delivered sufficiently early before the readers

leave for their work places. If there are frequent late

deliveries, readers would consider switching their

subscriptions to other newspapers. Timeliness is one of the

most important requirements in newspaper delivery. However,

since newspapers need to be printed as late as possible in

order to contain the most up-to-date news, there is usually

only a very short time available between printing and

delivery. Although the newspaper company tries to minimise

the printing time and to increase the printing capacity

nowadays, there are still chances of being late without an

efficient delivery scheme. Delivery plan is needed to

deliver newspapers to as many local distribution centres as

required using minimum number of vehicles in the stipulated

time (Ree and Yoon, 1996).

2.7.7 Recyclable Material Collection VRP

Large university campuses may face the problem of collecting

the waste / refuse produced by offices, classrooms,

laboratories, etc. This problem becomes especially complex

when there are many buildings spread out over a large area

(Bommisetty et al. 1998). In a given five-day work week, the

vehicles are needed to visit various buildings in order to

collect the waste in a prescribed pattern. The problem is to

minimise the distance travelled by the vehicle, the vehicle

and the collection time. The constraints could be required

collection frequency, number of vehicles, and volume in

terms of bins, vehicle capacity and time constraints.

2.7.8 Earthquake - Food Distribution VRP

Distribution of food to earthquake affected area is a

complicated distribution problem. This is because in general

there is neither an inventory system in place nor are their

plans for transporting goods to these regions. There is a

need to determine optimal patterns of food supply and

inventory allocation for earthquake affected area. It is

also essential to formulate the VRP incorporating inventory

allocation and the optimal distribution based on minimising

the amount of pains and suffering of the affected people

instead of travel distance. Time is also the main factor to

cover almost many affected areas. A longer route with fewer

vehicles and a high efficiency of supply is more attractive

proposition to this kind of problem (Hwang 1999).

2.7.9 Perishable Food Distribution VRP

Managing perishability of foods is a difficult problem in

distribution management. In food industries perishability

arises for products like milk, vegetables and meat. The time

period between preparation date and delivery of these

products is of a major interest for both producers and

retailers. Inefficient distribution process will cause a

serious decline in quality, loss of sale efficiency,

increase in storage cost, and decrease in order volume. The

problem is to find a set of optimal routes for heterogeneous

capacity of vehicles, with the constraints on number of

vehicles, capacity of vehicles, waiting and servicing time

for each customer, and the distance traveled (Tarantilis and

Kiranoudis, 2001b).

2.7.10 Bank Cheque Collection VRP

In the case of bank cheque collection, the problem is to

determine the routes and departure date / day and times for

vehicles from the head office to distribute the cheque and

other forms and to retrieve the cheques that accumulate at

the retail branch locations. The vehicle will return back to

head office after all collections and distributions are

over. The complication of the problem is to predict the

retrieval volumes. When the schedule result in delayed

retrieval, some cheques miss the deadlines and resulting in

handling fees. If the retrieval is too early, then there

will be large accumulations at the cheque processing

facility centre. (Anbuudayasankar et al 2008).

2.7.11 Vehicle Routing for Transporting Hazardous Material

The transportation of hazardous materials evolve greater

amount of risk. The problem becomes more acute because of

the increase in the quantity of such materials to be

transported these days. One way to overcome this risk is to

take appropriate routing decisions which can lead to the

determination of alternative routes, incase needed with

respect to truck driver’s decisions. The problem is to find

a risk minimised routes with the constraints of distance

travelled and time. The precedence can be given to truck

driver’s preference. (Tarantilis and Kiranoudis, 2001a;

Zhang et al 2005)

2.7.12 Share Auto VRP

The tendency of the people to engage share autos to reach

their destinations rather than using public transport system

increases the load on the environment and raises issues

about the quality of life. The use of share autos in the

main road of the cities creates high level of air pollution,

parking problems, noise pollution and traffic congestion.

So, designing alternative routes for the share autos with

less travel distance is the objective. The preferences of

the public and norms of the government should be taken care

of while designing the routes.

Table 2.1 Diverse Applications of TSP and VRPSl.No Application Variant Literature

1 School Bus Routing

Multi objective VRP and TSP Bowerman et al. (1995)VRP Braca et al. (1997)Multi objective VRP and TSP Li and Fu (2002)Open VRP with Time Dead Lines Ozyurt et al. (2005)VRP with Coupled Time Windows Fugenschuh (2006)VRP Park and Kim (2010)

2 Food Distribution

VRP with Time Windows (VRPTW) Hsu and Feng (2003)Ioannou et al. (2001)

VRPEvans and Norback (1985)Prindezis et al. (2003)Johansson (2006)

TSP with Time Windows Bräysy et al. (2009a)

3 Waste Collection Problem

VRP with Crew Constraints Angelelli and Speranza (2002)

VRP Tung and Pinnoi (2000)Nuortio et al. (2006)

VRPTW Kim et al. (2006)

4 Milk Collection and Distribution

VRP Boldon et al. (1996)Two-Period TSP Butler et al. (1997)

VRP Tarantilis and Kiranoudis (2000)

VRP with Heterogeneous Fixed Fleet Tarantilis et al. (2004)

VRPTW Marshall et al. (2006)

Periodic VRP Claassen and Hendriks (2007)

5 Container Transport

VRP

Cattrysee et al. (1996)Kim and Kim (1999)Koo et al. (2004)Imai et al. (2007)Stahlbock and Voβ (2008)Bandeira et al. (2009)

Period VRP Baptista et al. (2002)VRPTW Lee et al. (2003)VRP and TSP Steenken et al. (2004) Capacitated VRP Karlaftis et al. (2009)Multi-traveling salesman problem with time windows (m-TSPTW)

Zhang et al. (2009)

6 City Logistics Dynamic VRP Taniguchi et al. (2000)Montemanni et al. (2005)

Multi Depot VRPTW (MDVRPTW) Chiu et al. (2006)

VRPTarantilis and Kiranoudis(2002)Bräysy et al. (2009b)

7Company Employee Distribution Relations

VRP with Load Balancing Lee and Ueng (1999)

Table 2.1 Diverse Applications of TSP and VRP (Contd...)

