A BASIC EVOLUTIONARY ALGORITHM FOR THE PROJECT STAFFING PROBLEM

GHENT UNIVERSITY FACULTY OF ECONOMICS AND BUSINESS ADMINISTRATION YEARS 2013 2014 A BASIC EVOLUTIONARY ALGORITHM FOR THE PROJECT STAFFING PROBLEM Master thesis presented in order to acquire the degree of Master of Science in Applied Economics: Business Engineering Piet Peene Under the management of Prof. Dr. Broos Maenhout

GHENT UNIVERSITY FACULTY OF ECONOMICS AND BUSINESS ADMINISTRATION YEARS 2013 2014 A BASIC GENETIC ALGORITHM FOR THE PROJECT STAFFING PROBLEM Master thesis presented in order to acquire the degree of Master of Science in Applied Economics: Business Engineering Piet Peene Under the management of Prof. Dr. Broos Maenhout

Permission Undersigned declares that the content of this master thesis may be consulted and reproduced, when referencing to it. Piet Peene

I. Preface This master thesis denotes the end of my studies in Business Engineering, Operations Management at the University of Ghent. It is the conclusion of a fascinating road through the fields of knowledge in operations management. However, exploring these paths sometimes presented unforeseen challenges. Being able to overcome these challenges will only strengthen motivation and courage towards future trials. May that be an important lesson I have learned in the process. In my opinion, writing a thesis is a long-haul task in a subject of personal interest. My spark of interest in mathematical modelling was ignited when taking a 3 rd Bachelor class in Operations Research taught by Prof. Dr. Broos Maenhout. Further classes in the master s degree that broadened my interest in the subject of planning and scheduling included Project Management and Applied Operations Research, taught by Prof. Dr. Mario Vanhoucke. A thesis on the project scheduling and staffing problem is therefore a perfect match with my interests. This thesis required a lot of effort, but it would not be accomplished without the help and support of others. Special thanks go to my promoter Prof. Dr. Broos Maenhout, for guiding me through the process, offering advice and working material, not to mention his flexibility in scheduling of consultation meetings, even outside regular working hours. Furthermore I owe my parents, Annie Lips and Yves Peene, and my sister Tine and her boyfriend Geert Depuydt many thanks for the love and support they provided, not only during the making of this thesis but throughout the completion of my higher education. I also want to thank my friends, especially Annelies Deleersnyder, Arno Wallays and Michelle Vu, whom I got to know during my time at the university. They were always available for mental support and distractions. I

II. Table Of Content 1. Introduction... 1 2. Problem description and model formulation... 4 2.1. Project Scheduling Problem Area... 4 2.2. Problem description... 6 2.2.1. Project scheduling problem description... 6 2.2.2. Project staffing problem description... 8 2.3. Mathematical model formulation... 10 3. Methodology... 14 4. Literature Overview... 15 4.1. Genetic algorithms... 15 4.1.1. What are genetic algorithms... 15 4.1.2. Genetic algorithm framework... 16 4.2. Data representation... 19 4.2.1. Project Schedule representation... 19 4.2.2. Project Staffing representation... 24 4.3. Solution methods... 26 4.3.1. Initialization... 26 4.3.2. Selection... 29 4.3.3. Operation... 32 4.3.4. Local Optimization... 37 4.3.5. (Partial) Evaluation... 41 4.3.6. Reinsert... 44 4.3.7. Ending condition... 47 5. The algorithm... 49 6. Computational experiments... 50 6.1. Observation link AVGSQDEV Total cost... 50 6.2. Benchmark... 51 6.3. Results... 54 6.3.1. Basic cycles... 55 6.3.2. Stage contributions... 58 6.3.3. Sensitivity Analysis... 59 7. Conclusions and further research... 69 II

III. List of figures Figure 1 Extended Project Management triangle... 1 Figure 2 Flowchart Genetic Algorithm, framework 1... 16 Figure 3 Flowchart Genetic Algorithm, framework 2... 17 Figure 4 Activity-on-the-node project schedule activity network (AN2)... 20 Figure 5 Example project schedule, PS2 (based on AN2)... 23 Figure 7 Labor supply and demand for AN2, PS2... 25 Figure 6 Flowchart Genetic Algorithm, framework 2... 26 Figure 8 Pseudo code Initialization methods... 28 Figure 9 Example Roulette Wheel... 31 Figure 10 Pseudo code Roulette Wheel Selection... 31 Figure 11 Pseude code Tournament Selection... 32 Figure 12 Pseudo code Blend CrossOver... 35 Figure 13 Local search simplified... 38 Figure 14 Pseudo code LS1 Burgess and Killebrew simplified RLP... 39 Figure 15 Pseude code Double-Justification... 40 Figure 16 Pseudo code 2-exchange neighborhood... 41 Figure 17 Average squared deviation of resource consumption... 43 Figure 18 AVGSQDEV - Total Cost dispersion for all project lengths... 50 Figure 19 AVGSQDEV - Total Cost dispersion for project length of 11 days... 51 Figure 20 Benchmark setup... 52 Figure 21 Lower bound benchmark in function of project duration... 53 Figure 22 Total Cost evolution basic cycle AN2 Framework1... 56 Figure 23 Total Cost evolution basic cycle AN2 Framework2... 56 Figure 24 Total cost Vs number of operations for different population sizes... 60 Figure 25 Total cost Vs population size for different number of operations... 61 Figure 26 Trade-Off Total Cost Vs. Execution Time... 62 Figure 27 Efficient Frontier Total Cost Vs. Execution Time... 63 Figure 28 Efficient Frontier Total Cost Vs. Execution Time, with / without doubles.. 65 Figure 29 Total Cost Vs. Number of Iterations... 65 Figure 30 Total Cost Vs. Mutation Percentage... 66 Figure 31 BSP Vs number of operations... 67 Figure 32 Total cost Vs ending condition... 68 Figure 33 Performed operations Vs ending condition... 68 III

IV. List of tables Table 1 Example Duration Vector... 21 Table 2 Absolute Starting Times... 21 Table 3 Relative Starting Times... 22 Table 4 Resulting Starting Times... 22 Table 5 Work patterns forming labor supply... 24 Table 6 Total Cost evolution basic cycle AN2 Framework1... 55 Table 7 Total Cost evolution basic cycle AN2 Framework2... 56 Table 8 Best solution methods, total cost and lower bound per basic cycle... 57 Table 9Total cost comparison with and without doubles in the population... 64 IV

V. Abbreviations AN AVG AVGSQDEV BSP CP GA PS RACP RCPSP RLP RRP SPI TSP Activity network Average Average squared deviation Best schedule percentage Critical path Genetic algorithm Project schedule Resource availability cost problem Resource-constrained project scheduling problem Resource levelling problem Resource renting problem Serial parallel indicator Travelling salesman problem V

1. Introduction This thesis deals with a project scheduling and staffing problem. It fits in the functional area of project management. Many people and institutions have tried to give a meaningful definition of project management. The Association of Project Management (APM) came up with an apt definition stating project management is the planning, organisation, monitoring and control of all aspects of a project and the motivation of all involved to achieve the project objectives safely and within agreed time, cost and performance criteria. The project manager is the single point of responsibility for achieving this. (APM BOK, 1995) A project is a temporary endeavour undertaken to create a unique product, service or result. (PMBOK, 2004) This very short definition of a project is very meaningful in two of its words, i.e. temporary and unique. Temporary indicates that there is a well-defined start and end. Unique signifies that there is no predefined scheme for executing the project. However there may be similarities to previous projects. A project consists of multiple tasks or activities that need to be executed in a certain order, this order and the precedence relations between the activities are defined in an activity network. The actual timing of all activities is defined in a project schedule. A classic combination of criteria, measuring success or failure of a project is depicted in the iron triangle or the project management triangle. (Atkinson, 1999) We made an extended version of the triangle including the link of project management with project staffing and project scheduling in figure 1. The triangle has a performance measure on each of its corner-points. The scope contains the content of the project, what should be done. The cost refers to the budget of the project. Time refers to the amount of time to complete the whole project. It is often stated that if one of the measures is Figure 1 Extended Project Management triangle 1

altered, it will have an impact on the other two. For example, when extending the scope of a project, the cost and time are likely to increase as well. However, these relations are not necessarily strict meaning that a decrease in time does not necessarily mean an increase or decrease in cost (cfr. non-regular objectives of performance). The goal of the thesis is to find an intelligent way to construct a schedule of activities that has the lowest staffing cost. This construction of a schedule is called project scheduling. Every activity in the schedule has a certain need for resources. In our research, these resources are labor. Project scheduling creates a demand of labor over the span of the project and results in a project make span (time criterion in figure 1). However this demand cannot always be met exactly by the supply of labor, which is defined by project staffing. Once the schedule and resource needs are known, the project staffing can be executed. This project staffing results in the supply of resources and gives the eventual staffing cost. A close match between supply and demand of resources is more likely to result in a lower staffing cost. This matching of supply and demand and its link to project scheduling and project staffing is shown in the bottom part of figure 1. In order to attain the goal of constructing a good schedule, we implement a basic evolutionary algorithm, coded in C++. This research and its coded implementation are subject to various limitations. Since the focus here is mainly on the quantitative aspects of project staffing and project scheduling, the qualitative aspects are being neglected. An example of the qualitative aspect is job satisfaction or the loss of it resulting from irregular or acyclic working patterns, including overtime and idle time. Besides the lack of qualitative information, also some quantitative aspects show deficiencies. These deficiencies are mainly due to the many assumptions included into the modelling. Examples include the assumption that the time and resource consumption of each activity is exact and known a priori, and costs assigned to the different types of labor time is only an estimation. However these assumptions and estimations are vital to the construction of a mathematical model and are being set to resemble reality as close as possible. 2

There are also practical limitations towards the execution of the algorithm, in the sense that available computing power sets a boundary to this research. Although computing power is ever increasing, computationally testing the possible combinations to construct the algorithm is still limited. To conclude this introductory chapter, a brief overview of the structure of this thesis is given. Chapter two will dig deeper into project staffing, project scheduling and the interactions between them. The chapter concludes with an unambiguous definition of both problems and their mathematical model. Chapter three presents the methodology, i.e. the way in which the research is conducted. Chapter four provides a literature overview on the different types of solution algorithms and its building blocks. Chapter five describes the algorithm that proves to be the most performant. In chapter six, the computational results are given in combination with general observations and calculation of a benchmark. A conclusion and recommendations for further research are presented in chapter seven. 3

2. Problem description and model formulation The first chapter situates our problem in the functional area of project management. The second chapter will define the problem more in detail. In the first section, the problem is put into the bigger picture by describing similar problems. The second section gives the specific problem description of our problem that will be used throughout the remainder of this thesis. The third and final section translates the problem description into a mathematical model. 2.1. Project Scheduling Problem Area In project scheduling, a set of activities need to be scheduled meaning a start time has to be assigned to all activities. The project scheduling problem has been widely researched. Overviews and classification methods for the scheduling problem are given by Icmeli et al. (1993), Elmaghraby (1995), Herroelen et al (1997, 1998, 1999), Brucker et al. (1999) and Hartmann and Briskorn (2010). Different classifications can be made based upon the difference in characteristics between the problems. The use of a different type of resource, i.e. renewable or non-renewable leads to a different kind of problem as well as the activity characteristics and type of scheduling objective. Examples of objectives for the project scheduling problem are minimization of the duration of the project, levelling resources over the course of the project, minimizing resource idle time and maximizing the net present value. The most popular problems are the resource-constrained project scheduling problem or RCPSP. The goal of this problem is to minimize the total length of the project taking into account a certain renewable resource constraint. Other similar problems are the resource availability cost problem (RACP), the resource levelling problem (RLP), the time-constrained project scheduling problem (TCPSP) and resource renting problem (RRP). The RACP aims at minimizing the total cost of the unlimited renewable resources required to complete the project before a certain deadline. The RLP has the objective to schedule the activities such that the resulting resource demand over the span of the project is as levelled as possible. The TCPSP aims at meeting project deadlines, starting with a fixed capacity of resources. In order to meet the deadlines, decisions have to be made concerning working overtime and hiring additional 4

resources to enlarge the existing fixed capacity. The RLP has as objective to minimize the renting costs incurred by renewable resource, these costs concern both fixed and variable renting costs. After the project activities are scheduled, the staffing needs to foresee sufficient labor resources to carry out the resource demand of the schedule. The main objective is to minimize total staffing cost of the project. Total costs are the sum of the cost of regular personnel, cost of overtime, cost of idle time and cost of temporal personnel. This problem objective is often referred to as the deadline problem (brucker et al, 1999), meaning that there is a given deadline on the makespan of the project and the goal is to find a feasible schedule that minimizes the costs. This is opposed to the budget problem where one is given a certain budget and needs to find a feasible schedule that minimizes the makespan. The staffing problem has already been solved deterministically by Maenhout and Vanhoucke (2014). This thesis will focus on the project scheduling problem. The goal of this thesis is to develop an algorithm that generates a project schedule that minimizes the staffing costs. However, it is important to note that a shorter make span or more levelled resource usage does not necessarily mean a lower total cost of the project. Therefore, translating the global objective into an intermediate objective for the scheduling problem is not straightforward. 5

