Conclusions

This PhD work builds on Cuckoo Search (CS) and Flower Pollination Algorithms (FPA) algorithms and extends them to address challenges in optimising large- scale project schedule in an uncertain environment, subject to multiple constraints such as limited resource capacities and strict precedence relationships between activities. Several novel concepts have been proposed, implemented and tested during this PhD study.

a) Derivation of Discrete Cuckoo Search (DCS) and Discrete Flower Pollination Algorithm (DFPA), respectively, to solve RCPSP in discrete domain

DCS and DFPA were derived by adapting CS and FPA to solve discrete RCPSP rather than RCPSP in the continuous domain, which CS and FPA were originally developed, through reinterpretation of their key elements: solution representation scheme, solution improvement operators and Lévy flight. For the solution representation scheme activity list was chosen, as it is the most common representation scheme for the RCPSP and a large variety of operators has been developed. In the review on state-of-the-art heuristics for the RCPSP by Kolisch and Hartmann (2006) 23 out of 27 algorithms operated on activity list solution representation scheme. For solution improvement operators, pairwise exchange and shift operators were selected, as combination of these operators seemed to work best when integrated into Lévy flight: for smaller Lévy number (i.e. small step) a number of shift operations was performed on a solution, whereas for big numbers (i.e. large step) pairwise exchange was executed.

Both algorithms were evaluated in accordance to the standardised tests defined by Hartmann et al. (2000) by running all benchmark instances of J30, J60 and J120 sets from PSPLIB for 1000, 5000 and 50000 objective evaluations. Results showed that both algorithms were capable of solving discrete problems and both showed relatively good performance and managed to outperform some non-hybrid metaheuristics against which they were compared, such as Tabu

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Search (TS) and Genetic Algorithm (GA). In particular, DCS appeared in top 1 for J30, J60, and J120 sets, whereas DFPA was outperformed by TS in J30 tests by 0.01% and by GA in J120 by 0.25%. However, both algorithms suffered from several drawbacks that affected their performance: reliance on inefficient solution representation scheme (activity list) and utilisation of random-based solution improvement operators (pairwise exchange and shift operators). This leads to the development of the IDCS algorithm.

b) Improved Discrete Cuckoo Search (IDSC) algorithm

The main disadvantage of the activity list solution representation scheme in DCS and DFPA is the representation of a single schedule by multiple representation schemes. Because of that, if two activities have identical starting times, then interchanging their positions will not bring any changes to a solution, which, in turn, results in wasted operation. To address this issue, IDCS replaced activity list by event list. The main distinctive feature of event list from other solution representation scheme is that activities with identical starting times are grouped into events. In some circumstances, such events can comprise activities sharing common project characteristics, such as having the same predecessors and/or successors. Moreover, the introduction of the new solution representation scheme resulted in smarter and less random operators to be used to their advantages: event move exploits shared network characteristics of events for more efficient perturbations, whereas event crossover was used to combine useful problem-specific information extracted from the parents for generating high-quality offspring. The performance of IDCS was again evaluated on sets of benchmark instances from PSPLIB and compared against other algorithms for the RCPSP. The results showed that IDCS greatly improved the performance of DCS and DFPA. In addition, IDCS outperformed all other metaheuristics for J120 sets and its performance was among the top five for J30 and J60 sets.

During the validation of IDCS on single benchmark instances, it was found that IDCS is able to obtain multiple solutions for one problem instance. In fact other researchers, for example Czogalla and Fink (2009), also pointed out that RCPSPs exhibit complex multimodal fitness landscapes; hence, for one instance of the problem, several optimal solutions might exist. Multiple solutions are beneficial in that they can eliminate premature convergence to local optima and can sometimes lead to more innovative outcomes, such as more efficient or well- balanced schedules. Nevertheless, at the time of writing, this property had only

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been addressed in one publication (Pérez, Posada, & Lorenzana, 2015). As a result, research effort thereafter was devoted to explore multimodal optimisation techniques such as the Species Conservation (SC) technique to combine with IDCS to obtain multimodal solutions for RCPSPs.

c) Development of Discrete Species Conserving Cuckoo Search (DSCCS) In order to explore the potential of IDCS to obtain multimodal solutions, techniques for multimodal optimisation were reviewed. Among the different techniques, SC was chosen for its simplicity and effectiveness as it only relies on distance metric (Euclidean distance) to estimate the similarity between solutions. However, due to the specifics of the RCPSP and its discrete search space, SC could not be directly applied. Therefore, in order to be adapted, the Euclidian distance metric needed to be replaced with a similarity metric specific for the RCPSP. Czogalla and Fink (2009) in their analysis of the RCPSP fitness landscape reviewed various distance measure techniques. The authors conducted a series of experiments on the sets of benchmark instances from PSPLIB. As the conclusion of their analysis, the authors noted that algorithms that operated on the deviation distance measure (Reeves, 1999) tend to produce better results. Moreover, Chen et al. (2010) and Paraskevopoulos et al. (2012) used the abovementioned distance measure in their algorithms, which confirms its applicability and effectiveness. As the conclusion of these analyses, deviation distance measure was chosen as the main method of similarity measure in adaptation of SC to the RCPSP.

