Resumen del marco teórico

li k

=

{

max[h

- dik,l , Tik+w]

if k + w ::;

n

+ 1 ,

II -

dik,l

if k + w >

n

+ 1 ,

where w is a user-defined parameter which may be altered in order to vary the time window width. Therefore the time window of a customer depends upon the times at which the customers w positions before and after it in the tour are serviced. Precedence constraints were also generated so that the initial tour created is feasible in the presence of these given side constraints. The authors claimed that the test problems created by this method for the TSPTW have an initial solution, i.e., the randomly generated tour, that is on average more than 80% longer than the optimal solution and that the time windows are much wider than those found by previous

168 Problem Generation Methods

methods. Thus the initial solution is not a good bound on the optimal solution and there is a larger number of feasible solutions available, so these problems are more difficult to solve.

The Vehicle Routing Problem

The major difference between the generation of test problems for the TSPTW and for the Vehicle Routing Problem with Time Windows (VRPTW), is that for the TSPTW all customers must be serviced by the same vehicle, whereas with the VRPTW the fleet size is usually a decision variable. Therefore a feasible solution to the VRPTW will always exist, by using as many vehicles as are required to service all the customers. The feasibility concerns of the TSPTW aren't relevant to the VRPTW, so different generation methods may be used.

For the Vehicle Routing Problem with Time Windows (VRPTW), the standard problem set that has been used in many papers was created by Solomon [166] . He generated the customer locations in three ways: all customers randomly dis­ tributed, all customers distributed amongst clusters and semiclustered problems (in which customers are either clustered or randomly distributed). For each of these distributions of locations two problem sets were created, one with short time horizon and with service vehicles having low capacity, and the other with a longer horizon and higher capacities. These different time and capacity constraints allow more or fewer customers to be able to be included in a vehicle's route. Therefore there are six problem sets created, each with different location characteristics and different numbers of customers that can be included in a route.

For each of the problem sets, different instances were created by removing the time window constraint from certain customers, thereby altering the number of customers whose time window is active. With the random and semiclustered prob­ lem sets, the different instances were created by generating the width of the time window (either fixed width for all customers or different widths for the customers) and altering the proportion of customers whose time windows are active. For clus­ tered problems, a 3-opt heuristic was used to generate a route through the points of the cluster, with the middle of the customer's time window occurring at the time the customer is serviced in this route. The proportion of customers with time windows and the width of the windows were varied to create the different instances. The use of the 3-opt heuristic to generate the centre of the time windows creates

a near-optimal solution within each cluster, which can be obtained by replicating the 3-opt method.

The two clustered data sets are shown in Figure 7. 1 . The first clustered data set, Cl , clearly contains 10 clusters. Each cluster has been generated in such a manner that all the customers within the cluster may be serviced by the same vehicle. Therefore the time windows within the clusters allow a vehicle to serve all the customers with no waiting time required, and the total load for each cluster is less than, although close to, the capacity of the vehicle. This is designed so that there is a known benchmark solution, with as many vehicles as there are clusters, so any reasonable solution must attain this standard. This problem is not very flexible in terms of the number and "character" of feasible solutions allowed, as all reasonable solutions require each customer within a cluster to be serviced by the same vehicle. The different solutions are therefore created through varying the order of serving the relatively small number of customers in each cluster. The second clustered problem, C2, includes many of the same points, although some points are moved, and the resulting problem has the clusters less well-defined. In order to place the time windows, a solution route through the clusters had to be created, so the points were partitioned into three clusters, and, thus, all reasonable solutions are required to only use three vehicles. This problem is more flexible than the first clustered problem, as the time and capacity constraints aren't as restrictive in preventing a vehicle from being able to service customers from more than one of the clusters present, so the solution space is effectively larger.

An interesting feature of the problem instances is the dominance of the service times over the travel times. For the clustered set, Cl , Desrochers, Desrosiers and Solomon [32] found the optimal solution for the problem for which all time windows are active. This solution required a total time of 9827.3 units; since the time of servicing each of the 1 00 customers is 90 units, the optimal solution involves 827.3 units of travelling time and 9000 units of time spent servicing the customers. Therefore, with the time windows being relatively short compared with the service times, the allowed flexibility in the routing is reduced.

For the Vehicle Routing Problem with Soft Time Windows (VRPSTW), Kosko­ sidis, Powell and Solomon [99] randomly generated five problem sets. The impor­ tant characteristics they identified include the number of customers to be serviced, the size of the area in which the customers may be located, the size of the loads to

110 Problem Generation Methods C1 Data Set 90 90 ** * * r 80 * * * - 80 *** * 70

�

70 60 *A 60

m

* * ** 50 _,() 50 ** 4 - "'* 4 * *

_..

