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Multi-mode resource constrained project scheduling under resource disruptions
Accepted Manuscript Title: Multi-Mode Resource Constrained Project Scheduling Under Resource Disruptions Author: Ripon K. Chakrabortty Ruhul A. Sarker Daryl L. Essam PII: DOI: Reference:
S0098-1354(16)00007-7 http://dx.doi.org/doi:10.1016/j.compchemeng.2016.01.004 CACE 5349
To appear in:
Computers and Chemical Engineering
Received date: Revised date: Accepted date:
9-4-2015 3-12-2015 5-1-2016
Please cite this article as: Chakrabortty, R. K., Sarker, R. A., and Essam, D. L.,Multi-Mode Resource Constrained Project Scheduling Under Resource Disruptions, Computers and Chemical Engineering (2016), http://dx.doi.org/10.1016/j.compchemeng.2016.01.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
MULTI-MODE RESOURCE CONSTRAINED PROJECT SCHEDULING UNDER RESOURCE DISRUPTIONS
Research Highlights
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1. Formulated two discrete time based models to deal with disruption scenarios
2. A solution approach is proposed to generate a revised schedule after disruptions
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3. Proposed recovery options show better performance than simple right shifting
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4. This model is capable of dealing with a single, as well as multiple disruptions
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MULTI-MODE RESOURCE CONSTRAINED PROJECT SCHEDULING UNDER RESOURCE DISRUPTIONS
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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
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Over the last few decades, research on resource constrained project scheduling has focused on the development of mathematical programming based approaches for the generation of a
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nominal schedule under a deterministic environment. During the implementation phase, however, the nominal schedule may need to be revised when one or more resources are disrupted for a length of time. In this paper, we formulate two discrete time based models to
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deal with two different disruption scenarios for multi-mode resource constrained problems. We propose a reactive re-scheduling procedure for a single, as well as a series of disruptions,
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without having any disruption information in advance. To test the proposed approaches, sets of ten, twenty and thirty-activity multi-mode test instances from Project Scheduling Library
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(PSLIB) were used after introducing randomly generated disruption events. The experimental studies were also carried out to determine the effect of different factors related to the disruption recovery process.
Keywords: Multi-mode Resource Constrained Project Scheduling, Rescheduling, Disruption, Mixed Integer Linear Programming.
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MULTI-MODE RESOURCE CONSTRAINED PROJECT SCHEDULING UNDER RESOURCE DISRUPTIONS
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ABSTRACT
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Over the last few decades, research on resource constrained project scheduling has focused
8
on the development of mathematical programming based approaches for the generation of a
9
nominal schedule under a deterministic environment. During the implementation phase,
10
however, the nominal schedule may need to be revised when one or more resources are
11
disrupted for a length of time. In this paper, we formulate two discrete time based models to
12
deal with two different disruption scenarios for multi-mode resource constrained problems.
13
We propose a reactive re-scheduling procedure for a single, as well as a series of disruptions,
14
without having any disruption information in advance. To test the proposed approaches, sets
15
of ten, twenty and thirty-activity multi-mode test instances from Project Scheduling Library
16
(PSLIB) were used after introducing randomly generated disruption events. The experimental
17
studies were also carried out to determine the effect of different factors related to the
18
disruption recovery process.
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Keywords: Multi-mode Resource Constrained Project Scheduling, Rescheduling, Disruption,
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Mixed Integer Linear Programming.
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1. INTRODUCTION
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In Resource Constrained Project Scheduling Problems (RCPSPs), the objective is to
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minimize the makespan while satisfying the resource constraints and precedence relationships
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among the activities. The Multi-mode Resource Constrained Project Scheduling Problem
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(MM-RCPSP) is an extension of the conventional RCPSP, in which the duration of each task
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is a function of the level and type of resources committed to it, and the project interactions
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that result from the utilization of shared resources that are taken into consideration (Zapata et
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al. (2008). According to the classification scheme of Herroelen et al. (1999), this MM-
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RCPSP is denoted as
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renewable and nonrenewable| strict finish start precedence constraints with zero time-lag,
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activities that have multiple execution modes, the activity resource requirements are a
(i.e., m resource types which can be both
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discrete function of the activity duration| the objective is to minimize the makespan). The
2
resources used by project activities are generally of two types, namely: (1) renewable
3
resources with availability restrictions that may vary from one period to the next (e.g. the
4
number of workers per shift), (2) non-renewable resources with availability restrictions over
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the whole project horizon (e.g. raw material). As of the literature, the renewable resources are
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mainly considered for single mode RCPSP, however both renewable and non-renewable
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resources are considered simultaneously for MM-RCPSP. Other specific resource categories
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that have been considered for RCPSP are: partially (Nonobe & Ibaraki) renewable resources
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(Böttcher et al., 1999), dedicated resources (Bianco et al., 1998), spatial resources (Hans et
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al., 2007), cumulative resources (Neumann et al., 2003), reusable resources (Shewchuk &
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Chang, 1995), synchronizing resources (Schwindt & Trautmann, 2003), multi-skill resources
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(Néron, 2002), heterogeneous resources (Tiwari et al., 2009), and allocatable resources
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(Schwindt & Trautmann, 2003). The variants of traditional RCPSP include: Generalized
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RCPSP, RCPSP with generalized precedence constraints, RCPSP with time varying resource
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constraints, and Dynamic RCPSP (Węglarz et al., 2011).
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RCPSP has gained widespread attention for the last few years due to its practical importance
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and computational challenge. While some of the earlier endeavor was on refining the basic
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model, the majority of research has been aimed at developing better solution methods (Zhu et
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al., 2006). Blazewicz et al. (1983) have shown that RCPSP is an NP-hard problem.
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Moreover, when the process allows the choice of modes (in MM-RCPSP), further complexity
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is added by enlarging the search space (Kyriakidis et al., 2012). In solving MM-RCPSP,
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mixed integer linear programming (MILP) modeling is a popular choice. For finding optimal
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solutions for RCPSP (and also MM-RCPSP), copious algorithms and methods can be found
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in the literature. Among them, the branch and bound algorithm (Hartmann & Drexl, 1998;
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Sprecher & Drexl, 1998), branch and cut based algorithm (Zhu et al., 2006), tree based
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branch and bound algorithm (Hartmann & Drexl, 1998), self developed heuristics (Ballestín
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et al., 2008) and, linear programming based algorithm (Kopanos et al., 2014) are the most
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common approaches. According to Herroelen (2005), computational results indicate that
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many of the 60-activity and most of the 90- and 120-activity instances from the Project
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Scheduling Library-PSLIB (Kolisch & Sprecher, 1997) are still a good way off the solution
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capabilities of the exact methods.
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Industrial resource constrained problems have been considered as a significant challenge in
2
highly regulated industries, such as pharmaceuticals and agrochemicals, where a large
3
number of candidate new products must undergo a set of tests for certification (Choi et al.,
4
2004). In spite of that, in the recent past, different industrial resource constrained problems
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have been applied/addressed for process systems engineering, such as the application of
6
RCPSP in semi-continuous food industries (Kopanos et al., 2011), multistage batch
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processing (Méndez & Cerdá, 2003), automated wet-etch station (AWS) scheduling (Novas
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& Henning, 2012), and for varied set up times (Nadjafi & Shadrokh, 2008). A detailed
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discussion on earlier applications of planning and scheduling in the process industry can be
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found in Kallrath (2002). However, chemical process industries are dynamic in nature, and
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therefore different types of unexpected events occur quite frequently. The most frequent
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rescheduling factors in the chemical process industry are: machine failure, rush job arrival,
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job cancellation, due date change, inadequacy of raw materials, price changes, and
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overestimation (or underestimation) of processing time, set-up times, and equipment release.
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In particular, Adhitya et al. (2007) proposed a heuristic for rescheduling crude oil operations
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to manage abnormal supply chain events. Their proposed model gives some provision to
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refinery personnel to choose a suitable feasible schedule from amongst many identified
18
feasible schedules. Apart from that, Janak et al. (2006) also proposed a reactive schedule for
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a large-scale industrial batch plant in which the authors ignored full rescheduling in the
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current production horizon. Instead, they utilized an efficient mixed integer linear
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programming (MILP) mathematical framework to determine which tasks would not be
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affected by the unforeseen event, either directly or indirectly, such events were carried out as
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scheduled. However, reactive scheduling in case of RCPSPs is still insufficient. Keeping this
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in mind, this paper deals with reactive rescheduling techniques for real time based
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generalized MM-RCPSPs. The applicability of this research is highly diversified, as this
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paper conveys the dynamic features of machine or resource inadequacy/unavailability for a
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general MM-RCPSP case. This way of tackling such resource uncertainties can easily be
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applied for any real-time based chemical process industry.
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During the implementation phase, a project may face significant predicaments due to
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resource unavailability, unproven technology, unreasonable commitment and unrealistic or an
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unclear goal set up (Zhu et al., 2005). Due to these factors, a project may be delayed in
33
completion, so any such noteworthy deviation in a project schedule is considered as a
34
disruption. Because of disruption, the traditional deterministic project scheduling models 3
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must be revised and resolved to match with the changed environment (Deblaere et al., 2011).
2
That means, an initial optimal solution is only optimal during the execution of the schedule if
3
there is no disruption. Vieira et al. (2003) have classified the existing rescheduling strategies
4
into three primary types: (1) repairing a schedule that has been disrupted, often known as
5
reactive strategy (Deblaere et al., 2011); (2) creating a schedule that is robust with respect to
6
disruptions, known as proactive scheduling (Herroelen & Leus, 2004); and (3) studying how
7
rescheduling policies affect the performance of dynamic manufacturing systems. In the case
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of proactive (robust) scheduling, a degree of anticipation of variability during project
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execution is incorporated into the nominal schedule. Hence even if there is no variation in the
10
project run, this strategy always have some extra allowance and therefore gives suboptimal
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results. The use of a nominal schedule in combination with reactive scheduling procedures is
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sometime referred to as proactive-reactive scheduling, which is an iterative strategy. Reactive
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scheduling, on the other hand, is generally of two types: schedule repair, often known as the
14
right shift rule because it moves forward all the affected activities (Sadeh et al., 1993) and
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full rescheduling (Artigues & Roubellat, 2000) which differs considerably from the nominal
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schedule. However, determining the best rescheduling solution still remains an open research
17
issue, and consequently is the most difficult part of the rescheduling process (Vieira et al.,
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2003).
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The literature on handling disruption in MM-RCPSP is however scarce. To the best of our
21
knowledge, there are only two earlier works on handling disruptions for MM-RCPSPs. Zhu et
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al. (2005) have formulated an MILP model for a general class of reactive scheduling
23
problem, and solved it with a hybrid mixed integer programming or constraint programming
24
procedure. For recovering disruptions, they considered three different recovery options,
25
namely rescheduling, mode alternation and resource alternation. Deblaere et al. (2011)
26
considered activity duration variability and resource disruption explicitly, and evaluated some
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dedicated exact reactive scheduling procedures, as well as a Tabu search heuristic for
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repairing a disrupted schedule, under the assumption that no activity can be started before its
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baseline starting time. However, developing a proper mathematical programming model for
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multi-mode RCPSP, considering resource disruptions, is still a challenging research topic.
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When dealing with RCPSPs and their analysis, the two practical scenarios of preempt-repeat
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and preempt-resume are generally considered by researchers (Lambrechts et al., 2010). In the
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case of the preempt-repeat environment, interrupted activities must be started from scratch, 4
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because they assume that incomplete jobs cannot be continued for completion and are so
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counted as wastage. On the contrary, in a preempt-resume environment, only the residual
3
portion of any affected/interrupted activity will need to be restarted during its recovery
4
schedule.
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In this paper, we consider multi-mode resource constrained project scheduling under
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disruption. First, the problems with disruption of multiple renewable resources is discussed
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for two different practical scenarios, known as ‘preempt-repeat’ and ‘preempt-resume’, and
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the mathematical programming models, based on discrete time, for recovering from the
10
disruptions are developed. A solution approach is proposed, which can generate a revised
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schedule after a disruption event takes place, where the disruption information is not known
12
in advance. It is expected that the parameters of disruption follow a stochastic process. We
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deal with these stochastic parameters within a deterministic environment. The proposed
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solution approach is capable of dealing with a single, as well as a series of disruption events,
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for multiple resources and for multiple modes, on a real-time basis. To judge the performance
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of our proposed approach, we have generated a set of test problems and compared the
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solutions with their upper and lower bounds. In generating the test problems, we have
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selected sets of ten, twenty and thirty-activity benchmark instances from PSLIB and
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introduced randomly generated disruption scenarios into them. Experimental studies have
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also been conducted to analyse the effects of different factors relating to the disruption
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recovery process, such as changes in activity duration, changes in precedence relationship,
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and addition of new activity.
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The structure of the paper is as follows: in section 2, we define basic MM-RCPSPs and
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discuss the associative disruption recovery strategies. The terminologies, disruption recovery
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models, and their MILP formulations are described in section 3. In section 4, solution
27
approaches and relevant algorithm design are discussed. The experimental study, along with
28
the results of computational tests that measure the behavioural patterns of recovery schedules
29
are described in section 5. In the final section, we provide overall conclusions.
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2. PROBLEM DESCRIPTION
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In this research, we consider MM-RCPSPs under resource disruption. The objective of the
33
problem is to re-schedule project activities and their required resources, to recover from
34
disruptions as soon as possible. For this purpose, we consider the following assumptions: (i) 5
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the activities composing a project have known durations for each and every mode; (ii) all
2
predecessors must finish before an activity can start; (iii) resources are of both renewable and
3
non-renewable; (iv) activities are non-preemptive (i.e., cannot be interrupted when in
4
progress); (v) the activities cannot be started before their planned starting times (railroad
5
scheduling); and (vi) the objective is to minimise the project completion time.
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Here it is assumed that I is the number of activities to be scheduled. The activities
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constituting the project are represented by a set {0,..., I+1}, where 0 and I+1 are dummy
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nodes representing start and end respectively. The important notations and sets are depicted
10
in the next section as nomenclature. The basic consideration is to execute the recovery
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schedule immediately after the disruption experiences, and is therefore termed as a
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responsive strategy. If during the recovery time window of a disruption, if the problem faces
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another new disruption, then the later one can be considered as an independent disruption and
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a new recovery plan can be made in the similar way as for the previous one. This type of
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scenario, (i.e. that the problem is affected by a series of disruptions) makes the case more
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complex for recovery planning, as all previous and recent disruptions must be considered
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when generating the revised schedule. In this paper, two recovery options are considered,
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namely preempt-repeat (earlier work is lost) and preempt-resume (no work lost), as discussed
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earlier.
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3. THE MATHEMATICAL MODELS
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The mixed integer linear programming models (MILP-models) for our proposed MM-RCPSP
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models are discussed in this section. To do this, we first explain the mathematical model for a
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nominal schedule and then reformulate it for both recovery schedules.
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3.1 Nomenclature
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Sets I T W R Q
set of activities, i =0…I+1 set of time periods, t = 0…T represents the precedence set set of non-renewable resources, w = 1…W set of renewable resources, r = 1…R set of modes for activity i, m = 1… set of disruptions, q = 1…Q set of incomplete activities, including activities which start during a disruption repair period (affected activities) and activities which start after a disruption ends (non ] affected activities) 6
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set of affected and incomplete activities which start before disruption, l ( ] set of completed activities before disruption experienced, ( ]
Parameters
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Capacity of non-renewable resource w Usage of renewable resource r by activity i at mode m Usage of non-renewable resource w by activity i at mode m Duration of activity i at mode m Usage limit of renewable resource r on time slot
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set of time slots on which renewable resource r is constrained,
Resource usage of activity i for resource r Total planning horizon/upper bound of the project duration. Last job Finish time of activity i in the results from the nominal schedule. Start time of activity i from the nominal schedule. Represents the disruption start time Represents the disruption finish time (disruption start time + repair time) Disruption repair time Number of disrupted resources (e.g., breakdown machines) Very small amount of time span Incomplete portion of task i Used non-renewable resource ‘w’ up to time t Non-renewable resource ‘w’ required for completing portion of work Baseline makespan Recovery makespan after ‘qth’ disruption
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3.2 Basic MM-RCPSP formulations for Nominal Schedules The standard MM-RCPSP requires sequencing the project activities, so that the precedence constraints are met, determining the execution mode for each activity, and satisfying the resource limitations that minimize the project duration. From the literature, several mathematical models can be found for MM-RCPSP. Most of those models are somehow extensions of the model proposed by Talbot (1982). That is why the model from Talbot (1982) is considered here as the mathematical model for solving the nominal schedule. As the variables considered depend on discrete time parameters, this type of formulation is called a discrete type formulation. Here a binary decision variable
is defined to be 1 if activity
starts at mode m on time instant , and is 0 otherwise. Thus the formulation can be written as follows:
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(1) 1 2
Constraints: (2)
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(3) (4)
(6) (7)
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(5)
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As of many other MM-RCPSP models, the objective (equation 1) is to minimize the
5
completion time of the last activity of a project (i.e. the project completion time). Constraint
6
(2) ensures that each activity starts from exactly one time period in a given mode. That
7
means, no balking or priority violation is allowed (Non-preemptive case). The jobs which
8
have started must be completed first, depending upon their corresponding priority. Constraint
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(3) ensures the precedence relationship that means an activity cannot be started before the
10
completion of its preceding activity. Inequalities (4) and (5) represent the capacity constraints
11
for renewable and non-renewable resources respectively. For
12
resource r is called renewable. The usage of a renewable resource is limited in each time
13
period, but there is no limit on total usage. In contrast, the non-renewable resources are
14
available throughout the whole project tenure (T) and are consumed by all activities.
