Algorithms to solve the orienteering problem: A comparison

224 European Journal of Operational Research 41 (1989) 224-231 North-Holland

Theory and Methodology

Algorithms to solve the orienteering A comparison *

problem:

C. Peter K E L L E R Department of Geography, University of Victoria, Victoria, B.C., Canada

Abstract: The multiobjective vending problem (MVP) (Keller, 1985) is a generalization of the traveling salesman problem (TSP) where it is not necessary to visit all nodes in the problem definition. A special case of the MVP is the orienteering problem (OP) (Tsiligirides, 1984; Golden et al., 1985). The problems are NP-hard. A heuristic was designed to solve the general MVP (Keller, 1985). Three other heuristics were designed to solve the OP (Tsiligirides, 1984; Golden et al., 1985). The algorithms underlying these heuristics are outlined. Their performances are compared against three OP problems.

( Keywords: Traveling salesman problem, generalization, orienteering problem, heuristics, performance comparison

1. Introduction

The multiobjective vending problem (MVP) (Keller, 1985) is a generalization of the traveling salesman problem (TSP) where it is not necessary to visit all nodes in the problem definition. The objective is to examine the trade-off relationship between maximizing reward potential by visiting as many nodes as possible, at the same time minimizing penalty for traveling the links by visiting as few nodes as possible. A conceptually similar problem concerns the orienteering problem (OP) discussed by Tsiligirides (1984), and Golden et al. (1985). It has been demonstrated that both problems are NP-hard (Garey and Johnson, 1979; Keller, 1985; Golden et al., 1985).

The objective of this paper is to evaluate and discuss the performance of four algorithms written to solve the above two problems for three sets of OP data. Tsiligirides (1984), Keller (1985) and Goldert et al. (1985) have suggested that the MVP and the OP have numerous applications. The relative performance of algorithms to solve these problems are therefore of interest. The paper commences by outlining the two problem definitions. Next, the different solution approaches are summarized. The performances of the algorithms are compared using Tsiligirides (1984) three score orienteering problems.

2. Problem definition

* This research has been supported by NSERC grant no. A 6533.

Received November 1987; revised May 1988

The TSP is conceptually defined as follows: Given a set of n nodes, determine the shortest complete circuit that connects all nodes so that

0377-2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

C.P. Keller / Algorithms to solve the orienteering problem 225

every node is visited once and once only. An assumption underlying this definition is that the set of nodes to be visited is given and fixed. The problem is therefore reduced to a single objective, that of finding the optimal traveling salesman route that connects all nodes specified. This is not always realistic. Let us assume that there is a reward associated with visiting each node. There exist applications where it is of interest to examine the trade-off relationship between maximizing total reward to be collected and minimizing the penalty for traveling the links. Such a problem therefore contains two conceptual objectives, that of maximizing reward to be collected by visiting as many nodes as possible, and that of keeping the total link penalty to a minimum. If the two objectives can be defined in commensurable terms, say dollars, or if a trade-off relationship can be specified, then the problem can be solved as a single objective problem. In most cases the two objectives will not be commensurable, and a study of the trade-off relationship between them is of interest in itself.

Such a multiobjective definition has been discussed by Keller (1985) who refers to this problem as the multiobjective vending problem (MVP). The MVP is conceptually defined as follows. A set of N nodes or demand points, each with a known reward potential, is connected by links, each with a known travel penalty. The objectives are to find a circuit through a subset of the demand points in order simultaneously to maximize reward and minimize the accrued travel penalty. No relationship is defined a priori between reward and penalty, and the two objectives must therefore be treated as noncommensurable.

Keller (1985) has discussed a number of possible approaches that could be utilised to derive the noninferior solution set underlying such a multiobjective definition. He suggests that the constraint method (Marglin, 1967; Cohon, 1978) is the one most appropriate to handling the MVP definition. The constraint method transforms a multiobjective problem into a finite number of single objective problems by optimizing for one objective while constraining the other objective to a specified value. It is possible to examine the noninferior solution set by successively increment- ing the constraining objective, in this case the maximum penalty PMAX. With PMAX set to zero the circuit will be confined to the depot, and

the reward will be equal to the depot 's reward. At the other extreme a large value of PMAX will allow the circuit to include all demand points and to collect the total reward available: the MVP will in this case degenerate to the TSP.

