Robust optimization for the resource-constrained multi-project...

Scientia Iranica E (2020) 27(1), 361{376

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringhttp://scientiairanica.sharif.edu

Robust optimization for the resource-constrainedmulti-project scheduling problem with uncertainactivity durations

E. Nabipoor Afruzi�, A. Aghaie, and A.A. Naja�

Faculty of Industrial Engineering, K.N. Toosi University of Technology, Tehran 1999143344, Iran.

Received 13 June 2017; received in revised form 12 March 2018; accepted 6 August 2018

KEYWORDSMulti-projectscheduling problem;Resource sharingpolicy;Robust optimization;Resource constraint;Uncertain activityduration.

Abstract. This paper studies the multi-project scheduling problem that involves multipleprojects with di�erent importance weights, prede�ned assigned due dates, activities withuncertain durations, and renewable constrained resources. The resource sharing policyis applied to share resources among projects. Due to the environmental rapid changesand, also, the uniqueness of projects, the probability distribution function of uncertaindurations cannot be estimated with con�dence. Besides, the multi-project schedulingproblem with its large-scale investment dictates a conservative approach to deal withthe existing uncertainty. Therefore, the Robust Resource-Constrained Multi-ProjectScheduling Problem (RRCMPSP) is studied in this paper, while the maximum totalweighted tardiness of the projects should be minimized. A scenario-relaxation algorithmis implemented, which results in optimal solutions for the RRCMPSP. The aim is to �ndan optimal structure that contains all of the projects such that it transfers the resourcesbetween the activities based on the resource sharing policy, while the maximum weighteddi�erences between the projects' �nish times and their assigned due dates will be minimum.

© 2020 Sharif University of Technology. All rights reserved.

1. Introduction

The Resource-Constrained Project Scheduling Problem(RCPSP) aims to minimize the project makespan whileconsidering precedence and resource constraints [1].This problem is one of the most well-known problemsto which researchers have devoted considerable e�ortsover the past decade.

The RCPSP is applicable in many areas suchas make to order industries, construction, software

*. Corresponding author. Tel.: +98 21 84063363;Fax:+98 21 88674858E-mail addresses: [email protected] (E.Nabipoor Afruzi); [email protected] (A. Aghaie);aanaja�@kntu.ac.ir (A.A. Naja�)

doi: 10.24200/sci.2018.20801

development, etc. In modern enterprises in whicha large number of projects are set up to achievethe product innovation, the key resource is mostlymanpower, which belongs to renewable resources. Incontrast to the importance of renewable resources andits role in project management success, the renewableresources have not attained su�cient consideration inthe literature [2]. As a brief de�nition, the renewableresources are those resources such as manpower, ma-chines, etc. that are constrained and that there is acertain available capacity of this kind of resources ineach time period. By �nishing one activity, its requiredrenewable resources can be released and applied toother activities. In this paper, the project schedulingproblem is investigated under the renewable resource-constraint condition.

The Resource-Constrained Multi-Project Sched-

362 E. Nabipoor Afruzi et al./Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 361{376

uling Problem (RCMPSP) as the extension of theRCPSP is considered as the simultaneous scheduling oftwo or more projects, which demand the same scarcesources [3]. Multi-project management is a major wayof doing business both in manufacturing and services,and being a large-scale complex problem constitutesan important research area [4]. According to the studyby Payne [5], up to 90% of all projects in the world areexecuted in a multi-project management environment.It is notable that the management of multipleprojects presents challenges that are fundamentallydi�erent from single project management [6]. Thus,managing the multi-project problem is not simply anaggregate of single project e�orts. In this paper, themulti-project management problem is investigated.

During the project execution in an indeterminateenvironment, the projects are subject to considerableuncertainty. In other words, due to unavailable re-sources, delays in the delivery of materials, absent em-ployees, bad weather conditions, accidents, and manyother uncontrollable factors, some project activitiesmay last longer than expected, threatening the opera-tional viability of the planned schedule [7]. Therefore,the obtained results of the project scheduling modelwith deterministic parameters are no longer valid. Inother words, when the project parameters take realizedvalues, the usability of any result of the deterministicmodels is under question. Therefore, it is conceivablethat as the data takes values di�erent from the nominalones, several constraints may be violated and theoptimal solution found using the nominal data may nolonger be optimal or even feasible [8]. In this paper,the uncertainty of the activities' duration is understudy.

