# Can Bio-Inspired Swarm Algorithms Scale to Modern Societal Problems?

###### Abstract

Taking inspiration from nature for meta-heuristics has proven popular and relatively successful. Many are inspired by the collective intelligence exhibited by insects, fish and birds. However, there is a question over their scalability to the types of complex problems experienced in the modern world. Natural systems evolved to solve simpler problems effectively, replicating these processes for complex problems may suffer from inefficiencies. Several causal factors can impact scalability; computational complexity, memory requirements or pure problem intractability. Supporting evidence is provided using a case study in Ant Colony Optimisation (ACO) regards tackling increasingly complex real-world fleet optimisation problems. This paper hypothesizes that contrary to common intuition, bio-inspired collective intelligence techniques by their very nature exhibit poor scalability in cases of high dimensionality when large degrees of decision making are required. Facilitating scaling of bio-inspired algorithms necessitates reducing this decision making. To support this hypothesis, an enhanced Partial-ACO technique is presented which effectively reduces ant decision making. Reducing the decision making required by ants by up to 90% results in markedly improved effectiveness and reduced runtimes for increasingly complex fleet optimisation problems. Reductions in traversal timings of 40-50% are achieved for problems with up to 45 vehicles and 437 jobs.

Can Bio-Inspired Swarm Algorithms Scale to Modern Societal Problems?

Darren M. Chitty, Elizabeth Wanner, Rakhi Parmar and Peter R. Lewis Aston Lab for Intelligent Collectives Engineering (ALICE) Aston University, Aston Triangle, Birmingham, B4 7ET, UK darrenchitty@googlemail.com

## Introduction

The natural world is filled with a wealth of differing animals and ecosystems. Many of these organisms display collective behaviours which they use to overcome problems within their ecosystem such as ants foraging for food or bees communicating locations of nectar. These organisms have inspired many computing algorithms to assist in solving difficult real-world problems. Much of this inspiration comes from the exhibition of collective behaviours whereby thousands of organisms work together for the benefit of a colony, flock or hive. Each organism is simplistic in nature and by itself cannot survive but as part of a collective, problems such as finding sources of food can be solved. Nature has been used as a source of inspiration for the direct design of meta-heuristic algorithms that are moderately successful in solving optimisation problems of human consideration such as routing problems, information management and logistics to name a few. Examples of bio-inspired collective behaviour algorithms include Ant Colony Optimisation (ACO) (Dorigo and Gambardella, 1997) inspired by how ants forage for food; Artificial Bee Colony (ABC) (Karaboga and Basturk, 2007) based upon the way bees communicate sources of nectar; and Particle Swarm Optimisation (PSO) (Eberhart and Kennedy, 1995) which models the complex interactions between swarms of insects. These algorithms can be grouped under the term swarm intelligence through their use of hundreds or thousands of simulated digital organisms.

However, the types of problems that are tackled in nature by these organisms such as finding sources of food can be considered much more simplistic than the complex societal problems facing the human world. In an increasing digital world whereby the available data is growing considerably alongside inter-connectivity and joined up thinking, the size and complexity of the problems that require solving are increasing rapidly such as with smart city planning (Batty, 2013; Murgante and Borruso, 2015). Moreover, unlike the natural world, restrictions exist on modern computers in terms of compute capability and available memory to be able to simulate many thousands of collective organisms.

In regards to the literature of swarm algorithms most implementations of collective behaviour algorithms are applied to relatively small problem sizes. However, there have been some works in the field addressing scalability. For instance, Piccand et al. (2008) found applying PSO to problems of greater than 300 dimensions resulted in failing to find the optimal solution more than 50% of the time. Cheng and Jin (2015b) noted that PSO fails to scale well to problems of a high dimensionality potentially as a result of problem structure. However, the authors employ a social learning implementation whereby many particles act as demonstrators and present promising results on problems of sizes up to 1,000 dimensions. Cheng and Jin (2015a) later propose a modification to PSO whereby instead of using local and global best solution to update particle positions a pairwise competition is performed with the loser learning from the winner to update their position. The technique demonstrated improved results over PSO on benchmark problems of up to 5,000 dimensions although it was noted this was very computationally expensive. Cai et al. (2015) applies greedy discrete PSO to social network clustering problems with as many as 11,000 variables. For further reading Yan and Lu (2018) provide a review of the challenges of large-scale PSO.