Given that it is possible to break the multiobjective problem definition into a set of single objective problems, the issue that remains is how to solve each single objective problem. One possible solution approach, the MVP heuristic, has been outlined by Keller (1985), and has been evaluated against a 25 node problem. The logical steps underlying this heuristic are summarized further on.

Three algorithms developed to solve the OP, that have also been suggested as being applicable for solving MVP type problems (Golden et al., 1985), are the S-algorithm and the D-algorithm developed by Tsiligirides (1984), and the Knap-. sack algorithm by Golden et al. (1985). The logic underlying these algorithms, too, is summarized further on.

The OP concerns score orienteering where a start point and end point that are very close to each other are specified, along with other loca- tions termed control points. Each control point has associated with it a score, with control points further away given higher scores. Competitors are given a fixed amount of time in which to find as many control points as possible summing their scores. The highest score total collected within the maximum time allocation wins.

Although apparently different, the OP is equiv- alent to a special case of a single objective MVP if the distance between the starting point and the end point in the OP is set to zero, and an additional constraint is added enforcing that both points must be part of any feasible solution. De- fined this way, the OP is argued to be a special case of the MVP since the distribution of rewards in the OP has an underlying symmetrical structure not necessarily true for the general MVP. In the OP, rewards are distributed akin to the shape of the surface of a saucer, where the x and y direc- tions represent the cartesian coordinate space and the z dimension represents reward potential. Points near the start and finish, the centre of the saucer, have low associated scores, and points far away on the periphery have highest scores.

The following section briefly summarizes the logic underlying the four algorithms.

226 C P. Keller / AIgorithms to solve the orienteering problem

3. The algorithms

3.1. MVP algorithm

The first step in the MVP heuristic (Keller, 1985) is to identify the maximum penalty PMAX. The heuristic will then identify that subset of all possible nodes, j , that can be visited directly from the depot, d, without exceeding the maximum penalty constraint. This is defined as

P~j + 6~ ~ PMAX.

This step ensures that any node that can not be reached within PMAX is excluded from further consideration. The next step is to generate some feasible starting solution. Two alternative approaches are evaluated for each problem. The first approach selects in sequence those nodes that promise highest possible reward while staying within a reasonable distance of the node last visited. The measure used is a simple ratio of the reward potential of a node Rj to the penalty that must be accepted to get there from the last node visited, Plast,j" The second approach commences by standardizing and stacking the reward poten- tials of all nodes that are still feasible to be visited so that their sum equals unity. A random number selected between zero and one will subsequently determine which node will enter the starting solution.

Once a feasible starting solution has been generated the heuristic implements two routines that at tempt to reduce total link penalty by altering the route sequence while maintaining the present route membership. The first routine searches for and eliminates self-crossing paths, which by the trian- gle property can not be contained in the optimal route. The second routine sequentially drops every member of the starting solution out of its present position and inserts it at every alternative position within the route sequence. Each cycle of the second routine identifies and implements that change which will result in the largest decrease in penalty.

Once these two routines detect no further possible reduction in penalty the heuristic proceeds to search for possible increases in total reward while remaining within the limit imposed by PMAX. This is achieved by attempting to alter the route membership using three strategies: 'one in-zero out'; 'one in -one out'; and 'one i n - ' t w o out' in

which the two nodes moved out are adjacent to each other in the route sequence.

A combination of these three simplest node exchange methods tends to detect most of the possible improvements to a given route. If any one of them results in an improvement, the heuristic reverts to calling the first two penalty reduction routines. If all five routines fail to make improvements two further routines are called. Experiments with the set of five routines described so far showed that the heuristic favours the inclusion of large and sometimes remote nodes with a high associated penalty over a cluster of small nodes in close proximity to each other. This was found to be true even when the sum of all the rewards of the clustered nodes was larger than the reward potential of the individual large node. A further routine was therefore added to alleviate this problem. It operates by successively dropping every node in turn out of the present route sequence, but temporarily not returning it to the set of feasible nodes not presently included in the route. Each time a member of the existing route is thus dropped the remaining route is evaluated using the five routines described previously. If an improvement is detected that yields a higher total reward potential than the route that included the member presently dropped, then the improvement is accepted as the new solution.