SC technique was then integrated into IDCS to form DSCCS. In DSCCS the whole population was divided into smaller subpopulations (i.e. species). Each subpopulation had devoted region of search space and was centred on the fittest solution (i.e. dominating individual). Having multiple sub-population ensures that diversity of population is kept high at all stages and significantly reduces chances of falling into local optima trap.

Similarly to other developed algorithms, the performance of DSCCS is assessed on the sets of benchmark instances for RCPSP. In comparison to IDCS, the performance was slightly degraded for J120 and the algorithm moved from top 1 to top 5. For J30 and J60 sets its performance stayed on the same level. The lower performance can be explained by the fact that due to having a need to maintain several sub-populations in parallel and being restricted by the amount of objective evaluations that can be performed, the amount of times Lévy

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flight can be applied to improve one solution has decreased. Nevertheless, such trade-off in performance is understandable and has been discussed in many publications (Wolpert & Macready, 1997). However, DSCCS achieved what other algorithms could not achieve in finding multiple solutions for RCPSPs.

During algorithm testing, depending on the test instance, the number of candidate solutions varied from 1 to 22, 1 to 60, and 1 to 21 for J30, J60, and J120 test instances, respectively. The number of solutions primarily depended on the value of the species distances σs: for smaller value of σs the algorithm was

capable to obtain more solutions and vice versa. It is also worth mentioning that the DSCCS was able to find multiple optimal (or best-known) solutions in the majority of all test instances with high success rate.

d) Extension of the RCPSP by introduction of efficiency and learnability factors to resources

In order to take advantage of the RCPSP multimodal properties, the HARNet Project Management Problem (HPMP) is proposed. HPMP can be characterised as a special case of the RCPSP and it represents a model for scheduling large and complex projects that are new to their execution environment. The main difference of HPMP from other variants of the RCPSPs is that throughout the project execution, as the resources are consumed by activities, the resources gain experience which then can influence the durations of activities. The rate at which resources acquire experience is defined by their learnability coefficient. The maximum amount of time by which the duration of the activity can be reduced is defined by the effectiveness coefficient. To take the advantage of RCPSP’s multimodal property, for successful completion HPMP considers optimisation of two objectives: primary – makespan minimisation, and secondary – resource efficiency balancing. The optimisation of these objectives is achieved as follows: firstly, the applied algorithm acquires a set of candidate solutions with the shortest makespan; secondly, out of the acquired set, the algorithm chooses the most suitable solution with the lowest resource efficiency-balancing coefficient. Nevertheless, HPMP can also be solved by the methodology developed for standard RCPSP; however, in this case, only one objective can be optimised at a time.

A application scenario HPMP was devoted to the planning and management of the large-scale projects (i.e. software development) where activities were executed through different resources (i.e. groups of researchers and developers)

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that have no, or have very little of, relevant experience. In the beginning of the execution of such projects, there is a high level of uncertainties caused by the lack of knowledge and expertise. However, as the project goes on, people practice relevant skills and, as the result, the level of uncertainties vanishes and the duration of activities reduces.

To assess applicability of the proposed optimisation model in real-life scenario and further evaluate performance of the DSCCS two case studies are created. First case study represents a scheduling of the practical project. The considered project is the real-world project that was undertaken a year ago. It consisted of 51 interrelated activities execution of which required 6 types of resources. Four of these resources are groups of people, other two types of resources are specialised equipment. For this case study, two sets of experiments are conducted, which cover the production of normal deterministic and variable stochastic schedules. The results of these experiments are then compared and analysed. For the second case study, several most popular methodologies for the RCPSP are implemented. Further, the performances of the implemented methodologies, along with the DSCCS are evaluated on sets of the edited benchmark instances. The main difference of these benchmarks from their standard variants is the inclusion of two additional parameters needed for the proposed optimisation model: learnability and effectiveness coefficients. As the results of evaluation, the DSCCS has greatly outperformed all other implemented algorithms, however, its computational time the worst among all. Nevertheless, these case studies demonstrated the operability and effectiveness of the proposed optimisation model: showed how the activity durations may decrease depending on the time of execution and applied resource, and verified the relative ease of application RCPSP methodologies to it. Moreover, they confirmed the great performance of the DSCCS, its capability to obtain multiple solutions, and its applicability in scheduling of real-life projects.