Resource constrained project scheduling with uncertain activity durations

Accepted Manuscript Resource constrained project scheduling with uncertain activity durations Ripon K. Chakrabortty, Ruhul A. Sarker, Daryl L. Essam PII: DOI: Reference:

S0360-8352(16)30518-6 http://dx.doi.org/10.1016/j.cie.2016.12.040 CAIE 4592

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Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

15 March 2016 29 September 2016 30 December 2016

Please cite this article as: Chakrabortty, R.K., Sarker, R.A., Essam, D.L., Resource constrained project scheduling with uncertain activity durations, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie. 2016.12.040

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RESOURCE CONSTRAINED PROJECT SCHEDULING WITH UNCERTAIN ACTIVITY DURATIONS

Ripon K Chakrabortty*, Ruhul A Sarker and Daryl L Essam School of Engineering and Information Technology University of New South Wales, Canberra 2600, Australia *Corresponding email: [email protected]; [email protected] *Corresponding phone no: +61-420882041

ABSTRACT In this paper, we consider Resource Constrained Project Scheduling Problems (RCPSPs) with known deterministic renewable resource requirements but uncertain activity durations. In this case, the activity durations are represented by random variables with different probability distribution functions. To deal with this problem, we propose an approach based on the robust optimization concept, which produces reasonably good solutions under any likely input data scenario. Depending on different uncertainty characteristics, we have developed six different heuristics to incorporate the uncertain duration as a deterministic constraint in a robust optimization model. The resulting optimization model is then solved by using a Coin-Branch & Cut (CBC) solver. To judge the performance of the algorithm, we solved 30, 60, 90 and 120-activity benchmark problems from the project scheduling problem library (PSPLIB). Our proposed approach guarantees the feasibility of solutions and produces high-quality solutions, particularly for larger activity instances, compared to other existing approaches.

Keywords: RCPSP; SRCPSP; Branch and Cut algorithm; Robust Optimization; Heuristics. 1

RESOURCE CONSTRAINED PROJECT SCHEDULING WITH UNCERTAIN ACTIVITY DURATIONS

ABSTRACT In this paper, we consider Resource Constrained Project Scheduling Problems (RCPSPs) with known deterministic renewable resource requirements but uncertain activity durations. In this case, the activity durations are represented by random variables with different probability distribution functions. To deal with this problem, we propose an approach based on the robust optimization concept, which produces reasonably good solutions under any likely input data scenario. Depending on different uncertainty characteristics, we have developed six different heuristics to incorporate the uncertain duration as a deterministic constraint in a robust optimization model. The resulting optimization model is then solved by using a Coin-Branch & Cut (CBC) solver. To judge the performance of the algorithm, we solved 30, 60, 90 and 120-activity benchmark problems from the project scheduling problem library (PSPLIB). Our proposed approach guarantees the feasibility of solutions and produces high-quality solutions, particularly for larger activity instances, compared to other existing approaches.

Keywords: RCPSP; SRCPSP; Branch and Cut algorithm; Robust Optimization; Heuristics.

1. INTRODUCTION The resource constrained project scheduling problem (RCPSP) aims to minimize the duration of a project, subject to satisfying the precedence and resource availability constraints. The RCPSP assumes complete information of resource usage and activity duration, and determines a feasible baseline schedule which encompasses a list of activity starting times while minimizing the makespan (Bruni et al., 2011). For many years, the RCPSP has been of interest for practitioners and researchers. There has been a tremendous increase in research for the RCPSP, both in terms of heuristic and optimal procedures over the last few decades. Among them, Hartmann and Kolisch (2000) considered an in-depth literature survey on stateof-the-art heuristics and analysed the behaviour of different heuristics with respect to their 3

components, such as priority rules and meta-heuristic strategy. Apart from that, from a modelling perspective, RCPSPs can be formulated by following discrete or continuous time (i.e., even based) depending upon the number of variables (Chakrabortty et al., 2014). Alongside of that, very recently Koné et al. (2013) have reported some important merits and demerits of different state-of-the-art exact approaches (i.e. mixed integer linear programming models-MILP) and concluded that the exact approaches are not competitive enough for higher activity instances for RCPSPs. Relevant surveys on different heuristic and exact approaches for RCPSPs can be found from Hartmann and Briskorn (2010), Huang et al. (2013) and Zhou et al. (2013). Among a few exact approaches, Zhu et al. (2006) presented an exact branch and cut procedure for solving RCPSP directly while using an integer linear programming (ILP) model. However, in their paper, they only considered 30 and 60 activity instances from the project scheduling problems library (PSPLIB). Therefore, B&B or B&C for higher instances is still an open area to perform research.

Because of a growing awareness that traditional deterministic project scheduling models are in plain conflict with a reality that is characterized by uncertainty, a recent research track focuses on the stochastic resource constrained project scheduling problem (SRCPSP). The SRCPSP is an extension of the RCPSP that involves the minimization of the expected makespan of a project with stochastic activity durations (problem

). As

the SRCPSP tries to incorporate uncertainty in the formulation itself by expressing the random parameters of the problems as random variables, the resulting stochastic problem is challenging from both theoretical and computational point of view (Bruni et al., 2015). Despite its applicability, the literature concerning SRCPSPs is virtually void (Van de Vonder et al., 2007). Uncertainties during project execution may lead to considerable schedule disruptions. This uncertainty may arise from a number of possible sources: activities may take more or less time than originally estimated (i.e., activity disruptions), resources may become unavailable (i.e., resource disruptions), material may arrive behind schedule, ready times and due dates may have to be changed, new activities may have to be incorporated or activities may have to be dropped due to changes in the project scope, weather conditions may cause several delays, etc (Herroelen and Leus, 2005). Existing research in project scheduling has considered activity disruptions (durational variability for activities) (Deblaere et al., 2011a, Bruni et al., 2011, Ma et al., 2015, Artigues et al., 2013, Lamas and Demeulemeester, 2014, Van de Vonder et al., 2008), resource disruptions (due to breakdowns) (Chakrabortty et al., 2016, 4

Lambrechts et al., 2007, Lambrechts et al., 2010) or even combination of both activity and resource disruptions (Deblaere et al., 2011b, Fu et al., 2015). Although the sources of variability in the project environment are diversiform, the main scheduling objectives are mostly pertinent to the activities’ starting (or ending) times, with project makespan being the single most studied objective (Artigues et al., 2013). This rationalizes a restriction to the study of uncertainty in activity processing times or durations only, although different sources may be the basis of this uncertainty.

Even though a significant amount of theoretical work has already been done to conceptualize and measure uncertainty in project scheduling, still the most proficient managers have difficulty in handling them. The lack of practitioner’s knowledge on adequate answers to uncertainty in RCPSP is surrounded by a relatively scant scientific interest for this hot topic. As a matter of fact, whilst RCPSPs have been widely investigated in the academic literature, the issue of the incorporation of uncertainty in project scheduling has received growing attention in the last decade (Bruni et al., 2015). The uncertainty that arises during project execution can be explicitly taken into account through preventive approaches, such as twostage stochastic programming, parametric programming, fuzzy programming, chance constraint programming and robust optimization techniques (Ye et al., 2014). Since for robust optimization approaches most of the material on scheduling policies developed for stochastic scheduling can be transferred without major alterations, and the whole uncertainty space can be easily addressed instead of using parametric scenarios, the authors restrict this paper work to that approach only.

