An Efficient Implementation Of Genetic Algorithms For ... - IEEE Xplore

AN EFFICIENT IMPLEMENTATION OF GENETIC ALGORITHMS FOR CONSTRAINED VEHICLE ROUTING PROBLEM

MINEA FILIPEC

DAVOR SKRLEC

SLAVKO KRAJCAR

University of Zagreb Faculty of Electrical Engineering and Computing Department of Power Systems Unska 3, I0000 Zagreb, Croatia email: caddin@zvne.€er.hr, httr,:!iwww.fer.~iricaddin

forecasting and also methods for geographical data extraction from GIS database (Geographical Information System), and the other two authors have worked together on the development of Genetic Algorithm applied to distribution network expansion planning. The present distribution system network arrangement is a natural starting point for planning purposes. The various types of equipment, their location, electrical loading and mechanical conditions are all factors to be taken into account when considering future developments. Therefore the GIS is the source of sufficiently accurate data, the topological information of distribution network elements as well as possible routes of new feeders in keeping with urban zoning plans, ecological and esthetic constraints.

ABSTRACT We propose a genetic algorithm based heuristic for solving the problem of open loop distribution network planning. The goal of power distribution system planning is to satisjj the growth and changing system load demand during the planning period and within operational constraints, with minimal costs. Although the algorithm was developed for spec$c real world problem, method is quite general and can be encountered in many planning contexts that can be correlated with well known Capacitated Vehicle Routing problem (CVRP). For the CVRPproblem, the influences of the respective control parameters were examined. Also the issues regarding the usage of different selection parameter have been examined in this paper in order to observe their impact on the optimization procedure. The results of experiments testing the solution procedures are reported in the paper.

Although the algorithm was developed for specific real world problem method is quite general and can be encountered in many planning contexts that can be correlated with well known Capacitated Vehicle Routing problem (CVRP) if the following equivalencies are presumed: the depots correspond to high-voltage substations, and consumers to the middlevoltage substations, the vehicle’s capacity constraint represents the maximum load that can be transferred through a given set of existing cables, and the vehicle’s reach constraint represents the two energetic constraints - the allowed voltage drop that is function of cables length, and the maximum number of middle voltage substations per one loop. While the travel costs are equivalent to the total cost of laid cables. This is the reason why the method is described in the paper in totally general manner as if used for solving the CVRP problem.

KEYWORDS: open loop distribution network planning, Capacitated Vehicle Routing Problem (CVRP), Genetic Algorithm (GA)

1. INTRODUCTION This paper presents an application of new genetic algorithm approach to long range distribution network planning of open loop arrangement. The process involves the selection of primary feeder configuration for given number of MV substations of a given location and capacity. The cost of installation along with the cost of energy losses should be minimal while maintaining the reliability of supply, predefined structure of the network (open-loop), and at the same time not violating thermal limit and maximal voltage drop constraints. The work reported comes as convergence of several years of research effort. One of the authors has originated the development of models and algorithms for small area load

0-7803-4778-1/98 $10.00 0 1998 IEEE

For many organizations distribution costs are significant portion of total expenditure. Finding good solutions to distribution problems is important. Unfortunately, many of these problems are difficult to solve. These two factors have lead to an enormous amount of work being published by operations research community on these problems. In special issue of joumal Interfaces, Golden and Wong [ 141 introduced nine papers describing real world applications in which

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to distribution problems faced by companies. This is just the tip some objective measure is optimized. Three objectives are frequently considered. They are:

operations research techniques have been successfully applied of the iceberg. There have been thousands of papers written about these problems, which, in the literature known as computers can significantly reduce distribution costs. For example, Fisher et al. [14] report that a computerized vehicle routing application was able to reduce company's distribution costs by over 15%. Sutcliffe and Board [16 ]analyzed the results of 36 empirical studies that reported some quantitative routing and scheduling problems. The use of mathematical modeling techniques and measure of the benefits of mathematical modeling over the actual previously used solutions. They found that for average sized problems savings of between 7% and 37% could be realized.

minimize the total distance traveled (or time taken) by all vehicles, minimize the number of vehicles, and for that minimal number minimize the total distance traveled, and minimize a combination of vehicle costs and distance traveled. Although the algorithm we use can be easily modified to use the third objective, what would better suit the model of most real life problems, for the results presented in this paper the first objective was used.