8 Furniture Transport

VRP with Heterogeneous Fixed Fleet Prins (2002)

9 Public Transport VRP

Bodin (1990)Sateesh and Ray (1992)Nurcahyo et al. (2006)Lam et al. (2009)

10 Soft Drink Industry

VRP Golden and Wasil (1987)Zeng et al. (2008)

TSP with Pick up and Delivery Gendreau et al. (1999)11 Brewing Industry VRP Eibl et al. (1994)

12

Land Transportation of air cargo forwarder

VRP with Backhauls and Time Windows with Heterogeneous Fleet of Vehicles

Cheung and Hang (2003)

VRP with Time Windows (VRPTW) Kritikos and Ioannou (2010)

13 Sugar Cane Transport MDVRPTW Abel et al. (1981)

14 RetailDistribution VRP Soehodho and

werdinigngsih (2003)

15 Fresh MeatDistribution Multi Depot VRP Tarantilis and Kiranoudis

(2002a)

16 Mail Carrier (or)Mail Delivery

VRP Tarantilis et al. (2002c)Period TSP Paletta 2002

17 Ship Routing

Multi-trip VRP Fagerholt (1999)

TSP with Time Windows Fagerholt andChristiansen (2000)

TSP with Time Windows Fagerholt (2001)

18 Emergency Planning VRP with Mixed delivery andpickup Ozdaman et al. (2004)

19 Gas distribution VRP Day et al. (2009)

20 AgriculturalTransport VRP

Osvald and Stirn (2008)Bochtis and Sørensen(2009a)Bochtis and Sørensen

(2009b)

21Live Stock(Animal)Distribution

VRP Grlbkovskaia et al. (2006)

22 Post BoxCollection TSP Laporte et al. (1996)

23 Rural PostmanProblem TSP Eiselt et al. (1995)

24 PerishableProducts

VRP Federgruen et al. (1986)Hsu et al. (2007)

VRP with Time Windows (VRPTW) Chen et al. (2009)

25Analysis of thestructure ofcrystals

TSP Bland and Shallcross(1987)

26 Overhauling of gasturbine engines TSP Plante et al. (1987)

27Drilling ofprinted circuitboards

TSP Ancău (2008)

28 Material handlingin a warehouse TSP Ratliff and Rosenthal,

(1983)

29 Clustering of dataarrays TSP Lenstra and Rinooy Kan,

(1975)

30Sequencing of jobson a singlemachine

TSP Gilmore and Gomory,(1964)

2.7.13 Vendor Managed Distribution Systems

In vendor-managed systems, distribution companies estimate

customer inventory level in such a way to replenish them

before they run out of stock. Hence, demands are known

before and all customers are static. Yet, because demands

are uncertain, some customers may run out of stock and have

to be served urgently. This is a real time VRP with varying

load and constant time. (Archetti et al 2007).

2.7.14 Taxi Cab services

In taxi cab services, almost every customer is dynamic and

demand rate is usually high. Sometimes vehicles become idle

from time to time. Using the idle vehicle to meet unexpected

demand can be thought of. Directing these idle vehicles to

customers, when the scheduled vehicle is unable to reach the

customers, due to traffic jam is an important issue to be

considered. (Holly, 2007).

2.8 OBSERVATIONS AND RESEARCH GAP

A careful analysis of literature on the variants,

methodologies and applications of TSP and VRP published

hitherto reveals the following: It is evident that there is

a need to address a variety of complex variants of the

classical TSP and VRP pertaining to Balanced Logistics,

Reverse Logistics, Distribution Logistics and Emergency

Logistics. TSPs and VRPs in the real world often need to

include additional factors such as multi depot, time window,

route length, heterogeneous capacity, sequential and

simultaneous loads etc. The underlying principle of the work

is to consider additional real-time factors and constraints

in the complex routing variants and to develop solutions for

the same. The impetus for the development of the work is an

urge to provide better algorithms for the logistics decision

makers for the complex routing variants. This research

rationale is a "living document" that will continue to

change as the research evolves in terms of different

variants and solution methodology. The contribution lies in

the development of effective and competitive new / derived

(devised based on leveraging the concepts from the existing

approach or solution methodology) / combination of new and

derived, unified solution methodologies (heuristics / meta-

heuristics) for the new / less researched complex routing

variants.

The aim of the study is of two folds:

To explore and identify the new / less researched

challenging, practical, complex variants of routing

problems in the current global logistics trend based on

academic and business literature and from the interview

conducted with chief officers of global supply chain.

To propose and develop new / derived and unified

methodologies to solve the above routing problems in order

to support the decision making ability of logistics

managers.

2.8.1 Problem in Balanced Logistics

According to the council of Logistics Management, logistics

is “the process of planning, implementing and controlling

the storage of goods, services and related information from

point of origin to point of consumption for the purpose of

conforming to customer requirements”. This reveals that the

importance is not only given to strategic level (or)

tactical level but also to the operational level in which

the actual implementation of the transfer of goods from one

point to various destinations comes in to picture. The

customer satisfaction can be improved with the increase in

the level of logistics service which in turn has a major

impact on revenues, especially in markets with homogeneous

low price products where competition is based on the

promptness in delivery of the products.

