Preliminary Experiments with Heuristic Methods

A heuristic method gives an approximate, or good, solution, but not necessarily an optimal one. However, as stated previously, a heuristic solution can be used as an initial upper bound that can be incorporated into optimal techniques to tighten the search and hence obtain the optimal solution relatively quickly.

For the case of the continuous p−centre problem, finding an upper bound to the opti- mal solution means that any covering circle (from the set of potential facility locations) with a radius larger than the upper bound can be discarded, as a solution has been obtained with p circles whose radii are equal to or smaller than the upper bound. Once these larger circles have been removed from the search, the number of calculations and overall computational time will decrease. This increases the chances of optimally solving larger location problems in a reasonable amount of time. Therefore, the better the quality of the heuristic solution, the tighter the upper bound found and therefore there is an increased chance of optimally solving larger problems.

Investigative work was completed for two different heuristics to help solve large contin- uous p−centre problems, namely the Cooper’s multi-start discrete method (M SDA), also known as Cooper’s discrete locate-allocate, and Cooper’s multi-start method in the continuous space (M SEHA) as previously discussed in Section 2.2.2, Chapter 2. As the M SEHA requires an established method to optimally solve the 1−centre prob- lem, the Elzinga & Hearn (1972) algorithm is used here, and is referred to as EHA.

The full description of EHA is given in Algorithm 1. For clarity, the full algorithm for the M SEHA is found in Algorithm 2 as it is used widely in this research. The M SDA shares many similarities to the M SEHA algorithm, and so, for completeness, the steps of this algorithm that differ from M SEHA can be found in Algorithm 3. Algorithm 1. The EHA Algorithm

1. Select two points P1 and P2 .

2. Construct a circle whose diameter is the line joining P1 and P2. If this circle

encompasses all the points, stop. If not, go to Step 3. 3. Select a point P3 outside the circle made from P1 and P2.

4. Determine if points {P1, P2, P3} form an acute, right-angled or obtuse triangle.

(i) Right-angled or obtuse triangle: Take the two points furthest away from each other and go back to step 2.

(ii) Acute triangle: Construct the circle passing through these three points. If this circle encompasses all the demand points stop. Else, go to Step 5.

5. Select a point P4 outside of the circle, and let H be the point from {P1, P2, P3}

that is furthest away from P4. Extend the diameter through the point H so the

plane is divided into two halves. Select a point L from {P1, P2, P3} that is in the

half plane opposite to the point P4. Set P1 = H, P2 = L and P3 = P4 and go to

Step 4.

Algorithm 2. The MSEHA Algorithm

1. Select the number of multi-starts, M ulStart. Set M ul = 0. 2. While M ul < M ulStart do:

(a) Set Mul=Mul+1; Select any p points from the m customer sites or on the plane to be temporary facility locations.

(b) Allocate all the demand points to their closest temporary facility. This cre- ates p clusters.

(c) For each cluster, find the optimal solution for the single facility problem using Algorithm 1.

(d) Reset the location of the facilities as the p circle centres found for each single facility problem.

(i) If the configuration of the facilities change, go back to Step 2 (b). (ii) Else, take the radius of the largest circle as the solution value. Go back

to Step 2 (a).

3. Take the smallest solution value found as the final value. Stop. Algorithm 3. The MSDA Algorithm

1. Steps 1-2 (b) in Algorithm 2.

2. (c) In each cluster, find the furthest Euclidean distance between any two points. This yields p distances.

(d) Take the largest of the p distances as the solution value. Go to Step 2(a). 3. Take the smallest solution value as the final value. Stop.

Note that as Step 2 (c) in Algorithm 3 is time consuming and Step 2 is repeated several times, the M SEHA requires more computational time than the M SDA.

Comparing the Heuristics

The two heuristics were tested on a generated data set where n = 100 (see Figure 3.1). The algorithm was written on a HP Elitebook 8570w with 12 GB of RAM. Table 3.1 shows the solution value found using each heuristic, and this is compared to the optimal solution to see how far each heuristic deviated from the exact answer.

Note that the deviation (%) was calculated by: Hp− Op

Op

× 100, (3.22) where Hp and Op refer to the heuristic and optimal solution for a given p respectively.

Table 3.1 is organised in the following way. The first column, labelled ‘p’, shows the number of facilities located. The second column, titled Z∗_{, displays the optimal}

solution for the corresponding p value. The next two columns displays each heuristic’s solution value, ZH, which could be used as an initial upper bound for optimal methods.

To test the quality of the solution given for each heuristic, the last two columns show the percentage deviation that the heuristic solution is from the optimal solution. Fig- ure 3.2 is also given to illustrate the solution values found for each heuristic compared to the optimal solution.

ZH Deviation (%)

p Z∗ _{MSDA MSEHA MSDA MSEHA}

2 85.566 113.071 85.566 32.1 0.00 5 49.649 79.158 49.649 59.4 0.00 10 29.921 55.000 29.921 83.8 0.00 15 21.319 44.553 24.910 109.0 16.8 20 16.101 42.426 21.319 163.5 32.4 25 13.210 30.870 16.867 133.7 27.7 30 11.161 28.231 14.603 152.9 30.8 Average 32.418 56.187 34.689 104.9 15.4 Table 3.1: Results for the Heuristic Investigation where n = 100

It is clear to see that the solution value found using the M SDA deviates much further from the optimal solution compared to the solution value found using the M SEHA. This is the expected result, as the M SDA can only choose possible facility locations from a discrete set of points. This restricts where the facilities can be located, thus restricting the overall solution value. The M SEHA does not have these restrictions, however, as facilities can be located anywhere in the plane. This therefore allows the solution value to be much tighter. It is worth noting that the solution of each run of the M SDA could be used as an initial solution for the M SEHA, rather than starting with p random points, in order to obtain an even tighter upper bound.

Figure 3.2: Heuristic solution values compared to the optimal solution value Although the importance of finding a good heuristic solution is practically demon- strated in Chapters 4 and 5, it is especially relevant in Chapter 6. This is because this chapter creates a matheuristic by combining the M SEHA mentioned above with the exact algorithm developed in Chapter 5.

Note that the generated data set can be collected from the author or accessed from the Centre for Logistics and Heuristic Optimisation (CLHO (2016)) website http://www.kent .ac.uk/kbs/research/research-centres/clho/datasets.html by following the links to ‘con- tinuous data sets’ and then to ‘random p−centre’.