There are several approaches to schedulingprojects under uncertainty. In order to select anappropriate approach for dealing with uncertainty inproblems, �rst of all, we should investigate the na-ture and characteristic of the studied problem. Thefundamental approaches to scheduling projects underuncertainty are reactive scheduling, stochastic schedul-ing, scheduling under fuzziness, proactive (robust)scheduling, and sensitivity analysis [9].

Considering the uniqueness of each project inthe real world, it is not uncommon that its activ-ities are seldom or even never have been executedbefore. Therefore, these indeterminacies cannot betreated as fuzziness, probability, roughness, ambiguityor entropy. Instead, uncertainty theory can be auseful tool [1]. Robust Optimization is an appropriateapproach that is totally compatible with the nature ofthe project scheduling problem and is applied in thispaper. Robust optimization belongs to an importantmethodology for dealing with optimization problemswith data uncertainty. In this type of the method, adeterministic data set is de�ned within the uncertain

space, and the best solution, which is feasible forany realization of the data uncertainty in the givenset, is computed through the solution of the robustcounterpart optimization problem [10].

The major advantage of robust optimization com-pared to stochastic programming is that no assump-tions regarding the underlying probability distributionof the uncertain data are required [11]. It is alsotrue when comparing the robust optimization approachwith the fuzzy approach because there is no need forRO to de�ne the membership function for the uncertainparameter.

On the other hand, in this paper, the multi-project scheduling problem is investigated that requirestime, cost, resources, etc. in a large-scale quantity.Therefore, it seems that a conservative approach isessential that can immunize the project schedulingproblem against data uncertainty. It is exactly thecharacteristics of the robust optimization approachthat is applied for dealing with uncertainty in thispaper.

In this paper, the robust optimization approach isapplied to the multi-project scheduling problem underresource constraint and uncertain activities' durationto cover some shortcomings in the existing multi-project models. The problem is represented in a two-stage model in which the objective function is tominimize the maximum total weighted tardiness of theprojects.

The structure of the paper is as follows: Sec-tion 2 describes the related literature review. Thede�nitions of the problem are presented in detail inSection 3. The proposed mathematical model andthe two-stage approach are explained in detail inSection 4. Section 5 describes one simple numericalexample with its results to clarify the proposed model.Computational experiments are explained in Section6. Finally, the conclusion and further research arepresented in Section 7.

2. Related works

The related works about the multi-project schedulingproblem, the resource management policies, and theproject scheduling problem under uncertainty are men-tioned in this section brie y.

2.1. Multi-project scheduling problemThe RCMPSP comes from practical multi-project envi-ronments, in which a number of projects concurrentlyshare limited resources in precedence or other con-straints [12]. In fact, the single project managementrarely occurs today, and companies usually managemore than one project simultaneously called \multi-project management". The importance of multi-project management has increased over the last decades

E. Nabipoor Afruzi et al./Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 361{376 363

and is still growing. In the middle of the last century,project and multi-project management gained momen-tum; the share of project work has increased sincethen, and the penetration of �rms by correspondingmanagement methods has not stopped at the beginningof this century [13]. The researchers concur thatthe literature of the project management problem isheavily biased towards the single project environment,while there are few studies on the multi-project prob-lem [14].

The main reason for the insigni�cant fruitionwith regard to the topic of multi-project schedulingcompared with the single project one comes from itshigh complexity, which is a�ected by many factorssuch as the huge solution space, intense contending forresources, various and con icting objectives, the inter-project dependence and priority, the high level of un-certainty, and so on [12]. Therefore, many researchershave studied recently the multi-project problem toovercome this identi�ed gap [15{18]. In addition, someheuristic priority rules and metaheuristics have beenstudied to solve the RCMPSP [19{22].

2.2. Resource sharing policyIn the literature of the multi-project problem, theprimary topic is the allocation of common resourcesto simultaneous projects since the resource-based re-lations de�ne the multi-project problem by joiningthe individual projects together. The characteristicsof resource usage by the individual project in themulti-project environment are described in accordancewith the resource management policy [4]. In themulti-project problem, there are several projects thatare executed in parallel, and they use the commonresource pool for one resource type, at least. There areseveral approaches to optimally allocating the resourcesto the activities of multiple projects, such as theresource sharing policy, the resource dedication policy,etc. [16,17,19]. Regarding the existence of di�erentexisting policies, in this paper, the most common one,i.e., resource sharing policy, is applied to determine howto allocate the common resources to projects.