The term robust optimization has come to encompass several approaches to protecting the decision maker against parameter ambiguity and stochastic uncertainty (Gabrel et al., 2014). Because of its applicability, robust optimization has been applied in diversified areas such as classical logistic problems (Remli and Rekik, 2013), facility location problems (Baron et al., 2011), inventory management (Song, 2010), finance (Fabozzi et al., 2010), revenue management (Davendralingam, 2011), stochastic systems (Bertsimas et al., 2013) and even in energy systems (Jiang et al., 2012). The detailed reviews on recent advances in robust optimization can be referred to (Gabrel et al., 2014). Although the concept of robust optimization has already been utilized in different practical areas, the application on RCPSPs is still meagre. Apart from the earlier work of Artigues et al. (2013), the authors could not find out any works on robust optimization approach for RCPSPs. Therefore, the application 5

of a robust optimization approach for RCPSPs that consider uncertain activity durations and renewable/non-renewable resource constraints is a completely new and unique contribution of this work.

Artigues et al. (2013) proposed a few project scheduling models with considerable uncertainty in their activity durations, for when a decision maker does not have sufficient confidence in the subjective probabilities that can be attributed to different duration scenarios. They developed a scenario-relaxation algorithm, based on the robust optimization concept, and proposed two heuristic procedures for solving up to medium sized instances. Our work differs from this line of work in two aspects. Firstly, our approach is capable of solving not only small scale problems but also large activity instances. Secondly, instead of using either mean durations or omitting them for heuristic procedures, we propose six different heuristic models that depend on different possible uncertainty levels, and solve them with the proposed algorithm. The contributions in this paper are threefold: (i) depending on various forms of parametric uncertainty levels, we have developed six different heuristics to incorporate the uncertain activity durations as deterministic constraints in a robust optimization model; (ii) we propose a robust optimization based approach for solving resource constrained project scheduling under durational uncertainties; (iii) we apply a distinct branch-and-cut algorithm (B&C) for solving those robust optimization models for stochastic-RCPSPs .

Primarily, the focus of this paper is on the application of a newly defined branch-and-cut algorithm (B&C) for solving deterministic RCPSP models particularly for those which cannot be solved by using traditional exact approaches. In this case, a constrained logic programming (CLP) technique was employed with a branch-and-cut algorithm to satisfy the constraints and to facilitate a computing capability. The deterministic RCPSP model was solved by using a commercial optimization solver from the OPTI toolbox that comes with Matlab (R2012b). The proposed algorithm is capable of dealing with large activity instances of RCPSPs within a reasonable computing time (1000 seconds). To demonstrate the performance of B&C algorithm, the benchmark instances with 30, 60, 90 and 120 activities from the PSPLIB were solved. The solutions were compared with traditional heuristic and exact approaches for some randomly chosen 30-activity instances. Later, we studied the stochastic-RCPSP with the possibility of activity duration uncertainty. Depending on different durational uncertainty levels/forms, we propose six different heuristic procedures to 6

determine the realized activity durations and to incorporate the deterministic constraints into the robust optimization model. To develop different deterministic robust counterpart optimization models for those uncertain activity durations, we then employed a robust optimization framework (Janak et al. (2007), Li et al. (2011), Li and Floudas (2014), Lin et al. (2004)). Finally, those robust optimization models were solved by using our B&C algorithm for different benchmark instances from PSPLIB. The solution from the robust optimization framework was guaranteed to be feasible for the whole space of the uncertain duration parameters.

The remainder of this paper is organizes as follows: in section 2, we define the deterministic RCPSP and stochastic RCPSP under varied activity durations with a brief discussion of the scheduling methodology. The traditional deterministic mathematical formulation for RCPSP is shown in section 3. In section 4, an overview of the robust optimization approach/framework is discussed with different uncertainty levels. Our robust optimization approach for durational uncertainty is addressed in section 5, followed by a detailed discussion on our proposed B&C algorithm and its solution methodologies (section 6). The results of our computational experiments on different PSPLIB benchmark instances for both deterministic and stochastic RCPSPs are presented in section 7. The computational behavioural patterns exhibited by the different heuristic procedures are described in section 7.2.4. Finally, we provide conclusions in the last section.

2. PROBLEM DESCRIPTION The RCPSP can be stated as follows. A single project consists of

activities where each

activity has to be processed in order to complete the project. The dummy activities 0 and correspond to the ‘project start’ and to the ‘project end’, respectively. The activities are interrelated by two kinds of constraints. First, precedent constraints force activity j not to be started before its immediate predecessor activities have been finished. Second, all the activities resource consumption at each particular time should satisfy their maximum resource availabilities. In general there are two types of resources, i.e. renewable and nonrenewable. Renewable resources are limited for a particular time period and can be replenished (for example, machine and manpower). On the contrary, non-renewable resources are limited throughout the project tenure like money. The RCPSP under study in this paper is based on the following assumptions (i) the activities composing a project have certain and known durations for the deterministic part only; (ii) all predecessors must be 7

finished before an activity can start; (iii) although resources can be either of renewable or non-renewable, here we consider only the renewable resources; (iv) activities are nonpreemptive (i.e., cannot be interrupted when in progress); (v) the main objective is to minimize the project completion time. The proposed formulations for scheduling are discrete in manner, i.e., it depends on discretization of the time horizon T, which is represented as time period t

). Each resource type has a certain capacity limit which should be

utilized throughout the project life without exceeding that limit.

Meanwhile for SRCPSPs, we associate with each activity

a set

possible realizations of the uncertain duration of activity (with reals). The set of

can be a discrete set

In the first case, we also write activities, 0 and

containing the the set of non-negative

or an interval and

.

. The dummy

have zero duration. Moreover, to handle uncertainty, the set of

is

also assumed to potentially follow different general distributions (i.e., uniform, normal and Poisson). Carrying out of a project with uncertain activity durations is a dynamic decision process. Under this scenario, a solution is a policy which defines actions at decision times. A schedule is constructed gradually through time . In this paper, we focus on activity duration uncertainty which is considered as: (a) bounded, (b) symmetric and bounded, (c) following a known distribution such as uniform, Poisson or normal distribution, and (d) following an unknown probability distribution. It should be noted that the random duration parameters considered in this paper are assumed to be fully independent.

To the best of our knowledge, apart from the work of Artigues et al. (2013), none of the methods proposed in the literature of SRCPSPs considered a robust optimization framework that can represent uncertain probabilistic parameters. Moreover, unlike the literature, instead of assuming that only one activity at a time disturbs the starting time of a successor activity, we considered that any precedence activities may hamper the starting time of their successors. Secondly, our standpoint is rather unusual in the literature on stochastic resource constrained project scheduling, since our aim is to construct a proactive schedule under uncertainty, partially bridging the gap between the stochastic scheduling literature and the robust one. In uncertain environments, especially from a practical context, constructing a proactive schedule, with associated vectors of predictive starting and completion times, to

8

attempt to limit the risk of future perturbations, is still a challenging research area for practitioners.