15
Constraint (6) defines the baseline finish time of each activity.
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3.3 Disruption Recovery Models
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Project disruptions are the events that prevent project managers from completing the work as
19
planned. The possible reasons for these disruptions encompass diversified causes, such as
20
normal errors, biased or parsimonious estimations of activity durations or usage rate, and
21
sudden customization. The consequences of these disruptions are project delay or cost
22
overruns, that consequently demands some reschedule of the nominal schedule. To
23
reschedule the project activities under any disruption, it is necessary to reformulate the
24
nominal MM-RCPSP model by incorporating disruption or unavailability information. In this
8
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1
research, we have formulated two models, for a single disruption, under both the preempt-
2
repeat, and the preempt-resume conditions. These two models are presented below.
3
3.3.1 Preempt-Repeat Condition
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As mentioned earlier, for the preempt-repeat case, the affected activities should start from
6
their very beginning, which requires perfect knowledge of the disruption start time. Having
7
some prior information regarding the disruption start time and repair or replacement time
8
may facilitate a rescheduling strategy. But, as indicated earlier, the disruption start time is not
9
known in-advance; rather it is unknown and random. So the reschedule plan will be generated
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just after the real disruption takes place.
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3.3.1.1 Objective function
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The objective is to minimize the completion time of the last job (i.e. makespan time) plus the
14
weighted deviations of all activity’s finish times in the revised schedule. Where
15
are given penalty weights for the deviation of activity completion times and
16
positive value that is equal to max{Y, 0}. Penalty values are used to minimize the instability
17
of activity starting times in the recovery schedule, as project instability may hamper project
18
profitability. This has also been used by Kopanos et al. (2008) for maintaining schedule
19
stability. Here, the first term represents the project completion time. The second and third
20
terms represent the weighted deviations between the finish times of each activity in the
21
recovery and initial schedules.
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and
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represents a
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(8)
3.3.1.2 Constraints
1. Start time limitation: In the case of rescheduling, we assumed that the recovery period starts immediately after the disruption experienced and it is to be continued
26
until the project ends, and is defined as the recovery window. In the recovery
27
schedule, the affected or incomplete activities
28
disruption start time, must be started within its recovery window. Meanwhile
29
activities
whose finish time is greater than the
should consider their best suitable mode for execution, and have a unique
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start time. Here, T represents the whole project horizon within which the project must
2
be accomplished. (9)
3
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2. Precedence relationship: Same as constraint (3), where we assumed zero-lag finish-
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4
start precedence relations.
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3. Resource capacity: Here, constraints (10) and (11) represent the capacity limitations
8
of non-renewable resources, while equations (12) – (14) represent the capacity
9
constraints for renewable resources. For resource utilization, three different cases can
10
occur for renewable resources in a recovery schedule. For the first case, it is
11
assumed, that for time spans up to the disruption start time, the maximum capacity of
12
how much renewable or non-renewable resources is available, is as defined in
13
constraint set (12). For the second case, we assumed that after the disruption start
14
time, the capacity is appropriately reduced (measured by
15
back in service (defined in constraint set (13)). Note that the capacity can be reduced
16
to zero when there is no resource to perform a job. Later on, for the third case, it is
17
assumed, that after repairing, overhauling or replacement, the resource will achieve
18
the actual capacity level, as defined in constraint set (14). Here,
19
time period depending on disruption scenario and is taken tentatively. For constraint
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) until the full capacity is
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sets (12), (13) and (14),
is any specific
is depicted from three different conditions as mentioned
there. Here in these proposed models, at first we have considered that the disruption could happen for the renewable resources only. Later on, some experimental analysis has also been executed under both renewable and non-renewable resource disruptions. Again for non-renewable resources, the amount of resource consumed by the activities
, whose finish time is less than or equal to the disruption start time, is
26
measured by equation (10). During the recovery schedule from the disruption start
27
time to project completion, the earlier consumed non-renewable resources are
28
subtracted from the available resources. The renewable and non-renewable resource
29
constraints for different time periods are shown below. (10) (11)
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(12) (13) (14)
binary constraints:
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(15)
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3.3.2 Preempt-Resume Condition
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In the case of the preempt-repeat option, the partial completed portions of affected activities
5
are considered as wastage. This wastage has an adverse effect upon project completion, as it
6
consumes both time and resources. Unlike of the preempt-repeat condition, following the
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preempt-resume condition will certainly save those wastages, as in the recovery schedule the
8
affected activities start from that portion of work it left from before the disruption. As a
9
result, this option is more lucrative, in terms of completion time reduction and cost savings,
10
and is more appropriate than the preempt-repeat option (Lambrechts et al., 2007). The
11
mathematical model for the preempt-resume option is presented below.
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3.3.2.1 Objective function
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The objective function for the preempt-resume option is similar to that of the preempt-repeat
15
option, as shown in equation (8).
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3.3.2.2 Constraints
1. Start time limitation: In the recovery schedule, as shown in constraint sets (16), the affected or incomplete jobs
whose start time is greater than or equal to the
disruption start time, must be started within the recovery window following their best suitable mode and have unique start times. Again for constraint (17), the same condition should be followed by the activities l, whose start time is less than the
23
disruption start time but that have a finish time that is greater than the disruption start
24
time: (16) (17)
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1
2. Precedence relationship: For any predecessor activity
, other than the affected
or incomplete activities should sum up its usual duration time to allow the execution
3
of its successor j and should satisfy constraint sets (18). Again constraint sets (19)
4
define that the successor j can only be started when its predecessor (which has a start
5
time less than the disruption start time, but have a finish time that is higher than that)
6
completes all its incomplete portions among all the available execution modes:
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(19)
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(18)
3. Resource capacity: All the resources utilised by the activities must not exceed the
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resource’s capacity limitation. Here for the renewable resources, the resource capacity
10
constraints are the same as the constraints (12) to (14). But for the non-renewable
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resources, it has three different parts. For the activities
, which have finish time
lower than or equal to the disruption start time, the amount of resources consumed is measured first by following equation (20). Once again, the amount consumed for the affected activities l, which have its start time less than the disruption start time but where its finish time is higher than that, is also measured separately, as shown in equation (21). Then for the remaining activities
, whose start time is greater than the
17
disruption start time, the available non-renewable resources are measured by
18
subtracting all the upper mentioned resources by following constraint (22). (20) (21)
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(22)
4. Incomplete task: The incomplete portion of any task l, which has a start time that is
2
less than the disruption start time, and a finish time that is greater than the disruption
3
start time, should consider the following constraint (23):
ip t
1
(23)
4
6
8
3.4
Examples for the Recovery Planning Problem 3.4.1 Single Disruption Case
an
7
(24)
us
cr
binary constraints:
5
In this section, we demonstrate the recovery planning for a single disruption of a project with
10
10 activities (in total 12 activities including the start and end dummy nodes) each having
11
three different modes, and each mode with two renewable resources (R1 and R2) with a
12
capacity of 9 and 4 units respectively. The network of the project is shown in figure 2,
13
whereas the modes along with their duration and usage combinations are given in table A.1 in
14
the appendix. As shown in the Gantt chart in Figure 3, the makespan of the project is 20.
15
Now suppose, at the end of period 4, disruption is experienced, due to which, five units of
16
resource R1 and 2 units of resource R2 are not available from time periods 4 to 6 (as of the
17
estimated repair time of two units) that consequently reduces the resource availability of R1
18
from 9 to 4 units and of R2 from 4 to 2 units. However, after the end of period 6 (from the
19
beginning of period 7), the resources will be back in service. As the resource usage for R2 is
20
very small, it does not have any significant impact upon makespan, as a result, we will only
21
discuss the scenarios for R1 hereafter. As the resource requirements during the repair time
22
(period 4 to 6) is higher than what is available, we cannot execute two activities (4 and 5) at
23
the same time as scheduled in figure 3. After rescheduling with the new resource limit, the
24
revised Gantt charts for the preempt-repeat and the preempt-resume condition are shown in
25
figures 4 and 5 respectively. For the preempt-repeat condition, as shown in figure 4, the
26
recovery that started from scratch wasted the work of 1 time unit for activity 4 and 1 time unit
27
of activity 5, that is a total of 2 units of work (hatched boxes in figure 3) and therefore the
28
overall makespan was increased by 1 time unit as compared to the nominal schedule (Figure
29
3). On the contrary, for the preempt-resume option, as shown in figure 5, the amount of work
Ac ce p
te
d
M
9
13
Page 15 of 38
completed for any affected activities (shown as hatched boxes in figure 5) until disruption
2
takes place is not lost and that provides a lower makespan of 20. In the later scenario, only
3
the incomplete portion of the work for any affected activities (shown as dotted boxes in
4
figure 5) should be completed within the recovery window. The synopsis of this example is
5
that the availability of multiple modes means that the makespan only slightly increases, in
6
spite of resource disruptions. 2 7
3
6
8
9
4
Maximum Resource Line for R 1
5
8
[2]
6
8 [1]
4
2 [1]
0
D_s
4 3
D_e
5
Ac ce p
Resource use (unit)
d
2
9
9 [1] 15
D_s
18
20
18
20
Maximum Resource Line for R 2
11 [1] 10 [2]
D_e
5
10 Time
15
Fig 3: Gantt chart of optimum nominal schedule under initial conditions (here [a] represent selected activity mode ‘a’)
Resource available after disruption
Resource use (unit)
14 15
6 [3]
10 Time
Maximum Resource Line
9
13
7 [1]
2 [ 1] 0
11 12
4 [2]
te
Resource use (unit)
9
10
an
Fig 2: Project Network for the considered MM-RCPSP example under recovery planning
M
7 8
12
us
10
cr
11 5
1
ip t
1
8
4 [2]
6 4
6 [1]
2 [1 ]
2
8 [1]
7 [1]
9 [1]
5 [2] 0
D_s
5
D_e
10 Time
15
20
21
Fig 4: Gantt chart of optimum Recovery schedule under Preempt-repeat condition (here [a] represent selected activity mode ‘a’)
14
Page 16 of 38
Resource available after disruption
Maximum Resource Line
4
2 [1]
4_ i [2]
6 [1]
2
8 [1]
7 [1]
9 [1]
5_R [2] 0
D_s
5
D_e
10 Time
15
20
Fig 5: Gantt chart of optimum Recovery schedule under Preempt-resume condition (R: Remaining activity)
3.4.2 Multiple Disruptions Case
cr
3
4_R [2]
6
1 2
5_ i [2]
8
ip t
Resource use (unit)
9
For better understanding, the earlier example is further considered for multiple or a series of
5
disruptions. For the preempt-repeat option, we now assume another disruption takes place,
6
with only five units of resources of R1 being unavailable, from time period 10 to 12, which
7
affects activities 4 and 5, as shown in figure 6(a). Due to the second consecutive disruption,
8
the hatched portions of affected activities (3 units for activity 5 and 2 units for activity 4) are
9
being wasted. The revised schedule for recovering from the 2nd disruption is shown in figure
10
6(b), where the makespan is increased to 27 units from 23. Now suppose a third disruption
11
takes place from time period 14 to 16 that also affects activities 4 and 5. The revised schedule
12
is shown in figure 6(c) with an increased makespan of 31. During all the recovery planning,
13
for the activities which finished before the disruption take place, they naturally have the same
14
start and end times as before. For the readers, with the same disruption scenarios, the Gantt
15
charts for the preempt-resume option are presented in Appendix-A. For that preempt-resume
16
option, under the same disruption scenario results lower makespan values. In spite of
17
disregarding the earlier completed portions (2 units for activity 5 and 2 units for activity 4 as
18
shown in hatched boxes), preempt-resume option schedules the incomplete portion of each
19
work within its recovery window. As a consequence, the makespan of the project after 3rd
20
disruption is only 25 instead of 31 for preempt-repeat condition. Therefore, although for the
21
preempt-resume case, computational complexity may be higher than the repeat case, it saves
22
time.
Ac ce p
te
d
M
an
us
4
Resource available after disruption
Maximum Resource Line
Resourceuse(unit)
9
23 24
8
4 [2]
6 4
2 [1 ]
2
6 [1] 0
st
1
5
D_s
5 [2] 1s t D_ e
10 Time
7 [1]
15
8 [1] 9 [1] 20
21
23
Fig 6 (a): Optimum recovery schedule after first disruption 15
Page 17 of 38
Resource available after disruption
Maximum Resource Line
8 4 [2]
6 4
2 [1 ]
2
6 [3] 0
1
5
1s t D_ s
8 [1]
7 [1]
9 [1]
5 [2] 1s t D_ e
2nd D_e D_s
10 nd
2
15 Time
20
25
27
Fig 6 (b): Optimum recovery schedule after 2nd disruption Resource available after disruption
Maximum Resource Line
cr
2
8 4 [2]
4
2 [1 ]
3
8 [1]
2
6 [3] 0
us
Resourceuse(unit)
9
6
5
1s t D_s
5 [2] st
1
D_e
nd
10 2 2 nd D_s
D_ e
rd
3r d
15 3 D_s
D_e
20
25
7 [1]
9 [1]
30
31
an
Time
ip t
Resourceuse(unit)
9
Fig 6 (c): Optimum recovery schedule after 3rd disruption
4
6
4. SOLUTION APPROACH
M
5
In this section, the model for the nominal schedule is solved. A flowchart is shown to
8
describe the solution procedures used in solving disruption recovery models under both single
9
and a series of disruptions. Both the recovery preempt-resume and preempt-repeat strategies
te
d
7
10
are considered for generating the reschedules.
11 12 13
4.1
14
programming model that can be solved using an exact optimization algorithm, such as the
15
Branch and Bound algorithm (B&B) algorithm if the problem size is not big (Zhu et al.,
16
2006). The nominal model was solved by using the commercial optimization software
17
LINGO v10.0, and was executed on an Intel core i7 processor with 16 GB RAM and 3.40
18
GHz CPU.
19 20 21
4.2
22
series of disruptions on a real time basis. The recovery models further split into two
23
scenarios, known as preempt-repeat and preempt-resume. At the beginning, all the relevant
24
project parameters and disruption scenarios, such as the number of activities including
Ac ce p
Solution approach for nominal schedule The mathematical model presented in section 3.2, for MM-RCPSP, is an integer linear
Solution approach for disruption models This section proposes a solution approach to generate a recovery plan for both single and a
16
Page 18 of 38
1
dummy ones (I+1), duration of each activity (i), disruption start time (
2
time (
3
machine breakdown), disruption severity (i.e., number of machines breakdown), resource
4
requirement by each activity, precedence relations,
5
provided. If there is only one disruption (single disruption) then the model should follow the
6
initial start and end times from the nominal schedule provided.
), number of resources (R), capacity of resources, disruption condition (e.g.,
ip t
values, and time periods should be
start
Stop
Input the project parameters
Set Z* = min{z, ∞ }
Yes
No Does a disruption occur?
M
Yes
Is it the first or a single disruption ? No
Solve the integer model by using optimization software
Input start and finish time from baseline schedule
Update disruption volume and time
d
Determine incomplete tasks
te 10
Update the start and finish times
Input start and finish time from 1st recovery schedule
Ac ce p
9
Is there any other disruption?
an
Yes
us
No
cr
7
8
), disruption end
If the recovery strategy is pre-empt repeat, then formulate the model following section 3.3.1
If the recovery strategy is pre-empt resume, then formulate the model following section 3.3.2
Fig 7: Flowchart of proposed solution methodology for single or a series of disruptions
11
But if after finalizing the recovery plan, another disruption occurs (in the case of a series of
12
disruptions) within the recovery time window, the recovery plan needs to be revised again.