2.3. Project scheduling under uncertaintyThere are many studies in which the deterministic envi-ronment is considered for the project scheduling prob-lem [15,16,18,19,23]. However, in the real world, un-certainty during the project execution exists. In orderto consider uncertainty in problem modeling, di�erentassumptions can be applied. In some researches, thecosts of activities are considered uncertain [24,25] whilestudying the project scheduling problem. However, themost often objective function in the project schedulingproblem is the optimization of the project duration [9].Thus, the duration of activities with direct in uence on

the makespan of the project is studied as an uncertainparameter in the following studies.

2.3.1. Stochastic project scheduling problemThe stochastic RCPSP or Stochastic RCPSP (SR-CPSP) is the optimization problem that is solved whenthe deterministic durations in RCPSP are replacedby stochastic variables. While the goal in the classicRCPSP is to �nd a schedule with a minimum schedulelength or makespan, the goal in SRCPSP is to minimizethe expected makespan [26]. For more information,please refer to many studies that apply the stochasticapproach to uncertainty in the project schedulingproblem [27{29]. The serious challenging point forstochastic RCPSP is that, according to the maincharacteristic of the project, i.e., uniqueness, thereare di�culties accessing enough historical data to �ta probability distribution for an uncertain parameter.Therefore, applying the stochastic approach to theproject scheduling problem is susceptible to limitationsfrom the practical point of view.

2.3.2. Fuzzy project scheduling problemThe fuzzy project scheduling approach is based onthe concept of fuzzy activity duration, produces fuzzyschedules, and requires the membership function ofthe uncertain activity duration [30]. In this approach,the duration of the activities is estimated by experts,and the project manager deals with imprecise andvague judgment. For more information about fuzzyRCPSP, please refer to [31{34]. Therefore, similarto the determination of the distribution function foractivities' duration in the stochastic approach, thereare some challenges for project managers to determinethe membership function for fuzzy activity durations.

Thus, �tting distribution function with its pa-rameters or de�ning fuzzy membership function forthe activities' duration has challenges from a practicalpoint of view. In other words, this can seriously limitthe application of these two approaches to the projectscheduling problem.

2.3.3. Robust project scheduling problemThe robust optimization approach can immunize theproject scheduling problem against uncertainty. Thereare only three studies that apply this approach tothe RCPSP with uncertain duration in the singleproject problem, as will be mentioned in the following.Chakrabortty et al. [35] studied the RCPSP in whichthe activity durations were represented by random vari-ables with di�erent probability distribution functions.They proposed a robust optimization-based approachthat produced reasonably good solutions under anylikely input data scenario. Their proposed approachguarantees the feasibility of solutions and produceshigh-quality solutions. Bruni et al. [7] proposed anadaptive robust optimization model to derive the re-


source allocation decisions that minimized the worst-case makespan under general polyhedral uncertaintysets, assuming that the activity durations were subjectto interval uncertainty. Moreover, a general decomposi-tion approach was proposed by them to solve the robustcounterpart of the RCPSP, further tailored to addressthe uncertainty set with the protection factor. Artigueset al. [36] proposed models for project scheduling whenthere was considerable uncertainty in the activity du-rations. They developed and implemented a scenario-relaxation algorithm and a scenario-relaxation-basedheuristic. The �rst algorithm produces optimal solu-tions, but requires excessive running times even formedium-sized instances; the second algorithm produceshigh-quality solutions for medium-sized instances andoutperforms two benchmark heuristics.

The above-mentioned studies have been done inthe area of the single project scheduling problem.According to the large-scale multi-project schedulingproblem, the e�ect of uncertainty can be more de-structive. In the multi-project scheduling problem,some projects are related to each other by the commonresources, and the investment of time, cost, resources,etc. is done on a large scale. Therefore, the ap-plication of the robust optimization approach as amore conservative approach that can immunize theproblem against uncertainty is totally necessary. Tothe best of our knowledge, there is no research on theapplication of the robust optimization approach in themulti-project scheduling area. In the present paper,the robust optimization approach is applied to themulti-project scheduling problem under the resourceconstraint and uncertain duration of activities. Inthis research, the resource sharing policy is considered.Each project has a determined due date. In addition,the importance weight of the projects is di�erent. Theaim is to obtain an optimized structure for all ofthe projects in such a way that the maximum totalweighted tardiness of the projects will be minimum. Inthis study, the development of the existing models canbe demonstrated in two ways according to Figure 1.