Regarding ACO, Li et al. (2011) noted the scaling issues of the approach proposing a DBSCAN clustering approach to decompose large Travelling Salesman Problems (TSPs) of up to 1,400 cities into smaller sub-TSPs and solve these. Ismkhan (2017) also noted the computational cost and memory requirements and considered the use of additional heuristics or strategies to facilitate the scaling of the technique to larger problems. Improvements such as considering the pheromone matrix as a sparse matrix and using pheromone in a local search operator enabled ACO to be applied effectively to TSPs of over 18,000 cities. Chitty (2017) also noted computational issues with ACO and mitigated them with a non-pheromone matrix ACO approach which only made partial changes to good solutions applying the technique to TSP instances of up to 200,000 cities.

Therefore, it can be ascertained both ACO and PSO have issues in terms of scaling to high dimensional problems, the curse of dimensionality. Consequently, the question explored in this paper is can nature inspired, collective intelligence techniques scale up to the size and complexity of problems that the modern world desires solving? If not, what are the potential limiting causal factors for this and what mitigating steps could be taken? These questions will be investigated using a case study based on ACO to provide an illustration of the problems faced in scaling up a collective behaviour meta-heuristic and the hypothesized causal limitations by applying to a real-world fleet optimisation problem with steadily increasing complexity. The second aspect of this paper will attempt to mitigate ACO for these scalability issues using the novel Partial-ACO approach and enhance the approach further to assist scalability.

## Ant Colony Optimisation: An Exemplar Case

A popular swarm based meta-heuristic is based upon the foraging behaviours of ants and known as Ant Colony Optimisation (ACO) (Dorigo and Gambardella, 1997). Essentially, the algorithm involves simulated ants moving through a graph probabilistically visiting vertices and depositing pheromone as they move. The pheromone an ant deposits on the edges of graph is defined by the quality of the solution the given ant has generated. Ants probabilistically decide which vertex to visit next using this pheromone level deposited on the edges of graph plus potential local heuristic information regarding the edges such as the distance to travel for routing problems. An evaporation effect is used to prevent pheromone levels building up too much and reaching a state of local optima. Therefore, ACO consists of two stages, the first solution construction, simulating ants, the second stage pheromone update. The solution construction stage involves ants constructing complete solutions to problems. Ants start from a random vertex and iteratively make probabilistic choices using the random proportional rule as to which vertex to visit next. The probability of ant at point visiting point is defined as:

(1) |

where is the pheromone level deposited on the edge leading from location to location ; is the heuristic information from location to location ; and are tuning parameters controlling the relative influence of the pheromone deposit and the heuristic information .

Once all ants have completed the solution construction stage, pheromone levels on the edges of graph are updated. First, evaporation of pheromone levels upon every edge of graph occurs whereby the level is reduced by a value relative to the pheromone upon that edge:

(2) |

where is the evaporation rate typically set between 0 and 1. Once this evaporation is completed each ant will then deposit pheromone on the edges it has traversed based on the quality of the solution found:

(3) |

where the pheromone ant deposits, is defined by:

(4) |

where is the quality of ant ’s solution . This ensures that better solutions found by an ant result in greater levels of pheromone being deposited on those edges.

### Consideration of the Scalability of ACO

From a computational point of view, implementing an ant inspired algorithm on computational hardware to solve large-scale problems suffers from three potential limitations regarding overall performance. The degree of memory required, the computational costs of simulating thousands of ants and the sheer intractability of the problem itself.

#### Memory Requirements

A key aspect of ACO is the pheromone matrix used to store pheromone levels on all the edges in the graph . This can require significant amounts of computing memory. For instance, a fully connected 100,000 city Travelling Salesman Problem (TSP) will have ten billion edges in graph . Using a float data type requiring four bytes of memory will need approximately 37GB of memory to store the pheromone levels, considerably greater than available in standard computing platforms. In the natural world storing pheromone levels is not an issue with an infinite landscape to store them. A secondary memory requirement arises from ants only updating the pheromone matrix once all ants have constructed their solutions necessitating storing these in memory too. For a 100,000 city TSP a single ant will require 0.38MB of memory using a four byte integer data type. If the number of ants equals the number of vertices an additional 37GB of memory would be required.