13
This may happen repeatedly, which both affects the project makespan and needs to be
14
rescheduled. The revised recovery plan should be within the disruption start time and project
15
completion time. This scenario depicts that every time a disruption occurs, the optimization
16
model developed earlier remains the same but with some modifications of the activity start
17
times and project makespan. These changes are emulated in the objective function and
18
constraints for re-optimization. That means, to re-optimize the recovery plan the algorithm
19
must run, every time a disruption takes place. If there is no disruption, the system should
20
follow the nominal schedule, as shown in figure 7. For each recovery strategy, the model will 17
Page 19 of 38
1
follow the models presented in section 3.3.1 and 3.3.2 for the repeat and resume conditions
2
respectively. The recovery models were also solved by using LINGO with the same computer
3
configuration and a similar B&B algorithm.
4
5. EXPERIMENTAL RESULTS AND ANALYSIS
ip t
5
In this section, we have first analysed the results of single and multiple disruptions cases for
7
MM-RCPSP. To analyse the proposed approaches, multi-mode benchmark instances from
8
different activity numbers ranging from lower to higher activities (10 to 30 activities)
9
available in Project Scheduling Library (PSLIB) were randomly chosen. For choosing those
10
instances it was provided that those instances already have feasible optimum results
11
according to PSLIB. After then, those instances were introduced with randomly generated
12
disruption start times and duration of repair to them. In the later portion of this section, we
13
have explained the solutions and computational effort required with respect to the problem
14
size and complexity.
15
5.1 Problem classification
16
We have chosen benchmark test instances from the popular test library PSPLIB-Project
17
Scheduling Problem Library. The network construction procedure and hardness for the MM-
18
RCPSP instances largely depends on constraints on the network topology (known as network
19
complexity, NC), a resource factor (Merkle et al.) that reflects the density of the coefficient
20
matrix or the average number of resource required, where RF = 1 means all resources are
21
required by the job, and resource strength (RS) which measures the availability of resources
22
(Kolisch et al., 1995). For the 20-activity instances from PSLIB, for both the preempt-repeat
23
and resume conditions, it has been observed that the computational time increases with the
24
increase of the resource factor of both of the renewable (RFR) and non-renewable resources
25
(RFN) from 0.5 to 1. Here the solution times are far more susceptible to RFN than RFR.
26
Moreover, with increasing resource strength of non-renewable resources (RSN) from 0.2 to
27
1.0 the average solution time decreases. But for the renewable resources (RSR), the solution
28
time slightly increases. These scenarios were further investigated by Kolisch et al. (1995). To
29
mention, all those randomly chosen instances were chosen according to their RF and RS
30
values and were divided into four groups as shown in table 1. Considering the entire above
31
mentioned instance characteristics, it is assumed that the instance hardness increases from
32
type 1 to 4.
Ac ce p
te
d
M
an
us
cr
6
33 18
Page 20 of 38
Table 1: Problem type classification according to the multi-mode instance characteristics Problem type
RFN
RSN
RFR
RSR
Type 1 Type 2 Type 3 Type 4
0.5 0.5 1.0 1.0
0.2-0.5 0.7-1.0 0.7-1.0 0.2-0.5
0.5 0.5 1.0 1.0
0.2-0.5 0.7-1.0 0.2-1.0 0.2-1.0
Instance sets (Problems) 20-activity 30-activity 10-activity 9,10 2,10 9,10 19,20,27,28 19,20,27,28 19,20,27,28 53-56, 61-64 39,40,47,48,55 53-56, 61-64 37-40, 45-48 37,38,45,46,62 37-40, 45-48
ip t
1
2
5.2 Disruption scenarios and recovery setup
4
The proposed recovery models were tested on 640 twenty-activity benchmark problems. Each
5
instance contains 22 activities (20 plus 2 dummy for start and end) each having three modes,
6
two renewable and two non-renewable resources. As our proposed recovery strategy is
7
responsive in nature, it is assumed that the recovery will start immediately after a disruption
8
is experienced and will continue until the end of the recovery window. Here we consider an
9
arbitrary random disruption start and end time, from period end of period 10 to end of period
10
12 (repair time = 2) for all the selected test instances under a single disruption. This scenario
11
is employed for both the preempt-repeat and preempt-resume conditions. Again for multiple
12
or a series of disruptions, we used three consecutive and independent disruptions, such that
13
the most recent disruption affects the recovery schedule of its immediate predecessor
14
disruption. In this case, we arbitrarily use the following disruption scenarios for all test
15
problems: first disruption from period 5 to 7, second one from 10 to 12 and third one from 14
16
to 16. Moreover, for our experiments, for both recovery conditions, we selected randomly
17
two disruptions on the first resource and five more on the second resource. As the non-
18
renewable resources are available throughout the project tenure, disruption for the non-
19
renewable resource is insignificant and therefore is not considered here. As the primary focus
20
is to reduce project completion time, the instability weight of the dummy end activity is
21
essential for meeting the project due date, which is usually more important than starting each
22
activity at the planned starting time. For this reason, we considered higher instability weights,
23
or penalty values, for the dummy end activity, in comparison to the penalty values of the
24
other
Ac ce p
te
d
M
an
us
cr
3
activities.
The
penalty
coefficients
we
used
were:
25 26 27
5.3 Results for single disruption
19
Page 21 of 38
Before experimenting with single and multiple disruptions, all the randomly chosen instances
2
of activities 10, 20 and 30 were solved by using LINGO optimization solver. As mentioned
3
earlier on section 3.2, the formulations of Talbot (1982) is considered here for solving the
4
nominal schedule, all the results found was optimum and same as in the PSLIB (shown in
5
table 2). Later on, considering the above mentioned single disruption scenarios, the results
6
for both the preempt-repeat and preempt-resume conditions are presented in table 2. From
7
that table, it can be seen that the average deviation of makespan (average recovery makespan
8
– average baseline or initial makespan) is always higher for the preempt-repeat condition than
9
the preempt-resume condition. Although the deviation does not depends on the problem
10
hardness, rather it depends on the amount of work wasted due to the repetition of affected
11
activities during the preempt-repeat condition, which engenders higher makespan than the
12
resume case. Under both preempt-repeat and preempt-resume options, the CPU time or
13
solution time for all randomly chosen 10, 20 and 30 activity instances were within reasonable
14
span of time comparing with other direct approaches. Meanwhile for justifying the proposed
15
disruption recovery algorithm, further modifications of those disruption scenarios have also
16
been executed. In this case, both renewable and non-renewable resources were considered
17
under disruption scenario. For doing this, 5 units of each non-renewable resource and 2 and 5
18
units for renewable resource 1 and 2 respectively were considered as disrupted or unavailable
19
during the repair period. Later on, some random instances were picked up under all three
20
types of activity sets ranging from 10 to 30 and the results are shown in table 3. Here in table
21
3,
22
disruptions and
23
only. As anticipated, recovery makespans under both resource disruptions always show
24
greater than or equal values than single renewable resource disruption only.
26
cr
us
an
M
d
te
represents recovery makespan under both renewable and non-renewable resource
Ac ce p
25
ip t
1
represents recovery makespan under renewable resource disruptions
Table 2: Results for group-wise benchmark problems under single disruption Project Type
10 activity
20 activity
Problem type
Initial makespan (average)
T/1 T/2 T/3 T/4 T/1 T/2 T/3 T/4
17 14.2 18.8 26 20.6 22.8 25.6 29.4
Preempt-repeat Recovery Average CPU time Makespan deviation (sec) (average)
19.75 17.8 22 28 24 25.8 28.4 32.6
2.75 3.6 3.2 2.0 3.4 3.0 2.8 3.2
0.425 0.5 0.8 0.9 0.7 0.7 0.8 0.9
Preempt-resume CPU Recovery Average time Makespan deviation (sec) (average)
19.5 15.4 21.2 27 22.4 23 27.4 30.2
2.5 1.2 2.4 1.0 1.8 0.2 1.8 0.8
0.425 0.6 0.6 0.9 0.6 0.8 2.5 1 20
Page 22 of 38
30 activity
T/1 T/2 T/3 T/4
31.4 34 29.75 35.5
34.8 35.4 32 37
3.4 1.4 2.25 1.5
1.25 1.6 3.0 46.0
32.4 34 29.75 35.5
1 0 0 0
1.4 2.5 3.0 6.2
1
ip t
2 3
cr
4 5
20activity
7 8 9
22 18 17 19 21 22 27 22 26 23 33 35 29 30 38
an
16 12 17 19 17 18 23 17 26 22 31 33 29 29 36
Ac ce p
30activity
Preempt-repeat
Initial Makespan
M
10activity
Randomly Chosen Instance sets J102_6 J1020_1 J1048_1 J1055_1 J1062_1 J209_8 J2010_5 J2027_1 J2048_9 J2064_10 J309_1 J3010_5 J3027_10 J3055_1 J3064_10
d
Problem Set
us
Table 3: Results for both Renewable and Non-renewable resource disruptions
te
6
23 20 18 20 23 25 29 24 28 23 34 36 30 32 39
Preempt-resume
22 15 17 19 18 22 26 17 26 22 32 33 29 29 36
22 15 17 20 19 23 27 19 26 22 32 33 29 30 37
5.4 Results for a Series of disruptions
10
For better understanding, we have calculated the percentage gap due to increasing disruptions
11
for both the preempt-repeat and preempt-resume conditions. For all three consecutive/series
12
of disruptions, the percentage gaps were measured with respect to the baseline makespan
13
using the equation
14
table 4 for all randomly selected instances of activities 10, 20 and 30. It can be seen from that
15
table that the percentage gap increases with increasing number of disruptions for the preempt-
16
repeat case. The rationale behind that is with increasing disruption numbers, the amounts of
17
wasted duration due to repetition are also increased, which leads to higher and higher
. The detailed results are provided in
21
Page 23 of 38
1
makespan. On the contrary, that is not always true for the preempt-resume case, as shown for
2
the T/2 instances of activity 30 in table 4. Moreover, the percentage gaps may vary due to the
3
conditions and precedence constraints of the affected activities at the time of disruption,
4
although it had the same disruption period.
5
ip t
6 7
cr
8 9
Project
Problem
nd
rd
% gap under preempt-resume
1st disr.
2nd disr.
3rd disr.
10.29
27.94
42.64
76.05
14.08
22.53
26.76
69.14
17.02
28.72
40.42
29.80
3.84
5.92
10.57
20.38 27.19
33.98 44.73
4.85 0
10.68 2.63
18.45 3.51
24.24
0
0
1.01
2 disr.
3 disr.
T/1
19.11
42.64
66.17
10
T/2
29.57
47.88
activity
T/3
20.21
50.00
T/4
4.80
16.34
T/1
M
an
1 disr.
T/2
activity
T/3
8.08
T/4
32
49.32
66.68
21.32
10.68
17.32
8.28 7.06
2.54 0
3.18 0
7.00 0
d
20
7.77 10.53
40.40
17.20 16.47
29.94 25.88
T/3
5.88
13.45
21.85
0
0
0.84
T/4
8.45
15.50
23.94
0
2.81
2.81
T/1 T/2
Ac ce p
30 activity
12
st
type
Type
11
% gap under preempt-repeat
us
Table 4: Results for group-wise benchmark problems under series of disruptions
te
10
13 14
5.5
15
From the literature, there is no benchmark for MM-RCPSPs with disruptions that can be used
16
for validating our proposed algorithms. For this reason, we have derived the upper and lower
17
bounds of solutions for both single and a series of disruptions and compared that with our
18
solutions. Here we used the baseline makespan time as a lower bound for all the recovery
19
stages. Then we derived upper bounds for both the preempt-repeat and preempt-resume
20
conditions. We also measured tighter upper bounds for the series of disruptions case. Let us
21
illustrate the upper bound, for the preempt-repeat condition, with an example as shown in
Lower and Upper Bounds under
Disruptions
22
Page 24 of 38
figure 8. Here, suppose activities 6 and 7 are being affected due to a resource disruption at
2
period 16. As for the preempt-repeat condition, the affected activities should start from
3
scratch in the recovery schedule, it is required to identify the maximum waste duration of the
4
affected jobs, which is five time units for activity 6 (greater than the waste duration for
5
activity 7). The upper bound can then be calculated as (base makespan + maximum waste
6
duration + disruption duration), because shifting all the affected and incomplete jobs to the
7
right by an amount equal to (maximum waste duration + disruption duration) will provide a
8
feasible schedule that may be reduced by rescheduling the remaining (affected and
9
incomplete) jobs. Maximum duration wastage for repetition
us
cr
ip t
1
Maximum Resource Line
8 2 6
4 4
2 0
7
5 10
5
15 16
18 20
Dis_start Dis_end
10
22 Time
Fig 8: Upper bound calculation for Single disruption preempt-repeat condition
d
11
8
an
3
6
M
Resource use (unit)
10
Lower bound for all conditions,
13
Upper bound for single disruption under the preempt-repeat condition,
15 16 17
Ac ce p
14
te
12
Upper bound for a single disruption under the preempt-resume condition,
Upper bound for a series of disruptions under the preempt-repeat condition,
18 19
Upper bound for a series of disruptions under the preempt-resume condition,
20
23
Page 25 of 38
Tighter upper bounds (
, here q represents number of disruptions and repeat
2
represent preempt-repeat condition) for a series of disruptions under the preempt-repeat
3
condition can be derived by repeating the calculations after each disruption as follows.
4
After first disruption (q=1),
ip t
1
5
After second disruption (q=2),
cr
6 7
After third disruption (q=3),
us
8
an
9 10
Tighter upper bounds for a series of disruption under the preempt-resume condition are as
12
follows.
13
After first disruption (q=1),
14
After second disruption (q=2),
15
After third disruption (q=3),
16
Here,
17
disruption respectively. The average recovery makespan, along with their bounds for selected
18
instances from each group and activity numbers, are summarized in table 5. From that table,
19
it can be seen that for both single and a series of disruptions, the recovery makespan is always
20
within the above mentioned upper and lower bounds.
21
makespan after the final (3rd) disruption has been reported. For all cases, the deviations
22
between the obtained makespan and their upper bounds are always higher for the preempt-
23
repeat condition than the same for the preempt-resume condition.
24 25
Table 5. Comparison of Make span time for group-wise benchmark problems (LB – lower bound; UB – upper bound; MS – makespan; TUB – tighter upper bound)
te
d
M
11
Ac ce p
represent the recovery makespan after the first, second and third
For a series of disruption, the
Project
Disruption
Problem
Preempt-repeat condition
Preempt-resume condition
Type
scenario
Type
LB
MS
UB
TUB
LB
MS
UB
TUB
10 activity
Single
T/1
17
19.75
21.5
-
17
19.5
20
24
Page 26 of 38
14.2
17.8
20.8
-
14.2
15.4
17.2
-
T/3
18.8
22
26
-
18.8
21.2
21.8
-
T/4
26
28
34
-
26
27
29
-
Series of
T/1
17
28.25
29
28.75
17
24.25
26
24.75
disruptions
T/2
14.2
25
28
27.8
14.2
18
23.2
20.4
(after final
T/3
18.8
31.8
33.2
31.8
18.8
26.4
27.8
26.8
disruption)
T/4 T/1
26 20.6
34.6 24
40 27
37.6 -
26 20.6
30.4 22.4
35 23.6
31.4 -
Single
T/2
22.8
25.8
29.6
-
22.8
23
25.8
-
disruption
T/3
25.6
28.4
33.2
-
25.6
27.4
28.6
-
T/4
29.4
32.6
36.2
-
29.4
30.2
32.4
-
Series of
T/1
20.6
27.6
33
disruptions
T/2
22.8
33
(after final
T/3
25.6
35.2
disruption)
T/4
29.4
41.67
T/1
31.4
34.8
Single
T/2
34
35.4
disruption
T/3
29.75
T/4
35.5
Series of
T/1
31.4
disruptions
T/2
(after final
T/3
disruption)
T/4
Ac ce p
1
cr
us
24.4
29.6
25.8
37
36.6
22.8
23.4
31.8
26.6
39.2
37.6
25.6
26.8
34.6
29.2
29.4
29.43
37
29.67
39.4
-
31.4
32.4
34.4
-
39.6
-
34
33.8
37
-
M
an
20.6
42.33
32
37
-
29.75
29.75
32.75
-
32.5
44.5
-
35.5
32.0
38.5
-
40.8
45
43.8
31.4
33.6
40.4
35.4
d
30 activity
31
44
34
42.8
47.6
45.8
34
33.8
43
36.8
29.75
36.25
43.25
40.75
29.75
30
38.75
32.25
35.5
44
48.5
47
35.5
36.5
44.5
39.5
te
20 activity
ip t
T/2
disruption
2
5.6
3
In this section, we examine the behaviour of the disruption recovery models with respect to
4
different parameter changes.