In uncertain environments, especially from a practical point of view, project managers are mainly interested in the generation of a proactive schedule with a quality that does not degrade during project execution because of future perturbations. In this paper, we used the unit penalty cost approach in addition to minimizing the schedule makespan, as shown in Eq. (1), where

represents the realized starting time of any activity after uncertainty,

represents the actual starting time of any activity before uncertainty is considered and the penalty cost or instability weight

and

, allocated to each activity

and

denotes

the marginal costs of deviating the start time of activity from its planned completion time in the baseline schedule. Hence, the overall objective for this SRCPSP is to find precedence feasible and resource feasible schedule

that minimizes both the project makespan

and the sum of deviation penalties of all activities under uncertain activity duration set

(as shown in Eq. (1)). Here we mention that, for the formulation of RCPSP,

we use a binary decision variable otherwise. Hence,

, which equals 1 if activity starts at time period , and 0

represents the binary decision variable which is equal to 1 if the last

activity (i.e., the dummy end activity,

) starts at time period , and

otherwise.

(1)

Uncertainty in activity duration is usually represented by random variables with known probability distribution functions. As the realized completion time is directly affected by duration uncertainty, to determine the deterministic robust counterpart for uncertain activity duration, we employed a robust optimization framework only on durational uncertainty.

3. DETERMINISTIC MATHEMATICAL FORMULATIONS In this section we present the Mixed Integer Programming model for the RCPSPs under study. For the formulation of RCPSP, we use a binary decision variable

which is equal to

1 if activity starts at time , and 0 otherwise. This formulation requires at most decision variables and

binary

constraints, where n is the number of jobs, m is the

number of arcs, and T is the maximum time required to complete the project. In spite of various alternative models (Bianco and Caramia, 2011, Brucker and Knust, 2000, Brucker et 9

al., 1998, Koné, 2012), mainly based on the representation of decision variables, the initial mathematical model from Pritsker et al. (1969) and Talbot (1982) for the RCPSP is still in use and is hence used in this paper. The basic formulation is presented below.

Sets I T

set of activities, i =0…I+1 set of time periods, t = 0…T Represents the precedence set set of renewable resources, r = 1…R

R

Parameters Capacity of resource at time Resource usage of activity for resource Duration of activity

Minimize Z = Constraints: (2) (3) (4) (5)

In the above model, the objective is to minimize the completion time of the last activity of the project, which represents the project completion time (this objective function is for only our deterministic RCPSP). Constraint (2) indicates that every job or activity must be started exclusively in one time period. That means, no balking or priority violation is allowed (Nonpreemptive case). The job which has started must be completed first, depending upon their corresponding priority. Constraint (3) ensures the precedence relationship that ensures that an activity cannot be started before the completion of its preceding activity. Finally, constraint (4) represents the capacity constraints of all renewable or non-renewable resources.

4. OVERVIEW OF THE ROBUST OPTIMIZATION FRAMEWORK The robust optimization framework developed by Janak et al. (2007), Li et al. (2011), Li and Floudas (2014), Li et al. (2012) is used to address duration uncertainty in this paper. Before going into detail, we present a brief robust optimization approach, which explicitly takes into account the various forms of parameter uncertainty within constraints and/or objective

10

functions. As shown in the previous section on RCPSP, the optimal solution of an MILP program may become infeasible, that is, one or more constraints are violated substantially, if the nominal data is slightly disrupted. Our objective here is to develop a robust optimization methodology to generate “reliable” solutions to the MILP program, which are less vulnerable to uncertainty. This means that the optimal solution should provide the best possible value of the original objection function while also being guaranteed to remain feasible in the range of the uncertainty set considered (Ben-Tal and Nemirovski, 2000). The robust optimization framework has the advantage that it not only explicitly takes into account the various forms of parameter uncertainty, but also ensures that the obtained solution is feasible for the nominal system conditions. In the robust optimization framework, a solution

is called

robust if it satisfies the following conditions: (i) (ii)

is feasible for the nominal problem; whatever are the true values of the coefficients and right-hand-side parameters, must satisfy the -th inequality constraint with an error of at most where

,

is a given infeasibility tolerance (Ye et al., 2014).

For further study, more elaborate explanation and proofs can be found in the work of Ye et al. (2014). Let us assume a generic deterministic MILP problem:

(6)

Let ascertain that the left-hand-side coefficients

and , and the right-hand-side parameters

of the inequality constraints are uncertain parameters. The true realization of an uncertain parameter is represented by,

(7)

where,

is an uncertain parameter with

uncertainty level, and

being its true realization,

represents a given

stands for a random variable. Then, according to the robust

optimization conditions, any inequality constraint in Eq. (6) becomes: 11

(8)

where is the index of the uncertain inequality, is the index of the binary terms. Thus,

is the index of the continuous terms, and

and

are the subsets that encompass those

uncertain parameters for the given inequality constraint

During project tenure, there may

arise numerous unavoidable uncertainties such as resource capacity fluctuations, infrequent arrival of raw materials, man-power uncertainty, activity duration uncertainty etc. Among them resource uncertainty and activity’s processing time uncertainty, are most common and can be described using continuous or discrete distributions. In this paper, we focus on activity duration or processing time uncertainty which is considered as: (a) bounded, (b) symmetric and bounded, (c) following a known distribution such as uniform, normal or poisson distribution, or (d) following an unknown probability distribution. The considered random duration parameters in this paper are assumed to be fully independent.

4.1 Bounded uncertainty Let us assume that the uncertain parameters vary in a bounded interval and they are represented by,

Then according to Ye et al. (2014), the deterministic robust counterpart for Eq. (8) derived as follows,

(9)

where,

and variable

is an auxiliary positive variable. Meanwhile,

is the index of that uncertain inequality constraint, terms, and

is the index of the continuous

is the index of the binary terms. Then, the deterministic robust counterpart

of the original uncertain MILP problem can be derived by adding constraint (6) and the limiting constraint:

12

4.2 Bounded and symmetric uncertainty Again, if the uncertain parameters are randomly and symmetrically distributed around their nominal values, then they are represented as follows:

where,

,

are random variables distributed symmetrically in the interval [-1,1].

Now we can define a probabilistic version of condition (ii), as described in the previous section for a robust solution. For the -th inequality, the probability of the event of constraint violation, i.e. where

, is at most

is a given feasibility tolerance and

,

is a given “reliability level” (Lin

et al., 2004).

Lemma 1. Let

be given reals and

distributed in [-1, 1]. Then for every

be independent random variables symmetrically the following inequality holds: (10)

For the sake of better understanding, we provide here the proof of this Lemma presented by Ben-Tal and Nemirovski (2000). By homogeneity arguments, it suffices to consider the case of

. Now, in this case

13

With (a) Tschebyshev inequality; (b) independence of

; (c) Taylor’s expansion and

; (d) Taylor’s expansion and (e)

symmetric distribution of

.

