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A Bi-Criteria Vehicles Routing ProblemTitiporn Thammapimookkul

Peerayuth CharnsethikulIndustrial Engineering Department

Kasetsart University, Bangkok, Thailand

AbstractThis paper is concerned with the Vehicle Routing Problem (VRP) arisen for

an ATM scheduling, a real-world routing-scheduling problem. Generally, the main

objective for the VRP is to minimize the total cost or distance of travel. From the

perspective of planning and control, this objective is sometimes insufficient to

provide a good practical solution. In this research we attempt to reformulate the

problem with simultaneous consideration of two objectives; minimizing the total

traveling time and minimizing traveling time of the longest tour. Many heuristic andoptimization approach have been reviewed. A new heuristic is developed and

proposed. The heuristic is an appropriate extension and modification of Clark-

Wright procedure. The heuristic performance was found to be efficient through a

number of tests. Results obtained for the case study illustrate some improvement in

both total cost and workload balancing for each tour.

Introduction Nowadays, commercial banks are trying to change customer behavior from

doing banking transactions through branch channel to electronic channel.

Consequently, there is an increasing number of Automated Telling Machines

(ATM). Two types of ATMs need to be addressed, one of which is the branch ATM,the other being the out-of-branch ATM. The branches will take care of the ATM

located in their respective branches, while the out-of-branch ATMs such as those

located in department store will be taken care of cash centers. Each cash center has

ATMs under its responsibility. Everyday cash center need to set up a route for

carrying money to fill its ATMs. The number of out of branch ATMs is very large

and every day we need to randomly fill up cash. Depending on the current capacity

of each ATM, many alternative decisions can be made. Now the work process

decision is made by operators. Thus, the problem of ATM routing is significant.

In this study, methodology “Re-routing ATMs” is proposed in order to

maximize efficiency measured by two objectives, minimizing the total traveling

time and minimizing traveling time of the longest tour simultaneously. This project

will support the operator in terms of decision-making of appropriate routing, by

solving a sequence of the classical “ Vehicle Routing Problem (VRP)” to achieve

both goals.

VRP sometimes called Vehicle Dispatching or Delivery Problem, like its

well-known cousin, the Traveling Salesman Problem (TSP), is fascinating. It is easy

to describe, but difficult to solve. VRP occurs in many service systems such as

delivery, customer pick-up, repair and maintenance. A fleet of vehicles, each with

fixed capacity, starts at a common depot and returns to the depot after visiting

locations where service is demanded. A further fascinating feature of VRP is that

the basic problem can be extended into an untold number of variations, which do

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occur in the real-world problem. In general, it takes a longer time to find the best

routes when the number of locations is larger.

Generally, the main objective for the VRP is to minimize the total cost or distance of travel. From the perspective of planning and control, this objective is

sometimes insufficient to provide a good practical solution. For instance, if all

vehicles have the same size and capacity, unbalanced workloads may produce

unbalanced performance of the system resulting in excessive delays in the delivery

of cash. The longest tour in the unbalanced workloads may produce the problem in

the delivery delay. Routing the transportation problem by VRP in the objective,

minimizing the total distance, can give a solution that has unbalanced workloads. In

order to solve this problem, we must embrace another objective of minimizing the

distance of longest tour.

In terms of multi-objectives decision making, these two objectives are very

difficult to optimize simultaneously because they are inconsistent in that a goodsolution for one objective may perform badly for the others. The major goal of this

research is to develop an efficient heuristic, which solves the VRP and produces

solution closed to the optimal solution of each individual objective.

Problem StatementIn several physical distribution problems, goods must be delivered using a

fleet of vehicles from central depot to a set of geographically dispersed customers.

The problem is to cluster customers into a route at minimizing the routing cost and

the cost of the maximal tour and satisfies every customer’s demand. In addition,

each vehicle must leave from and return to the central depot within the time limit.

Each customer is visited exactly once by a vehicle, and the total customer demandof each route must not exceed the capacity of the vehicle operating on this route.

ObjectivesThe overall objective of the research is to develop an efficient heuristic,

which solves the Bi-Criteria VRP, with applications to the ATM scheduling

problem.

Significance of this StudyThe cost of the distribution and logistics accounts for a sizable part of the

total operating cost of a company. However, the cost associated with operating

vehicles and crews for delivery purposes form an important component of totaldistribution costs. Small percentage saving in these expenses could result in a large

amount of savings over a number of years. This research will provide a robust

problem solving technique for a real-world, large-scale vehicle routing problem to

reduce operating cost.

ScopeIn this research, we focused on the Vehicle Routing Problems with multiple

homogeneous vehicles, single depot, deterministic demands, capacity restrictions

and multi-objectives. Since time windows, mixed pickups and delivery, etc. are not

applicable in this problem, we do not consider these constraints in this study.

Literature Reviews

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1. Vehicle Routing-Scheduling Problem1.1 Vehicle Routing Problem (VRP): The classical routing problem is

defined on graph G = (V,A) where V = {v1, … ,vn} is a set of vertices and A ={( v1,v j ): i ≠ j, v1, v j ∈ V} is the arc set . Vertex v1 is a depot at which is located a

fleet of m identical vehicles of capacity Q. The remaining vertices represent

customers. Each customer is associated with a nonnegative demand d j to be

collected or delivered by a vehicle. A matrix C = (cij) is defined on A; each edge (v1,

v j) is associated with a distance or travel cost cij.