5
5.6.1
6
Behavioural Patterns
Impact of disrupted quantity and
repair times
7
In real life MM-RCPSPs, there may be copious reasons for disruptions and their
8
characteristics may also vary depending on the disruption types and severity. Among many,
9
resource disruption and amount of that disruption is an important parameter to judge recovery
10
models. To figure out the possible impacts of disrupted quantity and repair times, we have
11
investigated our recovery models with some of our randomly chosen instances. Here the
12
disrupted quantity is being quantified by the amounts of resources disrupted and the repair 25
Page 27 of 38
1
time which is the duration of resource unavailability due to recovering from that disruption or
2
maintenance. Irrespective of the available modes, for both the preempt-repeat and resume
3
conditions, we considered that all the renewable resources are disrupted up to 20%, 40%,
4
60% and 80% of their available levels, with different repair times, such as
5
and with different disruption start times (D_s) (ranging from 10% to 70% of the project due
6
date PDD). For different disruption start times, the average recovery makespan (for randomly
7
selected instances from different activity sets: MJ1020_1, MJ1038_1, MJ2019_8, MJ2010_5,
8
MJ3010_5 and MJ3064_10) for different resource breakdown percentages are presented in
9
figures 9(a) to 9(e). In those figures, MS_repeat_3 represents preempt-repeat makespan for
10
repair time equals 3 whereas MS_resume_3 represents the preempt-resume makespan for
11
repair time equals 3.
12
All the figures show that the recovery makespan increases with increasing percentage of
13
resource breakdown. But the rate of increment is not so straightforward. From all the figures
14
for different disruption start times, under both the preempt-repeat and resume conditions, it
15
has been observed that the rate of increment for recovery makespan for up to 40% resource
16
breakdown (with smaller repair time, say 3 and 5) is almost flat or tapered off. But after that,
17
the increment rate is linear. It depends on the activity usage rate in that particular time period.
18
For example, as of instance MJ2010_5, if the
19
affected activities are 10, 11, 12, 13, 15 and 16 for repair times of up to 9. But they need only
20
50% or less of the available resource to complete, as they have lower resource strength values
21
(RSR and RFR equals 0.5). Moreover, the flexibility of multiple modes allows the activities
22
to switch in the best possible ways (duration and resource usage combination) to overcome
23
disruption affects. As a result, once the makespan is increased to 28, it remains the same until
24
40% of resources are broken down. Then it increases linearly as more resources are disrupted.
25
From figures 9(a), 9(c) and 9(d), it can be seen that for different disruption start times with
26
different repair times, the average recovery makespan for the preempt-repeat condition
27
remains same if only 20% of each resource is disrupted, the only exception was where for
28
= 30% of PDD. This is because one of the instances has low resource strength
29
(RSR=0.5), and so there is some resource redundancy, which allowed the makespan to
30
remain same. Meanwhile, from figure 9(e), it shows that the recovery makespan increased
31
quite linearly for preempt-repeat case from
32
then decreased linearly from
= 50% of PDD (time period = 12), the only
Ac ce p
te
d
M
an
us
cr
ip t
3, 5, 7 and 9,
= 10% of PDD to
= 30% of PDD to
= 30% of PDD and
= 50% of PDD. After that, from
26
Page 28 of 38
1
= 50% of PDD to
= 70% of PDD the recovery makespan for that repeat case increased
2
again. This is another interesting aspect of the recovery schedules. As with increasing
3
values, the probable numbers of affected jobs are increased, and as a result the makespan at
4
the beginning also increases. But for larger
5
reduced significantly (i.e. most of the jobs are completed). As a result, the recovery schedule
6
for larger
7
makespan. Meanwhile for much higher D_S values, the recovery schedule could not get that
8
much of time to recover efficiently. As a result, for higher D_S values (i.e., D_S= 70% of
9
PDD), the recovery makespan increased again. On the contrary, for the preempt-resume
10
option, because of minimum work lost, the recovery makespans always show some linearly
11
increment pattern.
ip t
values, the activities left to complete are
12 13
Ac ce p
te
d
M
an
us
cr
values have lower number of jobs to deal with, which also reduces the
Fig 9a: Makespan for different breakdown percentage & repair times (
= 10% of PDD)
Fig 9b: Makespan for different breakdown percentage & repair times (
= 30% of PDD)
14 15
27
Page 29 of 38
ip t cr Fig 9c: Makespan for different breakdown percentage & repair times (
= 50% of PDD)
4
Fig 9d: Makespan for different breakdown percentage & repair times (
Ac ce p
3
te
d
M
an
2
us
1
= 70% of PDD)
5 6
Fig 9e: Recovery makespan for different disruption start times
7
28
Page 30 of 38
1 2
5.6.2
3
Activities in the projects can also be disrupted by many multifarious reasons. For example,
4
due to some unexpected customization of end products or some modifications of process
5
plan, an activity can be disrupted. So, if any of the activity’s duration or resource usage
6
departs from the planned values, then that activity is said to be disrupted. The deviations can
7
be of any form (negative or positive) depending upon the disruption situations. To interpret
8
the effects of the activity disruptions on the recovery schedule, we have further distinguished
9
them as activity duration variability and activity resource disruptions. The later one will be
10
analysed in the next section. In this section, we have investigated the impact of the variability
11
of activity durations. This was carried out by multiplying each activity’s duration time on
12
each mode by a multiplication factor, MF (MF= 1+beta distribution of random numbers with
13
parameters 2 and 5). This implies that the duration of each activity for our randomly selected
14
instances (3 of each problem type of varied activity number ranging from 10 to 30) was
15
increased, without giving any restrictions to the project due date, and was treated with the
16
disruption scenario mentioned in section 5.2. The results for both the recovery schedules are
17
shown in figure 10, from which it can be observed that the percentage increase of recovery
18
makespan (ms) time is always lower than the duration variability, independent of problem
19
hardness. Meanwhile, the percentage increase of the recovery makespan under the preempt-
20
resume condition is lower than the repeat condition for the harder problem types (T/3 and
21
T/4), which proves that the resume option, in comparison to the repeat option, is more
22
vulnerable for harder instances for recovering from disruption. For easier instances, the
23
duration variability has less impact on the repeat makespans than for the resume ones. This
24
supports the intrinsic characteristics of the instances, while the solution time also increases
25
with increasing duration variability and problem hardness.
Impact
of
activity
duration
Ac ce p
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M
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cr
ip t
variability
26 29
Page 31 of 38
1
Fig 10: Relationship among duration variability and % increase of recovery makespans
2 3
5.6.3
4
To analyse the possible impacts of activity resource disruptions, again we have considered
5
those randomly selected instances from our problem groups of 10, 20 and 30 activities. An
6
activity resource disruption occurs when for any particular time span, the corresponding
7
activity takes extra usage for its completion on all its available modes. Here in this paper,
8
we assumed that activity 3 used
9
taken from 1 to 5. Under the same disruption scenario as mentioned in section 5.2, the
10
average penalty values that resulted are summarized in table 6. From table 6, for both
11
recovery options, it is quite clear that the recovery penalty values for any particular problem
12
type increases almost linearly with the increasing resource usage rate. Despite the fact that
13
the activity resource disruptions have only a minor impact on the recovery makespan
14
(remains almost unchanged with increasing resource usage rate), it has a significant impact
15
on penalty values. As with increasing usage rate, the activity start time in the recovery
16
schedule may be shifted, and this causes higher penalty values. For example, under the
17
preempt-repeat condition for instance MJ2056_6 with
18
activities 5 and 6 were shifted from the initial start time by 3 and 4 units. But for
19
deviation between the activities start time was increased to 5 and 7 units respectively. As a
20
result, the penalty values also increased. But the excess slack time of the project did not allow
21
the makespan time to increase in the recovery schedule. In table 6,
of
activity
resource
ip t
disruptions
was
M
an
us
cr
more of renewable resource k, while the value of
d
, the recovery start time of the
te
Ac ce p
22
Impact
and
represents the recovery penalty values for the preempt-repeat and resume
23
conditions respectively.
24
Table 6. Recovery results for different activity-resource disruptions Excess Resource Usage
1
2
3
4
5
32.33
34
36.67
37
39
10
11
11.33
12
13.33
25 26
5.6.4
27
Same disruption scenario has been considered to understand the effects of length of delay on
28
our proposed recovery problem. Here the basic assumption was to delay the execution of
29
some activities for certain time periods, say to delay the activity 3 of each randomly
Impact of Length of Delay
30
Page 32 of 38
1
considered instance between 1 and 10 time periods without restricting the makepsan. A series
2
of experiments were carried out and the results are reported in table 7. As anticipated, the
3
delay of execution has a significant impact on recovery makespan and as the delay increases
4
the average makespan for all instances also increases almost linearly for both the preempt-
5
repeat and resume conditions. Here
6
makespan under the preempt-repeat and resume conditions respectively for all those
7
randomly selected instances of activities 10, 20 and 30. The repeat condition needs more
8
time, because it must repeat previously done work. As a consequence, compare to the
9
preempt-resume condition, the makespan of the preempt-repeat condition increases at a
10
higher rate with increasing length of delays. This analysis helps to understand the effects on
11
makespan if there are any delays for executing activity during the recovery process.
12
Table 7. Recovery results for different delays 2
3
4
26
28.5
31
an
1 25.25
5
6
7
8
32.25
33
36
37
24.25 25.25 25.75 26.25 27.25
28
28.25
29.5
M
Delay
us
cr
ip t
represent the average recovery
13
9
10
37.75 42.25 34
38
5.6.5
15
In real life projects, as of many other interruptions, it is possible that some new activities may
16
be added or removed during project execution, and that may change the precedence
17
relationships of some activities. In this regard, two more important scenarios, namely the
18
addition of new activities (with deducting the same number of existing activities to keep the
19
same total activity number) and a change of precedence relationships can be analysed. In this
20
research, we considered four different cases with randomly generated new activities along
21
with their resource usage on each of three mode, precedence relationships and mode
22
dependent activity durations. Taking different instances of MM-RCPSPs from each of these
23
problem types, it was observed that the recovery makespan for both the repeat and resume
24
condition increased quite significantly over the initial one. We also considered four different
25
randomly generated new precedence relations, instead of the existing ones, and measured the
26
recovery makespan. Similar conclusion can be drawn for this case, as with increasing number
27
of precedence modifications, the recovery makespan also increased. Note that the deviations
28
may be positive or negative depending upon the characteristics of new activity and
29
precedence relationships. So, it is very difficult (sometime impossible) to build up a profound
30
relationship about the effect of precedence modifications on recovery makespan.
Impact of other parameters
Ac ce p
te
d
14
31
Page 33 of 38
1 2
6.
CONCLUSION
The objective of this research was to develop a real time disruption recovery plan for multi
4
mode, multiple resource constrained project management problems. For two newly proposed
5
recovery options, preempt-repeat and preempt-resume, we have developed mathematical
6
models to deal with a single disruption, as well as a series of disruptions, on a real time basis.
7
Optimization software was employed to solve the mathematical model for generating new
8
revised schedules under changed or disrupted conditions. A detailed stepwise procedure has
9
also proposed to deal with a series of disruptions, where repeated solutions from the
10
mathematical model are required after each change takes place. As mentioned earlier, we
11
have made a pragmatic assumption that the disruption information is known only after its
12
occurrence and the disruption recovery algorithm is then run with the information related to
13
the disruption. Such a schedule can be recognized as a real-time schedule. For our
14
experimental study, we have considered sets of multi-mode 10, 20 and 30-activity static
15
benchmark instances form PSLIB and introduced random disruption scenarios with them.
16
The experimental results clearly demonstrate that the re-optimization process with these two
17
recovery options can reduce the recovery makespan, as compared to simple right shifting of
18
affected activities (or upper bound solution), and the level of improvement depends on the
19
duration of the affected activities, their resource requirements, their relationships with
20
forwarding activities, disruption duration (repair time) and the amount of disrupted resources.
21
That means the proposed approaches add certain value to the project management body of
22
knowledge.
23
In this paper, we have also considered a few other realistic variations that can happen during
24
project execution, such as: change in activity duration, resource requirement, precedence
25
relationships, repair time, and addition of new activities which may have some significant
26
impact upon the recovery schedules for MM-RCPSP. These variations can be considered as
27
disruption events and the revised schedule can be generated using the disruption recovery
28
approaches proposed in this paper. The impact of such variations on the revised makespan
29
was analysed using sensitivity analysis.
30
The proposed approaches are easy to implement and therefore can be employed by
31
practitioners in different ways to generate revised schedule under disruption scenarios on a
32
real time basis. Moreover, significant financial and time loss can be lessened by applying
Ac ce p
te
d
M
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cr
ip t
3
32
Page 34 of 38
these approaches if any disruption is experienced. Here we have assumed static behaviours of
2
activity duration, but in real life that may not happen every time. Involving more developed
3
approaches for large projects under disruptions, and solving them by using meta-heuristic
4
algorithms, will be our future research.
5
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ip t
Table A.1: Task duration and Resource Requirement for the Motivating Example (R1 and R2 are the two renewable resources) Duration (Months) Mode 2
Mode 3
0 3 1 3 4 2 3 4 2 1 6 0
0 9 1 5 6 4 6 10 7 1 9 0
0 10 5 8 10 6 8 10 10 9 10 0
d
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12
Resource available after disruption
Maximum Resource Line
Resource use (unit)
9
19
8
4_i [ 2]
6 4
6 [3]
2 [1]
2
st
1 D_e
5
1s t D_s
10 Time
Resource available after disruption
Resource use (unit)
9 [1]
5_R [2]
5_i [ 2]
0
4 _R [2 ]
8 [1 ]
7 [1 ]
15
20
22
Maximum Resource Line
9 8 4_i [ 2]
6 4
6 [3]
2 [1]
2
5_R [2]
5_i [ 2]
0
20
Renewable Resource Requirements (Units) Mode 1 Mode 2 Mode 3 R1 R2 R1 R2 R1 R2 0 0 0 0 0 0 6 0 5 0 0 6 0 4 7 0 0 4 10 0 7 0 6 0 0 9 2 0 0 5 2 0 0 8 2 0 5 0 0 7 5 0 6 0 3 0 4 0 2 0 1 0 1 0 4 0 0 2 4 0 0 2 0 1 0 1 0 0 0 0 0 0
M
Mode 1
te
Activities
18
cr
APPENDIX
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12 13 14 15 16 17
39. Węglarz, J., Józefowska, J., Mika, M., & Waligóra, G. (2011). Project scheduling with finite or infinite number of activity processing modes – A survey. European Journal of Operational Research, 208(3), 177-205. 40. Zapata, J. C., Hodge, B. M., & Reklaitis, G. V. (2008). The multimode resource constrained multiproject scheduling problem: Alternative formulations. AIChE Journal, 54(8), 21012119. 41. Zhu, G., Bard, J. F., & Yu, G. (2005). Disruption management for resource-constrained project scheduling. Journal of the Operational Research Society, 56(4), 365-381. 42. Zhu, G., Bard, J. F., & Yu, G. (2006). A Branch-and-Cut Procedure for the Multimode Resource-Constrained Project-Scheduling Problem. INFORMS Journal on Computing, 18(3), 377-390.
an
1 2 3 4 5 6 7 8 9 10 11
5
1s t D_s
st
1 D_e
7 _i [1 ] nd 10 2 D_e
2 nd D_s
4_R [2]
8 [1]
15
7_R [1]
20
9 [1] 23
Time
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Resource available after disruption
8 4_i [2]
6 4 2
5_ R [2]
5_i [2]
5
1s t D_s
1s t D_e
4_Ri [2]
7 _i [1 ]
6 [3]
2 [1]
0
7_ R [1]
8 [1]
4_ Rr [2]
9 [1 ]
nd rd 10 2 D_e 3 rd D_s15 3 D_e 20 Time
2 nd D_s
25
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Fig A1: Gantt Charts of optimum recovery schedule under preempt-resume condition for series of disruptions (R: Remaining activity; Ri: Initial work of remaining activity; Rr: Last portion of remaining activity)
Ac ce p
1 2 3 4
Maximum Resource Line
ip t
Resource use (unit)
9
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S0098-1354(16)00007-7 http://dx.doi.org/doi:10.1016/j.compchemeng.2016.01.004 CACE 5349
To appear in:
Computers and Chemical Engineering
Received date: Revised date: Accepted date:
9-4-2015 3-12-2015 5-1-2016
Please cite this article as: Chakrabortty, R. K., Sarker, R. A., and Essam, D. L.,Multi-Mode Resource Constrained Project Scheduling Under Resource Disruptions, Computers and Chemical Engineering (2016), http://dx.doi.org/10.1016/j.compchemeng.2016.01.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
MULTI-MODE RESOURCE CONSTRAINED PROJECT SCHEDULING UNDER RESOURCE DISRUPTIONS
Research Highlights
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1. Formulated two discrete time based models to deal with disruption scenarios
2. A solution approach is proposed to generate a revised schedule after disruptions
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3. Proposed recovery options show better performance than simple right shifting
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4. This model is capable of dealing with a single, as well as multiple disruptions
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MULTI-MODE RESOURCE CONSTRAINED PROJECT SCHEDULING UNDER RESOURCE DISRUPTIONS
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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
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Over the last few decades, research on resource constrained project scheduling has focused on the development of mathematical programming based approaches for the generation of a
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nominal schedule under a deterministic environment. During the implementation phase, however, the nominal schedule may need to be revised when one or more resources are disrupted for a length of time. In this paper, we formulate two discrete time based models to
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deal with two different disruption scenarios for multi-mode resource constrained problems. We propose a reactive re-scheduling procedure for a single, as well as a series of disruptions,
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without having any disruption information in advance. To test the proposed approaches, sets of ten, twenty and thirty-activity multi-mode test instances from Project Scheduling Library
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(PSLIB) were used after introducing randomly generated disruption events. The experimental studies were also carried out to determine the effect of different factors related to the disruption recovery process.
