There are two optimal type approaches used to solve the continuous p−centre prob- lem in this study, namely the maximal circles-based method and the relaxation method. (i) The Maximal Circles-Based Method
Drezner (1984a) proposed an interesting exact method to find the optimal solution value to the continuous p−centre problem using a subset of potential facility covering circles called maximal circles. The process begins with an initial upper bound, Z, which was found using an established heuristic method, such as Cooper’s locate-allocate method or the H2 heuristic proposed by Drezner (both previously described in Section 2.2.2).
A circle is defined as maximal based on the upper bound Z. The set of maximal circles is treated as the set of potential facility locations, as Drezner proves that the largest covering circle in the optimal solution is a maximal circle. By narrowing the size of the set of potential facility locations, the problem size is efficiently decreased such that larger problems can be solved optimally. Drezner’s algorithm consists of a) finding the set of maximal circles and b) searching for a better solution (i.e. a feasible solution with a solution value smaller than Z) using this set. If a better solution is found, the upper bound Z is updated as the new, improved solution value, a new set of maximal circles is attained and the process begins again. If a better solution value cannot be
found with the set of maximal circles, the optimal solution is the current upper bound Z. A detailed description of Drezner’s algorithm can be found in Figure 4.1, Chapter 4. This method shows much potential as it yields a relatively small problem size for large problems, and so it may be beneficial to be investigated further. Drezner’s method is examined thoroughly in Chapter 4.
(ii) The Relaxation-Based Method
Relaxation is a simple method used to solve large problems by breaking them down into smaller sub-problems and successively solving them. The classic algorithm, first suggested by Handler and Mirchandani (1979), begins with an upper bound of infin- ity, and either updates the upper bound or adds demand points to the subset at each iteration until optimality is reached. Alternative variations to the classic relaxation algorithm have been researched and will also be described in this section. This method has great importance to this research, as it forms the backbone to the research given in Chapters 5, 6 and 7.
Chen & Handler (1987) were one of the first to propose a relaxation-based algorithm that solved the continuous p−centre problem. First, they explained that a finite set of potential facility locations (from the infinite set) could be attained by finding all the critical circles, where a critical circle has either a) three or more demand points on its circumference, b) two demand points forming the ends of the diameter or c) a null circle consisting of the single demand point (these points shall be referred to as critical points in this study). Therefore, the full set of potential facility locations can be calculated as n3 + n
2 + n, where n
3 refers to the number of circles created from three
demand points, n2 refers to the number of circles created from two demand points and n is the number of demand points. Furthermore, the number of critical circles can be decreased further when analysing the geometry of the demand points forming them. For example, all circles made from three points that form an obtuse or right−angled triangle can be discarded, as the circle created from the two points furthest from each other would incorporate all three points. This method of finding all critical circles formed from three demand points shall be referred to as the ‘angle method’. Chapter 3 proposes a new method that finds these critical circles more efficiently and in less
computational time.
The foundation of Chen & Handler’s relaxation-based algorithm is built on the well- known theorem stating that among all the optimal solutions to the p−centre problem, at least one of them consists of p critical circles encompassing all the demand points. Therefore, by finding the full set of possible facility locations, a finite set of potential fa- cility locations has been obtained which allows us to solve the problem optimally. Their method also incorporated the observation that the solution for the (p − 1)−centre so- lution yielded an upper bound for the p−centre solution. This allowed tight upper bounds to be obtained, and so meant many circles can be discarded if their radius size exceeded the upper bound. Therefore, this reduced the number of calculations and allowed the problem to require less memory and relatively less computational time to be solved. The authors gave a small example where n = 10 and p = 1, 2, . . . , 10. Their method starts by solving the 1−centre problem using established methods on a very small subset of demand points. An arbitrary point was then added and all the possible critical circles constructed from the subset of demand points are found. A solution is obtained for the relaxed problem (i.e. the subset of demand points), and feasibility for the full problem is checked. If the solution for the sub-problem is feasible for the full problem, another arbitrary point is added to the subset and the process continues until p facilities have been located. If not, the point farthest from its closest circle centre is added to the subset of demand points and another solution for the same number of facilities is found.
Chen & Chen (2009) proposed two new and interesting relaxation algorithms based on the classic relaxation algorithm to solve the discrete and the continuous p−centre problem optimally. The classic relaxation algorithm optimally solves large problems by breaking them down into smaller sub-problems that are easier to solve. Every time a feasible solution with a value less than the current upper bound is obtained for the sub-problem then, much like Chen & Handler’s (1987) approach, feasibility for the full problem is checked. If it is feasible for the full problem, then the upper bound is updated. Else, another demand point is added to the subset. If a feasible solution with a value less than the current upper bound cannot be found for the subset, then the current upper bound is the optimal solution value. The authors presented two
improvements for the classic relaxation method. The first improvement updates the upper bound more efficiently by treating every feasible solution for the relaxed prob- lem as a feasible solution for the full problem. The second improvement adds more than one point to the relaxed subset at a time to reduce the number of ‘uninformative iterations’ and therefore creates a more efficient algorithm. However, the number of demand points that are added to the subset needs to be carefully balanced between decreasing the number of iterations and keeping the number of circles created to a minimum. If too many demand points are added, this creates a large problem to solve and negates the use of the relaxation method. Chen & Chen state that this is an area that could be researched in more depth, and this gap of knowledge is investigated in Chapter 5.
Chen & Chen (2009) also suggested two new relaxation algorithms, namely the re- verse relaxation algorithm and the binary relaxation algorithm. In short, the reverse relaxation algorithm starts with a lower bound of 0, and constantly updates it until optimality is reached. The binary relaxation algorithm has “the best of both worlds” as it updates either the lower or upper bound at each step. The algorithm starts with an initial coverage distance LB+U B2 , where LB denotes the lower bound and U B de- notes the upper bound. The LB and U B values are updated throughout the algorithm depending on whether a solution can be found within the coverage distance for the sub- problem or for the full problem. A detailed description of these two algorithms can be found in Figures 5.2 & 5.3, Chapter 5, as both are tested in order to determine which one shows the most potential for further improvement. The results show justification for choosing the reverse relaxation for further development.