"*** .- * 30 *** *** 30 * 20 20 ,.. 1 0 * 1 0 .-* 0 0 0 20 40 60 80 1 00 0 * * ** C2 Data Set ** * * *** * t * * * ** * * * * **.*. ,() * * * * ** * * ** ** _{* * *} * * ** * * - * * ** * * * * * � "*** ,* * ** * * ** * ** ** .-* 20 40 60 80

Figure 7. 1 : Solomon's Clustered Data Sets

*

* * *

* ** *

1 00

be delivered, the planning horizon, the due date for a delivery and the length of the time windows. The number of customers, size of the service area, maximum load size and the planning horizon were used as inputs, and the other data was generated according to these values. The five problem sets involve a short time horizon with tight time windows, a medium horizon with three levels of time window strictness, and a long horizon with loose time windows. They also used Solomon's hard time window data in order to check the effectiveness of their solution method on these problems, by adjusting the penalties in order to reduce the number of customers serviced outside their time windows.

Van Landeghem [182] independently created a set of test problems for the VRPTW. He classified problems in terms of Sl, which is the average length of the window of availability for each customer, as a proportion of the total schedul­ ing horizon. The problem classes he identified are shown in Table 7. 1 . The most difficult problems identified have time windows between 10 and 40 percent of the scheduling horizon, and solution of these problems requires both scheduling and routing considerations. He generated 20 problem instances from a real-life problem with 31 customers, with the time windows randomly generated so as to vary the value of Sl. Two other problem instances were generated from a random, Euclidean problem containing 50 customers, and two more problems were adapted from an existing data set. Again the time windows were randomly generated with known length, thus, these problem instances control for one characteristic of the problem,

n-range n > 50% 30 < n < 40% 10 < n < 20% n < 8% Time windows very loose loose tight very tight Hardness moderate hard hard moderate Behaviour routing problem mixed mixed scheduling problem

Table 7. 1 : van Landeghem's Classification of VRPTW Problems

and can therefore be used to study how the solution methods are affected by this characteristic.

7. 1 . 2 Subset Selection Problem Generation

We have seen that the major difference between the generation of problems for variants of the VRP and the TSP is in the manner that feasibility is ensured. With most subset selection routing problems there is no concern with feasibility, as a solution route that contains no customers is a feasible solution. This is the case for the Orienteering Problem (OP) , as the inclusion of additional customers is beneficial for the goodness of a solution, although servicing no customers is a feasible option.

The Orienteering Problem

For the OP, the first test problems were created by Tsiligirides [176] . These prob­ lems have been used by subsequent authors for testing their solution methods (see [73] , [74]) . There are three problem sets, involving respectively 30, 1 9 and 31 customers available for servicing. For each set there is also an origin node, from which the solution route begins, and a separate depot where the route finishes. The customer locations appear to be randomly generated with the reward values being integer multiples of 5, which generally increase as the distances from the origin and destination nodes increase. The customers are therefore quite interchangeable, which creates multiple solutions of similar quality. In the case of the OP, many problem instances may be created from a given set of locations through varying the time limit, and Tsiligirides created 49 test problems from these three data sets. Keller and Goodchild [96] developed test problems for the OP using real geo­ graphical data. The customers are 25 selected cities from the former West Germany,

1 72 Problem Generation Methods

with their rewards represented by the total population in the cities and the inter­ customer distances being the road distances between the cities. This problem set therefore comes from an existing data set, which is assumed to be relevant for the OP.

Laporte and Martello

[108]

developed random, non-Euclidean test problems.

The direct distance between each pair of points was taken to be a random integer in the range

[1,100] '

with the actual distances used being the shortest path distances between the points. The rewards were also randomly generated in the range

[1,100] '

so this problem is purely random.

Sokkappa

[164]

developed a number of test problems for the OP. She identified a number of important characteristics of test problems and varied these in order to test the effectiveness of her solution methods under different circumstances. The characteristics that were varied are: the form of the distances between customers, i.e., Euclidean or non-Euclidean, the distribution of the customer locations, i.e. , random, clustered or with outliers, and the distribution of the rewards for the customers, i.e., all the same, uniform between various limits or exponentially dis­ tributed. The time limit was varied as a proportion of the approximate minimum distance required to service all the customers. She generated

18

classes of problems, each with

20

customers for testing her solution methods, and some larger problems

(

containing up to

50

customers

)

were generated for certain classes.

Kantor and Rosenwein

[91]

used the data of Tsiligirides for their version of the OP, which also contains time window constraints. The earliest time that customer

i

can be serviced, ei, was randomly generated over the interval

[0, Tmax/2]

with the latest time, di , randomly generated over the interval

r

ei, min

(

ei +

2/3Tmax, Tmax)].

Therefore the width of the time windows varied as the time limit varies, so these needed to be generated for each time limit. These time windows are wide, with expected width of

��Tmax'

or approximately 32% of the time limit. Therefore this paper only generates one type of problem, where the locations are those in Tsiligirides' data set and the time windows are quite wide.

Chao

[25]

used the data of Tsiligirides for testing his heuristics for the OP and the Team Orienteering Problem

(

TOP

)

. He also created two new test problem layouts for the OP, with the customer locations and rewards set out according to a systematic pattern, and higher rewards for customers further from the start and end points. These problems contained respectively

62

and

64

customers. For the

TOP he took the test problems for the OP and divided the time limits by the number of service vehicles available. He also created locations for two new test problems with 1 00 customers, one with random customer locations and customer rewards, and the other an adaptation of a problem set used for a different vehicle routing application.