Keywords: Multi-mode Resource Constrained Project Scheduling, Rescheduling, Disruption, Mixed Integer Linear Programming.
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1 2
MULTI-MODE RESOURCE CONSTRAINED PROJECT SCHEDULING UNDER RESOURCE DISRUPTIONS
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ABSTRACT
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Over the last few decades, research on resource constrained project scheduling has focused
8
on the development of mathematical programming based approaches for the generation of a
9
nominal schedule under a deterministic environment. During the implementation phase,
10
however, the nominal schedule may need to be revised when one or more resources are
11
disrupted for a length of time. In this paper, we formulate two discrete time based models to
12
deal with two different disruption scenarios for multi-mode resource constrained problems.
13
We propose a reactive re-scheduling procedure for a single, as well as a series of disruptions,
14
without having any disruption information in advance. To test the proposed approaches, sets
15
of ten, twenty and thirty-activity multi-mode test instances from Project Scheduling Library
16
(PSLIB) were used after introducing randomly generated disruption events. The experimental
17
studies were also carried out to determine the effect of different factors related to the
18
disruption recovery process.
19
Keywords: Multi-mode Resource Constrained Project Scheduling, Rescheduling, Disruption,
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Mixed Integer Linear Programming.
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1. INTRODUCTION
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In Resource Constrained Project Scheduling Problems (RCPSPs), the objective is to
24
minimize the makespan while satisfying the resource constraints and precedence relationships
25
among the activities. The Multi-mode Resource Constrained Project Scheduling Problem
26
(MM-RCPSP) is an extension of the conventional RCPSP, in which the duration of each task
27
is a function of the level and type of resources committed to it, and the project interactions
28
that result from the utilization of shared resources that are taken into consideration (Zapata et
29
al. (2008). According to the classification scheme of Herroelen et al. (1999), this MM-
30
RCPSP is denoted as
31
renewable and nonrenewable| strict finish start precedence constraints with zero time-lag,
32
activities that have multiple execution modes, the activity resource requirements are a
(i.e., m resource types which can be both
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discrete function of the activity duration| the objective is to minimize the makespan). The
2
resources used by project activities are generally of two types, namely: (1) renewable
3
resources with availability restrictions that may vary from one period to the next (e.g. the
4
number of workers per shift), (2) non-renewable resources with availability restrictions over
5
the whole project horizon (e.g. raw material). As of the literature, the renewable resources are
6
mainly considered for single mode RCPSP, however both renewable and non-renewable
7
resources are considered simultaneously for MM-RCPSP. Other specific resource categories
8
that have been considered for RCPSP are: partially (Nonobe & Ibaraki) renewable resources
9
(Böttcher et al., 1999), dedicated resources (Bianco et al., 1998), spatial resources (Hans et
10
al., 2007), cumulative resources (Neumann et al., 2003), reusable resources (Shewchuk &
11
Chang, 1995), synchronizing resources (Schwindt & Trautmann, 2003), multi-skill resources
12
(Néron, 2002), heterogeneous resources (Tiwari et al., 2009), and allocatable resources
13
(Schwindt & Trautmann, 2003). The variants of traditional RCPSP include: Generalized
14
RCPSP, RCPSP with generalized precedence constraints, RCPSP with time varying resource
15
constraints, and Dynamic RCPSP (Węglarz et al., 2011).
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RCPSP has gained widespread attention for the last few years due to its practical importance
18
and computational challenge. While some of the earlier endeavor was on refining the basic
19
model, the majority of research has been aimed at developing better solution methods (Zhu et
20
al., 2006). Blazewicz et al. (1983) have shown that RCPSP is an NP-hard problem.
21
Moreover, when the process allows the choice of modes (in MM-RCPSP), further complexity
22
is added by enlarging the search space (Kyriakidis et al., 2012). In solving MM-RCPSP,
23
mixed integer linear programming (MILP) modeling is a popular choice. For finding optimal
24
solutions for RCPSP (and also MM-RCPSP), copious algorithms and methods can be found
25
in the literature. Among them, the branch and bound algorithm (Hartmann & Drexl, 1998;
26
Sprecher & Drexl, 1998), branch and cut based algorithm (Zhu et al., 2006), tree based
27
branch and bound algorithm (Hartmann & Drexl, 1998), self developed heuristics (Ballestín
28
et al., 2008) and, linear programming based algorithm (Kopanos et al., 2014) are the most
29
common approaches. According to Herroelen (2005), computational results indicate that
30
many of the 60-activity and most of the 90- and 120-activity instances from the Project
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Scheduling Library-PSLIB (Kolisch & Sprecher, 1997) are still a good way off the solution
32
capabilities of the exact methods.
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Industrial resource constrained problems have been considered as a significant challenge in
2
highly regulated industries, such as pharmaceuticals and agrochemicals, where a large
3
number of candidate new products must undergo a set of tests for certification (Choi et al.,
4
2004). In spite of that, in the recent past, different industrial resource constrained problems
5
have been applied/addressed for process systems engineering, such as the application of
6
RCPSP in semi-continuous food industries (Kopanos et al., 2011), multistage batch
7
processing (Méndez & Cerdá, 2003), automated wet-etch station (AWS) scheduling (Novas
8
& Henning, 2012), and for varied set up times (Nadjafi & Shadrokh, 2008). A detailed
9
discussion on earlier applications of planning and scheduling in the process industry can be
10
found in Kallrath (2002). However, chemical process industries are dynamic in nature, and
11
therefore different types of unexpected events occur quite frequently. The most frequent
12
rescheduling factors in the chemical process industry are: machine failure, rush job arrival,
13
job cancellation, due date change, inadequacy of raw materials, price changes, and
14
overestimation (or underestimation) of processing time, set-up times, and equipment release.
15
In particular, Adhitya et al. (2007) proposed a heuristic for rescheduling crude oil operations
16
to manage abnormal supply chain events. Their proposed model gives some provision to
17
refinery personnel to choose a suitable feasible schedule from amongst many identified
18
feasible schedules. Apart from that, Janak et al. (2006) also proposed a reactive schedule for
19
a large-scale industrial batch plant in which the authors ignored full rescheduling in the
20
current production horizon. Instead, they utilized an efficient mixed integer linear
21
programming (MILP) mathematical framework to determine which tasks would not be
22
affected by the unforeseen event, either directly or indirectly, such events were carried out as
23
scheduled. However, reactive scheduling in case of RCPSPs is still insufficient. Keeping this
24
in mind, this paper deals with reactive rescheduling techniques for real time based
25
generalized MM-RCPSPs. The applicability of this research is highly diversified, as this
26
paper conveys the dynamic features of machine or resource inadequacy/unavailability for a
27
general MM-RCPSP case. This way of tackling such resource uncertainties can easily be
28
applied for any real-time based chemical process industry.
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During the implementation phase, a project may face significant predicaments due to
31
resource unavailability, unproven technology, unreasonable commitment and unrealistic or an
32
unclear goal set up (Zhu et al., 2005). Due to these factors, a project may be delayed in
33
completion, so any such noteworthy deviation in a project schedule is considered as a
34
disruption. Because of disruption, the traditional deterministic project scheduling models 3
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must be revised and resolved to match with the changed environment (Deblaere et al., 2011).
2
That means, an initial optimal solution is only optimal during the execution of the schedule if
3
there is no disruption. Vieira et al. (2003) have classified the existing rescheduling strategies
4
into three primary types: (1) repairing a schedule that has been disrupted, often known as
5
reactive strategy (Deblaere et al., 2011); (2) creating a schedule that is robust with respect to
6
disruptions, known as proactive scheduling (Herroelen & Leus, 2004); and (3) studying how
7
rescheduling policies affect the performance of dynamic manufacturing systems. In the case
8
of proactive (robust) scheduling, a degree of anticipation of variability during project
9
execution is incorporated into the nominal schedule. Hence even if there is no variation in the
10
project run, this strategy always have some extra allowance and therefore gives suboptimal
11
results. The use of a nominal schedule in combination with reactive scheduling procedures is
12
sometime referred to as proactive-reactive scheduling, which is an iterative strategy. Reactive
13
scheduling, on the other hand, is generally of two types: schedule repair, often known as the
14
right shift rule because it moves forward all the affected activities (Sadeh et al., 1993) and
15
full rescheduling (Artigues & Roubellat, 2000) which differs considerably from the nominal
16
schedule. However, determining the best rescheduling solution still remains an open research
17
issue, and consequently is the most difficult part of the rescheduling process (Vieira et al.,
18
2003).
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The literature on handling disruption in MM-RCPSP is however scarce. To the best of our
21
knowledge, there are only two earlier works on handling disruptions for MM-RCPSPs. Zhu et
22
al. (2005) have formulated an MILP model for a general class of reactive scheduling
23
problem, and solved it with a hybrid mixed integer programming or constraint programming
24
procedure. For recovering disruptions, they considered three different recovery options,
25
namely rescheduling, mode alternation and resource alternation. Deblaere et al. (2011)
26
considered activity duration variability and resource disruption explicitly, and evaluated some
27
dedicated exact reactive scheduling procedures, as well as a Tabu search heuristic for
28
repairing a disrupted schedule, under the assumption that no activity can be started before its
29
baseline starting time. However, developing a proper mathematical programming model for
30
multi-mode RCPSP, considering resource disruptions, is still a challenging research topic.
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20
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When dealing with RCPSPs and their analysis, the two practical scenarios of preempt-repeat
33
and preempt-resume are generally considered by researchers (Lambrechts et al., 2010). In the
34
case of the preempt-repeat environment, interrupted activities must be started from scratch, 4
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1
because they assume that incomplete jobs cannot be continued for completion and are so
2
counted as wastage. On the contrary, in a preempt-resume environment, only the residual
3
portion of any affected/interrupted activity will need to be restarted during its recovery
4
schedule.
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In this paper, we consider multi-mode resource constrained project scheduling under
7
disruption. First, the problems with disruption of multiple renewable resources is discussed
8
for two different practical scenarios, known as ‘preempt-repeat’ and ‘preempt-resume’, and
9
the mathematical programming models, based on discrete time, for recovering from the
10
disruptions are developed. A solution approach is proposed, which can generate a revised
11
schedule after a disruption event takes place, where the disruption information is not known
12
in advance. It is expected that the parameters of disruption follow a stochastic process. We
13
deal with these stochastic parameters within a deterministic environment. The proposed
14
solution approach is capable of dealing with a single, as well as a series of disruption events,
15
for multiple resources and for multiple modes, on a real-time basis. To judge the performance
16
of our proposed approach, we have generated a set of test problems and compared the
17
solutions with their upper and lower bounds. In generating the test problems, we have
18
selected sets of ten, twenty and thirty-activity benchmark instances from PSLIB and
19
introduced randomly generated disruption scenarios into them. Experimental studies have
20
also been conducted to analyse the effects of different factors relating to the disruption
21
recovery process, such as changes in activity duration, changes in precedence relationship,
22
and addition of new activity.
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The structure of the paper is as follows: in section 2, we define basic MM-RCPSPs and
25
discuss the associative disruption recovery strategies. The terminologies, disruption recovery
26
models, and their MILP formulations are described in section 3. In section 4, solution
27
approaches and relevant algorithm design are discussed. The experimental study, along with
28
the results of computational tests that measure the behavioural patterns of recovery schedules
29
are described in section 5. In the final section, we provide overall conclusions.
30 31
2. PROBLEM DESCRIPTION
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In this research, we consider MM-RCPSPs under resource disruption. The objective of the
33
problem is to re-schedule project activities and their required resources, to recover from
34
disruptions as soon as possible. For this purpose, we consider the following assumptions: (i) 5
Page 7 of 38
the activities composing a project have known durations for each and every mode; (ii) all
2
predecessors must finish before an activity can start; (iii) resources are of both renewable and
3
non-renewable; (iv) activities are non-preemptive (i.e., cannot be interrupted when in
4
progress); (v) the activities cannot be started before their planned starting times (railroad
5
scheduling); and (vi) the objective is to minimise the project completion time.
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1
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Here it is assumed that I is the number of activities to be scheduled. The activities
8
constituting the project are represented by a set {0,..., I+1}, where 0 and I+1 are dummy
9
nodes representing start and end respectively. The important notations and sets are depicted
10
in the next section as nomenclature. The basic consideration is to execute the recovery
11
schedule immediately after the disruption experiences, and is therefore termed as a
12
responsive strategy. If during the recovery time window of a disruption, if the problem faces
13
another new disruption, then the later one can be considered as an independent disruption and
14
a new recovery plan can be made in the similar way as for the previous one. This type of
15
scenario, (i.e. that the problem is affected by a series of disruptions) makes the case more
16
complex for recovery planning, as all previous and recent disruptions must be considered
17
when generating the revised schedule. In this paper, two recovery options are considered,
18
namely preempt-repeat (earlier work is lost) and preempt-resume (no work lost), as discussed
19
earlier.
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3. THE MATHEMATICAL MODELS
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The mixed integer linear programming models (MILP-models) for our proposed MM-RCPSP
23
models are discussed in this section. To do this, we first explain the mathematical model for a
24
nominal schedule and then reformulate it for both recovery schedules.
25 26
3.1 Nomenclature
27
Sets I T W R Q
set of activities, i =0…I+1 set of time periods, t = 0…T represents the precedence set set of non-renewable resources, w = 1…W set of renewable resources, r = 1…R set of modes for activity i, m = 1… set of disruptions, q = 1…Q set of incomplete activities, including activities which start during a disruption repair period (affected activities) and activities which start after a disruption ends (non ] affected activities) 6
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L
set of affected and incomplete activities which start before disruption, l ( ] set of completed activities before disruption experienced, ( ]
Parameters
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Capacity of non-renewable resource w Usage of renewable resource r by activity i at mode m Usage of non-renewable resource w by activity i at mode m Duration of activity i at mode m Usage limit of renewable resource r on time slot
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set of time slots on which renewable resource r is constrained,
Resource usage of activity i for resource r Total planning horizon/upper bound of the project duration. Last job Finish time of activity i in the results from the nominal schedule. Start time of activity i from the nominal schedule. Represents the disruption start time Represents the disruption finish time (disruption start time + repair time) Disruption repair time Number of disrupted resources (e.g., breakdown machines) Very small amount of time span Incomplete portion of task i Used non-renewable resource ‘w’ up to time t Non-renewable resource ‘w’ required for completing portion of work Baseline makespan Recovery makespan after ‘qth’ disruption
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3.2 Basic MM-RCPSP formulations for Nominal Schedules The standard MM-RCPSP requires sequencing the project activities, so that the precedence constraints are met, determining the execution mode for each activity, and satisfying the resource limitations that minimize the project duration. From the literature, several mathematical models can be found for MM-RCPSP. Most of those models are somehow extensions of the model proposed by Talbot (1982). That is why the model from Talbot (1982) is considered here as the mathematical model for solving the nominal schedule. As the variables considered depend on discrete time parameters, this type of formulation is called a discrete type formulation. Here a binary decision variable
is defined to be 1 if activity
starts at mode m on time instant , and is 0 otherwise. Thus the formulation can be written as follows:
7
Page 9 of 38
(1) 1 2
Constraints: (2)
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(3) (4)
(6) (7)
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(5)
3
As of many other MM-RCPSP models, the objective (equation 1) is to minimize the
5
completion time of the last activity of a project (i.e. the project completion time). Constraint
6
(2) ensures that each activity starts from exactly one time period in a given mode. That
7
means, no balking or priority violation is allowed (Non-preemptive case). The jobs which
8
have started must be completed first, depending upon their corresponding priority. Constraint
9
(3) ensures the precedence relationship that means an activity cannot be started before the
10
completion of its preceding activity. Inequalities (4) and (5) represent the capacity constraints
11
for renewable and non-renewable resources respectively. For
12
resource r is called renewable. The usage of a renewable resource is limited in each time
13
period, but there is no limit on total usage. In contrast, the non-renewable resources are
14
available throughout the whole project tenure (T) and are consumed by all activities.