The integration of inventory and transportation is also

a key aspect in reducing the total supply chain cost.

Particularly to reduce the operating cost the management

must determine the proper balance between inventory and

transportation costs. Frequent trips between depot and the

customer reduces the inventory cost but at the same time

transportation cost is high whereas infrequent trips leads

to high inventory cost and low transportation cost. In the

distribution of products from a depot to multiple customers,

assigning loads to vehicles is as important as deciding the

routes for the vehicles. In routing, the order in which the

customers are visited will determine how long the delivery

will take place and the time of returning to the depot.

There are times when inhuman weights are lifted, high

risks are taken and even acrobatics performed! But, at later

stages, the automation has invaded the in-bound logistics of

shop-floor. Set-up times have fallen drastically, precision

has improved, and quality is being talked of in the parts

per million ranges. Companies may not be interested so much

in human issues in out-bound logistics and distribution. A

burning issue today is the workload balancing between

drivers in transshipments of goods. Rarely do business

interests align with this kind of human issues. Nevertheless

it leads to the price of poor service and loss of goodwill

that may be the result of workers' fatigue. These factors

are bound to affect end-product pricing and constitute an

indirect cost that customer pays.

The original version of this problem is referred to as

multiple TSP (mTSP) in literature. The problem of finding

the minimum route length, in either time (or) distance, from

a depot through a set of customers to be served is

imperative for the competitive advantage which is known as

the classical TSP. Instead of single vehicle, if there are

multiple vehicles serving the customers, then it is the

mTSP. A generalisation of the standard mTSP is the well-

known TSP. The problem can be defined simply as the

determination of a set of routes for m salesmen, who all

start from and return to a single home city (depot).

In practice it is observed in many instances that more

than one salesman (or) vehicle starts from a single place to

serve the customers at different locations and returns to

the same place. This exactly resembles mTSP and hence it is

appropriate to model a real life situation with mTSP. Some

reported applications are Print press scheduling

(Gorenstein, 1970., Carter and Ragsdale, 2002), School bus

routing problem (Angel et al. 1972), Crew scheduling (Svestka

and Huckfeldt, 1973., Lenstra and Rinnooy Kan, 1975., Zhang

et al. 1999), Interview scheduling (Gilbert and Hofstra, 1992),

Mission planning (Brummit and Stentz, 1996, 1998, Yu et al.

2002., Ryan et al. 1998), Hot rolling scheduling (Tang et al.

2000), Design of global navigation satellite system

surveying networks (Saleh and Chelouah, 2004) etc.,

Now that it is observed that the balancing of the

workload along with the route optimization is essential. An

extension of mTSP with workload balancing (mTWB) is

developed. The objective of this model is equalising the

workload among the available drivers / vehicles with the

intention of finding out the optimal number of such entities

required to cover a set of nodes. The problem is addressed

under the conditions that all the drivers must return to the

node of origin (depot) and that every node will be visited

exactly once. This variant is applicable for both

manufacturing and service industries.

Many drivers may not object to drive more than 8 hours

a day, but they may not be happy visiting more vendors since

loading and unloading is a tiresome task. The logistics

manager is to find set of vendors for each driver for a

given day. Many drivers approach logistics manager

personally and request to allot them for fewer vendors. Now

the challenge of logistics manager is to allot a set of

vendors for each driver to balance the workload. It is also

assumed that the workload is almost equal at every vendor

location.

This variant mTWB is not a new variant. This is

already addressed by Chandran et al. (2006). The objective

is to develop new / derived / combination of new & derived

and unified solution methodologies to solve the variants

multiple traveling salesman problem mTSP and mTWB. Three

solution methodologies are proposed for the variants mTSP

and mTWB. The first heuristic is the combination of new and

derived approach. A new clustering approach is proposed and

it is combined with the derived approach of shrink-wrap

algorithm. The second heuristic is the combination of new

and derived approach, where as a new clustering approach is

proposed and it is combined with the derived approach of 2-

Opt algorithm. The third heuristic is the combination of new

and derived approach, where as a new min-max function with

prufer number concept is proposed and it is coupled with the

derived meta-heuristic process of Genetic Algorithm. The

contribution lies in the development of two heuristics

(combination of new and derived) and one meta-heuristic

(combination of new and derived) as a unified solution

methodology to solve both mTSP and mTWB.

2.8.2 Problem in Reverse Logistics

Sustainable supply chain is the management of goods from

suppliers to manufacturer / service provider and then to

customer and back. Here there is a need to reduce the carbon

footprint in the logistics. Reverse logistics is the

process of transshipment of goods / the containers from

their typical final destination to pre-destination or

origin, mainly for the reason of capturing value or for the

proper further use, re-use or disposal of the goods. Reverse

logistics is also applicable to the containers such as cans,

bottles etc. which carry the goods from the manufacturer to

the customers and supposed to bring back for re-supply of

the goods in the same container after the cleaning process.

Economic and environmental impacts are the main inspirations

for planning the reverse logistics of the channel for the

containers (Alshamrani et al. 2007).