Fischetti, Gonzalez and Toth [51] tested their optimization algorithm on a num­ ber of test problems from the literature. They defined four classes of problems; the first contained the data sets of Tsiligirides, as well as a number of Vehicle Routing Problem (VRP) instances from various sources, with the demand given in these problems being used as the customer rewards. The second class of problems in­ volved a number of published instances of the Travelling Salesman Problem (TSP), with OP instances being created by generating rewards in three ways: the same reward for each customer, randomly generated reward and reward given by a func­ tion of the distance from the depot. They state that the problems with the latter rewards are the most difficult to solve, and this is because there is the most equi­ tability in the attractiveness of the customers. In the third class of problems, the data was generated in the same manner as Laporte and Martello [108] , i.e. , with random rewards and random, non-Euclidean distances. The final class of prob­ lems involved customer locations being randomly distributed over a square, with Euclidean distances between the locations and the customer rewards randomly dis­ tributed. The authors claimed that this problem class appeared to be the most difficult to solve, where difficulty is measured by the computational time for their algorithm. This is because, for problems containing Euclidean distances, the paths between points are disjoint, so there are a greater number of feasible subpaths in these problems.

Gendreau, Laporte and Semet [64] randomly generated problem instances for the OP. They created instances containing Euclidean distances, in which customer locations were randomly generated, and non-Euclidean distances, in which the distances were randomly generated, and the reward values were also randomly generated. The time limit used was a proportion of the optimal TSP tour length around all the customers, for problems for which this length could be found, and an estimate of the TSP length was obtained via a heuristic method, for large instances. For their problem instances that required certain customers to be in the solution, the compulsory customers were randomly selected.

174 Problem Generation Methods

Other Subset Selection Routing Problems

The only previous problems generated for the MCPTDR have involved randomly generated problem instances. Brideau and Cavalier [20] generated examples with randomly generated distances between points and with the initial reward, the rate of decline of the reward and the service time at each point, being uniformly dis­ tributed. They stated that the range of values were such that some of the rewards became zero before all the customers were able to be be serviced, but they didn't give any more information about the parameter values used. Erkut and Zhang [45] randomly generated Euclidean locations for their points, with the other parameters uniformly distributed between given parameters. They tested the effect of increas­ ing the variability in the rate of decline of the rewards, and used this to speculate as to the effectiveness of their different heuristics under different conditions.

We believe that the problem instances that have been generated for the MCPTDR have only considered one scenario of the problem. All the problem instances are of similar form, with most of the customers being serviced in good solutions. We believe that other problem instances, which include lower time limits and different relationships between the initial reward and the rate of decline, are also worthy of investigation. We also believe that a more general form of the prob­ lem, which allows other forms of the reward function, e.g. , reward not necessarily just decreasing, should be investigated, with a further set of problem instances

being required to enable this investigation to take place.

For the Prize Collecting Travelling Salesman Problem (PCTSP), most of the published research has involved proving theoretical results, without requiring any data for testing (see [8, 9, 18, 70] ) . The problem instances which have been used for solving, have all been randomly generated. Fischetti and Toth [53] generated test problems for the PCTSP by randomly generating the prize values and using three different forms of distances: general asymmetric, triangularized asymmetric and Euclidean. For Euclidean distances, customers were randomly positioned, whereas the distances themselves were randomly generated in the asymmetric cases. There were no penalties for non-service, and different problem classes were obtained by varying the proportion of the available reward that needed to be collected. Pekny and Miller [135] also generated test problems for the PCTSP, by randomly gener­ ating general, asymmetric costs of travelling,

Cij

between each pair of customers,

from the tour. Gomes, Diniz and Martinhon [75] took layouts that had previously been used as TSP instances and added randomly generated rewards and penalties, in order to create their test problems for the PCTSP.

7 . 1 . 3 Generation o f Task Problems

The published problem instances for precedence constrained problems fall mostly into two categories: random Euclidean instances, with locations uniformly dis­ tributed over a square region of the Euclidean plane, and real data, adapted for the variation of the problem being studied in the article.

Euclidean examples have been generated for the Dial-a-Ride Problem (DARP) , for many papers including those by Psaraftis [142, 1 43] , Jaw, Odoni, Psaraftis and Wilson [89] , Kubo and Kasugai [101, 102] ' Bianco, Mingozzi, Ricciardelli and Spadoni [17] and Healy and Moll [82] . In each of these instances, n customers are generated by generating 2n nodes, with their locations coming from the uniform distribution, and randomly assigning one of these nodes to be the origin and another node to be destination for each customer. Bianco et al. [17] also allowed for general precedences, by generating a random tour through the nodes. Starting with the final node and working forwards, each unassigned node was assigned to be the final subtask of a customer, and a random number of preceding nodes in this tour were

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