3. Problem statement

The RCMPSP with uncertain activity durations isstudied in this paper. The considered multi-projectproblem contains de�ned projects, G = 1; 2; :::; q. Allof the projects are shown by activity-on-node network,Graph = (V;E), in which the nodes demonstratethe activities of projects and the arcs represent theprecedence relations between activities, E. The setof activities for each project is indicated by V =f0; 1; :::; n + 1g. For each activity i 2 V of projectg, there is a set Pig � R+ containing the possiblevalues for the duration of activity i of project g (R+is the set of non-negative real numbers). Therefore,in the discrete set of Pig =

�pig1; pig2; pig3; :::; pigjPij

,

the minimum and maximum durations for activity i ofproject g are Pmin

ig � minPig

Pigc and Pmaxig � max

PigPigc,

respectively. The durations of activities 0 and n + 1are considered zero: P0g = Pn+1;g = f0g ;8g. It is no-ticeable that when pig 2 Pig, pg = (p0g; p1g; :::; pn+1;g)shows one possible scenario for the activities' durationof project g. When jPigj = 1;8i 2 V; 8g 2 G, theproblem converts to the deterministic RCPSP.

As mentioned before, the precedence relationshipbetween activities is shown by the binary relation ofE � V � V . The activity i of project g can be startedafter all its predecessors are �nished. The projectsapply the resource sharing policy. It means that theyutilize common resources from the resource pool. Thereare bigk 2 N units of resource k required by activity i 2V of project g during its execution. In each project, therequired resources in any type for dummy activities of 0and n+1 are zero: b0gk = bn+1;g;k = 0; 8g 2 G; 8k 2 R.

A set of activities F � V is one \Forbidden Set"of a precedence relation A if it is an anti-chain of Aand at least for one type of resource k 2 R:

Pi2F

bik >

bk. Therefore, these sets can give rise to resourcecon icts during project execution. A subset-minimalforbidden set is called a \Minimal Forbidden Set" ormfs. The set of mfss for precedence relation A is

Figure 1. The model development.


written as F (A) [36]. Any of the resource con icts canbe removed by adding extra precedence relationshipsto the primary precedence graph for postponing someactivities such that the makespan can be determinedby applying an Early Start policy (ES-policy) onan extended graph. Therefore, the extra precedencerelationships X � (V � V )nE should be found in sucha way that the extended graph, Graph0(V; (E [ X)),is acyclic and F (T (E [ X)) = � [7]. Accordingto Balas [37], the set X containing pairs of activitiesthat lead to one feasible ES-policy can be called asu�cient selection. After de�ning one selection andadding the extra precedence relationship X to theprimary precedence graph E, the resource constraintscan be ignored according to the precedence relationshipin the EUX, and the makespan can be obtained bycalculating the critical path problem on the extendedgraph, Graph0(V; (E [X)) [7].

The binary decision variable, xigjg0 , is introducedin this paper to show the precedence relationshipbetween the activities. According to the characteristicsof the multi-project scheduling problem, one activ-ity and its predecessor activities are not essentiallywithin the same project, and it is possible that oneactivity becomes the predecessor of another activityfrom a di�erent project. Therefore, the precedencerelationship between the two projects is introducedin this paper based on two reasons. The �rst reasonis that, in many real-world multi-project schedulingproblems, the precedence relationship exists betweenthe activities of two projects. For example, considertwo projects in an area with low population density:(1) construction of the residential complex and (2)installation of the town gas station. In this example,the high-pressure equipment installation activity in thesecond project is the predecessor of the installation andtesting of the town gas system of the residential com-plex in the �rst project. In the cases with no precedencerelationship between two projects, the special case mayoccur with g = g0 in the xigjg0 notation.

The second reason is relevant to the applied cal-culation method. According to the minimal forbiddenset, any of the resource con icts can be removed byadding extra precedence relationships (X) to postponesome activities. Based on the resource sharing policyin the multi-project scheduling problem, the activitiesof di�erent projects utilize common resources from theresource pool. Therefore, the extra precedence rela-tionships (X) can be also created between two activitiesfrom di�erent projects. Therefore, the variable xigjg0should present both of the projects between which theextra precedence relationship (X) exists.