This problem aims at returning containers to the depot

that are associated with goods / products delivered

previously on a route. The pick-up load needs to be replaced

in the place of delivery load. This is termed as constrained

capacity. The research problem is to deliver the goods to

customers and simultaneously pick-up the used containers

such as bottles, cans etc in the same vehicle in the place

of the delivered loads. So, the loads which picked from the

customer’s needs to be adjusted in the place of the load

delivered. The requirement is to route the vehicle with due

consideration to the loads involved in delivery as well as

pick up with the constrained capacity. The pick-up and

delivery should be performed simultaneously so that each

node is visited only once by the vehicle. This new variant

is termed as simultaneous delivery and pick-up problem with

constrained capacity (SDPC).

The base variant of this new variant is Traveling

Salesman Problem with Simultaneous Delivery and Pick-up

(TSPSDP) addressed by Ganesh et al. (2007b). The objective is

to develop new and unified solution methodologies to solve

the variants TSP and SDPC. Three solution methodologies are

proposed for the variants TSP and SDPC. The first, second and

third heuristic is the combination of new and derived

approach where as a new construction approach based on

branch and bound concept is proposed, the first heuristic is

coupled with the derived meta-heuristic process of Genetic

Algorithm, second heuristic is coupled with the derived

meta-heuristic process of Simulated Annealing and third

heuristic is coupled with the derived meta-heuristic process

of Hybrid Genetic Algorithm and Simulated Annealing. The

contribution lies in the introduction of new variant to the

domain of logistics and the development of three

construction heuristic based meta-heuristics (combination of

new and derived) as a unified solution methodology to solve

both TSP and SDPC.

2.8.3 Problem in Distribution Logistics

Transportation is generally considered as being a

significant factor of economic activities in any company.

The problem of appropriate usage of the vehicle fleet

appears as a matter of restricted resources of the company

and expectations of customers. The classical VRP with

backhauls (VRPB) is an extension of the VRP where two types

of customers are served from a single depot by a fleet of

vehicles. The first type of customers is known as “linehaul”

customers who require delivery of goods to their specified

location and the second type is known as “backhaul”

customers who require pickups from their specified

locations. In recent years, it became more obvious that in

real-world applications allowing vehicles, which are

returning from linehaul customers, to visit backhaul

customers leads to significant saving in the distribution

cost. Therefore the classical VRPB and its variants have

attracted the attention of researchers. The feature “each

vehicle has to serve backhaul customers, if any, after all

linehaul customers are served” defines the fact that in the

classical VRPB deliveries after pickups are not allowed. In

theory this restriction reduces the complexity of the

problem and in practice it avoids the problems that may rise

because of rearranging goods on the vehicle and supports the

fact that linehaul customers have priority over backhaul

customers. However, it can be easily proposed that ignoring

this restriction may reduce the total travelling cost.

Therefore, the mixed VRPB is defined as an extension of the

classical VRPB where the constraints and the objective are

the same as in the classical VRPB but deliveries after

pickups are allowed. This difference makes the mixed VRPB

(MVRPB) more difficult to solve than the classical VRPB. The

main reason behind this difficulty is the need to check the

capacity constraints for possible violation for every arc of

each route before inserting a customer into a new position

on any route. In the classical VRPB it is enough to check

capacity constraints violations in the corresponding part of

the route (backhaul or linehaul parts) while in the MVRPB

these capacity constraints have to be checked for every link

between the customers. Therefore, although the capacity

constraints seem to be similar for both problem types they

become more restrictive in the MVRPB.

This research addresses the variant MVRPB for the

application of third party logistics (3PL) service provider

organisation. The 3PL service providers are playing an

important role in the management of supply chains. The

global and competitive business environment of 3PLs has

recognised the significance of a speedy and proficient

service towards the customers in the past few decades.

Particularly in warehousing, distribution and transportation

services, customers anticipate improved lead times, fill

rates, inventory levels, etc.

This variant MVRPB is a not a new variant. This is

already addressed by Goetschalckx and Jacobs-Blecha (1989).

The objective is to develop new solution methodologies to

solve the variant MVRPB. Three solution methodologies are

proposed for the variant MVRPB. The first heuristic is the

derived approach. A derived clustering approach is combined

with the derived approach of Or-opt algorithm. The second

heuristic is the combination of new and derived approach;

where as a concept of changeover is coupled with the derived

meta-heuristic process of Simulated Annealing algorithm. The

third heuristic is the combination of new and derived

approach, where as a new composite concept with local search

is proposed and it is coupled with the derived meta-

heuristic process of Genetic Algorithm. The contribution

lies in the development of one heuristic (derived) and two

meta-heuristics (derived and combination of new and derived)

to solve MVRPB.

2.8.4 Problem in Urgency Logistics

The majority of existing supply chain research focuses on

managing and/or optimising the commercial supply of goods

and services. The supply process that deals during emergency

situations is an important domain for supply chain

management that has so far received little attention. The

unpredictability of the nature of the emergency and the

stake of adequate and timely delivery dominate this unique

and challenging material flow problem. Planning is a

critical process in those emergency and urgency situation.

Route planning systems are driven by choices of

objectives and the system picks the suitable algorithm based

on the need to provide the route. However, some experienced

drivers may choose the own route and the solution by the

system is much useful for inexperienced driver (Husdal,

1999).

At the same time, there are some special urgency /

forced requests and pressure from some of the customers to

take-back the used empty containers in order to free their

inventory space. Many customers are in high demand of space

and the demand of orders is also highly volatile. They would

like to move the empty containers out of the company as

early as possible in order to meet their emergency orders.

In the delivery and pick-up problem, the route planning

is normally made with respect to optimal cost. But, if there

is a specific urgency / forced requests, then the optimality

should be compromised with this aspect of service. In this

situation, some pick-up sequence is forced in the vehicle

routing and the route planning should be designed to satisfy

the urgency requests.