An ant inspired algorithm that addresses this memory overhead is Population-based ACO (P-ACO) (Guntsch and Middendorf, 2002) whereby the pheromone matrix is removed with only a population of ant solutions maintained. From this population, pheromone levels are reconstructed for the available edges by finding the edges taken within the population from the current vertex and assigning pheromone to edges based on the solution quality.

#### Computational Costs

A second aspect to consider with ACO is the time it will take to simulate ants through the graph . At each vertex an ant needs to decide which vertex to next visit. This is performed probabilistically by looking at the pheromone levels, and possibly heuristic information, on all available edges. This requires computing probabilities for all these edges. As an example, take a 100,000 city TSP, at the first vertex an ant will have 99,999 possible edges to take all of which require obtaining probabilities from. Once an ant has made its choice it moves to the chosen vertex and once again analyses all available edges, now 99,998. Thus, for the 100,000 city TSP an ant will need to perform five billion edge comparisons. If a processor is capable of 100 GFLOPS (billion floating point operations per second) and assuming an edge comparison takes one floating point operation it will require at least 0.05 seconds to simulate an ant through graph . If using a population of ants equivalent to the number of vertices in graph then to complete one iteration of solution construction would require nearly 90 minutes of computational time. For ants in nature, compute time is not an issue since each ant can act independently although, the actual time it would take real ants to move through a network of this size would still be problematic.

The simulation of ants is inherently parallel in nature and therefore can easily take advantage of parallel computing resources to alliviate the computational costs. In recent years, speeding up ACO has focused on utilising Graphical Processor Units (GPUs) consisting of thousands of SIMD processors. DeléVacq et al. (2013) provide a comparison of differing parallelisation strategies for - ACO on GPUs. Cecilia et al. (2013) reduced the decision making process of ants using a GPU with an Independent Roulette approach that exploits data parallelism and Dawson and Stewart (2013) went a step further introducing a double spin ant decision methodology when using GPUs. These works have provided speedups ranging from 40-80x over a sequential implementation, a considerable improvement. Peake et al. (2018) used the Intel Xeon Phi and a vectorized candidate list methodology to achieve a 100 fold speedup. Candidate lists are an alternative efficiency method of reducing the computational complexity of ACO whereby ants are restricted to selecting a subset of the available vertices within its current neighbourhood. If none of these vertices are available then the full set are considered as normal. Gambardella and Dorigo (1996) used this approach to solve TSP instances whereby speedups were observed but also a reduction in accuracy due to sub-optimal edges being taken.

#### Problem Intractability

A final scalability issue with ACO involves the amenability of the problem under consideration to be tackled by ACO. The key issue is the probabilistic methodology ACO employs to decide which edge to take next by utilisng the pheromone levels on the available edges to influence the probabilities. Computationally, an ant will take the pheromone level on each edge, and if available multiply by the heuristic information, and multiply this by a random value between zero and one. The edge with the largest product is selected as the next to be traversed.

As an example consider a simple decision point whereby an ant has two choices available, one being the correct, optimal selection, the other suboptimal. If the pheromone levels on each edge are equal then there is a 0.5 probability the ant will take the optimal edge. However, consider ten independent decision points each with two possible choices akin to a binary optimisation problem such as clustering a set of items into two groups. Probabilistically this is equivalent to ten coin flips. With equal pheromone on all edges, there is only a probability of an ant making the optimal choices, approximately one in a thousand. Conversely, an ant has a 0.999 probability of a sub-optimal solution so 1,000 ants would need to traverse graph to obtain an optimal solution. For a much larger problem of 100,000 decision points this would be requiring ants to find the optimal solution.