2.2. Problem description This section handles the problem description of both the project scheduling and project staffing problem. There is not a single formulation of these problems. The basic idea is always common, but subtle deviations can be made to the goal or the constraints of the problem, making it seemingly result in a totally different problem. 2.2.1. Project scheduling problem description The basic idea of project scheduling is to determine a start time for each activity in the project activity network. The assignment of these start times are not random but should serve a goal. The overall goal is to produce a schedule that provides the lowest personnel staffing cost. This cost determination is not part of the scheduling process but part of the staffing process. There is no direct translation of this staffing cost goal to a goal for the scheduling problem. As an intermediate goal that gives an approximation for the staffing goal, we set a resource levelling objective for the scheduling problem. This makes the project scheduling problem resemble the resource levelling problem (RLP) as discussed in section 2.1.1. The RLP has a non regular measurement of performance; it has no early completion measure. (Neumann and Zimmermann, 1999) Besides the objective of the problem under consideration, the scheduling constraints that are active have an influence on the problem definition. These constraints can be derived from either activity characteristics or resource characteristics. (Herroelen et al., 1997) Activity characteristics: No pre-emption (1) Finish-start precedence relations (2) Fixed and discrete duration per activity (3) Predefined project deadline (4) Acitivity resource needs: constant and discrete (5) Single execution mode (6) 6

Pre-emption or splitting of an activity is not allowed.(1) Pre-emption means that if an activity is started, it can be interrupted at some point in time, to resume later. Preemption brings more flexibility in the schedule and thus adds complexity. The scheduling of the activity is constrained to precedence relations. (2) This means that a certain order of execution of the activities needs to be maintained. This order is determined by the activity network. The only type of precedence relation used, is the basic PERT/CPM finish-start precedence relation. This means that the previous activity in the network has to finish first before the next activity can be started. Other precedence relations, referred to as generalized precedence relations, such as startstart, start-finish and finish-finish precedence relations are not used. Also the use of minimal and maximal time lags is omitted, for the sake of complexity. A start-start precedence relationship with a minimal time lag of three days means that the next activity can start three days or later than the start of the previous activity. The duration of an activity is known in advance and has an integer value. This means that the duration does not depend on a stochastic process or events on prior activities. Forcing the activities to have integer durations simplifies the calculations. A predefined project deadline of 21 days is applied, for technical reasons not functional. (4) We did not use a deadline as a relative percentage of the critical path since the critical paths of the activity networks under consideration differ heavily. This would result in strangling the solution space for the activity network with small critical path and creating an abundant solution space size for the activity network with a large critical path. The activity resource needs are constant, meaning that over the course of an activity, the resource demand for each time unit is equal. (5) The activity resource demand is integer for the same reason the activity durations are integer values. Contrary to discrete and constant, resource needs could be continuous and the amount necessary could be a function of the duration. There is only a single execution mode for the activities. (6) Multiple activity modes would imply the possibility of executing one activity or a subset of activities to be executed in different ways, possibly incurring different costs. Resource characteristics: Single resource (7) One resource type: renewable resource (8) Variable availability of resources (9) 7

The resource used for executing the activities is labor. Every unit of labor is assumed to be equal, the labor units do not require different skill levels.(7) Concerning the resource constraints, we consider only one resource type in our problem. This resource type is a renewable resource. (8) A renewable resource is a resource that gets renewed from period to period. Besides labor, machines are another example of a renewable resource. Examples of non-renewable resources are materials, energy and money; once they are used, they are gone. The availability of resources is variable, and defined by the staffing problem. (9) The amount of labor available at each time unit depends on the number of workers employed and their different working patterns. 2.2.2. Project staffing problem description The basic idea of project staffing is to find the combination of working patterns that covers the resource needs of an activity schedule. Each work pattern is a serial string of work days and days off and is executed by a single worker. The work patterns are non-cyclic, meaning that there is no predefined and reoccurring pattern of days off and days on. Not all patterns are allowed however, there are minimum and maximum constraints defined on the number of consecutive days off and consecutive days on. Minimum consecutive days on 2 Maximum consecutive days on 6 Minimum consecutive days off 1 (does not result in an actual constraint) Maximum consecutive days off 2 The goal of the staffing problem is to find the combination of working patters that satisfies the resource needs of the activity schedule and minimizes the labor costs incurred by the staffing. An overview of the different costs and their weights are presented below. (Maenhout & Vanhoucke, 2014) Regular personnel time units 2 Overtime units 3 Temporal personnel time units 4 Idle time units 1 8

The cost of regular personnel time units is a variable cost, depending on the project makespan. It does not take into account the actual number of days worked. Each project day incurs a cost of 2. A work pattern is subdivided into several periods, each containing seven days. A regular period of seven days has five days on and two days off. If a work pattern has an extra day on, on top of these five days, an overtime unit cost is incurred. For every work pattern, its cost can be calculated based upon the regular time units and the overtime units. The cost of a work pattern is thus known a priori, before the actual staffing takes place. The other two costs, i.e. temporal personnel units and idle time units are costs that come as a result of the combination of several work patterns. If the combination of work patterns does not supply enough labor on a certain day, external labor has to be hired for that day, incurring a temporal cost per extra unit. If however the combination of work patterns supply excess labor on a certain day, a penalty cost per idle time unit is added. 9

2.3. Mathematical model formulation A mathematical formulation of the scheduling and staffing problem leaves no room for misconception and has the potency of representing the problem concisely. The notation will be explained in the form of sets, input data and decision variables (Maenhout and Vanhoucke, 2010). Sets are well-defined groups of elements that have common characteristics. An individual element in a set is recognized by its index. A set will be denoted by a capital letter. If W is the set of workers, w1 represents the first individual worker in the set. Input data is static data and known on beforehand. It is important that this data is as close to reality as possible since it will have a great impact on the behavior of the algorithm and ultimately on the results. Decision variables are the unknown factor in the model. Solving the mathematical problem means determining the value of these decision variables. Sets W set of workers (index i) T set of days in the scheduling horizn (index t) A set of activities in the project (index j) Input Data c r c o c x c l d j p pj r j PD cost per worker per day cost per worker per day of overtime cost per worker per day of outsourced labor cost per worker per day of idle time duration of activity j 1 if p is a predecessing activity of activity j number of resources necessary to execute activity j project deadline DO min minimum consecutive days off DO max maximum consecutive days off DW min minimum consecutive days working DW max maximum consecutive days working 10

Decision Variables st j PL starting time of activity j resulting parameter: a jt, 1 if activity j is performed on day t project schedule length r w it o w it x w t l w t 1 if worker i works a regular shift on day t 1 if worker i works an overtime shift on day t number of workers outsourced externally on day t number of workers in excess on day t The actual model consists of an objective function and constraints. The objective function represents the ultimate goal. This could be the minimization of the make span, the levelling of work load, the maximization of profits etc. In this case the objective is to minimize the total personnel staffing costs. Underneath the objective function, a number of constraints are formulated. These constraints find their origin in a regulatory domain, (e.g. the number of allowed consecutive working days) or are driven by feasibility boundaries (e.g. activity 1 needs to be completed before activity 2 can start). Objective function (1) Subject to constraints (2) (3) (4) (5) (6) (7) (8) 11

(9a) (9b) (10) (11) (12) (13) (14) The objective function (1) represents the total personnel cost of the project. It can be broken down into 4 parts. The first part is a cost incurred for each worker in a regular schedule. The cost is independent of the actual number of days worked but is completely dependent of the length of the project schedule. It represents a fixed cost per hired worker. The second part of the cost calculation accounts for the overtime units. A worker is supposed to work a normal schedule of five days per week. However when he works more, an extra cost is incurred on top of the regular cost. The third part is a cost related to outsourcing extra workers. Sometimes, the regular hires can t carry all the workload thus extra workers will be sourced from outside. This has the advantage that there is no fixed cost over the whole length of the project. However these units of work are usually more expensive than regular or overtime work units. The last part of the cost function is a cost that represents the excess supply of workers. When there are more workers available than necessary for the amount of work, an extra cost is incurred. The first constraint equation (2) shows the connection between the project scheduling and the project staffing. The project scheduling results in a demand of labor for each day of the project, which is represented on the left-hand side of the equation. The right-hand side of the equation represents the supply of labor for each 12

day of the project, which is the result of the project staffing. Supply and demand of labor need to be in balance on a daily basis. If no balance can be found between the demand of labor and the supply generated by regular and overtime work units of the hired workers, extra outsourced labor or excess labor will rectify the total balance. Equations (3) - (6) are constraints exclusively related to the project scheduling problem. (3) is the mathematical representation of the finish-start precedence relations. The fourth equation forces the non-preemptive nature of the activities in the project schedule. The project length (5) is defined as the end of the last activity. This project length is bound to a certain predetermined project deadline (6). Equations (7) - (12) are constraints exclusively related to the project staffing problem. The seventh equation enforces that a worker can execute a regular work unit or an overtime work unit but never both. On one day, either extra workers get outsourced or there is excess labor, or none of both. (8) It does not make any sense to attract external workforce while there are still regular workers available. Constraints (9) (12) represent working agreements between the employees and the employer. Constraint (9) and (10) ensures the minimum and maximum number of consecutive days off work, respectively while (11) and (12) ensures the minimum and maximum number of consecutive days a worker is allowed to work. Equation (13) limits certain variables to a binary value and other variables to positive integers (14). 13

3. Methodology The problem explained in the previous chapters shows a complex interaction of two subproblems. A structural approach towards the solution of the problem is essential, and basic assumptions are necessary to limit its complexity. As mentioned before, the project staffing problem is perceived as a given and focus will go almost entirely to the project scheduling problem. First, the project activity network dataset will be described in more depth. This is important to show that we are solving a real life problem and not an abstract theoretical problem. Furthermore, it must be noted that there is no single best method to solve all kinds of different project activity networks. The characteristics of these networks might play an important role in the final selected solution method. In a second phase, existing basic genetic algorithm methods from the literature will be discussed. These methods can be grouped into generation methods, selection methods, operation methods, optimisation methods, reinsert methods and population management methods. These are picked from a broad range of applications, not limited to the project scheduling problem. This phase is concluded by placing these methods into a framework of a genetic algorithm. The third phase consists of programming the methods in C++ and to connect them in a logical way. The resulting program will then be executed in several runs on three prototype datasets. Every run will narrow the number of methods by either excluding the weakest methods or by withholding only the best methods. For each prototype dataset, a genetic algorithm will be formulated. As genetic algorithms do not promise an optimal solution, the goal is to reach a good solution. It is possible to determine an upper and lower bound for the cost objective function. These two values will then be used for benchmarking purposes of the proposed genetic algorithms. Besides benchmark testing, we will also perform tests to determine the effectivity of each method. General parameters will be changed to check the influence of these parameters on the applied algorithm and its results. 14

4. Literature Overview In chapter four, the literature overview is given. The first section gives an introduction on genetic algorithms and presents the genetic algorithm frameworks. The second section shows how the data is represented. The third and last section will elaborate in depth on the different solution methods that are applied within the genetic algorithm framework. 4.1. Genetic algorithms 4.1.1. What are genetic algorithms A genetic algorithm (Holland, 1975) is a heuristic that imitates human nature of evolution to find good (but often suboptimal) solutions to a problem in a reasonable amount of time. To the contrary there are exact solution methods which will always come up with the optimal solution. The genetic algorithm however intelligently exploits the random search. It has been proven that genetic algorithms, in combination with local search, simulated annealing or tabu searchn provide very good solutions among Heuristics. (Brucker, 1999) We will add local search to genetic algorithm framework. Genetic algorithms are population based algorithms, i.e. they work on a set of solutions. The general procedure is described below. Firstly, an initial population of member solutions is generated (1). Out of this population, parents will be selected for mating(2). The parents will be combined in a certain way to generate new solutions (=mating), called the children (3). The children will enter the population and a new cycle can start from the selection procedure. Step 2 and 3 will be repeated until an ending condition is reached (4). The underlying principal is survival of the fittest. This means that the stronger members of the population will survive while the inferior members will be eliminated. The next sections will go deeper into this general setup. 15