The original version of this problem is referred to as

VRP with Backhauls (VRPB) (Süral and Bookbinder, 2003; Wade

and Salhi, 2002). But, in the proposed variant the sequence

of forced pick-ups is considered. A set of pick-up nodes are

forced to visit and is known prior. The pick-up nodes should

be visited in sequential manner. The sequence of visiting

those pick-up nodes is not an input and it is considered for

optimization. The variant includes the constraints to

restrict the inclusion of delivery customers until a given

set of pick-up customers are served. So, this new variant is

an extension of MVRPRB with forced backhauls with the

objective of serving the urgency / forced requests in the

middle of delivery routes. This variant is termed as VRP

with Forced Backhauls (VRPFB). But, looking at the reality,

two objectives in VRPFB are considered and it is called as

Bi-Objective Vehicle Routing Problem with Forced Backhauls

(BVFB).

This variant BVFB is a new variant. The base variant of

this new variant is Vehicle Routing Problem with Backhauls

(VRPB) addressed by Goetschalckx and Jacobs-Blecha (1989)

and the bi-objective was coined by Thammapimookkul and

Charnsethikul (2001). The objective is to develop new /

derived / combination of new & derived and unified solution

methodologies to solve the variants VRPB and BVFB. Three

solution methodologies are proposed for the variants VRPB and

BVFB. The first heuristic is the derived approach. The

concept of arc removal is coupled with the derived savings

algorithm. The second heuristic is the derived approach;

where as a concept of node swap is coupled with the derived

savings algorithm. The third heuristic is the combination of

new and derived approach, where as a new fitness concept is

proposed and it is coupled with the derived meta-heuristic

process of Genetic Algorithm. The contribution lies in the

introduction of new variant to the domain of logistics and

the development of two heuristics (derived) and one meta-

heuristic (combination of new and derived) as a unified

solution methodology to solve both VRPB and BVFB.

2.8.5 Common Observations

Many of the authors (Potvin et al. 1996; Duhamel et al. 1997;

Fisher and Jaikumar,1981; Toth and Vigo, 1999) have

suggested the use of a constructive heuristic to obtain good

initial solutions for a meta-heuristic so that its

convergence can be accelerated. Only a few authors have

considered the use of hybrid approaches to solve different

variants of VRP. Glover et al. (1995) and Osman and Kelly

(1996) have pointed out that hybrid approaches focus on

enhancing the strengths and compensating for the weaknesses

of two or more complementary approaches. The aim is the

generation of better solutions by combining the key elements

of competing methodologies.

The quality of solutions obtained by many of the

proposed heuristic methods has not been established through

comparative evaluation with optimal solutions. While meta-

heuristics can yield better solutions, the computational

effort required by them often inhibits their use. There is a

scope for the application of multi-phase heuristics that use

a combination of intuitive and classical methods to

construct good initial solutions which, in turn, serve as

inputs for an intensive search using meta-heuristics. This

could yield quality solutions at reasonable computation time

(Johnson et al. 1991).

2.9 REVIEW OF THE VARIANTS OF TSP ADDRESSED IN THIS BOOK

Two variants of TSP are addressed namely mTSP and Bi-

objective TSP with Simultaneous Delivery and Pick-up.

Literature on these variants addressed in this study is

presented in the next few paragraphs.

2.9.1 Multiple Travelling Salesmen Problem (mTSP)

The mTSP is an extension of TSP with more than one salesman

all of them starting from one city (depot), visiting

different cities, and coming back to the starting city. mTSP

consists of finding tours for all m salesmen, who all start

and end at the depot, such that each intermediate city is

visited exactly once and the total cost of visiting all

cities is minimised. It can also be measured in terms of

distance, time, etc.

Hong and Padberg (1977) transformed an mTSP with fixed

charges for the assignment of salesmen and with a symmetric

cost matrix to a standard symmetric TSP for the ease of

handling. Lenstra and Rinnooy Kan (1979) have showed that

only linear admissible transformations are obtained by

adding constants to the rows and columns of a scalar

multiple of the distance matrix. Berenguer (1979) dealt the

mTSP with the transformation of the distance matrix and

analysed it in the linear context. Mole et al. (1983)

established the route first-cluster second heuristic to the

mTSP given that each salesman can visit any number of

customers in a stated range.

Kalantari et al. (1985) extended the branch and bound

algorithm of Little et al. (1963) to the TSP with pickup and

delivery customers which included single and multiple

vehicle cases as well as infinite and finite capacity cases.

Gavish and Srikanth (1986) developed a branch-and-bound

method for solving large scale mTSP and developed lower

bounds through a Lagrangean relaxation. Desrosiers et al.

(1988) proposed Lagrangian Relaxation Methods for solving

the Minimum Fleet Size mTSP with Time Windows to find the

minimum number of vehicles required to visit once, a set of

nodes.

Ferland and Michelon (1988) formulated a vehicle

scheduling problem and developed a heuristic and exact

methods for a single type of vehicle and shown that the

methods can be extended in a straightforward fashion to the

multiple-vehicle-types problem. Jonker and Volgenant (1988)

improve the standard transformation of the symmetric,

single-depot, mTSP to one on a sparser edge configuration.

Okonjo-Adigwe (1988) proposed an effective method of

balancing the workload amongst salesmen which addressed

large size problems and the addressed VRP is an extension of

the mTSP.