The VRP is to design a set of m vehicle routes of minimum total cost, each

starting and ending at the depot, such that each vertex of V \ {v1} is visited exactly

once by one vehicle and satisfied some side constraints. A route is a sequence of

locations that a vehicle must visit along with the service it provides. The routing of

vehicles is primarily a spatial problem. It is assumed that no temporal or other

restrictions. Impact the routing decision except for (possibly) maximum routinglength constraints.

Depending on the nature of each application, VRP may possess different

characteristics, which in turn decide the VRP category that problem belongs to in

table 1, based on a similar enumeration by Bodin and Golden (1981), who present

some broad characteristic in which various VRP may differ. The entire table may be

used to provide a quick description of a routing problem. Taking different

combinations of options within various characteristics on the left hand side of a

table 1 result in a large number of possible problem settings.

Table 1 Characteristics of Routing Problem

Characteristics Possible Options

1. Size of available vehicle One vehicle

Multiple vehicle

2. Type of available fleet Homogeneous (only one type of vehicle)

Heterogeneous (multiple types of vehicle)

Special vehicle types

3. Housing of vehicle Single depot

Multiple depots

4. Nature of demand Deterministic (known) demand

Stochastic demand

Partial satisfaction of demand

5. Location of demand At nodes

At arcs

Mixed

6. Underlying network Undirected

Directed

Mixed

Euclidean

7. Vehicle capacity restrictions Imposed (all the same)

Imposed (different route)

Not imposed (unlimited capacity)8. Maximum route times Imposed (same for all routes)

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Imposed (different for each route)

Not imposed

9. Operation Pick ups onlyDrop-offs (delivery) only

Mixed (pickup and delivery)

10. Cost Variable or routing cost

Fixed operating or vehicle cost

Common carrier cost

11. Objective Minimize total routing cost

Minimize number of vehicle required

Minimize utility function based on service or

convenience

Minimize utility function based on customer

priorities

* source: Bodin and Golden (1981)

1.2 Vehicle Scheduling Problem (VSP) The Vehicle scheduling problem

can be thought of as a routing problem with additional constraints related to the time

various activities may be carried out. The routing problem gives special importance

to the spatial characteristics of the activity. In scheduling problem, however, a time

is associated with each activity. Thus, the temporal aspects of vehicle movements

now have to be considered explicitly. As a result, the activities are followed in both

space and time.

The feasibility of an activity is also influenced by both space and timecharacteristics, e.g. a single vehicle could not service two locations with identical

delivery or pickup time. The sequencing of vehicle activities in both space and time

is at the heart of the vehicle scheduling problem (Bodin et al. 1983).

Real-world constraints commonly determine the complexity of the VSP. The

restrictions are:

a) constraint on the length of total time or distance a vehicle may be in-service

before it must return to the depot;

b) the restriction that certain task can only be serviced by certain types of vehicles;

c) the time allowed for vehicles to service at each location (or time windows);

d) the precedence requirement of the service (such as pick-ups should be done

before a delivery);

e) the presence of variety of depots where vehicles may be housed.

1.3 Computation Burden: Important consideration in the formulation and

solution of routing and scheduling problems is the computation burden associated

with various solution techniques for these problems. The nature of the growth of

computation time as a function of problem size is an issue of both theoretical and

practical interest.

Most routing and scheduling problems may be formulated as network

problems, hence they belong to the general class of network optimization problems

(Golden et al. 1981). A measure of the problem size is then available in the numbers

(and/or arcs) of the resulting network. A large part of network and combinatorial problems falls into the so called “NP-hard” class. This class is characterized by the

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property that there does not exist a polynomially bounded algorithm for any

particular problem in the class. It is suggested that the effort required to solve an

“NP-hard” problem increases exponentially with the problem size in the worst case.In the area of routing and scheduling, Lenstra and Kan (1981) provided a

concise overview of complexity results known to date. As expected, most routing

and scheduling problems are NP-hard. Thus, when faced with an NP-hard routing

and scheduling problems, one frequently resorts to approximate heuristic procedures

to obtain near optimal solutions in lieu of seeking optimal solutions.

A heuristic algorithm is a procedure that uses the problem structure in a

mathematical (and usually intuitive) way to provide feasible and near-optimal

solutions. A heuristic is considered to be effective if the solution it provides are

consistently close to the optimal solution. The effectiveness of a heuristic is judged

based on the quality of its solutions and the corresponding computational time

required to produce those solutions.2. Basic Vehicle Routing Problem and Algorithms

2.1 The Traveling Salesman Problem (TSP): The traveling Salesman

problem (TSP) requires the determination of a minimal cost cycle that passes

through each node in the relevant graph exactly once. If costs are symmetric, that is,

if the cost of travelling between two locations does not depend on the direction of

travel, we have symmetric TSP, otherwise we have asymmetric or directed TSP.

The TSP is well known as one of the NP-complete problems. It implies that

a polynomially bounded exact algorithms are commonly used for the solution of

TSPs. Several authors have proposed Branch and Bound (B&B) algorithms based

on the assignment problem (AP) relaxation. In the Carpaneto and Toth (1980) B&B

algorithm, the problem solved at a generic node of the search tree is a modifiedassignment problem in which some Xij variables are fixed at 0 or 1.