Therefore, according to this aforementioned Lemma, it can be justified that the reliability level

.

Now, for an infeasibility tolerance

and a reliability level

), the following

- deterministic robust counterpart for Eq. (9) can be derived as follows:

(11)

Where, (12) (13) (14) (15) (16)

where

,

inequality,

,

are auxiliary positive variables,

is the index of the continuous terms,

is a positive parameter with

is the index of the uncertain

is the index of the binary terms and . Then, the deterministic robust

counterpart of the original uncertain MILP problem can be derived by adding constraint sets (11) to (16).

4.3 Known probability distribution The true values of the uncertain parameters are obtained from their nominal values by random variables: (17)

Now, if the probability distributions of the random variables

,

in the

uncertain parameters are known, then the earlier MILP can be re-written as follows:

14

where the data set

varies in a given uncertainty set Z,

represents the true value of the uncertain coefficients, and

and

is an infeasibility

tolerance. The inequality (17) can be written in expanded form as

(18)

Every constraint ,

,

and

are again the true values of the uncertain

coefficients. Substituting the expressions for the true values of the uncertain coefficients given in constraint (17), the uncertain inequality in (18) can be rewritten as follows

(19)

where

and

define the sets of uncertain parameters

and

, respectively for

constraint . Now to transform the constraint into a deterministic form, Janak et al. (2007) proposed the following formulation

(20)

This constraint enforces that the probability of violation of the uncertain inequality is at most

, where

is a given feasibility tolerance and

is a given reliability

level (i.e., the probability of violation of constraint ). Hence, if we know a probability distribution function for the sum of the random variables,

(21)

15

we can use this information in the probabilistic constraint (20), to write a deterministic form for the uncertain constraint which is reliable, depending on the value of . This case is demonstrated using the definition of a probability distribution function and the following relationship

(22)

Then the final form of the deterministic constraint (or robust counterpart problem) is simply determined using the inverse distribution function (quantile) of the random variable

(23)

Thus, the generic deterministic robust counterpart of the probabilistic constraint for random variables following any distribution can be formulated as follows:

(24)

Where

is the index of the uncertain inequality constraint,

continuous terms, and

is the index of the binary terms.

is the index of the

is determined from

using

constraint (22) and the probability distribution function for . For example, if we make the assumption that all the relevant random variables are independent and follow a normal distribution, the function

can be represented in the following

manner: (25)

where

,

and

are the standard deviations for the independent normal random

variables used to represent the uncertainty associated with the parameters respectively. Replacing the function

,

and

with its equivalent form for random

variables following independent normal distributions, the deterministic robust counterpart constraint can be written as seen in Eq (26).

16

(26)

4.4 Unknown probability distribution In this section, we discuss the scenario of uncertain right-hand side parameters which follow an unknown probability distribution. The nominal values for the right hand side parameters of Eq. (6) (i.e., which follow an unknown probability distribution) will be considered equivalent to the average values for the same right hand side parameters. The inequality constraints for this type are like,

(27) (28)

Equations (29) and (30) represent the realization of the uncertain parameters,

, where

represents the subset of the right-hand side parameters which are stochastic in nature for a given constraint . (29) (30)

The upper mentioned inequalities can be reformulated into probabilistic inequalities, like Equations (31) and (32), while

is defined as a reliability level.

(31) (32)

Now, an upper bound of the probability of the sum of the uncertainty right-hand side parameters taking on a value greater than

can be provided by means of the Markov

inequality (Verderame and Floudas, 2009), as follows:

(33)

To determine the value of

, .

So,

(34) 17

The upper bound on

represents the largest expected value that the term

can

take on for a given value of , and as a result, it can be utilized in order to formulate a relaxed deterministic robust counterpart constraint for Eq. (29) (see equations 35 and 36).

(35) (36)

For the sake of computational ease, we assumed that the unknown distribution is approximately symmetric and the extremes are equidistant from the mean. Then a lower bound on the expected value of the term of , and it is equivalent to the value of

can be determined for a given value defined by Eq (37). The max operator is used

in order to signify the fact that the underlying distribution is nonnegative, as the duration values are always non-negative. The given lower bound can then be used in order to formulate a relaxed deterministic robust counterpart constraint for Eq. (30), as seen in equations (38) and (39).

(37)

(38) (39)

Here,

and is the index of the uncertain inequality constraint. Meanwhile,

represent the uncertain parameters for element

, where

represents the subset

of right-hand side parameters which are stochastic in nature for a given constraint . For better understandings, interested readers are referred to the work of Verderame and Floudas (2009).

Therefore, the deterministic robust counterpart optimization model for the original uncertain MILP problem can be derived by adding constraint set (39). 5. ROBUST OPTIMIZATION APPROACH FOR DURATIONAL UNCERTAINTY 18

In the case of RCPSPs, activity duration uncertainty may lead to increases in the project completion time or makespan time. Hence, in the case of designing robust RCPSPs, handling those durational uncertainties is a must. In its practical aspect, activity duration uncertainty replicates processing time uncertainties and may have numerous forms to represent. Keeping that in mind, in this paper we present six heuristics, based on the way the activity duration uncertainties are measured or estimated. Here, the activity duration information is implicitly lumped into set

, which denotes the processing times for an activity set c. As mentioned

earlier, the overall objective for this SRCPSP is to find precedence feasible and resource feasible schedule

that minimizes both the project makespan

and the sum of

deviation penalties of all activities under uncertain activity duration set

(as shown in Eq.

(1)). However, since the realized project makespan increases with increases in activity durations, the cardinality of set

should be higher than or equal to any typical

. (40)

Let

represent the true realization of the uncertain demand parameter of activity . Then Eq.

(40) becomes: (41)

Now we can apply the robust optimization techniques to Eq. (41). For simplicity, universal values for ,

and

were adopted; however, as noted by Janak et al. (2007), the robust

optimization framework can easily be extended in order to take into account scenarios where ε can vary from parameter to parameter and κ and δ can vary from constraint to constraint. Here in this research, we set two different uncertainty levels (U.L), infeasibility tolerance

as 0%; and the reliability level

as 0.05 and 0.15; the

as 5%.

5.1 Bounded uncertainty If the activity duration parameter varies in a bounded interval

, then

according to Eq. (9), the robust counterpart for Eq. (41) is given by: (42) and Here we include the robust counterpart (i.e., Eq. (42)) into the deterministic formulation to generate the robust optimization formulation. To solve this robust optimization formulation, we need to predetermine set

. Recall that the objective function is to

minimize the makespan, which increases as processing time increases. Therefore, only the equality part of Eq. (42) needs to be considered, which stands when, 19

which is the maximum true realization of the uncertain duration parameters. Since the values of set

need to be integer, then

should be considered, where function

provides the smallest integer

that is greater than . Later on, we present how this robust optimization model was solved by using our proposed B&C approach. The first heuristic is referred to as Heuristic 1 and is described in pseudo code below.

