Chapter 3 Optimisation Model
3.5 Optimisation Problem
Herein, for optimisation of the objectives, the problem exploits multimodal properties of the RCPSP. First, the problem is solved via optimisation of the objective f1 by finding a set of best solutions with minimal makespan. Then, for
each of the solutions in this set, the second objective f2 is calculated.
Therefore, the best way of approaching this problem is via application of a metaheuristic algorithm specifically designed for tackling multimodal optimisation
problems and that is capable of obtaining multiple global solutions. For the reasons of having multiple objectives (one primary and one secondary), first, the algorithm will obtain a set of best schedules with minimal makespan, thus completing objective f1. Then, out of this set, the most efficient schedule is going
to be chosen, satisfying objective f2.
Following the above definition, the problem can be formalised as follows:
Min: P C P p p n
1 1 (18)
R k k 1 (19) Subject to:
t V j k jk R r Rk t0 (20) Sj Si pi (i, j)E (21)where Vt is as defined in section 3.2.
The presented problem is a variation of the RCPSP and can be regarded as a NP-hard combinatorial optimisation problem (Blazewicz, Lenstra, & Kan, 1983) with a complex multimodal fitness landscape (Czogalla & Fink, 2009).
3.6 Summary
Optimisation model, namely HPMP, presented in this chapter represents a special case of the RCPSP in which activity durations follow probability distribution model that is dependent on resource efficiency, experience, and learnability of resources, where resource in the HPMP case refers to human resource, i.e. members of the project team. Resource efficiency reflects the speed at which an activity can be implemented by the project team; experience is the total amount of time that members of the project team have previously spent on working on a similar problem; and learnability is the reflection of how quickly resource acquires its experience. As the result, the duration of an activity may be shortened with the increase of resource efficiency.
The HPMP considers optimisation of two objectives: minimisation of makespan (primary) and balance of resource efficiency (secondary).
Optimisation of these objectives exploited multimodal properties of RCPSPs and is achieved via application of metaheuristic algorithm tailored specifically for multimodal optimisation problems and that is capable of obtaining multiple solution candidates. First, the algorithm obtains set of best solutions with minimal makespan. Then, from this set, the algorithm selects the most efficiently balanced one.
Being variation of the RCPSP, the problem is NP-hard combinatorial optimisation problem with a complex multimodal fitness landscape. Therefore, the application of metaheuristic algorithms for multimodal optimisation problems is justified.
Chapter 4 Methodologies
This chapter presents four new algorithms developed during the course of this PhD study, namely
Discrete Cuckoo Search (DCS) algorithm Discrete Flower Pollination Algorithm (DFPA)
Improved Discrete Cuckoo Search (IDCS) algorithm
Discrete Species Conserving Cuckoo Search (DSCCS) algorithm
The Discrete Cuckoo Search (DCS) and Discrete Flower Pollination Algorithm (DFPA) are population-based metaheuristic algorithms adapted from the Cuckoo Search (CS) (Yang & Deb, 2009) and Flower Pollination Algorithm (FPA) (Yang X.-S. , 2012). Previously, in the majority of cases CS and FPA had only been applied to problems in the continuous domain and demonstrated to be very effective in finding global optima with high success rate and, in some cases, even managed to outperform such popular metaheuristics as Genetic Algorithm (GA) and Particle Swarm Optimisation (PSO) in terms of efficiency and success rate. Nevertheless, at the time of writing this thesis, CS and FPA had only limited number of applications to optimisation problems in the discrete domain (Yang X. , 2010). The most prominent examples are the travelling salesman problem (TSP) (Ouaarab, Ahiod, & Yang, 2013) and the annual crop-planning problem (Chetty & Adewumi, 2013). In both of these examples, the algorithms showed competitve levels of performance, which validated their applicability for optimation problems in the discrete domain.
Performance of DCS and DFPA is evaluted using benchmark instances from Project Scheduling Problems Library (PSPLIB) (Kolisch & Sprecher, 1997). The results of evaluation can be regarded as satisfactory as the algorithms were able to outperform such heuristics as GA and Simulated Annealing (SA). Nevertheless, their performance can be furtherly improved by addressing some of the limitations that these algorithms have: inefficienct solutions representaions scheme and use of context-free operators. To address these limitations, the Improved Discrete Cuckoo Search (IDCS) is introduced in the next section. IDCS introduces several changes to the original DCS paradigm:
addition of a new mechanic aimed at improving the quality of received results but with less iterations;
new solution representation scheme specific for RCPSP and its stochastic variant; and
novel local search and crossover operators, based on the newly- introduced solution representation scheme.
Similarly to DCS and DFPA, performance of IDCS is testing using benchmark instances from PSPLIB. This time, the results of performance evaluation are compared against state-of-the-art heuristics for RCPSP where IDCS was able to appear in top ranks. Nevertheless, one of the limitations of IDCS is the inability to obtain multiple solutions candidates at once, hence Discrete Species Conserving Cuckoo Search (DSCCS) is introduced in the next section.
DSCCS is the result of integration of the Species Conservation (SC) (Li, Balazs, & Parks, 2002) technique into IDCS. SC technique is a method of evolving parallel sub-populations integration of which allows the algorithm to obtain multiple global solutions. The technique is based on distributed elitism, achieved by identifying in each generation a set of prime individuals that are considered to be worth preserving into the next generation. The formation of species allows to divide the search space into smaller regions, making each species focused on searching for solutions within the specified region. This creates an opportunity for a finer search for a local best optimum, provides higher chances of finding global optima, as well as enables the algorithm to obtain multiple solution candidates, thus making it suitable for applications in multimodal scenarios.