Consequently, pheromone levels are there to help guide the ants to taking the optimal edge. Consider the previous 100,000 decision point example again but with high levels of pheromone on the edge to the optimal choice, say 0.99 vs. 0.01 on the suboptimal edge, then the probability of obtaining the correct solution will be or approximately ants required, still a significant number. In fact, to get to a manageable number of ant simulations the pheromone on the optimal edges would need to be of the order 0.9999 vs. 0.0001 on the suboptimal edge when only approximately 20,000 ants would need to traverse the network before an ant probabilistically takes the correct edges at each decision point. However, this means the pheromone level would need to be 10,000 times greater on the optimal edge than the suboptimal edge. Moreover, the pheromone levels would need to build up over time before reaching these levels.

Hence, it can be observed that applying ACO to ever larger problems results in increasingly reduced probabilities of optimal solutions being found unless the pheromone levels become increasingly stronger on the important edges. Candidate lists, as covered in the previous section, can reduce the number of decisions that ants need to make but with a potential error reducing accuracy and can only use if heuristic information is available to define the neighbourhood. Of course ants in the natural world do not have problems of this magnitude to solve and have the numbers if necessary without any undue computational cost to consider.

### An Illustration of the Scalability Issues With ACO

To highlight the potential drawbacks of ACO it will be tested against a complex set of real-world fleet optimisation problems of steadily increasing complexity. These problems have been supplied by a Birmingham based maintenance company which operates a fleet of vehicles performing services at customer properties within the city. Each vehicle starts from a depot and must return when it has finished servicing customers. Each customer is defined by a location and a job duration predicting the length of time the job will take and in some cases, a time window for when the jobs must be completed. The speed of travel of a vehicle between maintenance jobs is defined at an average 13kph to account for city traffic. There is also a hard start time and end time to a given working day defined as 08:00 and 19:00 hours. Fleet optimisation is essentially the classic Multiple Depot Vehicle Routing Problem (MDVRP) (Dantzig and Ramser, 1959) with Time Windows (MDVRPTW).

The MDVRP can be formally defined as a complete graph , whereby is the vertex set and is the set of all edges between vertices in . The vertex set is further partitioned into two sets, representing customers and representing depots whereby is the number of customers and is the number of depots. Furthermore, each customer has a service time associated with it and each vehicle has a fixed capacity associated with it defining the ability to fulfill customer service. Each edge in the set has an associated cost of traversing it represented by the matrix . The problem is essentially to find the set of vehicle routes such that each customer is serviced once only, each vehicle starts and finishes from the same depot, each vehicle does not exceed its capacity to service customers and the overall cost of the combined routes is minimised.

Total Job | Total Fleet | |||
---|---|---|---|---|

Problem | Vehicles | Jobs | Servicing | Traversal Time |

(hh:mm) | (hh:mm) | |||

Week_1 | 8 | 77 | 47:09 | 31:12 |

Week_2 | 8 | 79 | 48:24 | 22:49 |

Week_3 | 8 | 81 | 48:33 | 19:54 |

Fortnight_1 | 16 | 156 | 95:33 | 54:01 |

Fortnight_2 | 16 | 138 | 102:01 | 57:07 |

Fortnight_3 | 16 | 160 | 96:57 | 42:43 |

ThreeWeek_1 | 24 | 237 | 144:06 | 73:55 |

ThreeWeek_2 | 24 | 217 | 150:25 | 79:56 |

ThreeWeek_3 | 24 | 219 | 150:34 | 77:01 |

Month_1 | 32 | 298 | 198:58 | 99:50 |

Month_2 | 32 | 313 | 190:26 | 96:28 |

SixWeek_1 | 45 | 437 | 267:47 | 142:46 |

The worksheet data supplied by the company has been divided into a series of problems of increasing complexity and size which are described in Table 1. The manner in which the company assigns customer jobs to vehicles is known apriori enabling a ground truth for the optimisation process. Effectively, the company assigns geographically related jobs to vehicles based on postcode and then orders them such that the vehicle performs the job furthest from its depot first and then works its way back, time windows allowing.

To highlight the drawbacks of ACO in terms of scalability, the - Ant System (AS) (Stützle and Hoos, 2000) will be tested upon these fleet optimisation problems. AS simulates ants through the graph but, in contrast to standard ACO, only the best found solution provides pheromone updates. Additionally, minimum and maximum levels of pheromone on edges are defined. To solve the fleet optimisation problem the fully connected graph has vertices relating to the number of vehicles and customer jobs. Ants start from a random vehicle vertex then visit every other vertex once only resulting in a sequence of vehicles beginning from their specified depots followed by the customer jobs they will service before returning to their depot. This representation is shown in Figure 1 whereby V relates to a vehicle and J relates to a job. The first vehicle will undertake jobs 6, 5 and 9, the second jobs 3, 7, and 2 and so forth.