Heuristic 1 1: determine the realized activity duration vector considering bounded interval

for every involved activity by

.

2: apply robust optimization framework (i.e., Eq. (42)) to measure or identify the cardinality of set 3: solve the corresponding robust optimization based RCPSP model by using our proposed B&C algorithm as seen in sections 6.1 to 6.4 4: output the solution found and generate a robust schedule 5: end

5.2 Bounded and symmetric uncertainty For the case of RCPSPs, if the demand parameters are uncertain and distributed around the nominal values randomly and symmetrically as follows, where

are random variables distributed symmetrically in the

interval [-1,1]. Then by following a similar guideline from Eq. (11), the robust counterpart for Eq. (41) is derived as follows, (43) Here,

is a positive parameter with

reliability level is

and the violation probability or

. Similarly,

.

Meanwhile the second heuristic is named Heuristic 2 and is outlined below:

Heuristic 2 1: determine the realized activity duration vector that

for each activity , while assuming

.

2: apply robust optimization framework (i.e., Eq. (43)) to measure or identify the

20

cardinality of set 3: solve the corresponding robust optimization based RCPSP model by using our proposed B&C algorithm as seen in sections 6.1 to 6.4 4: output the solution found and generate a robust schedule 5: end

5.3 Robust counterpart with known probability distribution As of Eq. (24), the generic robust counterpart of the probabilistic constraints for our targeted RCPSPs can be formulated as follows for random variables following any distribution, (44)

Similar to that with bounded uncertainty, the constraint can be,

5.3.1 Uncertainty with uniform probability distribution Assume the duration parameters are uncertain and follow a uniform continuous distribution. Given an uncertainty level ( ), an infeasibility tolerance ( ), and a reliability level ( ), to generate robust solutions, the following (

) - robust counterpart of the

original uncertain MILP problem can be derived.

(45)

Therefore, for right-hand-side parameter uncertainty (i.e., duration uncertainty), Eq. (45) becomes, (46) Here,

for uniform continuous distribution.

5.3.2 Uncertainty with normal probability distribution 21

Again, we assume that the distributions of the random variables

,

and

in (17) are

all standardized normal distributions with zero as the mean and one as the standard deviation. Then the distribution of

defined in (21) is also a normal distribution with

mean zero and

as the standard deviation (Janak et

al., 2007). According to Eq. (22),

and

is the inverse distribution function of a

random variable with standardized normal distribution. Thus,

For this considered RCPSP, if the activity duration parameters follow a normal distribution, the deterministic robust counterpart (i.e., Eq. (44)) is represented by, (47) Therefore,

is determined as

.

5.3.3 Uncertainty with Poisson probability distribution On the other hand, when the uncertain activity duration parameters follow a Poisson probability distribution, then the deterministic robust counterpart for Eq. (44) is given by, (48) where

is the inverse distribution function of a discrete variable with a

Poisson distribution. Here we categorize Heuristics 3, 4 and 5 depending on different uncertainty distributions and are outlined below:

Heuristic 3 1: determine the realized activity duration vector

for each activity , while assuming

that the activity duration parameters are uncertain and follow a uniform continuous distribution. 2: apply robust optimization framework (i.e., Eq. (46)) to measure or identify the cardinality of set 3: solve the corresponding robust optimization based RCPSP model by using our proposed B&C algorithm as seen in sections 6.1 to 6.4 4: output the solution found and generate a robust schedule 5: end

22

Heuristic 4 1: determine the realized activity duration vector

for each activity , while assuming

that the activity duration parameters are uncertain and follow a normal distribution 2: apply robust optimization framework (i.e., Eq. (47)) to measure or identify the cardinality of set 3: solve the corresponding robust optimization based RCPSP model by using our proposed B&C algorithm as seen in sections 6.1 to 6.4 4: output the solution found and generate a robust schedule 5: end

Heuristic 5 1: determine the realized activity duration vector

for each activity , while assuming

that the activity duration parameters are uncertain and follow a Poisson probability distribution 2: apply robust optimization framework (i.e., Eq. (48)) to measure or identify the cardinality of set 3: solve the corresponding robust optimization based RCPSP model by using our proposed B&C algorithm as seen in sections 6.1 to 6.4 4: output the solution found and generate robust a schedule 5: end

5.4 Unknown probability distribution When the activity duration parameters are uncertain and follow an unknown probability distribution, then according to the Eq. (39), the robust counterpart for Eq. (41) is represented as, (49) The violation probability or reliability level is

and

is determined as

Under this circumstance, Heuristic 6 is carried out by following the pseudo code provided below:

Heuristic 6 23

1: determine the realized activity duration vector

for each activity , when the activity

duration parameters are uncertain and follow an unknown probability distribution 2: apply robust optimization framework (i.e., Eq. (49)) to measure or identify the cardinality of set 3: solve the corresponding robust optimization based RCPSP model by using our proposed B&C algorithm as seen in sections 6.1 to 6.4 4: output the solution found and generate a robust schedule 5: end

6. SOLUTION METHODOLOGY Branch and cut (B&C) is a generalization of B&B where the LP relaxation of the ILP is used to obtain a bound at each node in the search tree. If a node has a fractional solution and cannot be fathomed, this approach finds cuts that are violated by the fractional solution but are satisfied by all feasible integer solutions (Zhu et al., 2006). If no cuts can be found, branching is performed to create new nodes in the tree. B&C has proven to be effective in solving different combinatorial optimization problems (Bianco and Caramia, 2012, Naber and Kolisch, 2014, Zhu et al., 2006). Our solution approach makes use of the cut generating features built into the CBC solver that comes with Matlab (R2012b). In addition, for accelerating convergence, we provide problem specific cuts to tighten the LP bounds, and also alternative branching rules. In this section, we detail the overall solution techniques for the proposed RCPSPs. 6.1

Fixing variables

Here for each activity

, we define the decision variable

for t bounded by the earliest

finish time (EFT) and latest finish time (LTF) instead of considering the upper bound on the project completion time T, which drastically reduce the number of variables. The EFT of activity i is defined as the earliest time period t such that there exist a feasible schedule in which activity i finishes at time t. Similarly, the LFT is defined as the latest time period t in which activity i can finish for a given upper bound of the project makespan. However, finding those EFT and LFT values is an NP-hard problem. Although for any case, it is not necessary to know EFT and LFT exactly for all activities to formulate valid MILP models (2)-(6), as calculating lower and upper bounds on those values is sufficient to lead to smaller models (Zhu et al., 2006). Here, we employed an iterative critical path method (CPM) for obtaining better lower bounds for more reduction of the number of variables. 24

6.2 Cuts from resource conflicts As from constraint set (5), all renewable resource constraints are in the form of knapsack constraints with generalized upper bound (GUB) variables (Gu et al., 1998). That is, all variables are divided into mutually exclusive subsets, and in each subset one variable must be present. For details on the complexity and the computation of GUB cover cuts see Gu et al. (1998). These cuts were developed whenever there was a resource conflict and were generated automatically by the CBC solver through Matlab when requested. For a renewable resource, a GUB cover cut prevents two or more activities from being executed at the same time if the combined resource usage exceeds the maximum limit. 6.3