In the Graph0(V; (E [ X)), the start time ofactivity i of project g, si;g(X; p), is the longest pathfrom the scheduling time horizon 0 to activity i ofproject g. Thus, one should check the paths originating

from the start activities of all projects (not only thestart activity of the project g containing i) whilecalculating si;g(X; p).

The resource ows between the activities aredemonstrated by the transshipment networks [36],which can be called (resource) ow network. Thenumber of resource types k transferred from the end ofactivity i of project g to the start of activity j of projectg0 is represented by ow f(i; g; j; g0; k) � figjg0k 2 N.It is notable that, for each resource type, a separate ow network will be created. The resource ow shouldsatisfy the conservation constraints and, also, the lowerand upper bounds on the ow for intermediate (notstart or end) nodes [36].

There are several selections for the same schedule.Bruni et al. [7] presented a numerical example for twodi�erent selections in the project with 5 activities.They illustrated that when the activities' durationswere deterministic, the project makespan would be thesame for two di�erent selections. However, accord-ing to the uncertainty condition, when the delays ofthe activities are also considered, di�erent selectionscause di�erent makespan. This example shows theimportance of proper resource allocation policy underuncertainty. They also stated that, in some cases,especially in the multi-project scheduling problem, theresources cannot be easily transferred between theactivities; hence, the decisions about resource transfersshould be made with greater care and sensitivity.As mentioned earlier, to the best of our knowledge,there has been no research on the investigation of thisproblem in the multi-project scheduling environment.

In this paper, the Robust Resource-ConstrainedMulti-Project Scheduling Problem (RRCMPSP) isstudied as a two-stage robust optimization model. Inthis study, some projects are considered as a multi-project problem and should be scheduled while thedurations of activities are not certain. For each project,a due date, DDg, is determined by the global projectmanager as a deadline for �nishing each of the projectsand is noti�ed to the local project managers. The aimis to minimize the deviation of each project makespanfrom its due date, while the required resources are incommon and the activities' durations are uncertain. Ofnote, in the multi-project problem, the cost of deviatingfrom the due date is not equal for di�erent projects.Therefore, the degree of priority and importance ofproject g, demonstrated by wg as its weight, shouldbe considered in the calculation such that:X

g

wg = 1:

The question is how to allocate and share the com-mon resources between di�erent activities so that themaximum weighted tardiness for all projects, shapingmulti-project, can be minimized while the activities'


durations are uncertain. Therefore, this study seeksone su�cient selection of all projects in which themaximum weighted di�erence between the projects'makespan and their due dates is minimized.

4. Mathematical modeling of the problem

The resource-constrained multi-project schedulingproblem under uncertain durations of activities isformulated as a two-stage robust optimization model.In the following, the notations of indices, parameters,and variables used in the proposed models are repre-sented.

4.1. The notationsA list of the notations applied in the proposed modelsis as follows:

IndicesG The set of projects in the multi-project

problemV The set of activity nodesR The set of renewable resourcesE The set of precedence relations between

activitiesP The set of scenarios belonging to

activities' durations

Parameterswg The weight (priority degree) of project

gDDg The due date of project g

Phi;g The duration of activity i in project gunder scenario h

bigk The required resource type k forperforming activity i of project g

bk The capacity of resource type k

Pmini;g The minimum scenario value for the

duration of activity i in project gPmaxi;g The maximum scenario value for

duration of activity i in project g

VariablesTTa� The total weighted tardiness of

projectsTag The tardiness of project g

Shi;g The start time of activity i of projectg under scenario h

xigjg0 The decision variable with value onewhen activity i of project g is thepredecessor of activity j of project g0;otherwise, it takes the value zero

figjg0k The number of resource units of type ktransferred from the end of activity iof project g to the start of activity j ofproject g0

aig The decision variable with value oneif the duration of activity i of projectg takes the maximum value, and ittakes the value zero if the durationof activity i of project g takes theminimum value

LPg The longest path of project g in themulti-project network

'ming00igjg0'

maxg00igjg0The minimum and maximum ows

belonging to project g00 transferredfrom activity i of project g to activityj of project g0, respectively

Si;g The start time of activity i belongingto project g

4.2. The �rst-stage modelThe following is the mathematical formulation of the�rst stage model:

minTTa� =GXg=1

wg:Tag; (1)

s.t.