Recently, a group of researchers: Bob Bixby (Rice University) and David

Applegate (AT&T Bell Labs), Vasek Chvatal (Rutgers University), and William

Cook (Bellcore) (1993) designed an algorithm based on “ branch and cut

“ technique to determine an optimal route for the TSP of 3038 cities. The

calculations for 3038 cities problem were done using parallel processing on a

network of 50 workstations and required total one and on half of years of computer

time.

The Balas and Christofides (1981) algorithm uses a stronger relaxation than

the AP relaxation: sub-tour elimination constraints are introduced into the objective

function in a Langragean fashion. Using this procedure, Balas and Christopher reported optimal solutions to randomly generated 325-vertex problem in less than

one minute on CDC 7600.

Heuristic approaches for TSP mostly fall into three board classes – tour

construction procedures, tour improvement procedures, and composite procedures

(Bodin et al. 1983). Tour construction procedures generate an approximately

optimal tour from the distance matrix. They consist of Nearest neighbor procedure

(Rosenkrantz et al. 1977), Clarke and Wright Saving (Clarke and Wright 1964),

Insertion procedures (including nearest insertion, cheapest insertion, arbitrary

insertion, quick insertion, greatest angle insertion), Minimal spanning tree approach

(Kim 1975), and Christofides heuristic (1976).

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Tour improvement procedures attempt to find a better tour given an initial

tour. Tour improvement procedure mostly use branch exchange approach. The 2-opt

and 3-opt heuristics were introduced by Lin (1965) and the k-opt procedure (k>=3)was presented by Lin and Kernighan (1973).

Composite procedures construct a starting tour from one of the tour

constructive procedure and then attempt to find a better tour using one or more of

the tour improvement procedures. See, for example , Russell (1995).

The multiple traveling salesman problem (M-TSP) is a generalization of the

TSP that comes closer to accommodating real world problems where there is a need

to account for more than one salesman (vehicle). In M-TSP, M salesman are to visit

all nodes in the given network in such a way that the total distance traveled by all M

salesmen is minimum. An M-TSP can always be converted into an equivalent TSP

by creating M copies of the located at the same position but not connected with one

another. The equivalent TSP formulations of the M-TSP were derivedindependently by Bellmore and Hong (1974), Orloff (1974), Svestka and Huckfeltz

(1973). An overview of TSP algorithms (both exact and heuristic) can be found in

Laporte et al. (1992).

2.2 The single-depot multiple-vehicle node routing problem The single-

depot multiple-vehicle node routing problem (sometime called the vehicle routing

problem – VRP) prescribes a set of delivery routes for vehicle housed at a central

depot, which services all the nodes and minimizes total distance traveled. The

formulations of these problems were stated by Bodin et al. (1977). This problem can

be extended by adding side-constraints such as time windows. Most solution

strategies for the standard VRP can be classified as one of the following approaches:

• Cluster-first route-second procedures, which cluster demand nodes and / or arcsfirst, then design economical routes over each cluster as a second step. Example

of this idea was given by Gillet and Miller (1974).

• Route – first cluster - second procedures, which work in a reverse sequence in

comparison with cluster – first route – second. Golden et al. (1982) provided an

algorithm that typifies this approach for a heterogeneous fleet size VRP.

Bienstack et al. (1993) proved that no heuristic approach can be asymptotically

optimal for the generic VRP.

• Savings or insertion procedures, which build a solution in such a way that at each

step of the procedure, a current configuration that is possibly infeasible. The

alternative configuration is one that yield the largest saving in the term of some

criterion function, such as total cost, or that inserts least expensively a demand

entity in the current configuration into the existing route or routes. Examples of

these procedures can be found in Clarke and Wright (1964) or in Solomon

(1987).

• Improvement or exchange procedures, such as the well-known branch exchange

heuristic which always maintain feasibility and strive towards optimality. Other

improvement procedures were described by Potvin and Rousseau (1995),

including Or-opt exchange method in which one, two, three consecutive node in

a route will be removed and inserted at another location within the same or

another route; k-interchange heuristic in which k link in the current routes are

exchanged for k new links; and 2 – opt procedure which exchanges only twoedges taken from two different routes.

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• Mathematical programming approaches, which include algorithms that are

directly based on a mathematical programming formulation of the underlyingrouting problem. An example of this procedure was given by Fisher and

Jaikumar (1981). Christofides et al. (1981) discussed Lagrangean relaxation

procedures for the routing of vehicle.

• Interactive optimization, which is a general purpose approach in which a high

degree of human interaction is incorporated into the problem solving process.

Some adaptations of this approach to VRPs are presented by Krolak et al. (1970).

• Exact procedures for solving VRPs, which includes specialized branch and

bound (B & B), dynamic programming and cutting plane algorithms. For

instance, Agarwal et al. (1989) presented a set – partitioning – base exact

algorithms to solve VRP. Kolen et al. (1987) described a branch and bound (B &

B) method for the vehicle routing problem with time windows (VRPTW). Other applications of the above algorithms for VRPs were discussed in details by

Christofides et al. (1981) and Laporte (1992).