Cuts from precedence relations

Precedence cuts were obtained by explicitly enumerating the possible finish times for activities with precedence relations. For instance, if activity i precedes activity j, which has a duration of

, then according to constraint (3), if activity i finishes at time t then activity j

must finish after

. Each of the enumerated cases was written in the form of a cut which

was used to tighten the LP bounds. Meanwhile, it was not necessary to consider cuts from all of the precedence relations, because many were redundant according to the proposition mentioned in Zhu et al. (2006). 6.4 Branching rules and Bound tightening In this case, the bound on one variable at a time is fixed in the construction of the search tree. We avoided branching on a single variable, which is supposedly very inefficient, because it gives identical results for the two sub problems of the original model. In special ordered set (SOS) branching, the variables that make up each special ordered set were partitioned into two subsets and all the variables in one subset were set to zero on each branching. Predominantly, the branching was performed on the finish time of an activity. We employed the CBC solver defined default bound tightening scheme every time a SOS was used for branching. Despite having several techniques to accelerate the solution process, initial testing indicated that many problem instances were still difficult to solve due to the large number of binary variables. In response, we adopted a solver that defined a greedy heuristic approach for faster divergence while the neighbourhoods were defined by linear inequalities. However,

25

if the heuristic results do not yield a new solution with makespan less than the incumbent makespan, we switched the heuristic and directly solved with the solver (with its default local branching scheme). Benchmark problems developed by Kolisch and Sprecher (1997) have been widely used to test various exact and heuristic methods for RCPSPs and can be found in the PSPLIB. In this paper, the B&C approach was evaluated by solving instances from that library. For solving those MILP models, a commercial optimization solver from the Computational Infrastructure for Operations Research (COIN-OR) community, named as Coin-OR Branch and Cut (CBC) (version 2.9.3), was employed. It comes with Matlab (R2012b), and was executed on an Intel core i7 processor with 16 GB RAM and 3.40 GHz CPU. All relevant documents and solver specified information can be found from the OPTI toolbox. 7. COMPUTATIONAL RESULTS 7.1 For Deterministic RCPSPs To analyse the B&C algorithm, all benchmark instances from different activity numbers, ranging from lower to higher activities (30 to 90 activities), available in PSPLIB were chosen, except for 120-activity instances where some random instances were chosen. In table 1, we give the results of our B&C algorithm compared with the tightest bound on the KSD instances proposed by Brucker and Knust (2000) (BK: column 4), with the destructive lower bound (DLB: column 5) proposed by Demassey et al. (2005) and with the B&B algorithm solved using Lingo. Here, %Optimal represents the ratio of number of instances solved optimally among all 480-instances of different activity numbers (except for the 120-activity instances where we solved 53 randomly chosen instances). Despite the higher computation time, our B&C algorithm is comparable to, and even gives better percentage of optimal results than any other exact algorithms particularly for higher activity instances. Moreover, except for the 30-activity instances, our proposed algorithm shows the least average %GAP than the other approaches, even for larger activity instances. Although the average number of integer variables and generated nodes for higher activity instances are much higher than the lower ones, their average solution time is always lower for the B&C algorithm than for B&B and even BK and DLB which further proves the efficiency of our proposed algorithm in comparison to exact approaches, particularly for higher activity instances.

Table 1: Computational results on the KSD Instance Sets

26

No. of Act

Parameters

B&B

BK

DLB

B&C

KSD 30 (480)

% Gap(avg.) CPU time (avg.)

0.00 12.20

1.5 0.4

0.7 3.2

7.16 41.47

%Optimal

100

66.25

83.95

94.37

% Gap(avg.)

11.59

7.8

7.7

6.59

CPU time (avg.)

153.17

5.0

168

89.82

%Optimal

66.04

71.04

75.00

80.00

% Gap(avg.)

17.97

7.2

7.0

6.86

CPU time (avg.)

376.67

72

379

135.68

%Optimal

57.91

72.91

75.83

76.75

% Gap(avg.)

22.27

21.4

19.1

15.70

CPU time (avg.)

437.25

21,300

1,388

284.35

%Optimal

9.434(53)

34.67 (600)

38.17(600)

68.87(53)

KSD 60 (480)

KSD 90 (480)

KSD 120 (53)

For better comparison, three other algorithms were executed on the same computer configuration for those randomly chosen 30-activity instances (only for table 2). To do this, we used four types of indicators which have emerged to characterize the RCPSP instances: precedence-oriented; time-oriented; resource-oriented and hybrid (Artigues et al., 2008). Kolisch et al. (1995) observed that the hardness of the RCPSP instances decreases as network complexity (NC) increases from 1.5 to 2.1. Another important indicator is resource strength (RS), which is a hybrid indicator which mixes both resource and time parameters. After experimenting with the 30-activity instances of PSLIB, it has been observed that the average solution time reduces with increasing RS values from 0.20 to 1. Again the resource factor (RF) reflects the average portion of resources requested per job, where RF = 1 means all resources are required by a job. Considering all those important instance characteristics, all the randomly chosen instances (only for table 2) were chosen from those that has [NC, RF, RS] of [1.50, 0.25, 0.2-0.7], which reflects the difficult ones among the available instances.

For finding optimal results, at first all the MILP models were solved by using the commercial optimization software LINGO v10.0, which predominantly use the branch and bound (B&B) algorithm as default and is exact in nature. Meanwhile, all those instances were also solved by using a genetic algorithm (GA) and a Lagrange relaxation based genetic algorithm (GA_LR) under the Matlab (R2012b) platform. Here we mention that for solving with GA, the authors only employed the built-in GA toolbox with minor customization, while for GA_LR, all the equality constraints of the MILP model were relaxed and acted as a penalty

27

function in the objective values. To compare with GA and GA_LR, it was infeasible to use the large activity instances. So only a subset of the 30-activity instances was used. All the results for the four different algorithms, within or before 1000 seconds, are summarized in table 2. As instance hardness depends on its NC, RF and RS values, all the chosen instances were grouped in three different cases by varying their RS values. Here in table 1, LB represents the lower bound or best obtained results within 1000 seconds, t represents the average computing time in seconds, while % Gap is calculated by using the following equation.

Compared with GA and GA_LR, both B&C and B&B are more consistent for solving 30activity instances (0% Gap) with much lower computing time. When B&C and B&B are compared, the both have 0% Gap, and B&C is faster for group 3, and B&B is faster for groups 1 and 2. On the contrary, for higher activity instances (activity numbers 60 and 90), this is not always true.