15
Constraint (6) defines the baseline finish time of each activity.
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,
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4
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3.3 Disruption Recovery Models
18
Project disruptions are the events that prevent project managers from completing the work as
19
planned. The possible reasons for these disruptions encompass diversified causes, such as
20
normal errors, biased or parsimonious estimations of activity durations or usage rate, and
21
sudden customization. The consequences of these disruptions are project delay or cost
22
overruns, that consequently demands some reschedule of the nominal schedule. To
23
reschedule the project activities under any disruption, it is necessary to reformulate the
24
nominal MM-RCPSP model by incorporating disruption or unavailability information. In this
8
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1
research, we have formulated two models, for a single disruption, under both the preempt-
2
repeat, and the preempt-resume conditions. These two models are presented below.
3
3.3.1 Preempt-Repeat Condition
5
As mentioned earlier, for the preempt-repeat case, the affected activities should start from
6
their very beginning, which requires perfect knowledge of the disruption start time. Having
7
some prior information regarding the disruption start time and repair or replacement time
8
may facilitate a rescheduling strategy. But, as indicated earlier, the disruption start time is not
9
known in-advance; rather it is unknown and random. So the reschedule plan will be generated
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just after the real disruption takes place.
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11
3.3.1.1 Objective function
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12
The objective is to minimize the completion time of the last job (i.e. makespan time) plus the
14
weighted deviations of all activity’s finish times in the revised schedule. Where
15
are given penalty weights for the deviation of activity completion times and
16
positive value that is equal to max{Y, 0}. Penalty values are used to minimize the instability
17
of activity starting times in the recovery schedule, as project instability may hamper project
18
profitability. This has also been used by Kopanos et al. (2008) for maintaining schedule
19
stability. Here, the first term represents the project completion time. The second and third
20
terms represent the weighted deviations between the finish times of each activity in the
21
recovery and initial schedules.
23 24 25
and
d
represents a
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13
(8)
3.3.1.2 Constraints
1. Start time limitation: In the case of rescheduling, we assumed that the recovery period starts immediately after the disruption experienced and it is to be continued
26
until the project ends, and is defined as the recovery window. In the recovery
27
schedule, the affected or incomplete activities
28
disruption start time, must be started within its recovery window. Meanwhile
29
activities
whose finish time is greater than the
should consider their best suitable mode for execution, and have a unique
9
Page 11 of 38
1
start time. Here, T represents the whole project horizon within which the project must
2
be accomplished. (9)
3
5
2. Precedence relationship: Same as constraint (3), where we assumed zero-lag finish-
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4
start precedence relations.
6
3. Resource capacity: Here, constraints (10) and (11) represent the capacity limitations
8
of non-renewable resources, while equations (12) – (14) represent the capacity
9
constraints for renewable resources. For resource utilization, three different cases can
10
occur for renewable resources in a recovery schedule. For the first case, it is
11
assumed, that for time spans up to the disruption start time, the maximum capacity of
12
how much renewable or non-renewable resources is available, is as defined in
13
constraint set (12). For the second case, we assumed that after the disruption start
14
time, the capacity is appropriately reduced (measured by
15
back in service (defined in constraint set (13)). Note that the capacity can be reduced
16
to zero when there is no resource to perform a job. Later on, for the third case, it is
17
assumed, that after repairing, overhauling or replacement, the resource will achieve
18
the actual capacity level, as defined in constraint set (14). Here,
19
time period depending on disruption scenario and is taken tentatively. For constraint
21 22 23 24 25
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) until the full capacity is
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20
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7
sets (12), (13) and (14),
is any specific
is depicted from three different conditions as mentioned
there. Here in these proposed models, at first we have considered that the disruption could happen for the renewable resources only. Later on, some experimental analysis has also been executed under both renewable and non-renewable resource disruptions. Again for non-renewable resources, the amount of resource consumed by the activities
, whose finish time is less than or equal to the disruption start time, is
26
measured by equation (10). During the recovery schedule from the disruption start
27
time to project completion, the earlier consumed non-renewable resources are
28
subtracted from the available resources. The renewable and non-renewable resource
29
constraints for different time periods are shown below. (10) (11)
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Page 12 of 38
(12) (13) (14)
binary constraints:
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1
(15)
2
3.3.2 Preempt-Resume Condition
4
In the case of the preempt-repeat option, the partial completed portions of affected activities
5
are considered as wastage. This wastage has an adverse effect upon project completion, as it
6
consumes both time and resources. Unlike of the preempt-repeat condition, following the
7
preempt-resume condition will certainly save those wastages, as in the recovery schedule the
8
affected activities start from that portion of work it left from before the disruption. As a
9
result, this option is more lucrative, in terms of completion time reduction and cost savings,
10
and is more appropriate than the preempt-repeat option (Lambrechts et al., 2007). The
11
mathematical model for the preempt-resume option is presented below.
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3.3.2.1 Objective function
14
The objective function for the preempt-resume option is similar to that of the preempt-repeat
15
option, as shown in equation (8).
17 18 19 20 21 22
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3.3.2.2 Constraints
1. Start time limitation: In the recovery schedule, as shown in constraint sets (16), the affected or incomplete jobs
whose start time is greater than or equal to the
disruption start time, must be started within the recovery window following their best suitable mode and have unique start times. Again for constraint (17), the same condition should be followed by the activities l, whose start time is less than the
23
disruption start time but that have a finish time that is greater than the disruption start
24
time: (16) (17)
11
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1
2. Precedence relationship: For any predecessor activity
, other than the affected
or incomplete activities should sum up its usual duration time to allow the execution
3
of its successor j and should satisfy constraint sets (18). Again constraint sets (19)
4
define that the successor j can only be started when its predecessor (which has a start
5
time less than the disruption start time, but have a finish time that is higher than that)
6
completes all its incomplete portions among all the available execution modes:
ip t
2
(19)
7
d
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(18)
3. Resource capacity: All the resources utilised by the activities must not exceed the
9
resource’s capacity limitation. Here for the renewable resources, the resource capacity
10
constraints are the same as the constraints (12) to (14). But for the non-renewable
12 13 14 15 16
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11
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8
resources, it has three different parts. For the activities
, which have finish time
lower than or equal to the disruption start time, the amount of resources consumed is measured first by following equation (20). Once again, the amount consumed for the affected activities l, which have its start time less than the disruption start time but where its finish time is higher than that, is also measured separately, as shown in equation (21). Then for the remaining activities
, whose start time is greater than the
17
disruption start time, the available non-renewable resources are measured by
18
subtracting all the upper mentioned resources by following constraint (22). (20) (21)
12
Page 14 of 38
(22)
4. Incomplete task: The incomplete portion of any task l, which has a start time that is
2
less than the disruption start time, and a finish time that is greater than the disruption
3
start time, should consider the following constraint (23):
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1
(23)
4
6
8
3.4
Examples for the Recovery Planning Problem 3.4.1 Single Disruption Case
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(24)
us
cr
binary constraints:
5
In this section, we demonstrate the recovery planning for a single disruption of a project with
10
10 activities (in total 12 activities including the start and end dummy nodes) each having
11
three different modes, and each mode with two renewable resources (R1 and R2) with a
12
capacity of 9 and 4 units respectively. The network of the project is shown in figure 2,
13
whereas the modes along with their duration and usage combinations are given in table A.1 in
14
the appendix. As shown in the Gantt chart in Figure 3, the makespan of the project is 20.
15
Now suppose, at the end of period 4, disruption is experienced, due to which, five units of
16
resource R1 and 2 units of resource R2 are not available from time periods 4 to 6 (as of the
17
estimated repair time of two units) that consequently reduces the resource availability of R1
18
from 9 to 4 units and of R2 from 4 to 2 units. However, after the end of period 6 (from the
19
beginning of period 7), the resources will be back in service. As the resource usage for R2 is
20
very small, it does not have any significant impact upon makespan, as a result, we will only
21
discuss the scenarios for R1 hereafter. As the resource requirements during the repair time
22
(period 4 to 6) is higher than what is available, we cannot execute two activities (4 and 5) at
23
the same time as scheduled in figure 3. After rescheduling with the new resource limit, the
24
revised Gantt charts for the preempt-repeat and the preempt-resume condition are shown in
25
figures 4 and 5 respectively. For the preempt-repeat condition, as shown in figure 4, the
26
recovery that started from scratch wasted the work of 1 time unit for activity 4 and 1 time unit
27
of activity 5, that is a total of 2 units of work (hatched boxes in figure 3) and therefore the
28
overall makespan was increased by 1 time unit as compared to the nominal schedule (Figure
29
3). On the contrary, for the preempt-resume option, as shown in figure 5, the amount of work
Ac ce p
te
d
M
9
13
Page 15 of 38
completed for any affected activities (shown as hatched boxes in figure 5) until disruption
2
takes place is not lost and that provides a lower makespan of 20. In the later scenario, only
3
the incomplete portion of the work for any affected activities (shown as dotted boxes in
4
figure 5) should be completed within the recovery window. The synopsis of this example is
5
that the availability of multiple modes means that the makespan only slightly increases, in
6
spite of resource disruptions. 2 7
3
6
8
9
4
Maximum Resource Line for R 1
5
8
[2]
6
8 [1]
4
2 [1]
0
D_s
4 3
D_e
5
Ac ce p
Resource use (unit)
d
2
9
9 [1] 15
D_s
18
20
18
20
Maximum Resource Line for R 2
11 [1] 10 [2]
D_e
5
10 Time
15
Fig 3: Gantt chart of optimum nominal schedule under initial conditions (here [a] represent selected activity mode ‘a’)
Resource available after disruption
Resource use (unit)
14 15
6 [3]
10 Time
Maximum Resource Line
9
13
7 [1]
2 [ 1] 0
11 12
4 [2]
te
Resource use (unit)
9
10
an
Fig 2: Project Network for the considered MM-RCPSP example under recovery planning
M
7 8
12
us
10
cr
11 5
1
ip t
1
8
4 [2]
6 4
6 [1]
2 [1 ]
2
8 [1]
7 [1]
9 [1]
5 [2] 0
D_s
5
D_e
10 Time
15
20
21
Fig 4: Gantt chart of optimum Recovery schedule under Preempt-repeat condition (here [a] represent selected activity mode ‘a’)
14
Page 16 of 38
Resource available after disruption
Maximum Resource Line
4
2 [1]
4_ i [2]
6 [1]
2
8 [1]
7 [1]
9 [1]
5_R [2] 0
D_s
5
D_e
10 Time
15
20
Fig 5: Gantt chart of optimum Recovery schedule under Preempt-resume condition (R: Remaining activity)
3.4.2 Multiple Disruptions Case
cr
3
4_R [2]
6
1 2
5_ i [2]
8
ip t
Resource use (unit)
9
For better understanding, the earlier example is further considered for multiple or a series of
5
disruptions. For the preempt-repeat option, we now assume another disruption takes place,
6
with only five units of resources of R1 being unavailable, from time period 10 to 12, which
7
affects activities 4 and 5, as shown in figure 6(a). Due to the second consecutive disruption,
8
the hatched portions of affected activities (3 units for activity 5 and 2 units for activity 4) are
9
being wasted. The revised schedule for recovering from the 2nd disruption is shown in figure
10
6(b), where the makespan is increased to 27 units from 23. Now suppose a third disruption
11
takes place from time period 14 to 16 that also affects activities 4 and 5. The revised schedule
12
is shown in figure 6(c) with an increased makespan of 31. During all the recovery planning,
13
for the activities which finished before the disruption take place, they naturally have the same
14
start and end times as before. For the readers, with the same disruption scenarios, the Gantt
15
charts for the preempt-resume option are presented in Appendix-A. For that preempt-resume
16
option, under the same disruption scenario results lower makespan values. In spite of
17
disregarding the earlier completed portions (2 units for activity 5 and 2 units for activity 4 as
18
shown in hatched boxes), preempt-resume option schedules the incomplete portion of each
19
work within its recovery window. As a consequence, the makespan of the project after 3rd
20
disruption is only 25 instead of 31 for preempt-repeat condition. Therefore, although for the
21
preempt-resume case, computational complexity may be higher than the repeat case, it saves
22
time.
Ac ce p
te
d
M
an
us
4
Resource available after disruption
Maximum Resource Line
Resourceuse(unit)
9
23 24
8
4 [2]
6 4
2 [1 ]
2
6 [1] 0
st
1
5
D_s
5 [2] 1s t D_ e
10 Time
7 [1]
15
8 [1] 9 [1] 20
21
23
Fig 6 (a): Optimum recovery schedule after first disruption 15
Page 17 of 38
Resource available after disruption
Maximum Resource Line
8 4 [2]
6 4
2 [1 ]
2
6 [3] 0
1
5
1s t D_ s
8 [1]
7 [1]
9 [1]
5 [2] 1s t D_ e
2nd D_e D_s
10 nd
2
15 Time
20
25
27
Fig 6 (b): Optimum recovery schedule after 2nd disruption Resource available after disruption
Maximum Resource Line
cr
2
8 4 [2]
4
2 [1 ]
3
8 [1]
2
6 [3] 0
us
Resourceuse(unit)
9
6
5
1s t D_s
5 [2] st
1
D_e
nd
10 2 2 nd D_s
D_ e
rd
3r d
15 3 D_s
D_e
20
25
7 [1]
9 [1]
30
31
an
Time
ip t
Resourceuse(unit)
9
Fig 6 (c): Optimum recovery schedule after 3rd disruption
4
6
4. SOLUTION APPROACH
M
5
In this section, the model for the nominal schedule is solved. A flowchart is shown to
8
describe the solution procedures used in solving disruption recovery models under both single
9
and a series of disruptions. Both the recovery preempt-resume and preempt-repeat strategies
te
d
7
10
are considered for generating the reschedules.
11 12 13
4.1
14
programming model that can be solved using an exact optimization algorithm, such as the
15
Branch and Bound algorithm (B&B) algorithm if the problem size is not big (Zhu et al.,
16
2006). The nominal model was solved by using the commercial optimization software
17
LINGO v10.0, and was executed on an Intel core i7 processor with 16 GB RAM and 3.40
18
GHz CPU.
19 20 21
4.2
22
series of disruptions on a real time basis. The recovery models further split into two
23
scenarios, known as preempt-repeat and preempt-resume. At the beginning, all the relevant
24
project parameters and disruption scenarios, such as the number of activities including
Ac ce p
Solution approach for nominal schedule The mathematical model presented in section 3.2, for MM-RCPSP, is an integer linear
Solution approach for disruption models This section proposes a solution approach to generate a recovery plan for both single and a
16
Page 18 of 38
1
dummy ones (I+1), duration of each activity (i), disruption start time (
2
time (
3
machine breakdown), disruption severity (i.e., number of machines breakdown), resource
4
requirement by each activity, precedence relations,
5
provided. If there is only one disruption (single disruption) then the model should follow the
6
initial start and end times from the nominal schedule provided.
), number of resources (R), capacity of resources, disruption condition (e.g.,
ip t
values, and time periods should be
start
Stop
Input the project parameters
Set Z* = min{z, ∞ }
Yes
No Does a disruption occur?
M
Yes
Is it the first or a single disruption ? No
Solve the integer model by using optimization software
Input start and finish time from baseline schedule
Update disruption volume and time
d
Determine incomplete tasks
te 10
Update the start and finish times
Input start and finish time from 1st recovery schedule
Ac ce p
9
Is there any other disruption?
an
Yes
us
No
cr
7
8
), disruption end
If the recovery strategy is pre-empt repeat, then formulate the model following section 3.3.1
If the recovery strategy is pre-empt resume, then formulate the model following section 3.3.2
Fig 7: Flowchart of proposed solution methodology for single or a series of disruptions
11
But if after finalizing the recovery plan, another disruption occurs (in the case of a series of
12
disruptions) within the recovery time window, the recovery plan needs to be revised again.