Wacholder et al. (1989) developed an efficient neural

network algorithm for solving the mTSP. They have introduced

a new transformation of the N-city M-salesmen mTSP to the

standard TSP. This algorithm was tested by them on many

problems with up to 30 cities and five salesmen. Okonjo-

Adigwe (1989) addressed the adult training centre problem

which resembles the mTSP. He proposed both heuristic and

exact algorithms to derive the best routing for four

vehicles which provide a daily service from a depot to 38

locations.

Fogel (1990) proposed a parallel processing approach to

solve the mTSP using evolutionary programming which

considers two salesmen with an objective of minimising the

difference between the lengths of the routes of each

salesman, in which he practiced the inversion mutation as

the genetic operator. Exceptionally good near-optimal

solutions were obtained for the problems with 25 and 50

cities with his evolutionary approach. Self-organising

approaches have also been successfully applied to the mTSP

where in Goldstein (1990) developed an extended elastic net

approach.

Hsu et al. (1991) proposed a neural network approach based

on the self-organised feature map model to solve the mTSP.

Gilbert and Hofstra (1992) introduced a new multi-period

mTSP with a polynomial heuristic method to the scheduling of

tour brokers and vendors at conventions of the tourism and

travel industry. The heuristic method is capable of

producing non-conflicting set of salesmen's tours. Brummit

and Stentz (1996) explored a dynamic environment that

involves multiple mobile robots in determining the optimal

path for each robot and to achieve the goals of the mission.

Chan and Merrill (1997) addressed a probabilistic multiple-

travelling-salesman-facility-location problem with an

asymptotic analysis using space-filling curve heuristic

which responds to stochastic demands. They suggested that

this procedure can be executed in the field with minimal

computational requirements. Torki et al. (1997) were motivated

by the outstanding performance of adaptive Neural Network

approach in the TSP and hence devised an algorithm to extend

the domain of applicability of this approach to the mTSP.

Modares et al. (1999) approached mTSP with several algorithms

based on self-organising neural network. A comprehensive

empirical study was provided by the authors in order to

investigate the performance of the algorithms. They stated

that the proposed algorithm exemplify significant advances

in the quality of the solution as well as the computational

efforts for most of the experimented data. Somhom et al. (1999)

introduced a new algorithm in competition-based network to

solve the minmax mTSP with an objective to minimise the

maximum distance among all salesmen travelled. They applied

the revised 2-opt exchange heuristic algorithms and the

elastic net algorithm to the minmax mTSP problem. They also

tried the combination of the adaptive algorithm with a

simple improvement heuristic and compared it with the

recently adaptive TS.

Tang et al. (2000) formulated the model for hot rolling

production scheduling with a case study at Shanghai Baoshan

Iron & Steel Complex and solved it using a new modified GA.

Chan et al. (2001) formulated A multiple-depot, multiple-

vehicle, location-routing problem with stochastically

processed demands and suggested a solution method for these

class of problems. Sofge et al. (2002) compared a variety of

evolutionary computation algorithms and paradigms for

solving the mTSP. They used a neighborhood attractor schema

(a variation on K-means clustering), the “shrink-wrap”

algorithm for local neighborhood optimisation, PSO, Monte-

Carlo optimisation, and a range of GAs and evolutionary

strategies for solving the same. Wang and Regan (2002)

described a solution method for the mTSP with time window

constraints. Their model described an iterative solution

technique in which explicit time constraints were replaced

by binary flow variables. Calvo and Cordone (2003)

introduced the overnight security service problem which was

modeled as a single-objective mixed integer programming

problem. They decomposed the problem as two subproblems

wherein one is a capacitated clustering problem and the

other is an mTSP with time windows and solved the problems

by employing the heuristic approach. Chan and Baker (2005)

addressed a mixed integer multiple depot, mTSP facility-

location problem formulation, expanded to include vehicle

range and multiple service-frequency requirements. This was

used to validate a heuristic solution for location and

routing in which a combination of the minimum spanning tree

and a modified Clarke-Wright procedure was employed.

Bektas (2006) reviewed the mTSP and its practical

applications and highlighted some formulations and described

exact and heuristic solution procedures proposed for mTSP.

Jacobson et al. (2006) demonstrated the use of generalised hill

climbing algorithms to determine optimal search strategies

over multiple search platforms for the mTSP. Computational

results with this algorithms applied to the mTSP

demonstrated that near optimal search strategies over

multiple search platforms can be obtained more competently

when limited computing budgets are available. Their interest

was mainly on the military applications. Junjie and Dingwei

(2006) suggested that classical optimisation procedures were

not adequate for the mTSP problems and thus made an attempt

to show how the ACO can be applied to the mTSP with ability

constraint.

Chandran et al. (2006) proposed a clustering approach to

solve the mTSP. The proposed method found to achieve a good

balance of workloads among the clusters, each of which is

visited by a salesperson, when tested over a range of data-

sets. Carter and Ragsdale (2006) proposed a new approach to

solving the multiple travelling salesperson problem using

GA. In that a new GA chromosome and related operators were

proposed for the mTSP. Fügenschuh and Martin (2006)

addressed the mTSP with a multi-criteria approach for

optimising bus schedules and school starting times. They

discussed the legal framework for the school and trip

starting times and presented a multi-objective mixed-integer

linear programming formulation for the same. They also

developed a two-stage decomposition heuristic and applied it

to practical data sets from three different rural German

counties. Kara and Bektas (2006) extended the classical mTSP

by imposing a minimal number of nodes that a traveller must

visit as a side condition. They considered single and multi-

depot cases and proposed integer linear programming

formulations for both, with new bounding and sub tour

elimination constraints.