Table 2: Test results for some randomly chosen 30-activity instances from PSPLIB B&B

GA

GA_LR

B&C

Group

RS

1

0.2

47.6

0.00

14.8

48.4

1.53

830

48.2

1.11

145.9

47.6

0.00

68.43

2

0.5

45.2

0.00

1.0

45.2

0.00

913

45.2

0.00

74.8

45.2

0.00

5.27

3

0.7

64

0.00

20.8

64.6

0.76

1000

64.2

0.20

460.3

64

0.00

2.77

7.2 For RCPSPs with Uncertain activity durations As mentioned earlier, the second target of this work is to deal with SRCPSPs under uncertain activity duration scenarios. Depending on the uncertain characteristics, we proposed six robust optimization based heuristics, which are mainly based on different ways to represent uncertainties and have already been discussed in Section 6. For comparing those heuristics and different ways of handling uncertainties, computational experiments have been carried out on a set of randomly selected benchmark problems/instances from the PSPLIB, for 30, 60 and 120 activities [10 problems with 30 activities (j301_1, j302_1, j305_1, j306_1, j307_1, j308_1, j309_7, j3010_1, j3037_1 and j3044_1), 10 problems with 60 activities (j601_1, j602_1, j605_1, j606_1, j607_1, j608_1, j609_1, j6010_1, j6037_1, j6044_1), 10 problems

28

with 120 activities (j1201_1, j1202_1, j1205_1, j1206_1, j1207_1, j1208_1, j1209_1, j12030_1, j12037_1, j12060_1)].

For solving those deterministic robust counterparts of the robust optimization formulation (i.e., six heuristics), we employed our proposed B&C algorithm which was coded in Matlab (R2012b) under the built-in CPLEX platform, and was executed on an Intel core i7 processor with 16 GB RAM and 3.40 GHz CPU. Note that for the sake of computational swiftness, we used limits of maximum 10,000 nodes, 2000 iterations and 500 seconds solving time for our B&C algorithm. Note that the results obtained by our B&C algorithm may not sufficiently be optimal for all instances due to its solution time limitation. Moreover, for larger activity instances, some infeasible results were found due to this defined solution time limit. In that case, we excluded those results for our future analysis. Hence, the average value of makespan after durational variance for SRCPSP (i.e., MADV-makespan after durational variance) presented in tables 3-5, are only for those instances, for which we got optimal or at least feasible and integer results. For solving heuristic 1, we assumed that the uncertain processing times vary in a bounded interval, while the lower bound of that interval

was measured

by multiplying each activity’s deterministic duration by 0.5 and the upper bound

was

measured by multiplying by 2.25. For better understanding, we set two different uncertainty levels (U.L), as 0.05 and 0.15; the infeasibility tolerance

as 0%; and the reliability level

as 5%. Here to recall that the overall objective for this SRCPSP is to minimize both the project makespan

and the sum of deviation penalties (SODP) of all activities

under uncertain activity duration set

(as shown in Eq. (1)). The penalty coefficients are: =8, which are mainly motivated from Zhu et

al. (2005). Below, we present our computational results for KSD (Kolisch and Sprecher, 1997) instance sets with 30, 60 and 120 activities.

7.2.1 Comparison of different uncertainty handling techniques for 30-activity instances Table 3 displays for every heuristic, the mean value of makespan under the deterministic condition (MBDV-makespan before durational variance), the average value of makespan after

durational

variance,

variance

for

SRCPSP

(MADV-makespan

after

durational

), standard deviation of obtained MADV values (SADV), sum of

deviation penalties (SODP) of all activities and the average CPU time (second) needed to 29

solve the set of ten randomly chosen instances with thirty non-dummy activities. In the table, each value reported in the columns for MBDV, MADV, SODV and CPU is the average of ten values obtained, after applying the B&C algorithm for a maximum of 500 seconds. Table 3 shows that the makespan obtained after durational variance (MAVD) and SODV for heuristic 5 (i.e., uncertainty follows known and Poisson distribution) is the least among all the heuristics under both uncertainty levels, although the computational time is a little higher than heuristic 1. The rationale behind this is that for heuristic 1, the uncertain processing times vary in a bounded interval, which is independent of uncertainty or risk levels. As a result, the MADV values for heuristic 1 are the same for both uncertainty levels. Also for heuristic 6, which assumed that the uncertainty distributions of the activity durations are unknown, we obtained slightly higher MADV values, but compared to the other heuristics, the PSC values are significantly higher. This is perhaps because heuristic 6 deals with the most uncertain distribution. This efficacy of heuristic 5 further proves that, for KSD 30 instances under stochastic condition, selecting a Poisson distribution for representing uncertainties is a sound approach.

Table 3: Comparison for KSD 30 benchmark problem U.L

Para.

5%

MBDV (avg.) MADV (avg.) SADV

SODP

15%

(avg.) CPU (sec) MBDV (avg.) MADV (avg.) SADV

SODP (avg.) CPU (sec)

Heuristic 1

Heuristic 2

Heuristic 3

Heuristic 4

Heuristic 5

Heuristic 6

56.1

56.1

56.1

56.1

56.1

56.1

95.4

72.075

101.4

77.7

62.475

77.625

16.35

81.78

36.67

29.08

12.11

108.14

412.48

241.88

470.22

282.48

114.96

753.24

192.91

351.985

363.903

400.97

203.919

232.603

56.1

56.1

56.1

56.1

56.1

56.1

95.4

106.65

103.875

105.375

66.225

77.625

16.35

79.54

34.26

37.5

15.71

109.81

412.48

570.32

481.16

431.2

206.24

753.24

197.0207

424.907

382.689

374.912

255.126

233.209

7.2.2 Comparison of different uncertainty handling techniques for 60-activity instances

30

In this section, we focus on the set of instances with 60 non-dummy activities. As shown in table 4, our proposed B&C algorithm took almost all our dedicated time limit (500 seconds) for solving these instances. In spite of being larger activity instances than before in section 7.2.1, similar conclusion can be drawn, apart from those about heuristic 6. Surprisingly for this case, heuristic 6 produced the least MADV values in comparison to other heuristics, although a precise justification of this scenario cannot be given because of its’ uncertainty characteristics. Meanwhile for two different uncertainty levels, table 4 shows that both the MADV values and SODP values for uncertainty level 5%, is quite lower than values for 15%, which is expected. More detailed discussion on the impact of risk or uncertainty levels will be covered in section 7.2.4.

Table 4: Comparison for KSD 60 benchmark problem U.L

Para.

5%

MBDV (avg.) MADV (avg.) SADV

SODP

15%

(avg.) CPU (sec) MBDV (avg.) MADV (avg.) SADV

SODP (avg.) CPU (sec)

Heuristic 1

Heuristic 2

Heuristic 3

Heuristic 4

Heuristic 5

Heuristic 6

97.2

97.2

97.2

97.2

97.2

97.2

125.7

81.45

138.375

110.625

81.6

43.575

17.42

77.71

1.08

72.14

26.23

23.71

1007.44

443.2

1141.74

703.1

443.2

735.88

414.1

448.238

482.78

488.234

450.864

492.375

97.2

97.2

97.2

97.2

97.2

97.2

125.7

90.825

138.45

153.525

101.475

43.575

17.42

80.20

3.52

75.58

36.8

25.68

1007.44

578.12

1141.9

1295.08

830.82

735.88

387.9006

409.454

482.679

344.107

455.712

492.678

7.2.3 Comparison of different uncertainty handling techniques for 120-activity instances We have performed similar comparisons for the set of 120 non-dummy activities. While most exact algorithms failed to solve the KSD 120 instances within our dedicated time frame (500 seconds), our B&C algorithm produced feasible and integer optimal (6 out of 10 instances on average) results, even though it took higher CPU time.