4.1.2. Genetic algorithm framework This subsection discusses the integration of the project scheduling and project staffing problem and will mold it into a genetic algorithm framework. The project scheduling problem is leading throughout the execution of the genetic algorithm. The staffing problem is called at the appropriate time when evaluation of the resulting schedule is needed. The integration of the scheduling and staffing problem is molded into two slightly different forms of genetic algorithms, presented in two diagrams in figure 2 and figure 3. The first framework consists of an initialisation phase, a selection phase and an operation phase. The obtained child after the operation phase will undergo local optimization where the algorithm will look for incremental improvements in the neighborhood of the child. This local optimization comes with partial evaluation to evaluate every instance of the explored neighborhood. We call it partial evaluation because the actual objective function is never calculated in this phase. Another measurement, which is a close approximation of the objective Figure 2 Flowchart Genetic Algorithm, framework 1 function under certain conditions, will be calculated in this phase. The reason for this partial evaluation is that the regular evaluation, which includes calculating the objective function through calling the staffing algorithm, consumes a considerable amount of time. This would extend the execution time of the algorithm unnecessarily. After the local optimization, only one schedule which is assumed to be the best, based upon the partial evaluation, will undergo the complete evaluation phase, including the staffing part. After the evaluation, a decision has to be made whether the newly generated and optimized 16

child can enter the population. This decision is made in the reinsert phase. The last phase checks an ending condition. If a certain ending condition is reached, the execution will stop and the best schedule found up to that moment will be the output of the algorithm. If the ending condition is not yet reached, the phases described above will be repeated starting from the selection phase. The advantage of this form of the algorithm is that it does not allow deterioration of the resulting schedule, i.e. the outcome of the schedule at the end. This is because at the end of each cycle, the objective function is calculated and the best schedule is stored. The biggest disadvantage of this form of the algorithm is that still an evaluation is being performed during each cycle, which consumes a considerate amount of time. This is the reason why an alternative framework is formulated, represented in figure 3. The only difference between the second and the first framework is that the second framework postpones the evaluation phase until the very end. This second framework will not perform the evaluation in every cycle and thus save lots of execution time. In this Figure 3 Flowchart Genetic Algorithm, framework 2 case the quality of the schedules in the population is entirely controlled by the partial evaluation. When the ending condition is reached, the evaluation will be executed on every schedule in the population. The biggest advantage of this framework is the amount of time that can be saved in each cycle. The disadvantage however is that it is possible that the best schedule present in the population gets replaced by another one during the execution of the agorithm. Because only at the very end, we will know which schedule is the best. Before that, we rely on an approximation of the objective function to get an indication of which schedule will 17

probably be good, and thus should not be replaced and which schedule is bad and thus should be replaced. It is expected that the second framework yields an inferior quality of the schedules. 18

4.2. Data representation The way in which the data is represented can have a great influence on the range of methods that can be applied. In this section the data representation for both scheduling and staffing problem will be shown. 4.2.1. Project Schedule representation Good project management starts with a solid representation of the project schedule. A good tool for this is PERT, the Project Evaluation and Review Technique. (Cottrell, 1999) It is used for analyzing and representing activities in a project and was first developed in the late 1950s by the U.S. Navy as a tool for measuring and controlling the development progress for the Polaris Fleet Ballistic Missile program. (Malcolm, 1959) The method perceives a project as a network of activities and events. An activity network shows the activities and the relations between them, often referred to as precedence relations. There are two types of activity networks, an activity-on-the-node (AON) network and an activity-on-the-arc (AOA) network representation. Activity-on-the-arc In this representation, each arc or arrow represents an activity or a task. The nodes define a milestone which is achieved when all activities on the arrows leading up to this node are completed. Dummy arcs can be introduced to enforce additional precedence relations. Activity-on-the-node In this representation, each node represents an activity or a certain task that has to be executed. The arcs or arrows represent the precedence relation. Figure 4 shows an example of such a network. Each node gets an activity number inside the node, the duration of the node is put on top and the necessary labor to execute the activity is put below the node. The network clearly visualizes that activity five can only be executed when both activity three and four have been executed. Activity five is called the successor of activities three and four, activities three and four are a predecessor of activity five. The activity-on-the-node network has two dummy activities to start 19

and end the network, they don t consume any time nor resources. Their sole function is to have a clear single node at the beginning and end of the network. The network in Figure 4 will be used as an example in the next chapters. Figure 4 Activity-on-the-node project schedule activity network (AN2) In order to limit the complexity, it is assumed that the durations of the activities are deterministic. PERT often takes a certain variance on the duration of an activity into account when analyzing project schedules. Another complexity limiting factor is the type of relationships that are used. In this thesis, only finish-start relationships are considered. This means that the successor can only start when the predecessor has finished. When using generalized precedence relations, also start-start, finish-finish and start-finish relationships can be defined.(dawson, 1995) Their respective meanings are: the successor can start when the predecessor has started, the successor can finish when the predecessor is finished and the successor finishes when the predecessor starts. In combination with PERT, CPM or critical path method is often used. The critical path represents the group of activities that cannot be delayed without increasing the length of the project. It thus is the chain of activities that result in the minimal length of the project. Throughout this thesis we have used the activity-on-the-node network representation method since this emphasizes the activities rather than the milestones. Furthermore it is easier to interpret at first sight and there is no need to define any dummy 20

activities besides the start and end activity. Other advantages identified by Turner include the ease of drawing activity-on-the-node networks and the ability to write network software more easily and the independency. (Turner, 1993) Programming data representation The network can be translated or decoded into static and dynamic data. The static data includes the duration of the activities, the necessary resources for the execution of each activity and the precedence relations of the activities. These will remain identical throughout the scheduling process. The dynamic data are the starting times of each activity, these will change throughout the process and are the eventual outcome. Both the aforementioned static and dynamic data will be saved into vectors. i.e. a vector of durations, a vector of resource usages, a vector of successors and a vector of starting times. For example figure 4, this results into a vector represented in table 1. a1 a2... a12 d[aj] 0 5 1 4 2 2 2 2 3 4 4 0 Table 1 Example Duration Vector For the decoding of the activity starting times, there are two options. The first one consists of the absolute starting time of the activities and the second considers a relative starting time. (Wall, 1996) The first method is straightforward and states an exact starting time, independent of the starting time of other activities. Table 2 shows an example of a vector with absolute starting times. a1 a2... a12 st[aj] 1 4 6 8 9 14 10 18 8 17 19 19 Table 2 Absolute Starting Times Activity one starts at day one, activity two starts at day four and activity three starts at day six. The second method does not state an absolute starting time but rather a relative starting time of the activity i.e. the relative starting time indicates how many days there are between the start of an activity and the end of the latest predecessor. The vector of these relative starting times will be referred to as the float vector in the remainder of the thesis. 21

Table 3 shows how a float vector is represented. a1 a2... a12 fl[aj] 0 4 6 8 4 2 6 3 8 5 5 0 Table 3 Relative Starting Times This float vector has to be interpreted in combination with the precedence relations to result into the actual starting times of the activities. If you combine this with the network of figure 4, the resulting starting times are calculated in table 4. a1 a2... a12 st[aj] 0 4 15 8 20 11 6 19 30 27 27 33 Table 4 Resulting Starting Times The start time of activity three is the end time of the latest predecessor (activity two) which is nine. Add to this number the float value of six and you get the starting time for activity three, i.e. fifteen. We opted to go for the relative time representation for the simple reason that it is impossible to break any precedence relations constraint since it is imbedded into the definition of the float vector. If the absolute starting times are used, every schedule that is produced needs to be tested if any precedence relation is violated and if necessary repair these violations. An example of this is activity nine in table 2, which starts at day eight while its predecessing activity seven starts at day 10. table 2 is thus an example on an infeasible schedule. Dataset We execute the algorithm on three prototype activity network datasets. These activity networks contain the same activities, i.e. twelve activities including a dummy start and dummy end activity. Even the activity characteristics, concerning duration and resource demand are identical. The only aspect that is different between the three activity networks under research is the order in which the activities should be executed. This order is visually represented by the arrows as the precedence relations in the activity networks. Appendix A shows all three activity networks, further denoted as activity network AN1, AN2 and AN3. Note that AN2 is discussed previously in this section. These three networks are not chosen at random but have a distinctive topological structure. We want to test how the algorithm reacts to activity networks with a tendency to a very parallel structure compared to activity networks 22

with a tendency to a very serial structure. This is done by measuring the serial or parallel indicator as a topological indicator to measure the network structure. (Vanhoucke et al., 2008) This indicator has a value ranging from 0 to 1, with 0 meaning a complete parallel network structure and 1 meaning a complete serial network structure. This indicator (I) is calculated using the formula below. In this calculation, n indicates the number of activities excluding the dummy start and end node and m denotes the maximum progressive level of the network. (Elmaghraby, 1977) AN1, AN2 and AN3 have 0.11, 0.33 and 0.44 as respective values for the serial parallel indicator (SP indicator). These values seem very low. However when setting higher values, the tendency towards a serial structure is so overwhelming that there is very limited scheduling flexibility. When the SP indicator is 0, which means that all activities are in a parallel structure, the scheduling flexibility is maximal since there is not strict order in the activities. When the SP indicator is 1, which means that all activities are in a serial structure, there is no scheduling flexibility since the order of the activities is completely fixed. We assume that parallel networks have a broader solution space and offer more possibilities to the staffing of a project, possibly resulting in lower staffing costs. When the scheduling problem is solved, all activities received a start time and a project schedule can be printed. (Figure 5) The red line represents the labor demand per day, this is the demand that needs to be covered by the project staffing. Figure 5 Example project schedule, PS2 (based on AN2) 23

4.2.2. Project Staffing representation The project staffing representation revolves around the representation of the work pattern. This work pattern is a binary vector indicating whether a day in the pattern is a working day or a day off. An example of a work pattern and a combination of work patterns to form the labor supply is shown below. Pattern Workers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 w1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 w2 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 0 1 1 1 1 w3 2 1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 w4 2 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 w5 3 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 Staffing (Supply) 8 8 8 8 5 6 6 6 6 6 5 5 7 9 8 9 3 9 9 Scheduling (Demand) 8 8 8 8 5 6 6 6 6 6 5 5 9 9 9 9 10 10 10 Ext / Idle (Difference) 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 0 7 1 1 Table 5 Work patterns forming labor supply Table 5 shows an example of staffing executed on AN2 and PS2. Five work patterns are distinguished to carry out the labor defined by PS2. Nine workers are hired for the project, work pattern one and two are executed by one worker each, work pattern three and four are executed by two workers each and 3 workers have working pattern five. The center of the table indicates whether a work pattern is on or off work on a certain day. Work patterns three and four have an overtime unit on day four; their first week contains six days of work in stead of the regular five days. In the bottom part of the table, the total labor supply per day is calculated as the sum of the individual active work patterns. The row below you can find the labor demand per day as defined by PS2. The lowest row indicates whether external labor units (positive value) should be hired or idle time (negative value) occurs for every day. Towards the end of the project, external workers are hired. The difference of labor supply and demand can also be shown in the project schedule graph. Below, you can find an extended version of figure 5. The red line still shows the labor demand while the green dotted line indicates the labor supply as defined by the staffing. Demand exceeds supply towards the end of the project. 24

Figure 6 Labor supply and demand for AN2, PS2 25

4.3. Solution methods This section contains an overview of all the methods that were taken into consideration for the algorithm. The structure of this section is guided by the flowchart in figure 6. The flowchart represents the different phases in the algorithm. Each phase contains multiple methods that contribute to the solution of the problem. The topics will be discussed in this order: Initialization, Selection, Operation, Local Optimization, Partial Evaluation, Reinsert, Ending Condition an Evaluation. Figure 6 indicates the sections and subsections in which the different methods are discussed. Figure 7 Flowchart Genetic Algorithm, framework 2 4.3.1. Initialization To start the algorithm, we need to initialize a population of schedules in the form of float vectors. Although research has often neglected the importance of the initialization phase, a bad initial population can lead to increased time-to-solution or even getting trapped into local optima. A minimum of diversity in the population is necessary to avoid premature convergence of the solutions towards suboptimal regions of the solution space. To initialize our population, simple constructive heuristics will be used. To construct the schedule float vectors, we distinguish three groups of initialization methods, i.e. random, uniform and Gaussian initialization. 26