This last routine is still not capable of detecting the following situation. An isolated cluster of nodes is included in the route. This cluster comprises one node with a very large reward, with the rewards of other adjacent nodes relatively small. This isolated cluster of nodes could be replaced by another cluster of nodes closer to the depot, resulting in an increase in total reward without exceeding the maximum penalty constraint. To detect this improvement an additional routine searches for and temporarily drops all nodes in the isolated cluster out of the solution similar to the previous routine, then reevaluating the problem. The individual procedures are described in detail by Keller (1985).

3.2. S-algorithm

The S-algorithm (S(R - I ) ) (Tsiligirides, 1984) builds a large number of routes utilising Monte Carlo techniques, finally selecting the best one as the final answer. Each route is built utilising the following logic. Each node so far not included in


the route is assigned a measure of desirability or weighted ranking, Aj. This measure is a combination of Sj, the reward associated with node j , and T~ast.j, the penalty for travelling from the last node visited to j . Tsiligirides (1984) uses in particular the following measure:

A j = ( s i . / / ( l l a s t , j ) ) 4"0.

The four nodes with highest desirability values are subsequently identified and normalised so that their sum equals unity. A randomly generated number between 0 and 1 determines which of the four nodes gets selected as the next node visited. This procedure is repeated until it is not possible to include additional nodes in the route without exceeding the maximum penalty constraint. If fewer than four nodes are left from which to choose then the next node is selected from whatever is left.

Tsiligirides (1984) subsequently imposes a route improvement routine called the (R - I)-algorithm. The first step in this algorithm is similar to the second improvement routine in the MVP heuristic. An attempt is made at reducing penalty for traveling the links by shuffling the sequence in which the present route membership is visited. The second step tries to insert new members into the route by a 'one in-zero out' and a 'one in-one out ' move similar to those in the MVP heuristic.

3.3. D-algorithm

The D-algorithm ( D ( R - I )) (Tsiligirides, 1984) is based on a principle similar to that of Wren and Holiday's (1972) method for vehicle scheduling. It operates by dividing the study area into a number of separate sectors determined by concentric rings, that is the area between two defined radii and axes. Sectors are varied by changing the two radii a n d / o r rotating the axes. Tsiligirides examines 48 sectors for each run. A route is identified when all the nodes in a particular sector have been visited, or if visiting another node will exceed the maximum capacity constraint. An attempt at improv- ing each route thus derived is made by subsequently imposing the (R - / ) - a l g o r i t h m . The route with the largest possible score wins.

3.4. Knapsack algorithm

The algorithm proposed by Golden et al. (1985) is based on the Knapsack problem. Their algorithm is made up of five steps. These steps are sequentially an initial route construction step, a route improvement step, the knapsack step, the knapsack improvement step and a perturbation step.

The route construction step is based on a weighted ranking, W/ for each node so far not visited. Here the weighted ranking is however a function of Sj, the reward associated with node j , C/, the distance to a calculated center of gravity rank and Ej, the sum of distances to two defined foci of an ellipse rank. Golden et al. (1985) use in particular the following relationship:

W =a * S +b* Cj+c* E,

The sum of a, b and c is defined to equal unity, and for each problem definition 22 different combinations of these parameters are evaluated, producing 22 routes. The route with the highest score passes to the route improvement step.

This step takes the route generated and imposes an improvement algorithm similar to Tsiligirides' ( R - I ) algorithm but avoiding the 'one in -one out' step. Route membership therefore is initially held constant while shuffling the sequence, followed by a cheapest insertion procedure in which as many nodes as possible are added without violating the maximum penalty constraint.

The knapsack step commences by calculating the weighted centre of gravity for all nodes presently included in the tour. A list of nodes ordered by decreasing values of Sj/~,~ is constructed where Tj.g represents the penalty to reach the centre of gravity from node j . Nodes are sequentially selected from this list until the knapsack is filled.

The knapsack improvement step tries to add nodes to the best route so far identified by cheapest insertion if the maximum penalty has not yet been exceeded, or removes the last node inserted followed by cheapest insertion and an interchange procedure if the maximum has been exceeded.

The last two procedures are called repeatedly until no further improvement can be identified. The perturbation step comprises the final move. Here the centre of gravity for nodes presently not

228 C.P. Keller / Algorithms to solve the orienteering problem

a member of the route is calculated, and the knapsack and knapsack improvement steps are repeated using this new point.