Once a new solution has been generated its quality needs to be assessed. This is measured using two objectives, the first of which is to maximise the number of jobs correctly performed within their given time window. The second objective is the minimisation of the total traversal time of the fleet of vehicles. Reducing the number of missed jobs is the primary objective. Hence, comparing two solutions, if the first services more customer jobs than the second then the first solution is considered the better. If though they have equal customer job time serviced then the solution with the lower fleet traversal time is considered the better.

The pheromone to deposit is calculated using these objectives to be optimised. A penalty based function will be utilised for the first objective whereby any customers that have not been serviced due to capacity limitations or missing the time window will be penalised by the predicted job time. The secondary objective is to minimise the time the fleet of vehicles spend traversing the road network between jobs. Solution quality can then be described as:

(5) |

where is the total amount of time of jobs to be serviced, is the amount of job service time achieved by ant ’s solution and is the total traversal time of the fleet of vehicles of ant ’s solution. Clearly, if ant has achieved the primary objective of fulfilling all customer demand then becomes merely the total traversal time of the fleet.

Number of Ants | 192 |

Max Iterations | 1,000,000 |

1.0 | |

1.0 | |

0.02 |

A parallel implementation of AS is tested against the exemplar problems from Table 1 with experiments conducted using an AMD Ryzen 2700 processor using 16 parallel threads of execution. The algorithms were compiled using Microsoft C++. Experiments are averaged over 25 individual execution runs for each problem with a differing random seed used in each instance. The parameters used with AS are described in Table 2.

The results from these experiments are shown in Table 3 whereby the issue of scalability is abundantly clear. As the size and complexity of the fleet optimisation problems increases, the ability for AS to find a solution which satisfies all the customer demand reduces. Similarly, AS cannot obtain solutions with a lower fleet traversal time than the company’s own scheduling when the problem size increases. Therefore, it can be considered that these results support the hypothesis that a nature inspired swarm algorithm such as ACO suffers from scalability issues.

Problem | Job Time | Traversal | Execution |
---|---|---|---|

Serviced (%) | Reduction (%) | Time (mins) | |

Week_1 | |||

Week_2 | |||

Week_3 | |||

Fortnight_1 | |||

Fortnight_2 | |||

Fortnight_3 | |||

ThreeWeek_1 | |||

ThreeWeek_2 | |||

ThreeWeek_3 | |||

Month_1 | |||

Month_2 | |||

SixWeek_1 |

## Addressing the ACO Scalability Issues

Given that the evidence seems to support the hypothesis that ACO methods will struggle to scale to larger, increasingly complex problems the next step is to attempt to address the underlying reasons behind the poor performance. As has been previously discussed, a key problem is the degree of decision making required to form solutions vs. the probabilistic nature of ACO. Therefore, it can be theorized that if the degree of decision making is reduced, ACO may well scale better. A novel modification to the ACO algorithm known as Partial-ACO (Chitty, 2017) provides a mechanism to achieve this. Essentially, this technique minimises the computational effort required and the probabilistic fallibility of ACO by ants only considering partial changes to their solutions rather than constructing completely new solutions. In contrast to standard ACO algorithms, Partial-ACO operates in a population based manner much the same as P-ACO. Essentially, a population of ants is maintained each of which represent a solution to the given problem. Pheromone levels are constructed from the edges taken within this population of solutions with their associated qualities which are relative to the best found solution. Partial-ACO also operates in a pure steady-state manner to preserve diversity. An ant only replaces its own best solution with a new solution if it is of better quality. Hence, each ant maintains a local memory of its best yet found solution.