I1 Random Initialization In the random initialization method, a maximum float value (MFV) is determined. Then a value is randomly generated in [0, MFV]. The parameterisation for MFV leads to the following methods. I1a MFV = 1 x AVG duration of activities I1b MFV = 2 x AVG duration of activities I1c MFV = 3 x AVG duration of activities I2 Uniform Initialization In the uniform initialization method, a central value (CV) and a deviation value (DV) is determined. Then a value is uniformly generated in [CV-DV, CV+DV]. The biggest difference with the random initialization method is that this method does not necessarily include the value 0. If CV-DV would return a negative value, it is automatically initiated with a 0. If a large amount of float values are generated using this method, you will notice that they follow a uniform distribution. The parameters for CV and DV led to the following methods. I2a CV = 1 x AVG duration of activities DV= 0,5 x AVG duration of activities I2b CV = 2 x AVG duration of activities DV = 0,5 x AVG duration of activities I2c CV = 2 x AVG duration of activities DV = 1 x AVG duration of activities Note that there is no method where CV = DV = AVG duration of activities since this method is identical to method I1b. I3 Gaussian Initialization In the Gaussian method, a central value (CV) and a standard deviation value (SDV) is determined. Then a value is generated conform the gaussian distribution with a mean CV and a standard deviation SDV. This method differs from the two previous ones by the fact that it allows more extreme values sine it has no maximum value, i.e. there is no closed upper end. If this method would return a negative value, it is automatically initiated with 0. The parameters for CV and SDV led to the following methods. I3a CV = 1 x AVG duration of activities SDV= 0,5 x AVG duration of activities I3b CV = 1 x AVG duration of activities SDV= 1 x AVG duration of activities I3c CV = 2 x AVG duration of activities SDV= 0,5 x AVG duration of activities I3d CV = 2 x AVG duration of activities SDV= 1 x AVG duration of activities 27

I4 Combined Initialization This method makes a combination of the previous methods I1, I2 and I3 to become the initial population. The pseudo code for the initialization phase can be found in figure 8. Pick an Initialization method I1, I2, I3 or I4 (I) Determine initialization characteristics MFV, CV, DV, SDV While population not entirely filled Create new empty float vector While there are activities without float value Select a random activity a If activity a has no float value Initialize activity a using initialization method I Endif Endwhile Endwhile Figure 8 Pseudo code Initialization methods Other Initialization method Another initialization method that was considered finds its background in RACP. It is based of the maximum / minimum bounding strategy for determining the cheapest resource availability levels for a project. (Demeulemeester, 1995) The general idea is to calculate a resource availability constraint and use this constraint in combination with a scheduling rule to generate initial schedules. In a first step, calculate the minimum possible resource usage that would be necessary if there would be a constant availability of resources over the span of the project. This in fact, is equal to the resource usage of the most needy activity in the project. Applying this to our example in figure 4, the minimum possible resource usage is 10. This corresponds to the resource usage of activity 9. Secondly, calculate the maximum possible resource usage that would be necessary if there would be a constant availability of resources over the span of the project. For our example, this resource usage is 19 and is the maximum resource usage that is possible if activities 9, 10 and 11 would occur simultaneously. In a third step, the 28

activities will be scheduled using a basic priority rule and schedule generation scheme. Step 3 is repeated with different priority rules and a resource constraint ranging from 10 to 19. Each time step 3 is executed, this results in a schedule that is put in the initial population. 4.3.2. Selection Once an initial population of schedules is available, we need to find a way to select one or more schedules on which a certain operation will be performed later on. These selected schedules are called parents. The key idea of this selection phase is to select good parents, in order to give them an opportunity to pass on their good genes onto the next generation. Likewise, this phase should also prevent the worst solutions from passing on their inferior genes onto the next generation. (Sivaraj, 2011) A distinction can be made between two types of selection methods or schemes, a proportionate scheme and an ordinal-based scheme. (Sastry and Goldberg, 2001) Using an ordinal-based scheme, the chance of an individual to be selected depends on the ranking of the individual in the population based on a fitness measure. With a proportionate scheme, the chance of an individual to be selected depends on the relative fitness of the individual in comparison with the other individuals in the population. In other words, with an ordinal-based scheme, the chance of being selected merely depends on the fitness rank of the individual in the population, while a proportionate scheme also takes into account how much one solution is better than the other to determine the selection likelihood. The latter not only implies an order of the individuals in the population but also a scaling measure to determine the relative superiority of one individual to another. In this section, we will take a closer look at three selection methods, a random selection method, a roulette wheel selection method (proportionate scheme) and a tournament selection (ordinal-based scheme). 29

S1 Random Selection This selection method does not embed any intelligence. It merely selects 2 individuals randomly. This method does not give preference to individuals that are more fit than others and therefore this method is perceived to be inferior compared to other selection methods that make use of more intelligent criteria. S2 Roulette Wheel Selection As stated in the introduction, the roulette wheel selection method is a proportional scheme to select individuals out of a population. The first step is to calculate a fitness value for each of the individuals in the population. Depending on the algorithm being used, this fitness value could be either the total cost (figure 2) or the average squared deviation (figure 3). The second step assigns a probability to each individual based on the fitness value. These probabilities are set out on a roulette wheel, the bigger the probability of the individual to be taken, the bigger the circumference of that individual on the roulette wheel will be. In the third step, the wheel gets spun and wherever the wheel stops, this individual will be taken. To select another individual, repeat the procedure starting from step two. A small example will illustrate this method. Assume five individuals and their respective fitness values. An increasing value indicates a better individual. The circumference of the roulette wheel gets divided into five parts, each part belonging to the selection of one individual. An example of the division of the wheel is given in figure 9. The wheel gets spun and the individual is chosen where the roulette wheel comes to a standstill. 30

Figure 9 Example Roulette Wheel The pseudo code for the roulette wheel selection can be found in figure 10. Calculate the fitness value of each individual While necessary to select additional individual Determine selection probability for each individual based on fitness value Generate a random number, mimicking the spinning of a roulette wheel Translate the outcoming number into the underlying individual Endwhile Figure 10 Pseudo code Roulette Wheel Selection S3 Tournament Selection The tournament selection method is an ordinal-based scheme to select individuals out of a population. The first step consists of determining the rank-order of the individual based on a fitness value. This value will again be the total cost or the average squared deviation. This ranking will determine which individual wins in a tournament. In the second step, two individuals are selected randomly. Thirdly, the individual with the highest ranking wins the tournament and survives the selection stage. Repeat steps two and three until enough individuals are selected. The pseudo code for the tournament selection can be found in figure 11. 31

Calculate the fitness value of each individual Make a ranking of the individuals based on fitness value While necessary to select additional individual Select 2 individuals randomly Determine the winner based on the ranking Endwhile Figure 11 Pseude code Tournament Selection This tournament selection method can be extended by either adding additional tournament rounds, such that an individual has to win 2 or more rounds before it is selected or by adding more individuals competing in each round. A big advantage of this method over the roulette wheel is that there are no scaling issues. Since the tournament selection merely uses a rank, there is no need to translate the difference in fitness in a different selection probability. (Whitley, 1989) S4 Combined Selection Selection method S4 uses all three aforementioned methods to select individuals. Which method is used for each selection is determined on a random basis. 4.3.3. Operation On the (pair of) parents, operators will be executed in order to generate different solutions, called children. The solutions can change drastically or just slightly. The two main types of operations are crossover and mutation. (Luke and Spector, 1998) Crossover relies on the hypothesis that highly fit individuals in the population consist of fit building blocks that can be mixed in order to become even more fit individuals. It will push the population to converge into one or more local optima. This process of convergence is often called intensification or exploitation. Mutation on the other hand serves the goal of maintaining genetic diversity in the population. Mutation thus fulfils the task of exploring the solution space. The aspects of mutation and crossover are antagonists but are both equally important. On the one hand, we need to be sure to have covered the solution space as much as possible through genetic diversity. But on the other hand, we also want to make sure that we find the best solution in the researched solution space through convergence into the best areas. 32

The occurrence of crossover and mutation are reflected by their crossover rate and mutation rate respectively, indicating the chance this operation will be executed on a selected pair of parents. Fixing these rates to a general optimal value is very difficult since they are very problem specific and they even depend on the stage of the genetic algorithm. Research has been done on the determination of these values for mutation and crossover both as a static constant and as a dynamic value, changing over the course of the execution of the algorithm. (Lin et al, 2003) In this section the following crossover operators will be discussed, including 1-point crossover, 2-point crossover, blend crossover, mean crossover, extrapolation crossover, uniform crossover and mutation. C1 1-Point Crossover This crossover operator heavily relies on the building block hypothesis as it cuts the parents into two halves or blocks in order to recombine these blocks into the children. The point where the parents should be cut is determined randomly. (Spears and Anand, 1991) An example is presented, 2 parents as float vectors containing the float value for 12 activities. The cut-off point is after the fourth activity. a1 a2... a12 Parent 1 0 4 6 8 4 2 6 3 8 5 5 0 a1 a2... a12 Parent 2 0 2 1 4 3 9 5 7 3 10 2 0 These parents swap their float values after the cur-off point to create 2 children. a1 a2... a12 Child 1 0 4 6 8 3 9 5 7 3 10 2 0 a1 a2... a12 Child 2 0 2 1 4 4 2 6 3 8 5 5 0 Assuming that parent 1 has a very fit first part and parent 2 has a very fit second part, child 1 has a high probability of outperforming both parents. 33

C2 2-Point Crossover This crossover operator is identical to the previous one besides the fact that there is not a single cut-off point but 2 cut-off points that will chop the parent into 3 blocks to be recombined. An example with cut-off points after the third and seventh float value. a1 a2... a12 Parent 1 0 4 6 8 4 2 6 3 8 5 5 0 a1 a2... a12 Parent 2 0 2 1 4 3 9 5 7 3 10 2 0 Resulting children are constructed by swapping the middle block. a1 a2... a12 Child 1 0 4 6 4 3 9 5 3 8 5 5 0 a1 a2... a12 Child 2 0 2 1 8 4 2 6 7 3 10 2 0 C3 Blend Crossover Using the blend crossover operator, you do not copy and recombine genetic material but you blend the corresponding genetic material, based on the distance between them. (Eshelman and Schaffer, 1992) (Takahashi and Kita, 2001) The blending will result into two values which are the boundaries for the newly generated child value. The steps to be undertaken will be illustrated with an example. a1 a2... a12 Parent 1 0 4 6 8 4 2 6 3 8 5 5 0 a1 a2... a12 Parent 2 0 2 1 4 3 9 5 7 3 10 2 0 Since the first activity has an identical float value, we will take the second activity as an example. Firstly the distance between the two parents is calculated as the difference between the respective float values, i.e. distance d = 4-2. (1) In the second step, 2 boundary values (X 1 and X 2 ) are determined taking the lowest float value, highest float value and the distance into account. (2) 34

This would result in the following for our example. (3) In the third step, a new value gets generated randomly in the interval [X 1,X 2 ]. In our example, the interval is [1,5]. Do this for all activities until a entire child is produced. The pseudo code for the blend crossover operator can be found in figure 12. For every activity Calculate the distance d between the two parents Calculate the lower boundary, using distance d Calculate the upper boundary, using distance d Randomly generate a new float value between the boundaries Endfor Figure 12 Pseudo code Blend CrossOver C4 Mean Crossover The mean crossover operator will calculate the mean value of the two parents for each activity and take this mean value as the new float value for the child (Wall, 1996). In case a non-integer value gets generated, the value will be randomly rounded up or down. An example is shown below. a1 a2... a12 Parent 1 0 4 6 8 4 2 6 3 8 5 5 0 a1 a2... a12 Parent 2 0 2 1 4 3 9 5 7 3 10 2 0 Applying the mean crossover operator will result in this child. a1 a2... a12 Child 0 3 3 6 4 5 5 5 6 8 3 0 35

C5 Uniform Crossover The uniform crossover is a very straightforward operator. To construct the child, for each activity it will randomly take the float value of either parent. Extra intelligence could be added in the sense that there is no random selection of the float value for each activity but the fittest parent gets a higher probability. (Magalhães-Mendes, 2013) Applying uniform crossover can result in he child shown below. a1 a2... a12 Parent 1 0 4 6 8 4 2 6 3 8 5 5 0 a1 a2... a12 Parent 2 0 2 1 4 3 9 5 7 3 10 2 0 a1 a2... a12 Child 0 2 6 8 3 2 6 7 3 10 5 0 C6 Combined Crossover This crossover method combines the five aforementioned crossover operators. Every cycle, a new crossover operator gets chosen randomly. Mutation Mutation will be executed on a single individual. It does not combine 2 solutions but merely alters an individual in a certain spot. Mutation of an activity float value can be done neglecting the current value, meaning that a reinitialization occurs. Another option is to take the current value into account and mutate that value by adding or subtracting some value. Our mutation operator neglects the current value. This operator can be useful after a high number of generations, since at that point solutions can converge. As mentioned in the introduction, mutation will maintain some diversity that hinders the converging behavior of the population. As the algorithm proceeds, it can be interesting to let the mutation rate evolve as well. Modifying this rate inversely (proportional) to the population diversity could prevent convergence. (Bäck, 1993) 36