4. Performance evaluation

The performances of the four algorithms are compared using the three score orienteering problems introduced by Tsiligirides (1984). The results

for each problem are given in Tables 1, 2 and 3 respectively. The best solution for each value of PMAX has been emphasised bold. Route sequences were not published by Tsiligirides (1984), or Golden et al. (1985). Therefore the route sequences shown are those identified by the MVP heuristic, and do not necessarily represent the best solution for a given value of PMAX.

Running times are available only for the Knapsack and the MVP algorithms. Both al-

Table 1 Problem 1 results

TMAX D(R - I ) S ( R - I) Knapsack MVP

Score Dist. Score Dist. Score Dist. Time a Time a Score Dist.

5 10 15 20 65 19.60 25 90 24.82 30 110 28.80 35 135 34.08 40 150 38.02 46 175 44.51 50 190 44.53 55 200 52.86 60 220 59.65 65 240 63.82 70 260 69.13 73 265 70.73 75 275 74.66 80 280 77.73 85 285 81.33

TMAX Route

10 4.14 (0.43) 15 6.87 (0.66) 45 14.26 (0.96)

65 19.60 65 19.85 (1.29) 90 24.65 90 24.88 (1.76)

110 28.80 110 29.88 (2.07) 135 34.08 125 33.60 (2.60) 150 38.02 140 39.87 (3.49) 175 44.51 165 45.85 (4.20) 190 49.78 180 49.92 (5.06) 205 54.08 200 54.38 (7.82) 220 58.93 205 59.40 (11.27) 240 63.82 220 64.69 (11.32) 245 69.13 245 69.91 (13.02) 265 70.73 255 72.27 (11.81) 275 74.66 265 74.61 (13.02) 280 78.34 275 79.57 (9.62) 285 81.82 285 81.78 (7.73)

5.89 b

sequence for MVP solutions

(0.01) 10 4.14 (0.01) 15 6.87 (0.05) 45 14.28 (0.21) 65 19.85 (0.42) 90 24.88 (0.41) 110 28.80 (0.64) 130 33.69 (0.76) 155 39.91 (1.01) 175 45.86 (1.17) 185 49.54 (0.61) 200 54.32 (1.05) 225 59.89 (1.07) 240 63.93 (1.82) 260 69.53 (1.31) 265 72.65 (0.82) 270 73.80 (0.16) 280 78.18 (0.01) 285 81.78 0.65 b

5 10 15 20 25 30 35 40 46 50 55 60 65 70 73 75 80 85

1 28 32 1 28 18 1 27 31 1 28 27 1 27 31 1 28 27 1 28 27 1 28 27 1 28 27 1 28 18 I 28 27 1 2731 1 28 27 1 28 29 1 27 26 1 19 20 1 1920 1 1920

32 26 20 19 32 31 26 20 21 19 31 26 25 23 21 20 19 32 31 26 22 21 12 11 10 8 9 13 32 31 262221 1211 1 0 8 2 3 7 6 3 2 31 26 25 23 22 21 1211 1 0 8 2 3 7 1 3 3 2 31 2625232221 1211 1 0 8 2 3 7 4 5 6 3 2 5 4 3 2 8 9 10 11 12 21 22 23 25 26 31 27 20 29 32 26 31 30 25 24 23 22 21 12 11 10 9 8 2 3 4 5 18 32 26 25 23 22 21 12 11 10 8 2 3 7 6 5 4 14 15 16 17 28 32 16 15 144 5 6 7 3 2 8 9 10 11 12 21 22 23 25 26 31 27 20 19 32 17 16 15 1 4 4 5 7 3 2 8 9 1 0 1 1 12 21 22 23 24 25 26 31 27 20 19 32 31 30 25 24 23 22 21 2111 1 0 9 8 2 3 7 6 5 4 1 4 1 5 1 6 1 7 2 9 2 8 3 2 27 31 26 25 24 23 22 21 1211 1 0 9 8 2 3 7 6 5 4 1 4 1 5 1 6 1 7 2 9 2 8 1 8 3 2 27 26 31 30 25 24 23 22 21 12 11 10 9 8 2 3 7 6 5 4 14 15 16 17 29 28 18 32 27 26 31 30 25 24 23 22 21 1 2 1 1 1 0 9 8 2 3 7 1 3 6 5 4 1 4 1 5 1 6 1 7 2 9 2 8 1 8 3 2

a Running times in seconds. b Average time.