This memory enables an ant to consequently only partially change this solution to form a new solution. To partially modify its locally best found solution an ant simply picks a random point in the solution as a starting point and a random sub-length of the tour to preserve. The remaining aspect of the tour is rebuilt using standard ACO methodologies in a P-ACO manner. This process is illustrated in Figure 2. To highlight the computational advantage of this technique, consider retaining 50% of solutions for a 100,000 TSP problem. In this instance only 50,000 probabilistic decisions now need to be made and only 1.25 billion pheromone comparisons would be required, a reduction of 75%. An overview of the Partial-ACO technique is described in Algorithm 1.

### Evaluating the Partial-ACO Approach

To test the hypothesis that reducing the degree of decision making that ants need to perform will enable them to scale to larger problems, Partial-ACO will be tested upon the same problems as previously. The parameters used for the implementation of Partial-ACO are described in Table 4. Note the lower number of ants in contrast to AS. The original Partial-ACO work found a low number of ants was highly effective. To ensure the same number of solutions are evaluated, Partial-ACO will use six times more iterations.

The results for the MDVRP fleet optimisation problem are shown in Table 5 whereby it can be observed that now in all problem instances, reductions in the fleet traversal times are achieved by Partial-ACO over the commercial company’s methodology. In fact, in many cases the improvement in the reduction in fleet traversal time is significantly better than that from AS, especially regarding the larger, more complex, problems. Disappointingly though, the Partial-ACO technique was also unable to service all the customer jobs for the larger problems. In terms of execution timings, Partial-ACO is slightly slower than AS when evaluating the same number of solutions. This is caused by the requirement to construct the edge pheromone levels at each point as an ant moves through the graph .

Number of Ants | 32 |

Max Iterations | 6,000,000 |

3.0 | |

1.0 |

Problem | Job Time | Traversal | Execution |
---|---|---|---|

Serviced (%) | Reduction (%) | Time (mins) | |

Week_1 | |||

Week_2 | |||

Week_3 | |||

Fortnight_1 | |||

Fortnight_2 | |||

Fortnight_3 | |||

ThreeWeek_1 | |||

ThreeWeek_2 | |||

ThreeWeek_3 | |||

Month_1 | |||

Month_2 | |||

SixWeek_1 |

## Enhancing Partial-ACO

Although the results of the Partial-ACO approach seemed promising they did not significantly enforce the premise that ants are less effective with higher degrees of decision making. Analysing the Partial-ACO methodology, it could be postulated that modifying a continuous subsection of an ant’s locally best found tour could present problems in that individual points within the solution cannot be displaced a great distance. They are confined to a local neighbourhood as to how they could be reorganised.

An enhancement to Partial-ACO is proposed which will facilitate the movement of points in a given ant’s locally best solution. To achieve this, it is proposed that instead of one continuous segment of an ant’s solution being preserved and the remaining part probabilistically regenerated as is the norm, a number of separate blocks throughout the solution are preserved instead. In this way a point at one end of a given solution could be moved to points throughout the solution. This should help prevent the ants becoming trapped in local optima. This methodology is actually well suited to the fleet optimisation problem since each vehicle can be considered as a stand alone aspect of the solution. Each preserved block could in fact be a vehicle’s complete job schedule. Before attempting to construct a new solution an ant can simply decide randomly which vehicle schedules to preserve and then use the probabilistic behaviour of moving through the graph to assign the remaining customer jobs to the remaining vehicles as normally. Figure 3 demonstrates the principle whereby it can be observed that two sections representing vehicle schedules are preserved by an ant from its solution with the rest built up probabilistically.

Problem | Job Time | Traversal | Execution |
---|---|---|---|

Serviced (%) | Reduction (%) | Time (mins) | |

Week_1 | |||

Week_2 | |||

Week_3 | |||

Fortnight_1 | |||

Fortnight_2 | |||

Fortnight_3 | |||

ThreeWeek_1 | |||

ThreeWeek_2 | |||

ThreeWeek_3 | |||

Month_1 | |||

Month_2 | |||

SixWeek_1 |

To evaluate the enhancement to Partial-ACO, it will be tested against the same problems as previously using the same parameters as described in Table 4. The results are shown in Table 6 and when contrasted to those in Table 5 it can be seen that significant improvements have been made over the standard Partial-ACO approach. Now, for all problem instances including the most complex, all the customer jobs have all been serviced. Furthermore, significantly improved reductions in the fleet traversal times have been achieved. In fact, as much as an additional 26% reduction in fleet traversal time for the ThreeWeek_2 problem instance. With regards execution timings, block based Partial-ACO is slightly slower which is caused by the overhead of assembling blocks of retained solution rather than one continuous section. Consequently, from these results it can be inferred that when using solution preservation with Partial-ACO, smaller random blocks should be preserved rather than a continuous section to obtain improved results.