4.3.4. Local Optimization A regular genetic algorithm would, after a crossover operation, perform an evaluation of the newly created child and consider whether to reinsert or to discard the child from the population. However, we opted to insert local optimization or local search first. Local search can optimize the children by looking into the nearby neighbourhood in order to discover better solutions. Local search has an intensification function as opposed to the diversification function. Intensification means that you will further look into an area of the search space in which you have found a solution yet, but want to optimize it further. Diversification has the goal to look into unexplored search space in order to discover new valuable solutions. (cfr. Mutation) By applying local search iteratively on one solution, and by updating this solution by its best neighbour, also known as hill climbing (Pisinger and Robke, 2010), you will end up in a local optimum. The local search operator can be a very simple swap operation or a small heuristic that reschedules a part of the schedule. This is very abstract but can be explained using a very simplified example in figure 13. This figure represents a two-dimensional landscape where the x-value represents a location and f(x) represents the height of a certain location. The objective is to find the location of the valley, i.e. the location with the lowest height. You could check every location x going from 0 to infinite, calculate its corresponding height and then conclude that the lowest point is location b. Another method would be to take random location samples. These random locations are indicated by five arrows. Starting from these five locations, you can explore the neighborhood for better locations. Starting from location at arrow number two, you can search in two directions, right and left. These two directions are called neighborhoods. When going to the right side you will soon notice that the height is going up, so we will not explore that side.(hill climbing) However when we go to the left, we notice the height to go down. Repeat this move until you cannot go any lower. You will end up in point a. When following the same neighborhood search strategy starting from arrow 3, you will end up in location b etc. The great advantage of the second method, using local search, is that you need to do less effort in order to find the valley. The disadvantage, however, is that you are not sure whether you end up in the global optimum b or in a local optimum such as 37

location a and location c. But when the initial locations are strategically set out well, the solution found should be close to the optimal solution. Figure 13 Local search simplified This example can relate to our problem in the sense that every x represents a schedule or its float vector and f(x) represents the cost accompanying this schedule. We want to find the lowest cost, so what we will do in the local search stage is to alter a schedule a little bit in order to find better schedules. The project scheduling problem, however, is a lot more complex because the neighborhood is extremely large. Therefore good neighborhoods have to be defined in order to explore them efficiently. Three local search strategies will be discussed into further detail, including a simplified version of the Burgess and Killebrew heuristic, double-justification and 2- exchange neighborhood. L1 Burgess and Killebrew simplified RLP This first local search method finds its origin in the heuristic proposed by Burgess and Killebrew in 1962. The main reason for adopting this method is the observation that, when keeping the length of the project constant, a more levelled solution will return a lower cost. (observation in 6.1.) Since local search will hardly change the length of the project, this gives the opportunity to translate the original problem into a resource levelling problem. We simplified the heuristic of Burgess and Killebrew in the sense that we do not use any priority rule but merely drag activities back and forth in a random order. The 38

number of activities that will be considered for this dragging is determined by the neighborhood depth variable. Furthermore we will not make use of the total sum of squares but the average squared deviation (AVGSQDEV) as an indicator for tendency towards a levelled solution. The following steps are executed in this proposed method. Starting from a schedule, take a random activity and calculate its earliest start time and latest start time. In terms of float values, the values represent 0 and the free float values respectively. Free float is defined as the maximum amount of delay you can add to an activity without disturbing any subsequent activity. The second step consists of an iterative procedure in which you schedule the activity at each point in time between the earliest start and the latest start. At the end of each iteration, a new schedule is constructed which gets evaluated in the third step using the partial evaluation method, described in section 4.3.5. The essence of that section states that the evaluation is not done based upon the fitness i.e. total cost of the schedule but rather on the tendency towards a levelled solution. Based on this partial evaluation, the best schedule is retained for further local optimization. These three steps are repeated for a predetermined number of times that is comprised in the neighborhood depth variable. The pseudo code for L1 can be found in figure 14. For 1 to neighborhood depth Select a random activity Calculate the free float value or slack For f: 0 to free float Schedule the chosen activity using float value f Calculate the AVGSQDEV for the resulting schedule If Newly created schedule performs better Retain this schedule for further local optimization EndIf EndFor EndFor Figure 14 Pseudo code LS1 Burgess and Killebrew simplified RLP 39

L2 Double-Justification This method of double-justification comes from the area of the RCPSP in which the project makespan minimization is the objective. (Muller, 2009) Double-justification means that a schedule gets sequentially right-justified and left-justified. A rightjustified schedule is a schedule in which all activities are pulled as close as possible to the end of the project while a left-justified schedule is one In which all activities are scheduled as close as possible to the beginning of the project. A small adjustment that we made to this procedure is the addition of a resource constraint during the backward and forward scheduling that result into the right- and left-justified schedule respectively. The following steps explain the procedure into further detail. Start with calculating the maximum workload, this workload will be used later on as a resource constraint. The second step consists of backwards scheduling, resulting in a right-justified schedule. The order in which the activities are scheduled is determined by their finishing time; the latest finishing activity gets scheduled first, as late as possible and taking the resource constraint into account. The third step consists of the forward scheduling resulting in a left-justified schedule. The order in which the activities are scheduled is determined by their starting time; the earliest starting activity gets scheduled first, as soon as possible and taking the resource constraint into account. The pseudo code for this local optimization method can be found in figure 15. Calculate the maximum resource usage MAX, for resource constraint purposes Determine order of activities for backward scheduling (~finishing times) For every activity in order Schedule activity as late as possible taking MAX into account EndFor Determine order of activities for forward scheduling (~starting times) For every activity in order Schedule activity as soon as possible taking MAX into account EndFor Figure 15 Pseude code Double-Justification 40

L3 2-Exchange Neighborhood The 2-exchange neighborhood or 2-opt neighborhood is a local search application on the travelling salesman problem (TSP). In the travelling salesman problem, two tours are neighbors if one tour can be obtained starting from the other by exchanging 2 destinations. We try to apply this swapping mechanism to project scheduling. The following steps explain how it works. The first step is the random selection of an activity. The second step consists of determining all direct and indirect predecessors and successors of the activity. These activities are neglected, since a swap in time of the selected activity and these activities would result in an infeasible schedule due to precedence relations. If there are activities with which the selected activity can swap, execute this swap by exchanging the starting time of both activities. The pseudo code of this local search method can be found in figure 16. Select an activity a1 randomly Find (in)direct predecessors and successors of the chosen activity, add them to list l If there exist activities, not on list l Select an activity a2, not on list l, randomly Execute an exchange of the starting times of a1 and a2 Repair solution if necessary EndIf Figure 16 Pseudo code 2-exchange neighborhood L4 Combined Local Search During the execution of the genetic algorithm, all three of the previously described local search methods are used in a random fashion. 4.3.5. (Partial) Evaluation When new individuals are generated, it is interesting to evaluate them, i.e. calculate a fitness value. This section contains two measures for evaluation, i.e. the total cost in the evaluation phase and the average squared deviation of labor consumption (AVGSQDEV) in the partial evaluation phase. Both these evaluation measures are non-regular measures of performance. 41

4.3.5.1. Total Cost (Evaluation) To check the total cost of a schedule, we will let the staffing algorithm process the generated schedule. The input is thus a project schedule and the outcome is a set of workers with a certain work pattern. Based on this outcome, the total staffing cost of the project can be calculated. The staffing is formulated and solved as a linear programming problem. The solution of this problem is deterministic and optimal meaning that given a certain input, always the same output is generated (deterministic) and this output is the best possible (optimal). This linear problem was provided by Prof. Dr. Maenhout and encoded into C++ which calls the Gurobi Optimizer, a mathematical programming solver for different problems such as linear programming and mixed integer programming problems. This total cost calculation is the most important measure for fitness of a schedule, it embodies the objective function as defined in section 2.3. However this calculation takes a lot of computational effort, ranging from two to six seconds, depending on the project length. Therefore, a second genetic algorithm framework is designed to eliminate the calculation of the total cost as much as possible while affecting the quality of the final schedule as little as possible. 4.3.5.2. Average Squared Deviation (Partial Evaluation) When doing a partial evaluation, the staffing algorithm will not be executed. This performance measure is designed as an alternative for the total cost measure. It represents a resource levelling approach. Observations in section 6.1 confirm that the use of this measure is adequate for comparing schedules with an identical project length. In that case, a lower AVGSQDEV will on average result in a lower total cost. This assumption can be applied to the local optimization stage. At the end of each search cycle, a performance measure needs to be calculated to determine the best schedule in the neighborhood. Since these neighborhood search methods hardly change the length of the project, the link between AVGSQDEV and total cost is justified. This previously mentioned use of partial evaluation is present in both genetic algorithm frameworks. In order to further decrease computational efforts, the second framework goes one step beyond. The second framework also uses the AVGSQDEV measure to decide 42

upon the reinsert of a schedule into the population. This often requires a comparison of schedules with very different project lengths. Observations in section 6.1 do not justify the use of the AVGSQDEV in this case, however it does not necessarily prove that it should not be used. The big advantage of this method is the small computational effort required for this calculation. The biggest disadvantage of this method is that it is an approximation of the total cost measure, which means that a lower AVGSQDEV will not necessarily always result in a lower total cost. This could lead to situations where a schedule with a better AVGSQDEV but worse total cost will replace a schedule with worse AVGSQDEV but better total cost during the reinsert stage. The formula to calculate the AVGSQDEV is given in figure 17. Figure 17 Average squared deviation of resource consumption T set of days in the scheduling horizn (index t) A set of activities in the project (index j) r j PL number of resources necessary to execute activity j project schedule length average resource use This AVGSQDEV gives an impression of how levelled the resource demand of a project schedule is. A low AVGSQDEV means that the resource consumption throughout the project schedule is relatively equal and thus tends to be levelled. A high average squared deviation points out that there will be big differences in resource consumption between the different days in the schedule. Alternative formulations of the levelling measure are the sum of squared deviations, the weighted jumps in resource usage and the sum of absolute deviations. (Herroelen et al., 1997) 43

4.3.6. Reinsert When new members (children) are generated, they should have the possibility to enter the population. This section will handle this entering of the population. In the first part, reinsert conditions are discussed. These conditions will decide whether a newly created child is eligible to enter the population. The second part will handle the population management. Population management will dictate how the population will evolve throughout the generations. 4.3.6.1. Reinsert Conditions The algorithm can decide to let a child, that is generated out of existing parent individuals, enter the population or not. Different conditions can be distinguished. Some are relatively loose and will be passed very easily, making the population improve slowly; other conditions are rather strict and as a consequence the population fitness will improve either very fast or not at all. The insert condition always results in the comparison of two or more schedules based on a performance measure. For framework1, this performance measure is the total cost while for framework2, this performance measure is the AVGSQDEV. The reinsert conditions are described, assuming a steady-state population. This results in the replacement of an existing member schedule by a new schedule. However as 4.3.6.2. points out, also generational populations will be tested. In that case, there is no replacement of an existing member but merely the addition of a new member into a new population. R1 Outperform weakest schedule This reinsert condition firstly calculates the performance measure of the newly generated schedule and compares this to the performance measure of the weakest schedule in the population. If the new schedule outperforms the weakest schedule, this weakest schedule will be replaced by the new one. 44

R2 Outperform one parent This reinsert condition firstly calculates the performance measure of the newly generated schedule and compares this to the performance of both parents. If the new schedule outperforms at least one parent schedule, the weakest parent will be replaced by the new schedule. R3 Outperform both parents This reinsert condition firstly calculates the performance measure of the newly generated schedule and compares this to the performance of both parents. If the new schedule outperforms both parent schedules, the weakest parent will be replaced by the new schedule. R4 Outperform 25% of existing population This reinsert condition firstly calculates the performance measure of the newly generated schedule and compares this to the performance measure of all member schedules in the population. If the new schedule outperforms at least 25% of the schedules in the existing population, the weakest schedule will be replaced by the new one. R5 Outperform 50% of existing population This reinsert condition firstly calculates the performance measure of the newly generated schedule and compares this to the performance measure of all member schedules in the population. If the new schedule outperforms at least 50% of the schedules in the existing population, the weakest schedule will be replaced by the new one. R6 Outperform 75% of existing population This reinsert condition firstly calculates the performance measure of the newly generated schedule and compares this to the performance measure of all member schedules in the population. If the new schedule outperforms at least 75% of the schedules in the existing population, the weakest schedule will be replaced by the new one. 45

