In order to test the effectiveness of a solution method, there needs to be a set of problems available for testing purposes. When the application being modelled is a real situation, the real data may be used to determine the performance of the solution methods in practice. When there is no actual data present, or for checking robustness and predicting future behaviour in real situations, then it is necessary to generate the required data. An advantage of generating the data is that it is possible to vary the types of problems considered, in order to find out how the methods respond to different circumstances, and these results may be used for cataloguing problem types. We will study in this chapter, how researchers have previously generated problem instances that are related to the Mep and develop our own methods for problem generation for our particular problem.
7. 1 Previous Problem Generation Methods
We now consider methods that have been previously used to generate problem instances for related problems. Our overall problem involves time windows, prece dence constraints, reward functions, time limits and subset selection, so we will refer to problems containing some or all of these elements.
166 Problem Generation Methods
7. 1 . 1 Time Window Generation
The Travelling Salesman Problem
The Travelling Salesman Problem with Time Windows (TSPTW) involves finding an efficient tour that services a set of customers, with each customer subject to constraints on the time at which they may be serviced. Savelsbergh [154] has shown that finding a feasible solution to the TSPTW is an NP-complete problem, so the initial aim of the generation methods is to ensure that a feasible solution exists.
For the TSPTW, Baker [6] designed a set of problems based on data given for the Vehicle Routing Problem (VRP) by Eilon, Watson-Gandy and Christofides [44] ; the locations of the customers were taken directly from these data sets. In order to create the time windows, a nearest neighbour tour through the customers was created, with the time windows determined so that the nearest neighbour tour was feasible and none of the time windows overlapped. This problem generation method ensures that there is at least one feasible route possible, as the nearest neighbour tour is feasible. Different problem instances are created through varying the number of customers whose time windows are active. The time windows create an explicit order in which the customers may be serviced; hence, with all the time windows being active there is only one feasible route. Identifying the optimal route is therefore trivial, as a simple ordering of the time windows will obtain the only solution, so this problem instance may be easily solved. With the time windows of some customers being inactive, the problem becomes more flexible, in that there may be a number of feasible solutions from which to choose. The active time windows still ensure a partial ordering of the customers, so this restricts the effective size of the problem. This problem generation method is therefore effective in creating problem instances that have a small state space, which makes it a useful method for creating problems that may be solved optimally, e.g. , Baker uses branch and bound to optimally solve problems with up to 50 customers.
Langevin, Desrochers, Desrosiers, Gelinas and Soumis [106] created test prob lems for the TSPTW with distances either randomly generated (to obtain asym metric or symmetric random problems) or with customer locations randomly gener ated (creating Euclidean problems) . A second nearest neighbour tour was created through the customers, and the time windows were placed around the time each customer is serviced in this tour. Therefore, if customer
i
is serviced at time Tiin the second nearest neighbour tour, then the time window for customer i is uni formly random in the range
[Ti
-width,Ti]
andh, Ti
+ width] , where width is auser-defined parameter. This creates problems where there is a guaranteed feasi ble solution but the feasible tour is likely to be much different from the optimal tour. Dumas, Desrosiers, Gelinas and Solomon [41] used the method of Langevin et al. [106] to create test problems with larger numbers of customers. They also tested the effect of the density of customers, by increasing the number of customers and the size of the service area by the same proportion. They claimed that, with fairly wide time windows, only about 20% of arcs that are present in the second nearest neighbour tour are the same as in the optimal tour, and this percentage appears to remain constant as the number of customers increases. This indicates that the initial tour may be significantly improved upon.
Mingozzi, Bianco and Ricciardelli [124] created test problems for the TSPTW, with precedence constraints also present. They claimed that the test problems generated by Langevin et al. [106] and Dumas et al. [41] have tight time windows and that the initial tour the time windows are based on, i.e., the second nearest neighbour tour, is only about 10 percent longer than the optimal tour, which led to these problems being easily solved. Their method of generating test problems begins by randomly creating an initial tour, ( 1 , i l , i2, . . . , in+l ) ' with
Tik
being the time at which the kth customer in this tour is visited. The time window of customer i is given byrei , li
] with the time window of customer 1 , the depot, given by:el
= 0 and h =Tin+l .
The time window of the k'th customer, ik, is given by:_