TMAX D ( R - I ) S (R - 1) Knapsack

Score Dist.

MVP

Score Dist. Score Dist. Time a Time ~ Score Dist.

15 120 14.25 120 14.88 120 14.64 (0.89) (0.05) 120 14,37 20 200 19.88 200 19.88 200 19.88 (1.31) (0.20) 200 19~88 23 210 22.65 210 22.65 210 23.00 (1.32) (0.20) 210 22,65 25 230 24.13 230 24.26 230 24.13 (1.50) (0.11) 230 24,13 27 230 24.13 230 24.26 230 24.13 (1.72) (0.26) 230 24,13 30 265 29.85 260 29.49 260 29.22 (2.10) (0.16) 260 29.22 32 300 31.63 300 31.63 275 31.69 (2.14) (0.35) 300 31.62 35 320 34.51 320 34.51 305 34.51 (1.86) (0.53) 320 34,51 38 355 37.62 355 37.62 355 37.79 (2.80) (0.36) 360 37.84 40 385 39.56 495 39.78 380 39.89 (2.48) (0.13) 380 39.86 45 450 44.44 450 44.44 450 44.44 (2.08) (0.07) 450 44.89

1.82 b 0.22 b

TMAX route sequence for MVP solutions

15 20 23 25 27 30 32 35 38 40 45

1 7 6 5 1 2 1 1 1014 1321 1 1 2 7 6 5 3 2 8 9 1 0 1 1 13 1421 1 7 6 5 4 3 2 8 9 1 0 1 1 1 4 2 1 1 1 2 7 6 5 4 3 2 8 9 1 0 1 1 131421 l 1 2 7 6 5 4 3 2 8 9 1 0 1 1 131421 1 1 2 7 6 5 4 2 0 3 2 8 9 1 0 1 1 131421 1 7 6 5 3 2 8 1 7 1 6 1 5 9 1 0 1 1 1 3 1 4 2 1 1 7 6 5 3 4 2 0 1 9 1 8 1 7 9 1 0 1 1 13 1421 1 7 6 5 2 3 4 2 0 1 9 1 8 1 7 8 9 1 0 1 1 13 1421 1 1 2 7 6 5 3 4 2 0 1 8 1 6 1 5 1 7 8 9 1 0 1 1 13 14 21 1 1 2 7 6 5 2 3 4 2 0 1 9 1 8 1 6 1 5 1 7 8 9 1 0 1 1 13 14 21


gorithms were written in FORTRAN 77. The Knap- sack heuristic was run on a U N I V A C 1190, the MVP heuristic on an IBM 4381. Given the dif- ferences in computers, a direct comparison of running times is meaningless. More meaningful is a comparison of relative running times for the different values of PMAX for each of the three data set.

Table 4 shows the total number of wins for each algorithm, separating them into shared wins and outright wins. The table shows the MVP heuristic to be the overall winner. Performance does however vary with problem type. The D ( R - I ) and S ( R - I) algorithms prove to,be superior for Problem 1. The D ( R - I ) and the MVP algorithms appear best capable of handling Problem 2. The D ( R - I) and S ( R - I) algorithms never identified the best answer in Problem 3.

It is of interest to speculate why the different algorithms performed better on one problem than on another. One possible reason is that there is a notable structural difference between the three

score orienteering problems. The three problems can be differentiated according to a number of criteria.

One criterion concerns problem size. Problems 1 and 3 contain 32 and 33 nodes respectively and are therefore reasonably equally complex. Prob- lem 2 is simpler with only 21 nodes. However, there is no evidence that size controlled performance except for running time.

A second criterion concerns the physical distribution of points on the ground relative to each other, and relative to the start and finish. The start and finish are located relatively central in problems 1 and 3, but are located on the edge in problem 2. As noted earlier, the nodes are reasonably evenly distributed in all three cases, with highest scores systematically further away from the start and finish. It may be of interest that problem 2 was least favoured by any one algorithm, but there is no direct evidence that this is due to the difference in location of the starting and end point.

230


C.P. Keller / Algorithms to soloe the orienteering problem

TMAX D ( R - I )

Score Dist.

S(R - I) Knapsack MVP

Score Dist. Score Dist. Time a Time a Score Dist.