Doubles On top of the aforementioned reinsert conditions, an extra condition can be added. This is the so-called doubles condition. This condition prohibits a new individual to enter the population if that individual already exists in the population. It thus promotes diversity in the population and will prevent convergence into local optima. 4.3.6.2. Population management Population management or reproduction strategies dictate how the population evolves throughout the execution of the algorithm. Two alternative startegies, i.e. steady-state resulting in an overlapping population and generational reproduction (Syswerda, 1991) resulting in a non-overlapping population are considered and a hybrid form. (Noever, Baskaran, 1992) P1 Steady-state population This population management mechanism states that only a few individuals get replaced at a certain point in time, keeping the size of the population constant. Which individuals get replaced is defined by the reinsert condition. This results in the fact that populations will overlap or different generations will overlap. It means that a child (new generation) can enter the population of parents (old generation), and can mate with its parents generation. P2 Generational population On the opposite side of the steady-state population, there is the generational population. This population management mechanism will not replace individuals separately but replace an entire population at once. It means that no replacement or intermediate deletion of schedules from the population occurs. When enough children are produced that pass the reinsert condition, the existing population will be replaced by this new population of children. However, in order not to lose highperformance member schedules from the previous population, a form of elitism or elitist model is applied. (De Jong, 1975) We applied a simple elitist policy, stating that when replacing the old population by a new population, the top 20% best schedules of the old population should be forced into the new population. Depending on the 46

used framework, the performance measure to determine the best schedules is the total cost and the AVGSQDEV for framework1 and framework2 respectively. P3 Hybrid populations This population management mechanism integrates the two previous systems. In the beginning the population will act as a steady state population. After a certain amount of replacements (80% of the population size), the population will act as a generational population and it will be replaced as a whole. 4.3.7. Ending condition The cyclical character of the genetic algorithm makes it a never ending process. Even when a local optimum or good solution is found, it can still generate new children. A clear ending condition has to be stated to end the cycle. This ending condition is a limitation and will never improve the objective function but rather limit the time spent. Static Ending Condition This is a predefined, fixed ending condition which is not influenced by the course of the genetic algorithm. Examples of this kind of ending condition are summed below. The algorithm stops after X operations The algorithm stops after X time units The algorithm stops after X evaluations Dynamic Ending Condition A dynamic ending condition is a condition that is influenced by the course of the genetic algorithm. Examples of this are listed below. The algorithm will stop if no improvement is found in X operations The algorithm will stop if no improvement is found in X time units The algorithm will stop if the population contains X duplicates (if duplicates are allowed) The algorithm stops after the fittest value of the population is lower than X 47

Hybrid ending condition Dynamic and static ending conditions can be used in harmony. The static ending condition will carry out the upper limit of the execution of the algorithm, expressed in number of operations or time units etc. The dynamic ending condition would play the role of early showstopper in case the solution is either satisfying enough or there is no hope for improvement left and thus interrupting the execution is appropriate. 48

5. The algorithm Based on the computational results in table 8 of 6.3.1., a definite form of algorithm is chosen. The framework of choice is framework 2. Although this framework performs slightly worse than framework 1, the execution time is significantly shorter and thus better. Combination of methods and parameters: Population size 20 Number of operations 100 I1b Random Initialization o MFV = 2 x AVG duration of activities S3 tournament selection C5 Uniform crossover No mutation L1 Burgess and Killebrew simplified RLP o neighborhood depth =5 R1 reinsert weakest in population o P1 Steady state population o No doubles allozed No ending condition 49

6. Computational experiments In this chapter, the computational experiments of the methods described in chapter four will be presented. In order to obtain the results, the solution methods are coded into C++. The coding of this program took a couple hundreds of hours. This program is then run on an Intel i3 core processor 2,13 Ghz and 4 Gb RAM. Executing this program took several hundreds of hours of computational time. In section 6.1. we provide an observation when first executing test cycles and exploring the data it produces. Section 6.2. presents the benchmark for our test, this benchmark gives an indication of how good the results, generate by the algorithm, are. The last section contains the actual test results. 6.1. Observation link AVGSQDEV Total cost When executing first test cycles, we were looking for performance measures besides the total cost. Other measures, such as average float, project length and average squared deviation were monitored. Afterwards, we tried to find connections between these measures. One connection, between AVGSQDEV and total cost, is significant and very useful practically. We plotted all schedules on a dispersion graph with AVGSQDEV on the x-axis and Total cost objective on the Y-axis, which is shown in figure 18. No clear correlation can be observed. Average Squared deviation - Objective 270 250 Objective 230 210 190 170 0 50 100 150 200 250 300 Average Squared deviation Figure 18 AVGSQDEV - Total Cost dispersion for all project lengths 50

However if we plot that same dispersion graph, but grouped per project length, we notice a graph like figure 19. This graph clearly shows the dispersion for all schedules with a project length of 11 days and a positive correlation between the total cost objective and the AVGSQDEV can be observed. Average Squared deviation - Objective 270 250 Objective 230 210 190 170 0 20 40 60 80 100 120 140 160 180 200 Average Squared deviation Figure 19 AVGSQDEV - Total Cost dispersion for project length of 11 days More dispersion graphs for other project lengths can be found in appendix D. The conclusion of this observation is that the AVGSQDEV measure is a good relative approximation for the total cost objective when comparing schedules. 6.2. Benchmark In order to assess the quality of a schedule produced by the algorithm, a benchmark is necessary. The best benchmark is the comparison of a schedule to the optimal schedule. However the optimal schedule is unknown and therefore we cannot calculate its quality for benchmark purposes. As an alternative, we can relax the problem by eliminating some assumptions and restrictions to the extent that it is possible to find the optimal solution for this relaxed or simplified problem. Since the problem is relaxed, it can be assumed that the optimal solution of the relaxed problem will always outperform the heuristic solution of the original problem. This creates a lower bound to the cost minimization problem. Figure 22 represents the scheduling cost minimization problem. 51

The vertical downwards arrow in figure 22 depicts an axis on which schedules can be ordered from low quality to high quality, i.e. high staffing costs to low staffing costs for the scheduling cost minimization problem. The blue dot is the best schedule that is found by the genetic algorithm, this is the schedule we want to compare to a benchmark. Figure 20 Benchmark setup The preferred benchmark is the optimal schedule, which divides the search space into the feasible region and the infeasible region. We can calculate a lower bound, which is in the infeasible region because the problem is relaxed in this calculation. The goal is to find a lower bound, as close as possible to the optimal schedule and thus minimize the GAP in figure 22 as much as possible. This inhibits a trade-off. On one hand, increasing relaxation will increase the ease of calculating a lower bound but on the other hand, increasing relaxation will increase the gap between the lower bound and the optimal schedule and thus limit the quality of the benchmark. After the lower bound to the relaxed problem is found, assumptions or constraints can be added again in order to improve the lower bound quality i.e. bringing it closer to the optimal schedule of the initial problem. The relaxation executed on our problem seems drastic, but section 6.3 will prove that they are not overdone. The first and major simplification is neglecting the complete project scheduling part. This includes the structure of the activity network with its precedence relations as well as the possible resulting schedules concerning schedule make span and resource demand distribution over the duration of the project. We assume that the total resource demand, defined as the sum of resource demands of all activities, is spread out equally over the duration of the project. In order to be able to compare the three different project activity networks, the total resource demand for each network is equal and set to 143. This means that, if a project takes ten days, the daily resource demand is 14,3. The focus of the lower bound calculation is put on the staffing problem. Small eliminations and loosening of constraints will be applied there. The first relaxation in the staffing is that we will only 52

make use of regular time units which have a cost of 2, since these are the cheapest. This means that we do not make use of overtime units and external time units, incurring a respective cost of 3 and 4. Also the possible presence of idle time, which costs 1, is neglected. The relaxation implicitly defines every worker to work five days per week. He cannot work more, because that would imply an overtime unit and he shouldn t work less because idle time is neglected. The second relaxation is that we allow fractional employees to work. For example, if there is a workload that would require 9,5 workers to execute, the real problem would require to hire 10 workers to execute the load. The relaxed problem however accepts this 9,5 fractional value. Based on these relaxations, a lower bound for the cost of a schedule can easily be calculated. This lower bound is dependent on the duration of the project and given in figure 21. Lower Bound Total Cost 405 395 385 375 365 355 345 335 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Project Duration Lower Bound Figure 21 Lower bound benchmark in function of project duration An example of the calculation is given for a project duration of 17 days. (lower bound = 374) Total resource demand 143 units (A) Project duration 17 days (B) Max available days per worker 13 days (C) Amount of workers necessary (A/C) 11 workers (D) Cost per worker per day of project 2 (E) Total Cost (E*B*D) 374 53

The calculation of the max available days per worker is based on the fact that each worker works five days per week, meaning that he will have 4 days off in the span of 17 days. If the project would have a duration of 18 or 19 days a worker would also have 4 days off. However if the project would have a duration of 20 or 21 days, every worker is required to take 5 and 6 days off respectively. This evolution in ratio between days off and project duration explains why the lower bound graph goes down during the first five days of the week and makes an upwards spark during the last 2 days of the week. (figure 21) The evolution in that ratio also explains the general upwards trend of the cost in function of the project duration. The lower bound graph applies to all three project activity networks since the total resource demand of each project activity network is identical and the remainder differences are relaxed. The graph can be limited by cutting off an upper and lower bound to the project duration. The upper bound is the fixed project deadline of 21 days, this is identical for every activity network. The lower bound represents the critical path of the activity schedule and is different between activity networks. If, for example the critical path is 14 days, it Is impossible to attain a cost of 343, which can only be accomplished with a project duration of 12 days. 6.3. Results In the results section, three topics will be discussed. The first topic handles the basic cycles. These are the cycles necessary to determine the best combination of methods, described in section 4.3, for each different framework of the algorithm and for each different activity network. There are 2 frameworks of the algorithm and 3 different activity networks. The second section on the stage contributions will consist of an analysis that determines the importance of each stage in the genetic algorithm. By removing or altering these stages and measuring the consequences on the total cost, the importance is quantified. This will only be done for the best algorithm on AN2. The third section consists of tuning different parameters in the algorithm; this also includes a sensitivity analysis on some parameters, emphasizing their importance or worthlessness. This will also only be done for the best algorithm on AN2. 54

6.3.1. Basic cycles A basic cycle contains the process of finding the best algorithm for an activity network in combination with an algorithm framework. Since we have two frameworks and three activity networks, six basic cycles will be performed. A basic cycle consists of several phases. In each phase, the algorithm framework is run multiple times, each time the algorithm consists of different solution methods as defined by section 4.3. In the first phase or Base case, we allow all solution methods to be chosen. In the last or Best phase, only one solution method per stage remains. In the phases in between, we gradually eliminate the eligible solution methods by removing the least performing or by retaining the best performing methods. In phase Best X, we make sure that only about ten different combinations of solution methods remain. Elimination is based on the cost that is on average associated with the solution method. A summary of the elimination process of each basic cycle is given in appendix B, which states every eligible solution method at a certain phase in the basic cycle in combination with the average cost of that phase. A single exception to the elimination rule, which states that the least performant solutions methods get eliminated, is made for local optimization method L2. This method is often removed from the possibilities despite its seemingly high performance. The algorithms that made use of L2 show no or very limited diversity in its resulting best solution. This gives an indication that no real exploration of the neighborhood occurs in spite of finding a good solution. For each basic cycle, the minimum, maximum and average costs per phase are indicated in a table and a graph. The tables and graphs for AN2 are shown below, the tables and graphs for AN1 and AN3 are available in appendix C. AN2 Framework 1 Phase Average Min Max Base Case 402,06 383 450 1st Exclusion 391,04 380 410 2nd Exclusion 388,91 378 401 Best X 386,74 373 405 Best 385,34 373 400 Table 6 Total Cost evolution basic cycle AN2 Framework1 55

Total Cost evolution 460 450 440 Total Cost 430 420 410 400 Average Min Max 390 380 370 Base Case 1st Exclusion 2nd Exclusion Best X Best Phase Figure 22 Total Cost evolution basic cycle AN2 Framework1 AN2 Framework 2 Phase Average Min Max Base Case 403,51 383 458 1st Exclusion 389,67 373 416 2nd Exclusion 389,09 377 414 Best X 387,03 378 402 Best 386,29 378 403 Table 7 Total Cost evolution basic cycle AN2 Framework2 Total Cost evolution 460 450 440 Total Cost 430 420 410 400 Average Min Max 390 380 370 Base Case 1st Exclusion 2nd Exclusion Best X Best Phase Figure 23 Total Cost evolution basic cycle AN2 Framework2 56