• Heuristic approaches , e.g. simulated annealing (SA) or tabu search (TS). For

example, Brame and Simchi-Levi (1995) introduced the location-based heuristic

for general routing problem, which is based on formulating the routing problem

as a location problem – commonly called the capacitated concentrator location

problem. This location problem was subsequently solved and the solution was

transforms back into a solution to the routing problem. The method incorporates

many routing features into the model. Thangiah et al. (1996) presented a two-

phase heuristic approach to solve the VRP with backhauls and time windows

constraints. First, the initial solution was obtained by using Solomon’s insertion

heuristic procedure , and was improved through a combination of λ-interchanges

and 2-opt exchanges to find a good solution for the problem Taillard et al. (1997)

introduced a tabu search heuristic for the VRP with soft time windows. The

original problem was converted into the vehicle routing problem with hard time

windows by adding large penalty values, and then an exchange procedure was

used to swap sequences of consecutive customers between two routes. Finally, a

selecting procedure was used to find to best overall solution.

2.3 The multiple–depot multiple-vehicle node routing problem The multiple-

depot multiple-vehicle node routing problem, or Multi-Depot VRP (MDVRP), is a

generalization of the standard VRP in that the fleet of vehicles now must serve Mdepots rather than just one. All other constraints from the standard VRP still apply.

In addition, each vehicle must leave from and return to the same depot. The real-

world examples of this problem include delivery of meals, chemical products,

machines industrial gases, and packaged food.

Since the VRP is NP-hard, the MDVRP is also NP-hard and very difficult to

solve to optimality even for relatively small size instances (Bodin et al. 1983), and

only two exact algorithms were proposed for the MDVRP (Renaud et al. 1996). The

first, which was presented by Laporte et al. (1984), formulated symmetric problems

as integer linear programs containing degree constraints, sub-tour elimination

constraints, chain barring constraints (to prevent chains of cities linking two

different depots) and integrality constraints. The problems were solved by a branchand bound algorithm that initially relaxes the last three types of constraints. Optimal

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solutions were reported for problems containing up to 50 customers and eight

depots. The other which was proposed by Laporte et al. (1988) for asymmetric

MDVRPs, first transform into an equivalent constrained assignment problem.Optimal solutions were then found by a branch and bound algorithm in which the

sub-problems . Results were reported for instances containing up to 80 cities and

three depots.

Most existing algorithms for the MDVRP are heuristics. One of the first

method is due to Tillman (1969) and used the Clarke and Wright saving criterion.

The algorithm first assigns each customer to its nearest depot and constructs back

and fourth routes between depots and the customers. They are gradually merged into

larger routes using a saving criterion that takes into account the presence of several

depots.

Another heuristic based on Clarke and Wright exchange process was

proposed by Sumicharast and Markham (1995). This method was considered to dealwith a specific MDVRP in which the constraints of delivery of different raw

materials were included. the results obtained from this approach were comparable

with those obtain from this approach were comparable with those obtained using

LINDO package.

Gillet and Johnson (1976) proposed an assignment sweep approach which is

an extension of the sweep heuristic, and solved the multiple depot problem in two

stages . Firstly, customers were assigned to depots to from compact and disjoint

clusters; secondly, independent single-depot VRPs were solved using the sweep

heuristic.

Raft (1982) presented a solution technique that can handle objectives other

than route length minimization. This heuristic is a modular algorithm whichdecomposes the problem into smaller sub-problems. The algorithm starts with a

route assignment phase. After having estimated the number of vehicles needed, the

algorithm constructs of customers, each assigned to one vehicle. These clusters are

not assigned to any depot and are constructed to provide a small expected length. In

the next phase, each route is assigned to a depot , and then a 2-opt exchange

procedure is applied to each route.

Chao et al. (1993) described a multi-phase heuristic. Customers were first

assigned to their closest depots. A VRP was then solved for each depot using the

modified savings algorithm of Golden et al. (1977). This solution was then

improved by moving customers to different depots.

Renauld et al. (1996) applied a tabu search heuristic to MDVRP. Their algorithm contain two parts: (1) construction of an initial solution by assigning

customers to its nearest depots and then using a heuristic to find the best route

selection; and (2) using tabu search to improve the solution.

2.4 The Fleet Size and Mix Problem The question commonly faced in

distribution management is what is the best composition of the fleet in order to

respond to customer demand at minimal cost. In real life, the appropriate fleet need

not be homogeneous and a good vehicle fleet mix is likely to yield better results. In

determining the vehicle fleet composition, the fixed cost of vehicles has to be one of

the major factors since it accounts for approximately 80% of the total cost

associated with the vehicle (Eilon 1977).

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The combined vehicle routing and vehicle fleeting composition problem can

be formulated as a mixed integer linear program (Salhi and Rand 1993). Other

formulations can be found in Golden et al. (1984).

There are only few algorithms that have been published for this vehicle mix

problem. Golden et al. (1982) were among the first to address this problem. They

presented two heuristics based on the adaptation of existing routing algorithms. One

is the saving technique of Clarke and Wright (1964) and the other is the giant tour

algorithm developed by Gillett and Miller (1974).

Ferland and Michelon (1988) formulated the vehicle mix problem and

showed that the exact methods and heuristics develop for the vehicle scheduling

problem with time windows and with a single type of vehicle can be extended to the

multiple vehicle type problem. Three heuristic based on discrete approximation,

assignment method and matching method respectively were formulated. Two exactmethods used a column generation technique and a time window constraint

relaxation were also proposed.