15 100 13.82 20 140 19.25 25 190 24.66 30 240 29.60 35 280 34.15 40 340 39.70 45 370 44.04 50 420 49.58 55 440 54.61 60 500 59.07 65 530 63.81 70 560 68.75 75 600 74.32 80 640 79.42 85 670 83.61 90 700 89.14 95 740 94.96

100 770 99.09 105 790 103.65 110 800 106.19

TMAX

100 13.82 170 14.47 (1.54) (0.09) 170 14.47 140 19.25 200 19.79 (2.07) (0.18) 200 19.79 190 24.66 250 23.61 (2.25) (0.34) 260 24.95 240 29.60 320 29.19 (2.85) (0.56) 320 29.47 290 34.93 380 34.69 (2.71) (0.69) 370 34.48 330 39.65 420 38.19 (3.47) (0.98) 430 39.69 370 44.04 450 44.96 (3.42) (0.52) 460 44.80 420 49.58 500 49.33 (4.07) (1.11) 520 48.94 460 53.97 520 52.81 (4.97) (0.76) 550 53.56 500 59.07 580 58.52 (5.35) (1.17) 570 59.26 530 64.65 600 63.72 (7.21) (2.18) 610 64.95 560 69.39 640 69.19 (8.34) (2.29) 640 69.51 590 74.78 650 71.00 (7.95) (1.31) 670 74.62 640 79.80 700 79.43 (11.04) (0.69) 700 79.66 670 83.61 720 83.82 (11.07) (0.99) 740 84.86 700 89.13 770 89.31 (10.11) (0.34) 760 88.76 730 94.67 790 92.79 (10.45) (0.25) 790 92.58 760 99.10 800 97.08 (7.96) (0.22) 800 96.99 790 104.89 800 97.77 (7.78) (0.21) 800 97.77 800 108.31 800 97.08 (8.06) (0.23) 800 96.99

6.25 b 0.76 b

Route sequence for MVP solutions

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

100 105 110

1 24 22 7 5 14 4 27 23 33 1 24 2 2 7 5 2 8 1 4 4 3 2 7 2 3 3 3 1 24 2 2 7 5 2 8 1 4 4 2 0 1 3 3 3 3 1 24 22 7 5 28 14 4 20 17 13 3 27 23 33 1 24 2 2 7 5 2 8 1 4 4 2 0 1 7 1 6 1 3 3 2 7 2 3 3 3 1 24 22 7 5 14 4 20 17 16 15 13 3 6 2 32 23 33 1 24 22 7 5 14 4 20 17 16 15 13 6 2 8 29 30 26 33 1 24 22 7 5 28 14 4 20 17 16 15 13 3 6 2 8 31 12 29 26 33 1 24 22 7 5 28 14 4 20 17 16 15 13 3 6 2 8 31 12 29 30 26 32 33 1 24 22 7 5 27 14 4 20 17 16 15 13 3 6 2 8 31 12 11 30 29 26 33 1 24 2 2 7 5 2 8 1 4 4 2 0 1 7 1 6 1 5 1 3 3 6 2 8 3 1 1 2 1 1 103029163233 1 2 5 9 1 0 1 1 30 29 12 3 1 8 2 6 3 1 3 1 5 1 6 1 7 2 0 4 1 4 2 8 5 7 2 2 2 4 2 3 3 3 1 24 25 9 10 18 11 29 12 31 8 2 6 3 13 15 16 17 20 4 14 28 5 7 22 27 23 33 1 2 4 2 5 9 1 0 1 8 1 1 12 21 3 1 8 2 6 1 3 1 5 16 21 17 2 0 4 1 4 2 8 5 7 2 2 2 7 2 3 3 3 1 2 4 2 2 7 5 2 8 1 4 4 2 0 1 7 1 6 1 5 1 3 3 6 2 8 3 1 1 2 1 1 19 1 8 1 0 9 3 0 2 9 2 6 3 2 2 3 2 7 3 3 1 2 4 2 2 7 5 2 8 1 4 4 2 0 1 7 2 1 1 6 1 5 1 3 3 6 2 8 3 1 1 2 1 1 19 1 8 1 0 3 0 2 9 2 6 3 2 2 3 2 7 3 3 1 24 25 9 10 18 19 11 30 26 29 12 31 8 2 6 3 13 15 16 21 17 20 4 14 28 5 7 27 23 33 1 2 4 2 5 9 1 0 1 8 1 9 1 1 30 26 29 12 3 1 8 3 2 2 6 3 1 3 1 5 1 6 2 1 1 7 2 0 4 1 4 2 8 5 7 2 2 2 7 2 3 3 3 1 2 4 5 7 2 2 2 5 9 1 0 1 8 1 9 1 1 3 0 2 6 3 2 2 9 1 2 3 1 8 2 6 3 1 3 1 5 1 6 2 1 1 7 2 0 4 2 8 1 4 2 7 3 2 3 3 1 24 2 5 9 1 0 1 8 1 9 1 1 30 26 29 12 3 1 8 3 2 2 6 3 1 3 1 5 1 6 2 1 1 7 2 0 4 1 4 2 8 5 7 2 2 2 7 2 3 3 3