### Reducing the Degree of Modification

The enhanced Partial-ACO approach has provided a significant improvement over standard ACO techniques such as AS. However, recall that the original hypothesis supporting the development of Partial-ACO was that potentially, collectively intelligent meta-heuristics could fail to scale well to larger problems because of the degree of decision making that is necessitated. This hypothesis seems to be borne out by the results achieved by Partial-ACO to some degree. However, it is possible to test this hypothesis to a greater extent by reducing the degree of permissible modification an ant can make. Currently, an ant will randomly preserve any amount of its locally best solution and will modify the rest using the ACO probabilistic rules, approximately 50% of the solution on average. To avoid a large aspect of redesign, a maximum degree of modification could be imposed on an ant changing its locally best solution. This will firstly have the benefit of increasing the speed of Partial-ACO but also, if the hypothesis is correct, lead to improved optimisation. As such, the previous experiments will be rerun using a maximum modification limit ranging from 50% of the solution down to 10% in increments of 10%. The improved block preserving version of Partial-ACO will be used and additionally, to prevent ants becoming trapped in local optima with a small random probability (0.001) an ant can modify its locally best found solution to any degree.

The results from reducing the degree of permissible modification of ants locally best solutions are shown in Figure 4. These describe the reductions in fleet traversal times over the commercial company’s own scheduling and execution timings. The percentage of customer jobs serviced is not shown as in all cases 100% of jobs were serviced. A clear trend can be observed for improved reductions in fleet traversal times whilst reducing the degree of permissible modification. This further reinforces the hypothesis that due to the probabilistic nature of ants, the degree of decision they are exposed to must be reduced in order for the technique to scale. Moreover, the larger the problem, the more pronounced the effect as evidenced by the month and six week long problem instances. Remarkably, for the largest problem, reducing ants decision making by 90% yields the best results fully enforcing the hypothesis that ants significantly benefit from reduced decision making. A further added benefit from reduced decision making of ants is faster execution times. Not only does the Partial-ACO approach provide improved reductions in fleet traversal times but can also achieve these reductions much faster by reducing the probabilistic decisions that ants need to make. In fact, from these results, it can stated that Partial-ACO is more accurate, much faster and more scalable than standard ACO as a consequence of the reduced decision making of ants within the algorithm.

## Conclusions

This paper has posed the hypothesis that although algorithms inspired by the collective behaviours exhibited by natural systems have been effective for simplistic human level problems, they may fail to problems of much greater complexity. Evidence supporting this hypothesis is provided by applying Ant Colony Optimisation (ACO) to a range of increasingly complex fleet optimisation problems whereby degrading results are observed as complexity rises. A theory postulated is that the degree of decision making required by ants to construct solutions becomes too great. Given a small probability of an ant choosing poorly at each decision point, the greater decisions required to construct a solution and available choices, the greater probability of reduced solution qualities. Consequently, this paper applies the Partial-ACO approach to reduce the decision making of ants. Indeed, the Partial-ACO approach provided much improved results for a complex fleet optimisation problem enabling ACO to scale to much larger problems with reductions of over 50% in traversal times achieved with the subsequent savings in fuel costs for the given company and a similarly significant reduction in vehicles emissions and city traffic.

In fact remarkably, for the larger problems, reducing ants decision making by up to 90% yielded the best results. Consequently, this reinforces the posed hypothesis that for collective behaviour algorithms to scale effectively, the degree of decision making should be minimised as much as possible. However, further studies need to be performed with bio-inspired algorithms besides ACO such as PSO and ABC and to consider problem areas other than fleet optimisation to provide better supporting evidence to the hypothesis posed by this paper.

## Acknowledgement

Carried out for the System Analytics for Innovation project, part-funded by the European Regional Development Fund.

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