13
This may happen repeatedly, which both affects the project makespan and needs to be
14
rescheduled. The revised recovery plan should be within the disruption start time and project
15
completion time. This scenario depicts that every time a disruption occurs, the optimization
16
model developed earlier remains the same but with some modifications of the activity start
17
times and project makespan. These changes are emulated in the objective function and
18
constraints for re-optimization. That means, to re-optimize the recovery plan the algorithm
19
must run, every time a disruption takes place. If there is no disruption, the system should
20
follow the nominal schedule, as shown in figure 7. For each recovery strategy, the model will 17
Page 19 of 38
1
follow the models presented in section 3.3.1 and 3.3.2 for the repeat and resume conditions
2
respectively. The recovery models were also solved by using LINGO with the same computer
3
configuration and a similar B&B algorithm.
4
5. EXPERIMENTAL RESULTS AND ANALYSIS
ip t
5
In this section, we have first analysed the results of single and multiple disruptions cases for
7
MM-RCPSP. To analyse the proposed approaches, multi-mode benchmark instances from
8
different activity numbers ranging from lower to higher activities (10 to 30 activities)
9
available in Project Scheduling Library (PSLIB) were randomly chosen. For choosing those
10
instances it was provided that those instances already have feasible optimum results
11
according to PSLIB. After then, those instances were introduced with randomly generated
12
disruption start times and duration of repair to them. In the later portion of this section, we
13
have explained the solutions and computational effort required with respect to the problem
14
size and complexity.
15
5.1 Problem classification
16
We have chosen benchmark test instances from the popular test library PSPLIB-Project
17
Scheduling Problem Library. The network construction procedure and hardness for the MM-
18
RCPSP instances largely depends on constraints on the network topology (known as network
19
complexity, NC), a resource factor (Merkle et al.) that reflects the density of the coefficient
20
matrix or the average number of resource required, where RF = 1 means all resources are
21
required by the job, and resource strength (RS) which measures the availability of resources
22
(Kolisch et al., 1995). For the 20-activity instances from PSLIB, for both the preempt-repeat
23
and resume conditions, it has been observed that the computational time increases with the
24
increase of the resource factor of both of the renewable (RFR) and non-renewable resources
25
(RFN) from 0.5 to 1. Here the solution times are far more susceptible to RFN than RFR.
26
Moreover, with increasing resource strength of non-renewable resources (RSN) from 0.2 to
27
1.0 the average solution time decreases. But for the renewable resources (RSR), the solution
28
time slightly increases. These scenarios were further investigated by Kolisch et al. (1995). To
29
mention, all those randomly chosen instances were chosen according to their RF and RS
30
values and were divided into four groups as shown in table 1. Considering the entire above
31
mentioned instance characteristics, it is assumed that the instance hardness increases from
32
type 1 to 4.
Ac ce p
te
d
M
an
us
cr
6
33 18
Page 20 of 38
Table 1: Problem type classification according to the multi-mode instance characteristics Problem type
RFN
RSN
RFR
RSR
Type 1 Type 2 Type 3 Type 4
0.5 0.5 1.0 1.0
0.2-0.5 0.7-1.0 0.7-1.0 0.2-0.5
0.5 0.5 1.0 1.0
0.2-0.5 0.7-1.0 0.2-1.0 0.2-1.0
Instance sets (Problems) 20-activity 30-activity 10-activity 9,10 2,10 9,10 19,20,27,28 19,20,27,28 19,20,27,28 53-56, 61-64 39,40,47,48,55 53-56, 61-64 37-40, 45-48 37,38,45,46,62 37-40, 45-48
ip t
1
2
5.2 Disruption scenarios and recovery setup
4
The proposed recovery models were tested on 640 twenty-activity benchmark problems. Each
5
instance contains 22 activities (20 plus 2 dummy for start and end) each having three modes,
6
two renewable and two non-renewable resources. As our proposed recovery strategy is
7
responsive in nature, it is assumed that the recovery will start immediately after a disruption
8
is experienced and will continue until the end of the recovery window. Here we consider an
9
arbitrary random disruption start and end time, from period end of period 10 to end of period
10
12 (repair time = 2) for all the selected test instances under a single disruption. This scenario
11
is employed for both the preempt-repeat and preempt-resume conditions. Again for multiple
12
or a series of disruptions, we used three consecutive and independent disruptions, such that
13
the most recent disruption affects the recovery schedule of its immediate predecessor
14
disruption. In this case, we arbitrarily use the following disruption scenarios for all test
15
problems: first disruption from period 5 to 7, second one from 10 to 12 and third one from 14
16
to 16. Moreover, for our experiments, for both recovery conditions, we selected randomly
17
two disruptions on the first resource and five more on the second resource. As the non-
18
renewable resources are available throughout the project tenure, disruption for the non-
19
renewable resource is insignificant and therefore is not considered here. As the primary focus
20
is to reduce project completion time, the instability weight of the dummy end activity is
21
essential for meeting the project due date, which is usually more important than starting each
22
activity at the planned starting time. For this reason, we considered higher instability weights,
23
or penalty values, for the dummy end activity, in comparison to the penalty values of the
24
other
Ac ce p
te
d
M
an
us
cr
3
activities.
The
penalty
coefficients
we
used
were:
25 26 27
5.3 Results for single disruption
19
Page 21 of 38
Before experimenting with single and multiple disruptions, all the randomly chosen instances
2
of activities 10, 20 and 30 were solved by using LINGO optimization solver. As mentioned
3
earlier on section 3.2, the formulations of Talbot (1982) is considered here for solving the
4
nominal schedule, all the results found was optimum and same as in the PSLIB (shown in
5
table 2). Later on, considering the above mentioned single disruption scenarios, the results
6
for both the preempt-repeat and preempt-resume conditions are presented in table 2. From
7
that table, it can be seen that the average deviation of makespan (average recovery makespan
8
– average baseline or initial makespan) is always higher for the preempt-repeat condition than
9
the preempt-resume condition. Although the deviation does not depends on the problem
10
hardness, rather it depends on the amount of work wasted due to the repetition of affected
11
activities during the preempt-repeat condition, which engenders higher makespan than the
12
resume case. Under both preempt-repeat and preempt-resume options, the CPU time or
13
solution time for all randomly chosen 10, 20 and 30 activity instances were within reasonable
14
span of time comparing with other direct approaches. Meanwhile for justifying the proposed
15
disruption recovery algorithm, further modifications of those disruption scenarios have also
16
been executed. In this case, both renewable and non-renewable resources were considered
17
under disruption scenario. For doing this, 5 units of each non-renewable resource and 2 and 5
18
units for renewable resource 1 and 2 respectively were considered as disrupted or unavailable
19
during the repair period. Later on, some random instances were picked up under all three
20
types of activity sets ranging from 10 to 30 and the results are shown in table 3. Here in table
21
3,
22
disruptions and
23
only. As anticipated, recovery makespans under both resource disruptions always show
24
greater than or equal values than single renewable resource disruption only.
26
cr
us
an
M
d
te
represents recovery makespan under both renewable and non-renewable resource
Ac ce p
25
ip t
1
represents recovery makespan under renewable resource disruptions
Table 2: Results for group-wise benchmark problems under single disruption Project Type
10 activity
20 activity
Problem type
Initial makespan (average)
T/1 T/2 T/3 T/4 T/1 T/2 T/3 T/4
17 14.2 18.8 26 20.6 22.8 25.6 29.4
Preempt-repeat Recovery Average CPU time Makespan deviation (sec) (average)
19.75 17.8 22 28 24 25.8 28.4 32.6
2.75 3.6 3.2 2.0 3.4 3.0 2.8 3.2
0.425 0.5 0.8 0.9 0.7 0.7 0.8 0.9
Preempt-resume CPU Recovery Average time Makespan deviation (sec) (average)
19.5 15.4 21.2 27 22.4 23 27.4 30.2
2.5 1.2 2.4 1.0 1.8 0.2 1.8 0.8
0.425 0.6 0.6 0.9 0.6 0.8 2.5 1 20
Page 22 of 38
30 activity
T/1 T/2 T/3 T/4
31.4 34 29.75 35.5
34.8 35.4 32 37
3.4 1.4 2.25 1.5
1.25 1.6 3.0 46.0
32.4 34 29.75 35.5
1 0 0 0
1.4 2.5 3.0 6.2
1
ip t
2 3
cr
4 5
20activity
7 8 9
22 18 17 19 21 22 27 22 26 23 33 35 29 30 38
an
16 12 17 19 17 18 23 17 26 22 31 33 29 29 36
Ac ce p
30activity
Preempt-repeat
Initial Makespan
M
10activity
Randomly Chosen Instance sets J102_6 J1020_1 J1048_1 J1055_1 J1062_1 J209_8 J2010_5 J2027_1 J2048_9 J2064_10 J309_1 J3010_5 J3027_10 J3055_1 J3064_10
d
Problem Set
us
Table 3: Results for both Renewable and Non-renewable resource disruptions
te
6
23 20 18 20 23 25 29 24 28 23 34 36 30 32 39
Preempt-resume
22 15 17 19 18 22 26 17 26 22 32 33 29 29 36
22 15 17 20 19 23 27 19 26 22 32 33 29 30 37
5.4 Results for a Series of disruptions
10
For better understanding, we have calculated the percentage gap due to increasing disruptions
11
for both the preempt-repeat and preempt-resume conditions. For all three consecutive/series
12
of disruptions, the percentage gaps were measured with respect to the baseline makespan
13
using the equation
14
table 4 for all randomly selected instances of activities 10, 20 and 30. It can be seen from that
15
table that the percentage gap increases with increasing number of disruptions for the preempt-
16
repeat case. The rationale behind that is with increasing disruption numbers, the amounts of
17
wasted duration due to repetition are also increased, which leads to higher and higher
. The detailed results are provided in
21
Page 23 of 38
1
makespan. On the contrary, that is not always true for the preempt-resume case, as shown for
2
the T/2 instances of activity 30 in table 4. Moreover, the percentage gaps may vary due to the
3
conditions and precedence constraints of the affected activities at the time of disruption,
4
although it had the same disruption period.
5
ip t
6 7
cr
8 9
Project
Problem
nd
rd
% gap under preempt-resume
1st disr.
2nd disr.
3rd disr.
10.29
27.94
42.64
76.05
14.08
22.53
26.76
69.14
17.02
28.72
40.42
29.80
3.84
5.92
10.57
20.38 27.19
33.98 44.73
4.85 0
10.68 2.63
18.45 3.51
24.24
0
0
1.01
2 disr.
3 disr.
T/1
19.11
42.64
66.17
10
T/2
29.57
47.88
activity
T/3
20.21
50.00
T/4
4.80
16.34
T/1
M
an
1 disr.
T/2
activity
T/3
8.08
T/4
32
49.32
66.68
21.32
10.68
17.32
8.28 7.06
2.54 0
3.18 0
7.00 0
d
20
7.77 10.53
40.40
17.20 16.47
29.94 25.88
T/3
5.88
13.45
21.85
0
0
0.84
T/4
8.45
15.50
23.94
0
2.81
2.81
T/1 T/2
Ac ce p
30 activity
12
st
type
Type
11
% gap under preempt-repeat
us
Table 4: Results for group-wise benchmark problems under series of disruptions
te
10
13 14
5.5
15
From the literature, there is no benchmark for MM-RCPSPs with disruptions that can be used
16
for validating our proposed algorithms. For this reason, we have derived the upper and lower
17
bounds of solutions for both single and a series of disruptions and compared that with our
18
solutions. Here we used the baseline makespan time as a lower bound for all the recovery
19
stages. Then we derived upper bounds for both the preempt-repeat and preempt-resume
20
conditions. We also measured tighter upper bounds for the series of disruptions case. Let us
21
illustrate the upper bound, for the preempt-repeat condition, with an example as shown in
Lower and Upper Bounds under
Disruptions
22
Page 24 of 38
figure 8. Here, suppose activities 6 and 7 are being affected due to a resource disruption at
2
period 16. As for the preempt-repeat condition, the affected activities should start from
3
scratch in the recovery schedule, it is required to identify the maximum waste duration of the
4
affected jobs, which is five time units for activity 6 (greater than the waste duration for
5
activity 7). The upper bound can then be calculated as (base makespan + maximum waste
6
duration + disruption duration), because shifting all the affected and incomplete jobs to the
7
right by an amount equal to (maximum waste duration + disruption duration) will provide a
8
feasible schedule that may be reduced by rescheduling the remaining (affected and
9
incomplete) jobs. Maximum duration wastage for repetition
us
cr
ip t
1
Maximum Resource Line
8 2 6
4 4
2 0
7
5 10
5
15 16
18 20
Dis_start Dis_end
10
22 Time
Fig 8: Upper bound calculation for Single disruption preempt-repeat condition
d
11
8
an
3
6
M
Resource use (unit)
10
Lower bound for all conditions,
13
Upper bound for single disruption under the preempt-repeat condition,
15 16 17
Ac ce p
14
te
12
Upper bound for a single disruption under the preempt-resume condition,
Upper bound for a series of disruptions under the preempt-repeat condition,
18 19
Upper bound for a series of disruptions under the preempt-resume condition,
20
23
Page 25 of 38
Tighter upper bounds (
, here q represents number of disruptions and repeat
2
represent preempt-repeat condition) for a series of disruptions under the preempt-repeat
3
condition can be derived by repeating the calculations after each disruption as follows.
4
After first disruption (q=1),
ip t
1
5
After second disruption (q=2),
cr
6 7
After third disruption (q=3),
us
8
an
9 10
Tighter upper bounds for a series of disruption under the preempt-resume condition are as
12
follows.
13
After first disruption (q=1),
14
After second disruption (q=2),
15
After third disruption (q=3),
16
Here,
17
disruption respectively. The average recovery makespan, along with their bounds for selected
18
instances from each group and activity numbers, are summarized in table 5. From that table,
19
it can be seen that for both single and a series of disruptions, the recovery makespan is always
20
within the above mentioned upper and lower bounds.
21
makespan after the final (3rd) disruption has been reported. For all cases, the deviations
22
between the obtained makespan and their upper bounds are always higher for the preempt-
23
repeat condition than the same for the preempt-resume condition.
24 25
Table 5. Comparison of Make span time for group-wise benchmark problems (LB – lower bound; UB – upper bound; MS – makespan; TUB – tighter upper bound)
te
d
M
11
Ac ce p
represent the recovery makespan after the first, second and third
For a series of disruption, the
Project
Disruption
Problem
Preempt-repeat condition
Preempt-resume condition
Type
scenario
Type
LB
MS
UB
TUB
LB
MS
UB
TUB
10 activity
Single
T/1
17
19.75
21.5
-
17
19.5
20
24
Page 26 of 38
14.2
17.8
20.8
-
14.2
15.4
17.2
-
T/3
18.8
22
26
-
18.8
21.2
21.8
-
T/4
26
28
34
-
26
27
29
-
Series of
T/1
17
28.25
29
28.75
17
24.25
26
24.75
disruptions
T/2
14.2
25
28
27.8
14.2
18
23.2
20.4
(after final
T/3
18.8
31.8
33.2
31.8
18.8
26.4
27.8
26.8
disruption)
T/4 T/1
26 20.6
34.6 24
40 27
37.6 -
26 20.6
30.4 22.4
35 23.6
31.4 -
Single
T/2
22.8
25.8
29.6
-
22.8
23
25.8
-
disruption
T/3
25.6
28.4
33.2
-
25.6
27.4
28.6
-
T/4
29.4
32.6
36.2
-
29.4
30.2
32.4
-
Series of
T/1
20.6
27.6
33
disruptions
T/2
22.8
33
(after final
T/3
25.6
35.2
disruption)
T/4
29.4
41.67
T/1
31.4
34.8
Single
T/2
34
35.4
disruption
T/3
29.75
T/4
35.5
Series of
T/1
31.4
disruptions
T/2
(after final
T/3
disruption)
T/4
Ac ce p
1
cr
us
24.4
29.6
25.8
37
36.6
22.8
23.4
31.8
26.6
39.2
37.6
25.6
26.8
34.6
29.2
29.4
29.43
37
29.67
39.4
-
31.4
32.4
34.4
-
39.6
-
34
33.8
37
-
M
an
20.6
42.33
32
37
-
29.75
29.75
32.75
-
32.5
44.5
-
35.5
32.0
38.5
-
40.8
45
43.8
31.4
33.6
40.4
35.4
d
30 activity
31
44
34
42.8
47.6
45.8
34
33.8
43
36.8
29.75
36.25
43.25
40.75
29.75
30
38.75
32.25
35.5
44
48.5
47
35.5
36.5
44.5
39.5
te
20 activity
ip t
T/2
disruption
2
5.6
3
In this section, we examine the behaviour of the disruption recovery models with respect to
4
different parameter changes.