Estévez-Fernández et al. (2006) introduced multiple

longest TSP. Mitrović-Minić and Krishnamurti (2006) found the

vehicle bounds for the mTSP with time windows by covering

two precedence graphs with the minimum number of paths. Qu et

al. (2007) solved the mTSP with a columnar competitive model

of neural networks incorporating with a winner-take-all

learning rule. Malik et al. (2007) presented an algorithm with

an approximation factor of 2 for the mTSP with Multiple

Depot. Chung et al. (2007) developed mathematical models for

the container road transportation in Korean trucking

industries by utilising the standard formulations of well

known TSP and VRP. They also presented Heuristic algorithms

to solve the models.

Vallivaara (2008) proposed a Team ACO for the mTSP with

MinMax Objective in which a team of ants is allowed to

construct solutions to the problem. Muralidharan et al. (2008)

developed an efficient mechanism for multi-robot

coordination. Their work was on the robot team that works

together on tasks that should be made to share the workload

efficiently in a cost effective manner. They formulated an

mTSP model and presented a novel approach for multi-robot

coordination using combinatorial auctions in order to

maximise robot utilisation and at the same time minimising

incurred costs. They also used Visual Basic System to

simulate the system and they validated the robustness of the

mechanism. Contributions by Li and Lu (2010) with a self-

organizing algorithm, Yamamoto et al., (2010) with an adaptive

routing method and Ghafurian and Javadian (2011) with an ant

system are considered to be recent additions in the

literature to solve the mTSP.

2.9.2 TSP with Simultaneous Delivery and Pick-up

TSP with Simultaneous Delivery and Pick-up, also known as

TSPSDP, remains a less researched problem. The Vehicles such

as mail vans and buses used for public transport need to be

routed both for delivery and pick up. Here as the service

node is visited only once by the vehicle, both delivery and

pick-up should take place at the same time, i.e.

simultaneously. In the limited literature on this problem,

Tang and Galvao (2002) used four different heuristics, viz.,

initial node heuristic, cheapest feasible insertion

heuristic (Mosheiov, 1994), minimal spanning tree heuristic

(Anily and Mosheiov, 1994) and cycle heuristic (Gendreau et al.

1999). Ganesh et al. (2007b) developed a heuristic called CAGE

(Construction of initial solution by Agglomeration and GA

for Effective search) to solve TSPSDP.

2.10 REVIEW OF THE VARIANTS OF VRP ADDRESSED IN THIS BOOK

Two variants of VRP namely Mixed Vehicle Routing Problem

with Backhauls (MVRPB) and Bi-Objective Vehicle Routing

Problem with Forced Backhauls (BVFB) are addressed here.

Literature on these variants addressed in this study is

presented in next few paragraphs.

2.10.1 Vehicle Routing Problem with Backhauls (VRPB)

Heuristics are commonly used because of the complexity to

deal with the classical VRPB and it has been studied by

several researchers for more than two decades. The first

constructive method for the classical VRPB was proposed by

Deif and Bodin (1984) which is the extension of Clarke and

Wright’s (1964) savings algorithm. Goetschalckx and Jacobs-

Blecha (1989) formulated the first mathematical problem

explicitly dealing with the VRP with clustered Backhauls.

They developed a heuristic method for the multi vehicle case

in which the clustering as well as the routing part are

solved by means of a space filling curve heuristic. Toth and

Vigo (1996) proposed a cluster-first, route-second heuristic

for VRPB.

Anily (1996) developed a lower bound on the optimal

total cost and a heuristic solution for the VRPB. The routes

generated by the heuristic were formed such that the

backhaul customers are served only after the delivery

customers are served.

Goetschalckx and Jacobs- Blecha (1993) used a

clustering method in their second paper, which is based on

the generalised assignment approach proposed by Fisher and

Jaikumar (1981). In this approach line-haul customers and

the backhaul customers are sorted according to their

increasing distance from the depot and decreasing distance

to the depot respectively. By solving generalised assignment

heuristics, both customer sequences are divided into K

clusters. Then line-haul and backhaul routes are merged

according to the best combination of connections that has

the smallest distance at the same time not allowing any

backhaul customer before a line-haul customer is served. The

results obtained by this method are superior to their first

approach (Goetschalckx and Jacobs-Blecha, 1989). Thangiah et

al. (1996) described a route construction heuristic as well

as different local search heuristics to improve the initial

solutions.

Toth and Vigo (1999) proposed another two-phase method

for the classical VRPB. They solved both symmetric and

asymmetric VRPB problem using the cluster-first route-second

heuristic approach. In their approach, visiting backhaul

customers before line-haul customers and routes containing

only backhaul customers are not allowed. There are no

distance restrictions for the vehicles. In the first phase

of the heuristic, Lagrangian relaxation method is used to

cluster the line-haul and backhaul customers separately.

They tested and compared their approach on three different

sets of VRPB instances with respect to the optimal solutions

with the approaches proposed by Deif and Bodin (1984) and

Goetschalckx and Jacobs-Blecha (1989) and found to be

outperforming.

Mingozzi et al. (1999) and Toth and Vigo (1997) approached

VRPB with exact methods. Mingozzi et al. (1999) formulated the

VRPB as an integer programming problem and described a

procedure that computes a valid lower bound to the optimal

solution cost by combining different heuristic methods. The

proposed exact algorithm was claimed to solve problems up to

100 customers. Salhi and Nagy (1999) applied a cluster

insertion heuristic for single and multiple depot VRPB. Toth

and Vigo (1997) described a new (0-1) integer programming

formulation of the VRPB based upon a set-partitioning

approach. Wade and Salhi (2001) proposed an Ant System

Algorithm for the VRPB. Wade and Salhi (2002) investigated

the problem with insertion-type heuristic algorithms. Osman

and Wassan (2002) presented a reactive TS heuristic to solve

the VRPB. Ropke and Pisinger (2004) reviewed numerous ways

of modelling backhaul constraints and the various

restrictions on handling backhaul nodes.