31

Table 5: Comparison for KSD 120 benchmark problem U.L

Para.

5%

MBDV (avg.) MADV (avg.) SADV

SODP

15%

(avg.) CPU (sec) MBDV (avg.) MADV (avg.) SADV

SODP (avg.) CPU (sec)

Heuristic 1 Heuristic 2

Heuristic 3 Heuristic 4

Heuristic 5 Heuristic 6

133.9

133.9

133.9

133.9

133.9

133.9

161.1

100.65

137.175

133.5

97.575

37.2

14.85

3.56

2.58

5.00

3.53

1.82

2270.6

1511.8

1707.98

1820.5

1506.24

2338.4

273.508

468.034

326.937

500.00

500.00

500.00

133.9

133.9

133.9

133.9

133.9

133.9

161.1

125.55

138.075

200.025

143.1

44.7

14.85

4.85

3.67

7.42

2.98

2.83

2270.6

2196.2

1727.68

2481.8

1836.22

2432.6

279.366

460.964

436.825

375.922

500.00

500.00

As of section 7.2.2, similar conclusion can be drawn for all six heuristics. Table 5 shows, for uncertainty level 5%, that heuristic 5 shows better solution robustness (i.e., least SODP value) than any other heuristics, although it produces higher makespan than heuristic 6. These analyses further justify that, if someone considers that the activity duration parameters are uncertain and follow an unknown probability distribution, he/she can obtain better makespan performance than with the other five ways of uncertainty handling techniques. Therefore, for larger activity instances (i.e., KSD 60 and KSD 120), representing uncertainties by an unknown probability distribution gives better result than others, while for lower activity instances (i.e., KSD 30), a Poisson distribution is the better choice.

7.2.4 Impact of risk level on different heuristics To analyse the impact of risk levels on both expected/realized makespan after considering duration uncertainties (here MADV is termed as EXPMAK) and sum of deviation penalties (SODP), we considered all ten above mentioned randomly chosen KSD 30 instances from PSPLIB under four different risk levels,

(0.05, 0.10, 0.15, 0.20). Here, figures 1 and 2

reports the SODP and EXMAK under varied risk levels for all six different heuristics. As we can observe in figure 1, apart from heuristics 1 and 6, the average SODP increases with increasing values of up to a certain limit. This is an expected result, since for decreasing 32

values of

we impose a more prudent project manager’s position, and thereby impose a

higher risk aversion level. Mathematically speaking, as the value of

decreases, probabilistic

constraints are somehow more binding and the schedule is more robust, since it is less exposed to disruptions. Moreover, with decreasing risk levels, a project manager can decide more prudently, which also leads to lower SODP vales. The reader may notice a seemingly strange trend, particularly for heuristics 2 and 3, in that the SODP started to decrease after a certain point. In effect, the SODP seems to have a non-monotonic behaviour, with an increasing slope up to a maximum and a decreasing slope afterwards. This behaviour is more evident for the KSD 30 instances. The unforeseen descendant behaviour of the expected makespan is due to the influence of two opposing forces (i.e., the expected makespan whose values increases as

decreases and on the other hand the expected tardiness that drastically

reduces as long as the risk we are willing to bear decreases) (Bruni et al., 2011). However depending on the uncertainty distributions, heuristics 1 and 6 show constant SODP values for all four uncertainty levels, which further proves their independence concerning risk parameters.

A similar trend can be observed in figure 2, for EXPMAK, which also increases with increasing risk levels. However, the increasing trend here is also non-monotonic, particularly for heuristics 2 and 3. According to figures 1 and 2, the stability costs and EXPMAK show better performance at risk levels of 0.1 to 0.15, and so it should be preferred in terms of both stability cost and makespan performance. Therefore, the range [0.1, 0.15] constitutes a meaningful choice for moderately risk averse project managers who wish to avoid unnecessary extra cost.

SODP

800 Heuristic 1

700 600 500 400 300 200 100 0 0.05

Heuristic 2 Heuristic 3 Heuristic 4 Heuristic 5 Heuristic 6

0.1

0.15

0.2

Risk Level

Fig. 1: Risk level versus SODP 33

180 Heuristic 1

EXPMAK

160

Heuristic 2

140

Heuristic 3

120

Heuristic 4

100

Heuristic 5 Heuristic 6

80 60 0.05

0.1

0.15

0.2

Risk Level

Fig. 2: Risk level versus EXPMAK

8. CONCLUSIONS In the past, exact methods for solving the RCPSP have mainly focused on enumerating all possible activity sequences within a branch and bound framework. In the first part of this work, we proposed a B&C procedure for general linear objective functions, with integer variables and extended types of resource constraints. Maintaining all the precedence and resource constraints, the proposed algorithm was employed for small to large activity instances. The primary contribution of the research pivoted on the application of the B&C algorithm while using the CBC solver, which predominantly is focused on the RCPSP discipline. Empirical tests on benchmark problems showed that our approach gives markedly improved results with respect to conventional exact algorithms and even some meta-heuristic approaches, under minimum modifications. The Second aspect of this article covers more realistic project scheduling problems, which are termed as stochastic-RCPSPs. In practical project management, a project’s parameters, such as activity duration and resource requirements, are scarcely ever precisely known and are usually subject to estimation errors. In this paper we utilized the robust optimization framework to develop a deterministic robust counterpart optimization model for activity duration uncertainty. It was assumed that the uncertainty can be of different forms and show different statistical distributions. Depending on those different uncertainty characteristics, we have developed six different heuristic models for SRCPSP the under robust optimization framework. All those heuristic models were then employed for small to large activity instances, and were solved by using our proposed B&C algorithm. The robust solution from the robust optimization framework is guaranteed to be feasible for the nominal parameters. The results on benchmark problems

34

proved the dominance of certain heuristics (i.e., certain uncertainty handling ways) over others and helped to give proper guidance to risk reluctant project managers and practitioners.

Using a genetic algorithm or any other advanced meta-heuristic (i.e. simulated annealing, particle swarm optimization etc.) for local branching, could be an important future work for getting good feasible solutions in the early stages of the computations. For projects with relatively long makespan or activities whose finish time intervals are large, more efficient solution methods are still needed. Along with durational uncertainty, there may be other sorts of uncertainties in project execution. Therefore, designing novel disruption handling strategies for SRCPSPs with those uncertainties could be another important area for future study.

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RESOURCE CONSTRAINED PROJECT SCHEDULING WITH UNCERTAIN ACTIVITY DURATIONS

RESEARCH HIGHLIGHTS 1. Introduced RCPSPs with uncertain activity durations 2. Developed the robust optimization concept to deal with uncertain activity duration 3. Proposed six heuristics to incorporate uncertainty into a robust optimization model 4. Extensive experimental study carried out for medium and large-sized problems

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