An efficient heuristic for determining the composition of a vehicle fleet was

proposed by Salhi and Rand (1993). The heuristic base on a perturbation procedure

applied within the route in order to improve the vehicle utilization of the whole

fleet. The idea here is to reallocate some customers to other routes to allow a given

route to be better split or combine route to be served by a cheaper vehicle.

2.5 Multi-Criteria Vehicle Routing Problem Multi-Criteria Vehicle

Routing Problem is VRP with multi-objectives, these objectives are very difficult to

optimize simultaneously because they are inconsistent in that a good solution

against one objective may perform badly for the others.Dean and White (1975) studied balancing workloads in machine scheduling

with the approach of a modified Little’s algorithm. The procedure is to continue

searching until the best solution with the best balance is found (measured as

minimum range). Computational results of some small problems were reported.

Blair and Vasquez (1984) proposed a heuristic which is based on a node

exchanging procedure which transfer a node in the maximal tour to the remaining

tours in order to minimize the longest tour and the total distance. An application of

this method was made to solve the VRP in a Flexible Material Handing System

(FMHS). They assumed that all vehicles carried a single unit load at the time they

passed through a sequence of jobs, so the VRP becomes the MTSP. Later, Blair,

Charnsethikul and Vasquez (1987) developed an alternative stopping rule and adifferent rule for node selection in the maximal tour which is transferred to the

remaining tour. The algorithm was tested at three levels of 15 nodes and 3 vehicles,

35 nodes and 4 vehicles, and 5o nodes and 5 vehicles. One hundred replications

were generated at each level.

Husban (1985) formulated the problem of minimizing the maximal tour of

the MTSP as an integer linear program and solved it using a branch and bound

technique similar to that of Dean and White (1975) for small problems. He also

formulated the VRP in the FMHS as a transportation problem by an arc covering

approach. Two new heuristic methods were developed. The first heuristic is based

on a node covering approach, while the latter heuristic is based on an arc covering

method. He reported some numerical experience using the same lower bound

developed in Blair and Vasquez (1984) to measure the performance of both heurtics.

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2.6 Stochastic Vehicle Routing: The Stochastic Vehicle Routing Problem

(SVRP) is a generalization of the deterministic VRP. The following modification to

the VRP are required:• Customer demand is a random variable with a known probability distribution.

• Route must be design before the actual demand becomes know.

• The objective is to minimize expected travel distance.

Since they combine the characteristics of stochastic and integer programs,

SVRPs are often regarded as computationally intractable (Gendreau et al. 1996) .

SVRPs are usually modeled as mixed and pure integer stochastic programs, or as

Markov decision processes. All known exact algorithms belong to the first category.

Exact algorithms for a number of SVRPs have been proposed by Laporte et

al. (1989, 1992, 1994) and Gendreau et al. (1995).

Most algorithms proposed for SVRPs are heuristic, typically adapted from

methods originally designed for the deterministic case. Gendreau et al. (1996) proposed a tabu search heuristic for the SVRPs addressing both stochastic demands

and stochastic customers. Other illustrations can be found in Laporte et al. (1989),

or in Bertsimas (1992) and Gendreau et al. (1996) for more complicated case.

2.7 Vehicle Routing with Time Windows The Vehicle Routing Problem

with time Windows (VRPTW) is VRP with the added complexity of allowable

delivery times, or time window, stemming from the fact that some customers

impose service deadlines and earliest service time constraints. In these problems, the

spatial aspect of routing is blended with the temporal aspect of scheduling, which

must be performed to ensure the satisfaction of the time window constraints.

Solomon (1986) pointed out that VRPTW class is fundamentally more difficult than

the VRP.Solomon (1987, 1986) designed and analyzed a variety of route construction

heuristics for the VRPTW. The result of the extensive computational study reported

in Solomon (1987) indicate that a sequential time-space insertion algorithm (a

generalization of the Mole and Jameson (1976) approach to the VRP) has proven to

be very successful in a number of important VRPTW environments.

Several efficient implementations of branch exchange solution improvement

procedures for VRPTW were developed in Solomon et al. (1988). Furthermore, the

Or-opt procedure (Or, 1976) showed the obtainability of improved solutions of

equivalent quality to those produced by 3-opt whist requiring significantly less

execution time . Taillard et al. (1997) proposed a new genetic-tabu search heuristic

for the VRP with soft time windows called CROSS exchange. CROSS exchange

can be seen as a general form of the Or-opt and 2-opt (Potvin and Rousseau, 1995)

procedure.

Koskosdisidis et al. (1992) presented another methodological approach for

the time constrained VRP, based on the Generalized Assignment heuristic proposed

by Fisher and Jaikumar (1981). The heuristic starts with a mixed integer

programming formulation which treats the time window constraints, and develops

an optimization based heuristic algorithm for the resulting VRP with soft time

window constraints.

Ferland and Fortin (1989) introduced a heuristic approach for the vehicle

scheduling problem with sliding time windows. The technique was first developed

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for the problem with time windows and then extended to the case with sliding time

windows.

While heuristics have been found to be very effective and efficient in solvinga wide range of practical size VRPTW, optimal approaches have considerably

lagged behind (Solomon and Desrosier 1988). Kolen et al. (1987) extended the q-

path relaxation algorithm of Christofides et al. (1981) to the problem with time

windows. Jornsten et al. (1986) proposed a Langragean relaxation for the

computation of a lower bound for the VRPTW.

