El plan de trabajo de la presente investigación

where

Tj

is the reward of customer j , j), is a weighting factor and

dij

is the distance between customers

i

and j . This subgravity measure therefore takes into account not only the reward for the customer, but also the reward of the customers near it, with the effect on the subgravity score being a decreasing function of the distance. The solution method also included randomization, through randomly selecting one of the best candidate customers to be included in the solution.

Keller [94] developed a model for what he called the Multi-objective Vending Problem (MVP) , where the objective is to simultaneously minimize the total dis­ tance travelled and maximize the total reward received. To achieve these aims he attempted to create a non-inferior solution set, which is the set of solutions where the reward received cannot be increased without increasing the time required and the time used cannot be decreased without reducing the reward. The simplest version of this problem is the OP, where for a fixed time limit the objective is to maximize the reward. Keller and Goodchild [96] described their solution method to this problem, which involves constructing an initial solution either randomly or using bang-for-buck, and successively using a series of point and path moves and exchanges to improve upon the solution. One interesting improvement routine they developed, attempts to remove isolated clusters of customers from the solu­ tion. This was developed after they had identified a shortcoming of the solutions obtained, where clusters of customers could not be removed from the solution route by any of the other improvement routines. They demonstrate the effectiveness of their solution method on a single data set for the OP. Keller [95] compared the effec­ tiveness of the heuristic, with the effectiveness of the heuristics of Tsiligirides [176] and Golden et al. [73] . He stated that the solution quality of this heuristic com­ pares favourably with that obtained by previously published methods, as it is the

highest scoring method when applied to the most test problems considered. This conclusion doesn't agree with the results in the paper, as Keller and Goodchild's heuristic is only superior in one of the three problem sets considered. Since this problem set was the one with the greatest number of instances, they obtained their desired conclusion.

Kataoka and Morito [92] independently formulated the OP, which they call the Single Constraint Maximum Collection Problem. They used branch and bound on the relaxed linear programming problem to find the optimal solution to problems with up to 10 customers.

Laporte and Martello [108] gave a new name for the OP, the Selective Travelling Salesman Problem (STSP) , and they developed an exact algorithm. They devel­ oped an integer programming formulation for the OP, and created lower and upper bounds for the value of the solution. These were used within a branch-and-bound framework to find the optimal solution. A number of test problems with uniformly distributed distances were generated, with problems with 10 customers able to be

26 Subset Selection Routing

consistently solved, and with larger problems, with up to 90 customers, able to be solved with short time limits.

Sokkappa [164] gave the OP yet another name, the Cost-Constrained Traveling Salesman Problem (CCTSP) , and dedicated a PhD thesis to this problem. She developed a new exact branch-and-bound algorithm using improved bounds via the knapsack relaxation of the problem. She determined that the important char­ acteristics of a heuristic for the CCTSP are the methods of selecting which node to insert and where to insert it, the ability to remove customers from the solution, the use of intra-route improvement and the repetition of solutions. She adapted features of the heuristic of Golden et al. [74] to create a new improved heuristic, and tested the effect on the solution quality of a number of features of the heuristic. A number of new test problems for the CCTSP were created through random gen­ eration, and varying the reward values, the form of the distances and the relative locations of the customers.

Ramesh and Brown [148] developed a new heuristic for the OP, for a problem they called the Generalized Orienteering Problem (GOP) , where there may be a distinct origin and depot for the solution route, and where customers may be re­ visited, although without receiving any reward. The customer revisitation case is only relevant with non-Euclidean distances that don't satisfy the triangle inequal­ ity. The heuristic uses four phases, where the three phases of insertion, distance reduction and deletion are repeated until the stopping criterion is met, with a fi­ nal phase consisting of greedy insertion. The authors claimed that this heuristic quickly generates near optimal solutions.

Ramesh, Yoon and Karwan [149] introduced yet another name, this time for the problem where the origin and depot are the same, and they called this problem the Orienteering Tour Problem (OTP) . They solved this problem optimally, using Lagrangean relaxation within a branch-and-bound framework, and were able to solve random problems with up to 150 customers.

Leifer and Rosenwein [ 1 1 2] developed upper bounds for the solution of the OP. They improved upon the linear programming relaxation of the problem, by adding further constraints and re-solving. This is an iterative procedure, to improve the bounds, where these improved bounds may be used to improve the speed of an optimal solution algorithm.

The neural network is used to create candidate solutions, and local search routines of 2-opt, insertion and deletion required to ensure feasibility or to improve upon the solution obtained. This heuristic therefore has the ability to generate infeasible solutions which need to be altered to bring about feasibility. This method was tested on Tsiligirides' data, and was found to be as effective as the best previous heuristic for most of the problem instances.