Both frameworks for AN2 show a similar behaviour and return similar results whereby framework 1 slightly outperforms framework 2 in total cost for the best case. The most significant improvement is made during the first exclusion. This is because in the phase before, i.e. the base case, the local optimization method L1 shows that it is significantly better than all others. So in the first exclusion phase, all inferior local optimization methods are eliminated. The local optimization stage proves to be the most vital stage for the algorithm. (cfr. Section 6.3.2.) The average total cost of the solutions generated throughout the different phases decreases steadily. This is primarily achieved by eliminating the possibility of returning a bad result, which is represented in the maximum cost generated per phase. Initialization Selection Operation Local Optimization Reinsert Doubles Population management Average Total Cost Best Total Cost Lower bound Average Total Cost deviation from lower bound Best Total Cost deviation from lower bound Basic cycle AN1 framework1 I1a S2 C5 L1 R1 N P1 355 352 343 3,57% 2,62% AN1 framework2 I1a S2 C5 L1 R2 N P1 366 355 343 6,61% 3,50% AN2 framework1 I1b S2 C5 L1 R1 N P1 385 373 362 6,45% 3,04% AN2 framework2 I1b S2 C5 L1 R1 N P1 386 378 362 6,71% 4,42% AN3 framework1 I3b S2 C5 L1 R1 N P1 384 378 362 6,16% 4,42% AN3 framework2 I3b S2 C5 L1 R2 N P2 386 380 362 6,63% 4,97% Table 8 Best solution methods, total cost and lower bound per basic cycle Table 8 is the summary of the execution of the basic cycles. It shows the best combination of solution methods for each basic cycle. The best selection, operation and local optimization method is identical for all basic cycles. Also the use of doubles in the population is not recommended regardless the activity network or framework. The best initialization method is independent of the chosen framework and seems to rely on characteristics of the activity network. The best reinsert method for framework1 is R1, where the weakest member of the population gets replaced while R2, i.e. the replacement of a random parent, is the best reinsert method for framework2 in combination with AN1 and AN3. The most widely used population management mechanism is the use of steady-state populations (P1). Only for AN3 framework2, the use of generational populations (P2) seems more beneficial. 57

The best solution method combinations for all basic cycles are very similar and also yield similar results compared to the lower bound benchmark. The distance between the average total cost, which is the total cost the basic cycle using the best methods returns on average, and the lower bound benchmark is 6% - 7%. AN1 framework1 is an exception with a distance of only 3,5%. Knowing that the lower bound returns a better cost value than the optimal total cost, we know that the actual distance between the average total cost and the optimal total cost is even smaller than the 6%-7% and 3,5%. If the best total cost, which is the best cost the basic cycle using the best methods returns, is compared to the lower bound benchmark, a gap of 2%- 5% is noticed. The relative gaps between the total costs and the lower bound also illustrate that framework1 consistently outperforms framework2 in terms of fitness. 6.3.2. Stage contributions In this section, we will further investigate to which extent each stage in the genetic algorithm contributes to the total cost objective function. This contribution testing is done by either removing a certain stage from the genetic algorithm or by replacing the method by its worst alternative. This is only applied to AN2 framework2. Initialization Removing the initialization stage is simply impossible since an initial population needs to be constructed with some logic. Therefore we compare the best initialization method (I1b) to the worst tested initialization method. A deterioration of the total cost from 386,29 to 392,86 or 1,7% is determined. Selection The best selection method is the tournament selection (S2). When replacing this method by random selection, while keeping all other methods equal, only a small deterioration from 386,29 to 386,92 or 0,16% is assessed. Operation The best method for this stage is the uniform crossover operator. When executing the same algorithm but without this operator, the total cost goes up from 386,29 to 390,77 or 1,16%. 58

Local Optimization The best local search method L1, which searches in the neighborhood for more levelled schedules is the best local optimization method being tested. However, if we remove this stage from the algorithm while keeping all other stages equal, the total cost worsens drastically from 386,29 to 403,36 or 4,42%. Reinsert condition The best reinsert condition under consideration is the replacement of the weakest member in the current population (R1). When removing this condition and thus always reinsert a newly created child into the population, replacing an existing member randomly, the total cost increases from 386,29 to 391,35 or 1,31%. Population management The best population management mechanism is the use of steady-state populations (P1). However when applying the worst alternative method, i.e. generational populations (P2), the total cost only worsens from 386,29 to 387,76 or 0,38%. Conclusion The local optimization stage has the most significant impact on the total cost, it account for over 4% when removing this optimization and keeping all other stages equal. The initialization, operation and reinsert condition have a moderate impact of little over 1%. The selection stage and the population management have a very minor impact on the total cost. 6.3.3. Sensitivity Analysis In this section, the computational results of a sensitivity analysis are shown. The start case is the best algorithm associated with AN2 framework2. (also specified in chapter 5) In the remainder of the section, we will refer to the configuration of the algorithm as the start case. In this section we will focus on the influence of changing some parameters on the outcome of the algorithm. First of all, we will have a closer look at the population size and the number of cycles or operations performed. Then we will check the influence of allowing doubles to 59

enter the population, the number of iterations or neighborhood depth in local optimization, the use of mutation and an ending condition on the total cost of the resulting schedule and the execution times necessary to obtain these schedules. Population size and number of operations The start case considers a population size of 20 schedules and the execution of 100 operations. We now expand these possibilities to 10, 20, 50 and 100 schedules as population size and 50, 100, 250, 625, 1500 and 3000 as number of executed operations. Besides these two dimensions, which will guide us, the total cost and execution times are 2 other resulting dimensions that will be monitored. Increasing the population size and the number of operations to be executed will increase the execution time. However the relationship between the population size / number of operations and the total cost is not that straightforward. An assumption could be that more operations will result in better total cost. This is not necessarily true because of the setup of framework2. As section 4.1.2. states The disadvantage (of framework2) however is that it is possible that the best schedule present in the population gets replaced by another one during the execution of the algorithm. This is exactly what happens after a very large amount of operations. The AVGSQDEV in the population becomes extremely low and good solutions get replaced by worse. This is illustrated by figure 24. Total cost Vs number of operations for different population sizes 392 Total Cost 390 388 386 384 10 schedules 20 schedules 50 schedules 100schedules 382 50 100 250 625 1500 3000 Operations Figure 24 Total cost Vs number of operations for different population sizes 60

For small population sizes of 10 or 20 schedules, the total cost seems to deteriorate when operating a high number of operations. For population sizes of 50 and 100 schedules, this phenomenon is also assumed to happen, be it at much higher levels of executed operations. In general we can assume the graphs to be U-shaped, whereby the second leg of the U does not come that high. The optimal number of operations, i.e. the bottom of the U-shape, increases with the size of the population. Similarly, the influence of increasing population sizes on the total cost, when keeping the number of operations constant can be depicted. This is done in figure 25. Total cost Vs population size for different number of operations Total Cost 394 392 390 388 386 384 382 380 378 376 10 20 50 100 Schedules 50 operations 100 operations 250 operations 625 operations 1500 operations 3000 operations Figure 25 Total cost Vs population size for different number of operations For algorithms with a large number of operations, the increased size of the population is a positive influence on the total cost. This can be explained by the fact that a larger population size will accommodate a larger extent of diversity and thus better exploration of the solution space which increases the chance at finding better solutions. However for algorithms with a relatively low number of operations, a large population size had a negative influence on the total cost. This is graphically shown in figure 25 where the pink and yellow graphs go upwards when applying large population sizes. This can be explained by that fact that the number of operations is so low, in comparison with the population size, that the algorithm has no chance to nest itself into a (local) optimum. 61

For academic purposes, the interaction between the population size / number of operations and the total cost is interesting. However for practical purposes, the relationship between the execution time and the total cost is much more interesting. Therefore we combine the aforementioned two logic statements: The number of operations and the population size has an influence on the execution time (logic statement 1) The number of operations and the population size has an influence on the total cost (figure 24 and figure 25) (logic statement 2) We translate these into a relationship between execution time and the total cost. This logic and its four dimensions are embodied in figure 26 and represent a trade-off between total cost and execution time. Trade-off total cost Vs. execution time 392 Total Cost 390 388 386 384 10 schedules 20 schedules 50 schedules 100schedules 382 0 200 400 600 800 1000 1200 Execution time (seconds) Figure 26 Trade-Off Total Cost Vs. Execution Time Every dot in figure 26 represents a combination of population size and number of executed operations. For example, the blue graph represents the dots with a population size of 10 schedules. The first dot in the blue graph stands for 50 operations, the second for 100 operations, the third for 250 operations and so on. Every dot also has an according execution time (logic statement 1) and a total cost (logic statement 2) which are both dependent on the population size and number of operations. This results in a total cost versus execution time trade-off. The trade-off should be read as follows: When keeping the population size constant, an increasing execution time will on average improve the total cost and vice versa. Note the upwards trend in the graph for population sizes of 10 and 20 schedules, this has the same reasoning as the one explained by figure 24. 62

The trade-off in figure 26 can be molded into an efficient frontier by removing all inefficient dots from the graph. An inefficient dot is a dot that is outperformed by at least one other dot on both the time and cost dimension. Conversely we retain only the efficient dots, which are defined as every dot that is not outperformed by any other dot on both time and cost dimension. This efficient frontier is shown in figure 27. The color of the frontier graph indicates the population size as indicated in figure 26. A practical application of this efficient frontier is that the algorithm chooses the population size and number of operations based on an amount of time, that the algorithm is allowed to run, entered by the user. For example, I want the algorithm to give me best schedule in an amount of 250 seconds. By checking the efficient frontier, the best schedule that can be found, within 250 seconds of execution time is the algorithm with a population size of 20 and 625 operations to be executed. The expected total cost of the resulting schedule is slightly below 385. Efficient frontier total cost Vs. execution time Total cost 392 391 390 389 388 387 386 385 384 383 382 0 200 400 600 800 1000 1200 Execution time (seconds) Figure 27 Efficient Frontier Total Cost Vs. Execution Time Doubles In the start algorithm, we do not allow doubles to enter the population. This primarily inhibits premature convergence of the schedules in the population. However, if we allow doubles to enter, the total cost of 386,29 deteriorates to 390,62 or 1,12%. The influence of allowing doubles or disallowing doubles to enter the population is dependent on the size of the population and the number of operations. 63

Allowing doubles has as a direct implication that the population can converge faster towards a (local) optimum. Nevertheless this optimum seems to be of lower quality in most cases. This is because the converging of the population prevents the program from exploring the broader solution space in order to find better solutions. In table 9 we calculated the difference in total cost between the start algorithm with doubles and the start algorithm without doubles, for each combination of population size and number of operations. An example of how the table should be read: When using the start algorithm with population size 10 and 50 operations, the total cost of the algorithm that uses doubles is 5,24 higher than the algorithm version without doubles. Overall, the algorithm without doubles is the best however we indicated a region in yellow where the algorithm with doubles outperforms or returns similar results to the algorithm without doubles. This area has relatively big population sizes and a relatively small number of operations. The reason why the algorithm with doubles is able to perform well in this area is because it forces the population to converge faster into (local) optima while the algorithm without doubles will converge slower towards possible (local) optima. Population size 10 20 50 100 Operations 50 5,24-1,17 0,22-0,69 100 5,17 4,33 0,2 0,46 250 10,76 8,54 2,13-0,2 625 9,44 10,57 7,61 6,06 1500 8,75 11,13 11,69 12,86 3000 8,25 9,74 12,05 14,51 Table 9Total cost comparison with and without doubles in the population When plotting this algorithm as an efficient frontier, we assess that this new efficient frontier is entirely outperformed by the efficient frontier of the algorithm without doubles. We can conclude that none of the combinations of population size and number of operations in the yellow area of table 9 is on the efficient frontier of the algorithm with doubles. The newly added efficient frontier in figure 29 is plotted above the previously determined efficient frontier. 64

Efficient frontier Cost Vs. Execution time Cost 396 394 392 390 388 386 384 382 0 200 400 600 800 1000 1200 Execution time Figure 28 Efficient Frontier Total Cost Vs. Execution Time, with / without doubles Local search iterations For local optimization method L1, an extra parameter can be tuned. This parameter defines the number of local search iterations or the neighborhood depth of the method. As defined in section 4.3.4. this parameter determines how many times we consecutively check the neighborhood starting from a different activity. Logically, the more iterations, the better the outcome will be. Cost Vs. neighborhood depth Total Cost 394 393 392 391 390 389 388 387 386 385 1 3 5 7 9 Neighborhood depth Total cost Figure 29 Total Cost Vs. Number of Iterations Figure 29 shows a decreasing improvement of the total cost. While the improvements are significant between one and five iterations, the improvement at five or more iterations stagnates. 65