A n o t h e r c r i t e r i o n c o n c e r n s t h e c o m p l e x i t y o f

t he s c o r i n g s y s t e m a n d the i r f r e q u e n c y d i s t r i -

b u t i o n . A c lo se r e x a m i n a t i o n o f t h e t h r e e p r o b l e m

s t r u c t u r e s s h o w s t h a t t h e r e is c o n s i d e r a b l e va r i a -

t i on b e t w e e n t h e t h r e e p r o b l e m s w i t h r e s p e c t to

t he n u m b e r o f s co re c a t e g o r i e s a n d t he f r e q u e n c y

d i s t r i b u t i o n w i t h i n the c a t e g o r i e s . P r o b l e m 1 h a s

o n l y t h r e e c a t e g o r i e s , w i t h s c o r e s d i s t r i b u t e d fa i r ly

n o r m a l l y a r o u n d t he m i d d l e c lass . P r o b l e m 2 c o n -

t a i n s s e v e n c a t e g o r i e s , w i t h s c o r e s r e a s o n a b l y u n i -

f o r m l y d i s t r i b u t e d . P r o b l e m 3 fa l l s b e t w e e n t he

two. T h e t h r e e p r o b l e m s c l e a r l y d i f f e r in s t r u c t u r e .

I t is p o s t u l a t e d t h a t a c o m b i n a t i o n o f t he a b o v e

c r i t e r i a e x p l a i n s o m e o f t h e d i f f e r e n c e s in p e r f o r -


Table 4 Performance comparison

D ( R - I ) S ( R - 1) Knapsack MVP

Problem 1 Shared wins 8 8 2 3 Outfight wins 3 2 1 2

Total wins 11 10 3 5

Problem 2 Shared wins 6 4 4 5 Outright wins 2 1 0 2

Total wins 8 5 4 7

Problem 3 Shared wins 0 0 2 3 Outright wins 0 0 6 11

Total wins 0 0 8 14

Overall Shared wins 14 12 8 11 Outright wins 5 3 7 16

Total wins 19 15 15 26

mance, but a meaningful and thorough discussion would require an examination and comparison of inferior and best route sequences identified. Un- fortunately Tsiligirides (1984), and Golden et al. (1985) did not publish theirs. However, it should be noted that examination of some inferior solutions derived by experimentation show that, for a given value of PMAX, two very different routes will often yield the same total score, with difference in total penalty highly marginal (often less than 0.1%). Only a very complex shuffle of route membership would have allowed the inferior solution to converge to the superior.

5. Conclusion

The OP can be regarded as a special case of the MVP. Four algorithms to solve the OP have been discussed and their relative performances have

been evaluated. Three of the algorithms, the D(R - I ) , the S ( R - I) and the Knapsack heuristics were written specifically to solve the OP. The fourth, the MVP heuristic, was designed to solve the general case. The MVP heuristic scored the largest number of wins, and would therefore have won the orienteering trophy. It can therefore be argued to represent the algorithm with the highest likelihood of winning score orienteering competi- tions.

It has not been the objective of this paper to test the ability and performance of the D ( R - I), the S(R - I ) and the Knapsack heuristics to solve general MVPs. Given the symmetric nature of the OP, an examination of the relative performances of the four algorithms on a general MVP, a problem where nodes of varying scores are fairly randomly distributed, and where there are clusters of nodes in isolation, ought to be one of the next lines of enquiry. Keller's (1985) 25 node West Germany problem would represent one possible test data set.

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Documents

Algorithms to solve the orienteering problem: A comparison