Arkin, Mitchell and Narasimhan [3] described a version of the OP on networks. Their formulation requires the rewards to be integer, as their objective is to max­ imize the number of customers visited, and therefore the rewards are represented as repeated customers. They obtained approximation bounds for variants of the OP, where there may or may not be certain customers that need to be included in the solution.

Millar [122] described an application of the OP to fish scouting, which he called the Fish Scouting Problem. Here the rewards are the expected gain from sampling a given area of a fishing ground, and there is a time associated with sampling each area. He gave some basic heuristics and a procedure for determining the upper bound of the number of areas included in the optimal solution. Millar and Kiragu [123] basically replicated the above paper, although this time for an Operations Research audience, and demonstrated how the bounding procedure may be used on some randomly generated problems. Both papers mentioned that the bounding procedure may be used with multiple vehicles, although they didn't do so in their problem instances.

Gendreau, Laporte and Semet [64] developed an optimal branch-and-cut algo­ rithm for a version of the OP where there is a set of customers that must be in the solution. They created two heuristics that are used to obtain lower bounds on the value of the optimal solution. The first heuristic involves placing the compulsory customers via the generalized insertion heuristic GENIUS [60] . Pairs of customers and single customers are added to the solution, which is improved through remov­ ing pairs of customers, applying GENIUS, and then using the insertion phase again. The second heuristic creates a tour through all the customers using GENIUS, then removing customers until the distance of the path is less than 80% of the time limit. Insertion and swapping are then used to improve the solution. The swap­ ping involves removing one customer from the route and including one non-solution customer, such that the reward of the route would be increased. GENIUS is then

28 S ubset Selection Routing

applied to the solution set, in an attempt to obtain a feasible path.

The heuristics are heavily reliant upon the use of GENIUS to construct solu­ tions. Each time GENIUS is called, the route is started again from scratch, in order to obtain the shortest length tour through the selected customers. This is a computationally expensive approach that emphasizes the importance of obtain­ ing the minimum distance tour. The branch-and-cut algorithm uses a number of valid inequalities and the new heuristics to restrict the search. They solved Eu­ clidean problems with relatively short time limit, with up to 300 customers, and non-Euclidean problems with longer time limits and again up to 300 customers. In fact, they claimed that it is the heuristic, i.e., the memory requirements for GENIUS, that restricts the size of the problem that they were able to solve.

Fischetti, Gonzalez and Toth [51] developed an optimal branch-and-cut algo­ rithm for the OP. They use several valid inequalities to speed up the solution process and they introduce a conditional cut to remove sub-optimal solutions. The algorithm is used to solve a number of different classes of problems, with non­ Euclidean problems with up to 500 customers able to be solved. They tested their algorithm on the same Euclidean problems as Gendreau et al. [64] , and claimed that their method is able to solve a number of instances substantially larger than those solved by Gendreau

et al.

Kataoka, Yamada and Morito [93] developed a new relaxation for the OP, and developed new methods for finding a lower bound for this relaxation. They test the effectiveness of their bound on some random test problems and found that their bound was generally more effective than using a relaxation based upon the assignment problem.

Butt and Cavalier [21] extended the OP to cases where there is more than one service route available, and created the Multiple Tour Maximum Collection Prob­ lem ( MTMCP) . Here the objective is to maximize the total reward collected over all the solution routes. They created a new heuristic that builds up tours through finding pairs of customers that should appear in the same route, and attempting to include the pairs of customers in the same route. Repeated solutions are obtained through varying the parameter that is used to assess the viability of including the customers within a pair. The heuristic was tested on small, random problem in­ stances and obtained optimal solutions for most of the cases, and solutions were obtained within a reasonable amount of computational time for problem instances

containing up to 100 customers.

Butt and Ryan [22, 23] attempted to optimally solve the MTMCP using column generation. Their model allows for there to be differing time limits for the different service routes. The method selects candidate subsets of customers and assesses their viability through creating a TSP tour through the customers. One important technique that is used within the solution method is to use a tree structure to store the subsets of customers that can and can not allow a time feasible tour to be obtained. Scanning the trees, will prevent the unnecessary application of the TSP solution method. The column generation method obtained will obtain optimal solutions to the MTMCP when they find the optimal TSP solution. This phase can be computationally restrictive, so they suggested that a TSP heuristic may substantially decrease the computational time without degrading the solution quality too much. Optimal solutions were obtained for problems containing up to 100 available customers, but the problem instances were such that the maximum number of customers in each solution route was approximately five. The restriction in the number of customers per route enabled the TSP algorithm to be implemented quickly and enabled the TSP heuristic to obtain optimal solutions in most cases.