Mutation Total cost Vs Mutation rate Total Cost 387,8 387,6 387,4 387,2 387 386,8 386,6 386,4 386,2 386 0 2 5 10 25 Mutation rate (in %) Total cost Figure 30 Total Cost Vs. Mutation Percentage Figure 30 shows the evolution of the total cost when increasing the mutation probability. This graph does not provide us enough information to make a clear statement on the use of mutation. Nevertheless we can state that a high mutation percentage will probably not enhance the working of the algorithm. The reason could be that the operation and local search process probably already contain enough exploration possibilities. The start algorithm does not make use of mutation. As stated in chapter 5, the local search method is very dominant and has a very intensifying function. This could push the cross-over operator into the direction of an exploring function, in order to become better solutions. The used cross-over operator c5 is uniform cross-over, which is especially in the beginning of the algorithm execution very exploratory. Since the cross-over method bears the exploratory function, mutation is no longer necessary. We must note that this is a wild guess. Ending condition An ending condition will end the execution of the algorithm before it reaches the predefined number of operations and will thus shorten the execution time of the algorithm. However the goal is to shorten the execution time without affecting the objective function too much. Before fine-tuning this parameter, we created an indicator stating the moment when the best solution is found. This indicator is represented as a percentage, the best schedule percentage or BSP. A value of 40% means that the best schedule of the algorithm was found after 40 percent of the 66

execution time and that 60 percent of the execution time is wasted. The BSP however does not give any indication on the quality of the solution. Best solution Vs. Number of operations Best Solution Percentage 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 50 100 250 625 1500 3000 Operations 10 schedules 20schedules 50 schedules 100 schedules Figure 31 BSP Vs number of operations Figure 31 shows the influence of the number of operations on the BSP. An increase of the number of operations logically decreases the BSP. Overall, the size of the population increases the BSP. Also this is very logic since it takes more operations in a big population to become good solutions than in smaller populations. In this section, we will zoom in on the algorithm setup with a population of 20 schedules and 100 operations. This algorithm has a BSP of 57% and an expected total cost of 386,29. Since the objective function calculation, in framework2 of the algorithm, happens completely at the end of the algorithm, the BSP is only known afterwards when the whole algorithm is run. Therefore we cannot calculate whether improvement is made or not during execution. We will again rely on the AVGSQDEV as a relevant approximation for the total cost objective. When no improvement in this indicator is assessed, the execution can stop. We ran the algorithm on an ending condition of 5, 10, 20 and 30. This means that, if no improvement in AVGSQDEV is respectively found after 5, 10, 20 or 30 operations, no more operations will be executed and the termination of the algorithm will start by evaluating the schedules present in the population. Figure 32 shows the influence of the ending condition on the total cost of the schedule. A strict ending condition can lead to drastic deterioration of the cost of the schedule. 67

Total cost Vs ending condition 402 400 Total Cost 398 396 394 392 Total Cost 390 388 5 10 20 30 Ending condition Figure 32 Total cost Vs ending condition Figure 33 shows the influence of the ending condition on performed operations. The maximum possible number of operations is 100. Using a very strict ending condition of 5, we observe that only 11 operations are executed. Using an ending condition of 30, this amount increases to 65 which is still significantly below 100. Performed operations Vs ending condition 70 Performed Operations 60 50 40 30 20 Performed Operations 10 5 10 20 30 Ending condition Figure 33 Performed operations Vs ending condition We come to the conclusion that, for a strict ending condition, the gain in number of performed operations can be significant. Figure 33 shows the number of performed operations going down from 100 to 11, which is a decrease of 89%. 68

7. Conclusions and further research The first important conclusion is that a basic genetic algorithm with limited complexity is able to handle and manage the integrated project scheduling and project staffing problem. With a total cost expectation of maximum maximorum 7% higher than the lower bound benchmark, this GA proves it is able to handle this problem. The power of the resource levelling objective, as an approximation for the total cost has proven to be effective. The alternative GA framework, i.e. framework2 which heavily relies on that premise does not perform drastically worse than the original framework (except for AN1). The observation that the AVGSQDEV is a good approximation for the total cost is, especially when keeping the project length constant, is confirmed. Framework1 always yields better solutions than framework2 but framework2 outperforms framework1 drastically in computational effort. Extending the classical GA algoritm with local optimization provides a siginificant boost to the quality of the resulting schedule. While the cross-over operator guides the population to better solutions steadily, a local search optimization is actively and aggressively going to look for better solutions. Another important conclusion is the existence of a time-cost trade-off. Knowing that this exists and being able to predict the location of the efficient frontier, it is possible to optimize the expected total cost in function of the computational time one is willing to spend. To conclude, we can install intelligent ending conditions. These conditions will never improve the quality of the solutions since they are an extra constraint on the execution time spent. However smart ending conditions can determine when that algorithm no longer makes any progression and thus decide to stop earlier in order to save computational effort. 69

Further research topics, related to this thesis that should deserve consideration include a more in-depth analysis of the interdependencies. I conducted research on different GA methods in a one- or two-dimensional way. Statistical analysis is necessary to discover more complex interdependencies between the applied methods. Another interesting topic for further investigation is the application of hyperheuristics. This kind of heuristics will, depending on the kind of problem or data it receives, alter its way of working which fits the problem the best. Applied to the project scheduling problem, this could mean that characteristics of the activity network are measured and based upon these characteristics, different methods in the GA are applied. Examples of these characteristics could be the size of the network, the Serial parallel indicator and the activity distribution indicator. (Vanhoucke et al., 2008) Since the local optimization stage holds the most significant value for the complete algorithm, more research could be done on that in the sense that more complex and intelligent local search methods should be tested. Large neighborhood search or very large neighborhood search methods could be interesting options to consider. An interesting topic to further investigate would be to check the robustness of the presented algorithm. What happens to the solution quality when the problem definition is altered slightly? Is the GA still appropriate when executing it on datasets with larger amount of activities in its network? Framework1 is probably more robust towards changes in the problem since it is not able to, in the course of the execution of the algorithm, worsen the resulting solution. Framework2 is able to worsen throughout the execution of the algorithm and would thus be less robust for certain changes. A fifth and last point I would like to mention for further research is the application of adaptive systems in the GA. This would mean that both used methods and the parameters are auto-maintained by the algorithm. The algorithm is thus intelligent to the extent that it can distinguish the different circumstances in which each method or parameter is the most appropriate. 70

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VII. Appendices Appendix A: Datasets AN1 CP = 9 days SPI = 0.11 76

AN2 CP = 12 days SPI = 0.33 AN3 CP = 16 days SPI = 0.44 77

Appendix B: Method elimination in basic cycles AN1 Framework1 Stage Initialization Selection Operation Local Optimization Reinsert condition Doubles Population management Cost Phase I1a I1b I1c I2a I2b I2c I3a I3b I3c I3d I4 S1 S2 S3 S4 C1 C2 C3 C4 C5 C6 L1 L2 L3 L4 R1 R2 R3 R4 R5 R6 N Y P1 P2 P3 Base Case x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 400,10 1st Exclusion x x x x x x x x x x x x x x x x x x x x x 368,73 2nd Exclusion x x x x x x x x x x x x x x 363,96 Best X x x x x x x x x x x x 358,23 Best x x x x x x x 355,25 AN1 Framework2 Stage Initialization Selection Operation Local Optimization Reinsert condition Doubles Population management Cost Phase I1a I1b I1c I2a I2b I2c I3a I3b I3c I3d I4 S1 S2 S3 S4 C1 C2 C3 C4 C5 C6 L1 L2 L3 L4 R1 R2 R3 R4 R5 R6 N Y P1 P2 P3 Base Case x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 400,67 1st Exclusion x x x x x x x x x x x x x x x x x x x 378,34 2nd Exclusion x x x x x x x x x x x x x 371,36 Best X x x x x x x x x x x 366,32 Best x x x x x x x 365,67 AN2 Framework1 Stage Initialization Selection Operation Local Optimization Reinsert condition Doubles Population management Cost Phase I1a I1b I1c I2a I2b I2c I3a I3b I3c I3d I4 S1 S2 S3 S4 C1 C2 C3 C4 C5 C6 L1 L2 L3 L4 R1 R2 R3 R4 R5 R6 N Y P1 P2 P3 Base Case x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 402,06 1st Exclusion x x x x x x x x x x x x x x x x x x x 391,04 2nd Exclusion x x x x x x x x x x x x x x 388,91 Best X x x x x x x x x x x x 386,74 Best x x x x x x x 385,34 78

AN2 Framework2 Stage Initialization Selection Operation Local Optimization Reinsert condition Doubles Population management Cost Phase I1a I1b I1c I2a I2b I2c I3a I3b I3c I3d I4 S1 S2 S3 S4 C1 C2 C3 C4 C5 C6 L1 L2 L3 L4 R1 R2 R3 R4 R5 R6 N Y P1 P2 P3 Base Case x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 403,51 1st Exclusion x x x x x x x x x x x x x x x x x x x x x x x x 389,67 2nd Exclusion x x x x x x x x x x x x x x x x x x x x x 389,09 Best X x x x x x x x x x x 387,03 Best x x x x x x x 386,29 AN3 Framework1 Stage Initialization Selection Operation Local Optimization Reinsert condition Doubles Population management Cost Phase I1a I1b I1c I2a I2b I2c I3a I3b I3c I3d I4 S1 S2 S3 S4 C1 C2 C3 C4 C5 C6 L1 L2 L3 L4 R1 R2 R3 R4 R5 R6 N Y P1 P2 P3 Base Case x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 399,82 1st Exclusion x x x x x x x x x x x x x x x x x x x x x 391,16 2nd Exclusion x x x x x x x x x x x x x x x x 386,60 Best X x x x x x x x x x x 384,42 Best x x x x x x 384,29 AN3 Framework2 Stage Initialization Selection Operation Local Optimization Reinsert condition Doubles Population management Cost Phase I1a I1b I1c I2a I2b I2c I3a I3b I3c I3d I4 S1 S2 S3 S4 C1 C2 C3 C4 C5 C6 L1 L2 L3 L4 R1 R2 R3 R4 R5 R6 N Y P1 P2 P3 Base Case x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 404,79 1st Exclusion x x x x x x x x x x x x x x x x x x x x 388,14 2nd Exclusion x x x x x x x x x x x x x x 388,19 Best X x x x x x x x x x x 387,83 Best x x x x x x x 386,00 79

Appendix C: Cost evolutions of basic cycles AN1 Framework 1 Phase Average Min Max Base Case 400,1 355 554 1st Exclusion 368,73 351 390 2nd Exclusion 363,96 354 379 Best X 358,23 352 374 Best 355,25 352 359 550 Total Cost evolution 500 Total Cost 450 400 Average Min Max 350 Base Case 1st Exclusion 2nd Exclusion Best X Best Phase AN1 Framework 2 Phase Average Min Max Base Case 400,67 361 414 1st Exclusion 378,34 352 403 2nd Exclusion 371,36 355 405 Best X 366,32 355 396 Best 365,67 355 382 80

Total Cost evolution Total Cost 410 400 390 380 370 360 Average Min Max 350 Base Case 1st Exclusion 2nd Exclusion Best X Best Phase AN2 Framework 1 Phase Average Min Max Base Case 402,06 383 450 1st Exclusion 391,04 380 410 2nd Exclusion 388,91 378 401 Best X 386,74 373 405 Best 385,34 373 400 Total Cost evolution 460 450 440 Total Cost 430 420 410 400 Average Min Max 390 380 370 Base Case 1st Exclusion 2nd Exclusion Best X Best Phase 81

AN2 Framework 2 Phase Average Min Max Base Case 403,51 383 458 1st Exclusion 389,67 373 416 2nd Exclusion 389,09 377 414 Best X 387,03 378 402 Best 386,29 378 403 Total Cost evolution 460 450 440 Total Cost 430 420 410 400 Average Min Max 390 380 370 Base Case 1st Exclusion 2nd Exclusion Best X Best Phase AN3 Framework 1 Phase Average Min Max Base Case 399,82 382 439 1st Exclusion 391,16 382 404 2nd Exclusion 386,6 380 394 Best X 384,42 378 392 Best 384,29 381 391 82

Total Cost evolution Total Cost 435 425 415 405 395 385 Average Min Max 375 Base Case 1st Exclusion 2nd Exclusion Best X Best Phase AN3 Framework 2 Phase Average Min Max Base Case 404,79 383 514 1st Exclusion 388,14 378 404 2nd Exclusion 388,19 378 400 Best X 387,83 378 400 Best 386 382 390 Total Cost evolution 515 495 475 Total Cost 455 435 415 395 Average Min Max 375 Base Case 1st Exclusion 2nd Exclusion Best X Best Phase 83

Appendix D: Observation link AVGSQDEV Total cost All project lengths Average Squared deviation - Objective 270 250 Objective 230 210 190 170 0 50 100 150 200 250 300 Average Squared deviation Project length = 8 Average Squared deviation - Objective 270 250 Objective 230 210 190 170 0 50 100 150 200 250 300 Average Squared deviation Project Length =11 Average Squared deviation - Objective 270 250 Objective 230 210 190 170 0 20 40 60 80 100 120 140 160 180 200 Average Squared deviation 84

Project Length = 14 Average Squared deviation - Objective 270 250 Objective 230 210 190 170 0 20 40 60 80 100 120 140 Average Squared deviation Project Length = 17 Average Squared deviation - Objective 270 250 Objective 230 210 190 170 0 10 20 30 40 50 60 70 80 90 100 Average Squared deviation Project Length = 20 Average Squared deviation - Objective 270 250 Objective 230 210 190 170 0 10 20 30 40 50 60 70 80 Average Squared deviation 85