MethodologyGenerally, the main objective for the VRP is to minimize the total cost or

distance of travel. Sometimes the solution in above objective is good in theoretical

but may not be good in practical. For example if all vehicle have the same size andcapacity, the solution that has unbalanced workloads may produce unbalanced

performance of the system resulting in excessive delays in delivery of material.

In this research, we attempt to solve this kind of delay by formulating

mathematical model with two objectives. First, we minimize the total traveling time.

Second we minimize the traveling time of longest tour. The mathematical model formulation is as following.

Mathematical model formulation NotationThe following notation are used for the model formulation:

N = number of locations to be visitedM = number of tours or salesmen or vehicle

Q = vehicle capacity

T(L) = traveling distance for tour L

T = total distance traveled in the maximal tour

Cij = traveling distance from location i to location j

Di = demand at location i

Ui = a vector of positive integer values which gives a set of subtour

elimination constraints in constraint ( )

Xijk = { The formulationThe basic mathematical formulation for the Bi-Criteria VRP is an integer

program with two objectives.

Z1 = Min T (1)

N + M N + M

1 if link i,j is concluded in tour k,

0 otherwise

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Z2 = Min∑ ∑ Cij Xijk k = 1,…,M

(2)i = 1 j = 1

Subject to

N + M N + M

T ≥ ∑ ∑ Cij Xijk k = 1,…,M

(3)i = 1 j = 1

where

N + M N + M

T(L) = ∑ ∑ Cij Xijl l = 1,…,Mi = 1 j = 1

N + M M

∑ ∑ Xijk = 1 j = 1,…,N+M

(4)i = 1 k = 1

N + M M

∑ ∑ Xijk = 1 i = 1,…,N+M

(5)i = 1 k = 1

N + M N + M

∑ Xihk - ∑ Xhjk = 0 h = 1,…,N+M, k = 1,…,M

(6)i = 1 j = 1

N + M N + M

∑ ∑ Di Xijk ≤ Q k = 1,…,M

(7)i = 1 j = 1

M

Ui - U j + (M+N)∑ Xijk ≤ M + N – 1, i,j = 1,…,N+M, i ≠ j

(8)k = 1

In the above formulation

• Equation (1) shows the objective function of minimizing the maximal tour.

• Equation (2) shows the objective function of minimizing the total distance.

• Constraint (3) ensures that T represents the maximal tour.

• Constraints (4) and (5) ensure that each point is visited by one and only one

vehicle.

• Constraint (6) represents route continuity.

•

Constraint (7) implies that the total load of each vehicle does not exceed thevehicle capacity.

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• Constraint (8) represents subtour elimination constraints.

As mention, the combinatorial nature and complexity of multiple objectives in

the mathematical model proposed in the previous chapter. A goal program may beformulated and solved by integer programming code, but the running time is costly

and there are some problems of roundoff error and propagation when the problem is

large. This approach is not appropriate in the case of this application.

For large vehicle routing and scheduling problem (VRP), the most efficient

way to obtain the solution is using approximation method (commonly called

heuristic). There are several heuristics for VRP deploying different techniques. Most

heuristics, however, are the adaptation of some basic well-known procedures for

each special scenario. In this research, we consider the route with “saving heuristic”,

proposed by Clarke and Wright in 1964.

Saving AlgorithmClarke and Wright (1964): suggested a simple method for optimum routing

of a fleet of trucks of varying capacities used for delivery from a central depot to a

large number of delivery points. Clarke and Wright have modified the original

method by Dantzig and Ramser. The merchandise is homogeneous with respect to

the unit of capacity. The shortest route between every two points in the system is

given. It is desired to allocate loads to trucks in such manner that all the

merchandise are assigned and the total mileage covered is at minimum. The

procedure given is simple but effective in producing a near-optimal solution. The

heuristic is called the “saving algorithm” which is “exchange” procedure in the

sense that each step one node of tour is exchanged for the better set. First of all,

each tour simply connects depot and 1 customer. Then it combines any 2 customersinto 1 route if total demand does not exceed vehicle capacity. The saving cost due to

combination is calculated and the largest is selected. Then extent the route by

combining with each other customer when saving cost due to combine is calculated

and the largest is selected, repeating this step until demand exceeds vehicle capacity.

In the method of Clake and Wright, saving of combining 2 customers i and j

into one route is calculated as:

Sij = dio + doj - dij

Where dij denotes travel cost from customer i to j. Customer “0” stands for

the depot. An illustration of the calculated is in Figure (1).

dij

i j

d j0

i

di0

j

d0j

d j0 d0id0i

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Figure 1 Illustration of the saving calculated

From the previous procedure, the algorithm can be described as follows.

Step 1. If P (set of unserved customers) is empty then exit. Otherwise create

a new route with starting and ending point as the depot. Select the 2 customers,

largest cost saving from combination, into 1 route if total demand does not exceedvehicle capacity.

Step 2. If P is empty then exit. Otherwise, select one customer from P and

inserted that customer into the current route. When saving cost due to combine is

calculated and the largest is selected, total demand does not exceed vehicle capacity.

If no customer is selection, repeat step 1.

The algorithm for Saving Heuristic is presented in the Figure (2). In this

algorithm it satisfy only minimize total distance. So we produce two heuristic,

modified the saving heuristic, with satisfy the objective minimize the maximal tour.