5
5.6.1
6
Behavioural Patterns
Impact of disrupted quantity and
repair times
7
In real life MM-RCPSPs, there may be copious reasons for disruptions and their
8
characteristics may also vary depending on the disruption types and severity. Among many,
9
resource disruption and amount of that disruption is an important parameter to judge recovery
10
models. To figure out the possible impacts of disrupted quantity and repair times, we have
11
investigated our recovery models with some of our randomly chosen instances. Here the
12
disrupted quantity is being quantified by the amounts of resources disrupted and the repair 25
Page 27 of 38
1
time which is the duration of resource unavailability due to recovering from that disruption or
2
maintenance. Irrespective of the available modes, for both the preempt-repeat and resume
3
conditions, we considered that all the renewable resources are disrupted up to 20%, 40%,
4
60% and 80% of their available levels, with different repair times, such as
5
and with different disruption start times (D_s) (ranging from 10% to 70% of the project due
6
date PDD). For different disruption start times, the average recovery makespan (for randomly
7
selected instances from different activity sets: MJ1020_1, MJ1038_1, MJ2019_8, MJ2010_5,
8
MJ3010_5 and MJ3064_10) for different resource breakdown percentages are presented in
9
figures 9(a) to 9(e). In those figures, MS_repeat_3 represents preempt-repeat makespan for
10
repair time equals 3 whereas MS_resume_3 represents the preempt-resume makespan for
11
repair time equals 3.
12
All the figures show that the recovery makespan increases with increasing percentage of
13
resource breakdown. But the rate of increment is not so straightforward. From all the figures
14
for different disruption start times, under both the preempt-repeat and resume conditions, it
15
has been observed that the rate of increment for recovery makespan for up to 40% resource
16
breakdown (with smaller repair time, say 3 and 5) is almost flat or tapered off. But after that,
17
the increment rate is linear. It depends on the activity usage rate in that particular time period.
18
For example, as of instance MJ2010_5, if the
19
affected activities are 10, 11, 12, 13, 15 and 16 for repair times of up to 9. But they need only
20
50% or less of the available resource to complete, as they have lower resource strength values
21
(RSR and RFR equals 0.5). Moreover, the flexibility of multiple modes allows the activities
22
to switch in the best possible ways (duration and resource usage combination) to overcome
23
disruption affects. As a result, once the makespan is increased to 28, it remains the same until
24
40% of resources are broken down. Then it increases linearly as more resources are disrupted.
25
From figures 9(a), 9(c) and 9(d), it can be seen that for different disruption start times with
26
different repair times, the average recovery makespan for the preempt-repeat condition
27
remains same if only 20% of each resource is disrupted, the only exception was where for
28
= 30% of PDD. This is because one of the instances has low resource strength
29
(RSR=0.5), and so there is some resource redundancy, which allowed the makespan to
30
remain same. Meanwhile, from figure 9(e), it shows that the recovery makespan increased
31
quite linearly for preempt-repeat case from
32
then decreased linearly from
= 50% of PDD (time period = 12), the only
Ac ce p
te
d
M
an
us
cr
ip t
3, 5, 7 and 9,
= 10% of PDD to
= 30% of PDD to
= 30% of PDD and
= 50% of PDD. After that, from
26
Page 28 of 38
1
= 50% of PDD to
= 70% of PDD the recovery makespan for that repeat case increased
2
again. This is another interesting aspect of the recovery schedules. As with increasing
3
values, the probable numbers of affected jobs are increased, and as a result the makespan at
4
the beginning also increases. But for larger
5
reduced significantly (i.e. most of the jobs are completed). As a result, the recovery schedule
6
for larger
7
makespan. Meanwhile for much higher D_S values, the recovery schedule could not get that
8
much of time to recover efficiently. As a result, for higher D_S values (i.e., D_S= 70% of
9
PDD), the recovery makespan increased again. On the contrary, for the preempt-resume
10
option, because of minimum work lost, the recovery makespans always show some linearly
11
increment pattern.
ip t
values, the activities left to complete are
12 13
Ac ce p
te
d
M
an
us
cr
values have lower number of jobs to deal with, which also reduces the
Fig 9a: Makespan for different breakdown percentage & repair times (
= 10% of PDD)
Fig 9b: Makespan for different breakdown percentage & repair times (
= 30% of PDD)
14 15
27
Page 29 of 38
ip t cr Fig 9c: Makespan for different breakdown percentage & repair times (
= 50% of PDD)
4
Fig 9d: Makespan for different breakdown percentage & repair times (
Ac ce p
3
te
d
M
an
2
us
1
= 70% of PDD)
5 6
Fig 9e: Recovery makespan for different disruption start times
7
28
Page 30 of 38
1 2
5.6.2
3
Activities in the projects can also be disrupted by many multifarious reasons. For example,
4
due to some unexpected customization of end products or some modifications of process
5
plan, an activity can be disrupted. So, if any of the activity’s duration or resource usage
6
departs from the planned values, then that activity is said to be disrupted. The deviations can
7
be of any form (negative or positive) depending upon the disruption situations. To interpret
8
the effects of the activity disruptions on the recovery schedule, we have further distinguished
9
them as activity duration variability and activity resource disruptions. The later one will be
10
analysed in the next section. In this section, we have investigated the impact of the variability
11
of activity durations. This was carried out by multiplying each activity’s duration time on
12
each mode by a multiplication factor, MF (MF= 1+beta distribution of random numbers with
13
parameters 2 and 5). This implies that the duration of each activity for our randomly selected
14
instances (3 of each problem type of varied activity number ranging from 10 to 30) was
15
increased, without giving any restrictions to the project due date, and was treated with the
16
disruption scenario mentioned in section 5.2. The results for both the recovery schedules are
17
shown in figure 10, from which it can be observed that the percentage increase of recovery
18
makespan (ms) time is always lower than the duration variability, independent of problem
19
hardness. Meanwhile, the percentage increase of the recovery makespan under the preempt-
20
resume condition is lower than the repeat condition for the harder problem types (T/3 and
21
T/4), which proves that the resume option, in comparison to the repeat option, is more
22
vulnerable for harder instances for recovering from disruption. For easier instances, the
23
duration variability has less impact on the repeat makespans than for the resume ones. This
24
supports the intrinsic characteristics of the instances, while the solution time also increases
25
with increasing duration variability and problem hardness.
Impact
of
activity
duration
Ac ce p
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M
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cr
ip t
variability
26 29
Page 31 of 38
1
Fig 10: Relationship among duration variability and % increase of recovery makespans
2 3
5.6.3
4
To analyse the possible impacts of activity resource disruptions, again we have considered
5
those randomly selected instances from our problem groups of 10, 20 and 30 activities. An
6
activity resource disruption occurs when for any particular time span, the corresponding
7
activity takes extra usage for its completion on all its available modes. Here in this paper,
8
we assumed that activity 3 used
9
taken from 1 to 5. Under the same disruption scenario as mentioned in section 5.2, the
10
average penalty values that resulted are summarized in table 6. From table 6, for both
11
recovery options, it is quite clear that the recovery penalty values for any particular problem
12
type increases almost linearly with the increasing resource usage rate. Despite the fact that
13
the activity resource disruptions have only a minor impact on the recovery makespan
14
(remains almost unchanged with increasing resource usage rate), it has a significant impact
15
on penalty values. As with increasing usage rate, the activity start time in the recovery
16
schedule may be shifted, and this causes higher penalty values. For example, under the
17
preempt-repeat condition for instance MJ2056_6 with
18
activities 5 and 6 were shifted from the initial start time by 3 and 4 units. But for
19
deviation between the activities start time was increased to 5 and 7 units respectively. As a
20
result, the penalty values also increased. But the excess slack time of the project did not allow
21
the makespan time to increase in the recovery schedule. In table 6,
of
activity
resource
ip t
disruptions
was
M
an
us
cr
more of renewable resource k, while the value of
d
, the recovery start time of the
te
Ac ce p
22
Impact
and
represents the recovery penalty values for the preempt-repeat and resume
23
conditions respectively.
24
Table 6. Recovery results for different activity-resource disruptions Excess Resource Usage
1
2
3
4
5
32.33
34
36.67
37
39
10
11
11.33
12
13.33
25 26
5.6.4
27
Same disruption scenario has been considered to understand the effects of length of delay on
28
our proposed recovery problem. Here the basic assumption was to delay the execution of
29
some activities for certain time periods, say to delay the activity 3 of each randomly
Impact of Length of Delay
30
Page 32 of 38
1
considered instance between 1 and 10 time periods without restricting the makepsan. A series
2
of experiments were carried out and the results are reported in table 7. As anticipated, the
3
delay of execution has a significant impact on recovery makespan and as the delay increases
4
the average makespan for all instances also increases almost linearly for both the preempt-
5
repeat and resume conditions. Here
6
makespan under the preempt-repeat and resume conditions respectively for all those
7
randomly selected instances of activities 10, 20 and 30. The repeat condition needs more
8
time, because it must repeat previously done work. As a consequence, compare to the
9
preempt-resume condition, the makespan of the preempt-repeat condition increases at a
10
higher rate with increasing length of delays. This analysis helps to understand the effects on
11
makespan if there are any delays for executing activity during the recovery process.
12
Table 7. Recovery results for different delays 2
3
4
26
28.5
31
an
1 25.25
5
6
7
8
32.25
33
36
37
24.25 25.25 25.75 26.25 27.25
28
28.25
29.5
M
Delay
us
cr
ip t
represent the average recovery
13
9
10
37.75 42.25 34
38
5.6.5
15
In real life projects, as of many other interruptions, it is possible that some new activities may
16
be added or removed during project execution, and that may change the precedence
17
relationships of some activities. In this regard, two more important scenarios, namely the
18
addition of new activities (with deducting the same number of existing activities to keep the
19
same total activity number) and a change of precedence relationships can be analysed. In this
20
research, we considered four different cases with randomly generated new activities along
21
with their resource usage on each of three mode, precedence relationships and mode
22
dependent activity durations. Taking different instances of MM-RCPSPs from each of these
23
problem types, it was observed that the recovery makespan for both the repeat and resume
24
condition increased quite significantly over the initial one. We also considered four different
25
randomly generated new precedence relations, instead of the existing ones, and measured the
26
recovery makespan. Similar conclusion can be drawn for this case, as with increasing number
27
of precedence modifications, the recovery makespan also increased. Note that the deviations
28
may be positive or negative depending upon the characteristics of new activity and
29
precedence relationships. So, it is very difficult (sometime impossible) to build up a profound
30
relationship about the effect of precedence modifications on recovery makespan.
Impact of other parameters
Ac ce p
te
d
14
31
Page 33 of 38
1 2
6.
CONCLUSION
The objective of this research was to develop a real time disruption recovery plan for multi
4
mode, multiple resource constrained project management problems. For two newly proposed
5
recovery options, preempt-repeat and preempt-resume, we have developed mathematical
6
models to deal with a single disruption, as well as a series of disruptions, on a real time basis.
7
Optimization software was employed to solve the mathematical model for generating new
8
revised schedules under changed or disrupted conditions. A detailed stepwise procedure has
9
also proposed to deal with a series of disruptions, where repeated solutions from the
10
mathematical model are required after each change takes place. As mentioned earlier, we
11
have made a pragmatic assumption that the disruption information is known only after its
12
occurrence and the disruption recovery algorithm is then run with the information related to
13
the disruption. Such a schedule can be recognized as a real-time schedule. For our
14
experimental study, we have considered sets of multi-mode 10, 20 and 30-activity static
15
benchmark instances form PSLIB and introduced random disruption scenarios with them.
16
The experimental results clearly demonstrate that the re-optimization process with these two
17
recovery options can reduce the recovery makespan, as compared to simple right shifting of
18
affected activities (or upper bound solution), and the level of improvement depends on the
19
duration of the affected activities, their resource requirements, their relationships with
20
forwarding activities, disruption duration (repair time) and the amount of disrupted resources.
21
That means the proposed approaches add certain value to the project management body of
22
knowledge.
23
In this paper, we have also considered a few other realistic variations that can happen during
24
project execution, such as: change in activity duration, resource requirement, precedence
25
relationships, repair time, and addition of new activities which may have some significant
26
impact upon the recovery schedules for MM-RCPSP. These variations can be considered as
27
disruption events and the revised schedule can be generated using the disruption recovery
28
approaches proposed in this paper. The impact of such variations on the revised makespan
29
was analysed using sensitivity analysis.
30
The proposed approaches are easy to implement and therefore can be employed by
31
practitioners in different ways to generate revised schedule under disruption scenarios on a
32
real time basis. Moreover, significant financial and time loss can be lessened by applying
Ac ce p
te
d
M
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cr
ip t
3
32
Page 34 of 38
these approaches if any disruption is experienced. Here we have assumed static behaviours of
2
activity duration, but in real life that may not happen every time. Involving more developed
3
approaches for large projects under disruptions, and solving them by using meta-heuristic
4
algorithms, will be our future research.
5
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ip t
Table A.1: Task duration and Resource Requirement for the Motivating Example (R1 and R2 are the two renewable resources) Duration (Months) Mode 2
Mode 3
0 3 1 3 4 2 3 4 2 1 6 0
0 9 1 5 6 4 6 10 7 1 9 0
0 10 5 8 10 6 8 10 10 9 10 0
d
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12
Resource available after disruption
Maximum Resource Line
Resource use (unit)
9
19
8
4_i [ 2]
6 4
6 [3]
2 [1]
2
st
1 D_e
5
1s t D_s
10 Time
Resource available after disruption
Resource use (unit)
9 [1]
5_R [2]
5_i [ 2]
0
4 _R [2 ]
8 [1 ]
7 [1 ]
15
20
22
Maximum Resource Line
9 8 4_i [ 2]
6 4
6 [3]
2 [1]
2
5_R [2]
5_i [ 2]
0
20
Renewable Resource Requirements (Units) Mode 1 Mode 2 Mode 3 R1 R2 R1 R2 R1 R2 0 0 0 0 0 0 6 0 5 0 0 6 0 4 7 0 0 4 10 0 7 0 6 0 0 9 2 0 0 5 2 0 0 8 2 0 5 0 0 7 5 0 6 0 3 0 4 0 2 0 1 0 1 0 4 0 0 2 4 0 0 2 0 1 0 1 0 0 0 0 0 0
M
Mode 1
te
Activities
18
cr
APPENDIX
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12 13 14 15 16 17
39. Węglarz, J., Józefowska, J., Mika, M., & Waligóra, G. (2011). Project scheduling with finite or infinite number of activity processing modes – A survey. European Journal of Operational Research, 208(3), 177-205. 40. Zapata, J. C., Hodge, B. M., & Reklaitis, G. V. (2008). The multimode resource constrained multiproject scheduling problem: Alternative formulations. AIChE Journal, 54(8), 21012119. 41. Zhu, G., Bard, J. F., & Yu, G. (2005). Disruption management for resource-constrained project scheduling. Journal of the Operational Research Society, 56(4), 365-381. 42. Zhu, G., Bard, J. F., & Yu, G. (2006). A Branch-and-Cut Procedure for the Multimode Resource-Constrained Project-Scheduling Problem. INFORMS Journal on Computing, 18(3), 377-390.
an
1 2 3 4 5 6 7 8 9 10 11
5
1s t D_s
st
1 D_e
7 _i [1 ] nd 10 2 D_e
2 nd D_s
4_R [2]
8 [1]
15
7_R [1]
20
9 [1] 23
Time
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Resource available after disruption
8 4_i [2]
6 4 2
5_ R [2]
5_i [2]
5
1s t D_s
1s t D_e
4_Ri [2]
7 _i [1 ]
6 [3]
2 [1]
0
7_ R [1]
8 [1]
4_ Rr [2]
9 [1 ]
nd rd 10 2 D_e 3 rd D_s15 3 D_e 20 Time
2 nd D_s
25
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Fig A1: Gantt Charts of optimum recovery schedule under preempt-resume condition for series of disruptions (R: Remaining activity; Ri: Initial work of remaining activity; Rr: Last portion of remaining activity)
Ac ce p
1 2 3 4
Maximum Resource Line
ip t
Resource use (unit)
9
36
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