Chao [25] modelled a number of problems as examples of what he called the Multi-Level Vehicle Routing Problem (MLVRP). He developed a general solution metaheuristic based on Threshold Acceptance, and adapted it to be applicable to the different versions of these problems. Two such problems considered were the OP, and the multiple vehicle version of the OP which he termed the Team Orienteering Problem (TOP). The solution method for these problems consisted of creating a number of initial solutions through greedy insertion, and taking the best of these to be the initial solution. All eligible points are included in vehicle routes, with the best k routes, where k is the number of vehicles in the solution, considered to be the solution. The initial solution is improved using insertion and exchange techniques, with moves carried out if they improve the solution or if they don't deteriorate the solution by too much. Distance reduction is applied to the routes, before new initial solutions are created through removing a number of points from the current solution. The OP and TOP were further analyzed by Chao, Golden and Wasil [26, 27] . The Threshold Acceptance heuristic appears to outperform the previous heuristics over the existing test problems for the OP. For the TOP, the worst-performing of the published heuristics for the OP, i.e., the

30 Subset Selection Routing

stochastic algorithm of Tsiligirides [ 1 76] , was adapted for the multiple vehicle case. Not surprisingly, the new heuristic appears to improve upon this adapted heuristic.

Another extension of the OP that has been investigated is to apply restrictions on the time at which the customers may be serviced. These create time window constraints, which will be discussed in much more detail in Chapter 5. The problem with these time windows is known as the Orienteering Problem with Time Windows

(OPTW), and it was introduced by Kantor and Rosenwein [91] . Summary of the OP

For the OP a number of heuristic solution methods have been used. Most of these methods involve creating an initial solution, improving upon the solution and then creating a new starting solution, through either repeating the construction phase or making alterations to the current solution. The approach of Golden et al. [73] is an example of this second updating method, as one of the criteria used to create the new solution, is the distance from a candidate point to the weighted centre of the current solution. The method of Ramesh and Brown [148] , involves succes­ sively inserting customers into and deleting customers from the current solution. This process effectively creates new solutions through making small changes to the current solution.

The published heuristic for the OP which appears to be the most effective is that of Chao [25] . One of the reasons for its effectiveness is the advanced search structure employed, as the heuristic allows deterioration to the current solution in order to possibly obtain improved solutions at a later stage. A more advanced metaheuristic approach, e.g. , Tabu Search is likely to further improve upon the solutions obtained, although with probable increase in the computational time required. The other major difference between this approach and the previous heuristics is that this method assigns all customers to vehicles, and then takes the vehicle routes with the highest reward values to be the solution. In this manner, the heuristic is able to deal with cases that the other methods can not, i.e. , if there are whole clusters to be included in the solution, then this method will obtain the best solution regardless of the order in which the clusters are inserted, whereas the previous methods require the best cluster to be inserted first. We will use this technique, i.e. , including all customers in feasible routes, in our heuristic methods. This technique enables a greater variety of routines to be considered, as, for example, the Cross exchange

routine can be used to swap subpaths between different routes. The customers that are not currently in the solution are able to be ordered in a effective manner, which leads to an implicit consideration of a greater number of solutions. The possibility also exists for improving a non-solution route sufficiently so this route becomes part of the solution.

One of the interesting routines used in the solution methods is the subgravity approach. In this the aggregate score for a customer is given as the weighted sum of the rewards of all customers, where the weighting for a reward is given by a negative exponential function of the distance. This approach of Golden et al. [74] , was improved by Sokkappa [164] , so that the score only takes into account the eligible customers, i.e., those that could be feasibly included in the current solution. This measure is therefore updated for each new solution and each new time limit. Sokkappa also proposed that the parameter for the exponential function be a function of the average distance in the problem instance, which enables the method to be sensibly applied to a variety of problem instances. We also will employ the subgravity approach, as it implicitly considers the customers that may be included following an insertion, which seems to be a sensible upgrade over myopically considering the effectiveness of the candidate customer.

The optimal algorithms use variations of Branch-and-Bound within a linear programming environment, to identify the best solution. The algorithms have been able to solve larger problems, due to the development of tighter bounds through using tighter inequalities. Thus, the algorithm of Fischetti et al. [51 J was used to solve problem instances for the OP with Euclidean distances, containing up to 300 customers. The time limit in these cases was set equal to 5 hours, and even with this relatively long computational time, the methods could not solve all of the problem instances. Thus, while the state of the art in algorithms for the OP have been able to solve large problem instances, the development of fast heuristic methods and metaheuristics to obtain good solutions quickly, are still warranted. Examples of situations where faster computational times are required, occur III dynamic decision-making, which we consider further in Chapter 9.

3 . 1 . 2 Reward Constrained Subset Routing

We now consider problems where there is a lower bound on the total reward to be collected within the solution route, and the objective is to minimize the total cost

32 Subset Selection Routing

of a route. In this case the cost of the route includes the costs of travelling around the selected customers, as well as penalty costs for the customers that are left

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