1) Arc Deleting Procedure (ADP)

2) Node Exchanging Heuristic (NEH)

These heuristics attempt to reduce the longest tour and maintain minimization of

the total distance.

The Proposed Heuristic MethodsAn Arc Deleting Procedure (ADP) The ADP is an appropriate modification to Saving algorithm. For example,

consider a problem with six nodes two vehicles. Suppose that the solution from the

Saving algorithm is 0-1-2-3-4-0, and 0-5-6-0 where 0 represent the depot. This

solution is good in term of minimizing total distance but may not be good in

practical solution.To satisfy balancing workloads, a heuristic technique is applied to the

problem of minimum total distance solved by the Saving algorithm. This heuristic is

called the “arc deleting procedure”. The main goal is to reduce the maximal tour.

The procedure deletes each link (i,j) where (i,j) is a sequence of node in the

maximal tour, by assigning Cij (cost of traveling from node i to node j) equal to

infinity, and the VRP corresponding to each deletion of link (i,j) is solved by Saving

algorithm. Suppose there are k arcs contained in the maximal tour, thus producing

the new k solutions. We select the best improvement to continue searching in the

same way until no improvement n time.

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P (set of unserved

customers) << 1

No

YesEnd

Create a new route.

Select two customer that have

max saving between them.

demand in route <vehicle capacity

No

Yes

P empty ?Yes

No

Select the customer k

from L into route that

have max saving.

demand in route <

vehicle capacity

L (Set of customer which

can be linked in tour)

empty?

Add customer k in to route

and remove it from P

Add customer i and j in to route

and remove its from P

Yes

No

Yes

No

Start

Remove customer k

from L

Step 1

Step 2

Figure 2 Saving Heuristic

From the previous procedure, the algorithm can be described as follows.

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Step 1. Solve the VRP by Saving algorithm.

Step 2. From the solution in step 1 or 3, select the produced maximal tour

and start to delete each link (i,j) in that tour and resolve the VRP by Savingalgorithm corresponding to each deletion of (i,j).

Step 3. If there is no improvement in that maximal tour from step 2 n times,

stop and print the best solution, Otherwise, select the best solution and go to step 2.

The algorithm for Arc Deleting Procedure is presented in the Figure (3).

Node Exchanging Heuristic (NEH) To satisfy balancing workloads, our objective function is to minimize the

longest tour instead of the total distance. We applied a heuristic technique which, is

called the “node exchanging heuristic”. The procedure is to move each node in the

maximal tour to the set of the nodes in the other tours, and the VRP corresponding

each moving of node is solved by Saving algorithm twice. First solving containsnode in maximal tour and second solving contains all nodes in other tours and node,

which move from maximal tour. Suppose there are k nodes contained in the

maximal tour, thus producing the new k solutions. We select the best improvement

to continue searching in the same way until no improvement n times.

Step 1. Solve the VRP by Saving algorithm.

Step 2. From the solution in step 1 or 3, select the produced maximal tour

and start to move each node in that tour and resolve the VRP by Saving algorithm

corresponding to each move node.

Step 3. If there is no improvement in that maximal tour from step 2 n times,

stop and print the best solution, Otherwise, select the best solution and go to step 2.

The algorithm for Node Exchanging Heuristic is presented in the Figure (4).

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Start

Solve the VRP by

Saving heuristic

Set the solution from

Saving heuristic = Solution 1

Set Solution 1 = Solution 2

and set counter =0

counter > n ?

L ( set of arc between

customers of longest tour in

Solution 2) empty ?

Select one arc (arc i) from L

then delete it and delete all

arc in M

The solution better than

Solution 3 or frist

running?

Set solution from running


and set arc i = arc 1

Set Solution 3 =

Solution 2 and set

counter = counter +1

Solution 3 better

than Solution 1 ?

Set Solution 3 =

Solution 1

Finalize set

of routes

End

No

Yes

Yes

No

Yes

No

Yes

No

Run Saving heuristic and

remove arc i from L

Record arc 1 in set M

Figure 3 Arc Deleting Procedure (ADP)

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Start

Solve the VRP by

Saving heuristic

Set the solution from

running Saving heuristic =

Solution 1

Set Solution 1 = Solution 2

and set counter =0

counter > 5 ?

L ( set of customers of longest

tour in Solution 2) empty ?

Select one customer (customer i) from

L delete it from L and N (aset of

customers of longest tour in Solution2) and then record it in M (set of

customers in other tours)

The solution better than

Solution 3 or frist

running?

Set solution from running


and set arc i = arc 1

Set Solution 3 =

Solution 2 and set

counter = counter +1

Solution 3 better

than Solution 1 ?

Set Solution 3 =

Solution 1

Finalize setof routes

End

No

Yes

Yes

NoYes

No

Yes

No

Run Saving heuristic in set N

and then in set M and

remove customer i from M

Combine two solutions from

running Saving heuristic and

record customer i in set N

Figure 4 Node Exchanging Heuristic (NEH)

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Test results

In order to evaluate the quality of solutions obtained from the proposedheuristic, the Visual Basic program, was tested on six cash centers and three

instances were randomly generated from the computer program. The details of test

cases are shown in Table 2.