Ropke (2005) addressed the VRP with pickup and delivery

and solved using Adaptive Large Neighborhood Search

heuristic, Branch-and-Cut algorithm and Branch-and-price

algorithms for the VRPB problems with time windows. Ropke

and Pisinger (2006) improved their own version of the large

neighborhood search heuristic (Ropke and Pisinger, 2004) to

solve VRPB. Brand˜ao (2006) presented a new TS algorithm

which was able to match almost all the best published

solutions and also found many new best solutions

particularly for a large set of benchmark problems. In a

nutshell Ropke and Pisinger (2006) and Brand˜ao (2006)

proposed competing results for the benchmark data sets

addressed so far. An extensive survey on VRPB and its sub

classes is available in Ropke (2005) and Parragh et al. (2008).

Currie and Salhi (2004) presented new TS algorithms for the

VRPB whereas Nagy and Salhi (2005) proposed modified

heuristic algorithms.

Crispim and Brandao (2001) applied the reactive TS and

variable neighbourhood descent to the VRPB. Tavakkoli-

Moghaddam et al. (2006) proposed a memetic algorithm which used

different local search algorithms (inter and intra-route

node exchanges) to solve the VRPB. Wassan (2007) proposed a

heuristic approach based on a hybrid operation of reactive

TS and adaptive memory programming to solve VRPB. Alshamrani

et al. (2007) developed a heuristic procedure for developing a

route design-pickup strategy planning, a reverse logistics

problem, motivated by blood distribution of the American Red

Cross. Imai et al. (2007) developed a sub-gradient heuristic

based on Lagrangian relaxation which consists of two sub-

problems: the classical assignment problem and the

generalised assignment problem. Gajpal and Abad (2009) used

multi-ant colony system to solve VRPB.

2.10.1.1 Mixed VRP with Backhauls (MVRPB)

The MVRPB is an extension of the classical VRPB where

deliveries after pickups are allowed and line-hauls and

backhauls are sequence-independent. There are a very few

papers addressing this problem. Golden et al. (1985) developed

an approach based on inserting backhaul nodes into the

routes formed by linehaul nodes. Casco et al. (1988) obtained

better results with a load-based insertion procedure which

considers the cost of inserting backhaul nodes. Mosheiov

(1994) investigated the TSP with delivery and pick-up (TSPDP)

and proposed a methodology to satisfy the load constraint.

Anily and Mosheiov (1994) presented a minimal spanning tree

approach for solving TSPDP. Salhi and Nagy (1999) extended

the method of Casco et al. (1988) by allowing backhauls to be

inserted in clusters.

Wade and Salhi (2003) introduced an enhanced ACO for

the MVRPB. Crispim and Brandão (2005) presented a hybrid

algorithm which comprises TS and variable neighbourhood

descent for the MVRPB. Wassan et al. (2008) implemented a

metaheuristic based on reactive TS for the problem. Tütüncü

et al. (2009) described a new visual interactive approach based

on Greedy Randomised Adaptive Memory Programming Search for

the mixed VRP with backhauls. Tütüncü (2010) also proposed

the same approach to solve the heterogeneous fixed fleet

vehicle routing problem with backhauls.

2.10.2 VRP with Forced Backhauls

In practice, in routing, it is indispensable to force the

vehicles to visit a set of nodes, for many reasons. This

important variant of VRPB can be thought of as forced VRP

with backhauls. Such type of problems has not been addressed

so far. Here the force is on the pick-up nodes in that the

vehicle is required to visit a set of pre-defined cluster.

The sequence of visiting the pick-up nodes are also

considered for optimisation.

Wade and Salhi (2002) coined the word restricted VRPB

(R-VRPB) in which mixed linehaul and backhaul customers are

permitted with a constraint of serving the backhaul

customers first. R-VRPB includes the constraints to restrict

the inclusion of backhaul customers until a given percentage

of the total linehaul load has been delivered.

They used a simple constructive heuristic, the R-INS

method, which used a greedy insertion heuristic (Salhi and

Nagy, 1999) to illustrate the practicality of this

restricted version of the mixed VRPB. In this algorithm, is

asked to set a restriction percentage on the insertion of

backhaul customers. In their procedure the linehaul

customers were routed first and then the backhaul customers

are inserted into the route and the RP was used to control

these insertions.

Tutuncu et al. (2009) proposed an inexpensive decision

support system based a new Greedy Randomised Adaptive Memory

Programming Search algorithm to solve the classical VRPB,

the mixed VRPB and the restricted VRPB in a visual

interactive environment. He claimed that the computational

results on VRPB benchmark test problems indicated that the

proposed visual interactive approach is effective towards

finding a compromise amoung the mixed, restricted and the

classical VRPB problems. The survey on literature reveals

that the forced backhauls is a new variant and it is not

addressed directly in the literature.

2.11 SUMMARY

This chapter presents a survey of literature on the selected

and complex variants of TSP and VRP that pertain to balanced,

reverse, distribution and urgency logistics. It also

highlights a report on variants, methodologies and

applications that call for further investigations. The gaps

identified from the literature provided the motivation for

the issues addressed in this book.