This has led researchers to develop heuristics that find approximate solutions in a reasonable amount of computer time. Surveys on classification and applications of the VRP can be found in Ball et al. [l], Fisher [5], Desrosiers et al. [4], Bodin [2], Osman [9], Laporte [7], Golden [6], Christofides [3], Taillard [Ill, Skrlec [lo]. The VRP is NP-hard problem. This is the reason why VRPs have first been tackled using heuristic approaches (Bodin et al. [2], Christofides [3], Osman [9] ). During the past fifteen years, exact algorithms have also been developed to solve VRPs of reasonable size to optimality. Some exact approaches are presented in the surveys of Christofides [3] and Laporte [7].

A precise mathematical programming based formulation of the VRP is given in [2]. The formulation recast so that the depot is node 0 and customers are numbered from 1 to n is as follows: n

n

K

Although these methods can provide significant savings there is still room for further improvements. The methods are usually heuristic in nature, only providing approximate solutions. However, the field is an area of active research, with faster exact algorithms and better quality heuristics constantly being developed. We propose a genetic algorithm based heuristic capable of solving the VRP, Skrlec [12]. To overcome the difficulties in handling the vehicle constraints, we introduce a technique to embed the maximum capacity and reach constraints in the decimal coding of GA. The paper demonstrates the successful use of GA in solving the VRP.

2. DESCRIPTION OF THE PROBLEM n

Consider the case where n customers each demand a certain quantity, q,, of goods, from a depot. The goods are to be delivered by a fleet of homogeneous vehicles. Each vehicle starts at the depot, delivers goods to a subset of the customers, fully satisfymg the demand of each customer it visits then returns to the depot. The route that each vehicle is assigned must satisfy a number of constraints. For instance, the quantity of goods delivered must not exceed the capacity of the vehicle, and there may be a limit on the length of the route. The Vehicle Routing Problem (VRP) consists of deciding which vehicles should deliver to which customers, and in what order, such that all customer demands are met, no constraints are violated, and

Ex:, 11

v=l,..K

(2.7)

Ex,;r l

v=l,..K

(2.8)

x,; E {OJj

for all I, j, k.

(2.9)

]=1

n

i=l

Where: n=number of customers (depot is node 0) K=number of vehicles

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BEGIN -

cl;=cost of travelling from node i to nodej

1 build the edge roble

w),for every X

E

(0,.n)

2 put all Cnodes in toble ofnot-selected ones

x& =I, if vehicle travels from node i to nodej

3 route number = I

0, otherwise

4N=n-l 5.WHILE (N>O)DO 6,11,12,13,21

qi =order size of customer i

6.IF ( edge roble ( 0 ) = empty ) THEN

Qv =capacity of vehicle v

7 select next C node from edge roble ( 0 )

dvY=travel distance from node i to nodej

ELSE 8 IF (roble closest (O)= m p y ) THEN

D, =maxirxim allowed route distance for vehicle v Constraints 2.2 and 2.3 ensure that each customer is serviced exactly once. Route continuity is enforced by constraint 2.4, where if a vehicle arrives at a delivery point it must also leave that point. Constraint 2.5 is the vehicle capacity constraint; each route must not service more than the capacity of a vehicle. Similarly constraint 2.6 limits the maximum route length. This can be modified to limit route cost or else both route length and cost can be limited by including an additional constraint. Constraints 2.7 and 2.8 ensure that each vehicle is used no more than once.

9 select randomly next C node from roble closerr ( 0 )

ELSE 10 select randomly next C node from roble of nor-selected ones

I 1 clear the selected Cnode from all robles 12. N = N - 1

13 WHILE (vehicles load