Table 2 Detail of test cases

No. Source Number of the demand node Vehicle capacity No. of Vehicles

1 Cash center 10 22 30 5



4 Cash center 5 22 30 55 Cash center 6 26 30 5


7 Random 25 30 5

8 Random 75 30 5

9 Random 100 30 5

For testing of six cash centers problem, we used the travel time instance

instead of the travel distance because it does contain some traffic factors. Next, the

ADP,NEH and saving heuristic of Clark & Wright are coded and tested using

Visual-Basic. In all cases, the running time is vary depending upon the CPU speedof computers. For the largest problem (100 demand nodes), the problem can be

solved within one hour using CPU Pentium III. The result of the test can be

summarized as shown in the following table 3.

Table 3 The best results of all of test cases.

Total traveling time Traveling time of longest tour No. Source

Number of demand

node ADP NEHSaving

heuristicADP NEH

Savingheuristic

1 Cash center 10 22 455 470 490 195 170 245

2 Cash center 15 28 433 433 452 154 135 182

3 Cash center 11 21 311 318 343 150 143 223

4 Cash center 5 22 305 267 346 139 113 168

5 Cash center 6 26 350 375 375 180 160 205

6 Cash center 13 21 171 172 177 66 69 77

7 Random 25 456 472 486 192 184 242

8 Random 75 1431 1458 1482 182 178 195

9 Random 100 1895 1904 1946 178 172 196

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Comparing of ADP and NEH, from Table 3 if we reach in criteria of

minimizing total traveling time, the solutions from ADP are almost better than the

solutions from NEH. On the contrary, if we approach in criteria of minimizingtraveling time of longest tour, NEH can provide the solution better than those

obtained from ADP.

DiscussionsBased upon our 9-problem test, in general, the proposed heuristic

outperforms saving heuristic in terms of the total distance and longest tour. We

discuss strengths and weaknesses of all algorithms in all perspectives. Generally the results from the proposed methods, the ADP and NEH, are

more effectively than the existing method, saving heuristic, in terms of total

traveling time and traveling time of longest tour. The reason is the ADP and NEH

heuristics finds the initial solution by using the saving method then repeatedly tofind the better solution. Due to the objective of minimizing the longest tour is added

into the proposed methods, ADP and NEH provide a good practical solution by

solving the delay problem in the delivery. Also the ADP and NEH can be adapted to

handle different priority using weighting factors assignment between two criteria.

In the real world, to use the results from the proposed method, we have to

run all of the weight ratios shown previously. The operators should select a suitable

result depending upon the situation. These situations have two case studies as

following. First, the depot has only one vehicle so that the objective of the

minimizing traveling time of longest tour is useless. The result, which lowest total

traveling time, should be selected. Second, the depot has fleet vehicles so that the

results, which have traveling time of the longest tour exceed the time limit, should

not be selected. To select the remaining results, if the total cost of travel is

concerned. The result, which has lowest total traveling time, should be selected. In

case of the workload balancing is considered. The result, which has lowest traveling

time of the longest tour, should be selected.

From the result, it cannot be concluded which heuristic between ADP and

NEH is the better methods. Each problem cannot enumerate which heuristic is best

before we test all of cases. Consequently if we use the concept of hybrid, using the

result from one method for initial solution in another method, among these methods

it might give a better solution. More research following this approach should be

further conducted.In general, ADP spent more computational time than NEH. But to compare

with saving heuristic both methods spent much computational time. So that in

practical in case of the method providing the best solution but spending much time

for computation may not be a good alternative. Accordingly to choose the suitable

method, computation time should be concerned.

ConclusionsThe main purpose of this study was to develop an efficient procedure for

routing problem, which can be daily used by ATM scheduling. We have reviewed

many existing algorithms for the Vehicle Routing Problem. The best existing

heuristic algorithms are studied and tested extensively. The best existing heuristicsis saving algorithm by Clarke and Wright (1964). This heuristic it gives the

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minimum total distance but in this ATM scheduling problem we can not solve the

problem with this single criteria. Thus, we have to add one more the criteria, which

is minimize longest tour.We have devised a new heuristics that we call Arc Deleting Procedure

(ADP) and Node exchanging procedure (NEH). The ADP and NEH are an

appropriate modification to Saving algorithm. These heuristics tried to reformulate

the problem with simultaneous consideration of two objectives; minimizing the total

traveling time and minimizing traveling time of the longest tour. The result of nine–

test problems shows that the ADP and NEH are more effectively than the existing

method, Saving heuristic, in term of total traveling time and traveling time of

longest tour. Due to the objective of minimize the longest tour is added in to

propose methods, therefore ADP and NEH provide a good practical solution by

solve the problem delays in the delivery.

Recommendation for further studySeveral aspects of routing problem for the ATM scheduling that remained

unsolved in this study will form interesting topics for further study. The following

recommendations are made for further studies:

It is observed that the travel time between any two points in the distribution

system is an importance parameter, which considerably affects the service time of

each job. More accurate estimation of travel time leads to more precise solution

obtained from the model. In this study, the travel time between any two points in the

deterministic value. For more realistic, the travel time between any two points

should be in term of stochastic then the traveler time could be any value in the

distribution defined.In this research we focus on homogeneous vehicle (only one type of vehicle)

but in the real world, some vehicle routing problems may have multiple types of

vehicle, which have different capacities. For further study, we should apply the

proposed method to multiple types of vehicle.

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