Tag � Shn+1;g �DDg; 8g 2 G;h = 1; :::; jP j; (2)

Shj;g0 � Shi;g + Phi;g �M(1� xigjg0);8(i; j) 2 V � V; 8g; g0 2 G�G; i 6= j or

g 6= g0; h = 1; :::; jP j; (3)Xg0

Xi2V;i6=0

Xg

f0gig0k = bk; 8k 2 R; (4)

Xg

Xj2V;j 6=n+1

Xg0fjgn+1g0k = bk; 8k 2 R; (5)

Xg02G

Xj2V;j 6=n+1(j 6=i or g 6=g0)

fjg0igk = bigk;

8i 2 V n f0; n+ 1g ; 8k 2 R; 8g 2 G; (6)Xg02G

Xj2V;j 6=0(j 6=i or g 6=g0)

figjg0k = bigk;

8i 2 V n f0; n+ 1g ; 8k 2 R; 8g 2 G; (7)


figjg0k � min fbigk; bjg0kg :xigjg0 ;8(i; j) 2 V � V; 8(g; g0) 2 G�G; 8k 2 R;i; j 6= 0; n+ 1; (8)

xigjg0 = 1; 8(i; g; j; g0) 2 E; (9)

S0g = 0; 8g 2 G; (10)

Tag � 0; 8g 2 G; (11)

Shi;g � 0; 8i 2 V; 8g 2 G; h = 1; :::; jP j; (12)

figjg0k � 0; 8(i; j) 2 V � V;8(g; g0) 2 G�G; 8k 2 R; (13)

xigjg0 2 f0; 1g ; 8(i; j) 2 V � V ; 8g; g0 2 G�G:(14)

The minimization of the total weighted tardinessof the projects is displayed in Eq. (1) as the objectivefunction. The tardiness of each project is the di�erencebetween the project's makespan and its determined duedate and is obtained by Constraint (2). Constraint (3)demonstrates the precedence relationships between theactivities, where M is a big number. Therefore, basedon this constraint, the successor activity, j, cannotstart earlier than the �nish time of its predecessorsunder each scenario. The sum of resource ows (type k)sent from dummy start nodes 0 is equal to the availablecapacity of resource (type k), as mentioned in Eq. (4).In addition, based on Eq. (5), the sum of resource owstype k sent from the activities of all projects to thedummy �nish nodes n + 1 of projects is equal to theavailable capacity of resource type k.

The sum of incoming resource ows (type k) fromother activities to activity i of project g is equal to therequired resource type k for performing the activity iof project g, which is described in Eq. (6). Similarly,Eq. (7) ensures that the sum of resource ows (type k)exiting from activity i of project g to other activities isequal to the required resource type k for executing theactivity i of project g. Constraint (8) ensures that theresource ow (type k) transferred from the activity i ofproject g to the activity j of project g0 is quite equal tothe minimum value of fbigk; bjg0kg. In addition, thisequation prevents resource transferring between twoactivities, where there is no precedence relationshipbetween them.

According to Eq. (9), the binary variable, x,is equal to 1 for the two activities with precedencerelationship between them. The start time of (dummy)activities 0 for all projects is zero (the start point ofthe scheduling horizon), as demonstrated in Eq. (10).

Based on Constraint (11), the tardiness of projectscannot be negative. Constraints (12) and (13) intro-duce the nonnegative decision variables of the starttime of activities and the resource ow between theactivities, respectively. At last, the binary variable, x,is presented in Eq. (14).

In this stage, the best structure E[X is obtainedfor the existing scenarios regarding the precedence re-lationships and resource requirements. This structureis the output of the �rst-stage model, which is requiredas an input for the second-stage model. In fact, thisstructure is achieved while the total weighted tardinessof projects as an objective function is minimized.

4.3. The second-stage modelIn this section the mathematical formulation for thesecond stage model is presented:

maxTTa� =GXg=1

wg:Tag; (15)

s.t.

Tag � (LPg �DDg); 8g 2 G; (16)

LPg00 =X

(i;g;j;g0)2EUX(pminig :'min

g00igjg0 + pmaxig :'max

g00igjg0);

8g00 2 G; (17)X(i;g;j;g0)2EUX

'maxg00igjg0 � aig;

8i 2 V n f0; n+ 1g ; 8g 2 G; 8g00 2 G; (18)X(i;g;j;g0)2EUX

'ming00igjg0 � 1� aig;

8i 2 V n f0; n+ 1g ; 8g 2 G; 8g00 2 G; (19)X(i;g;j;g0)2EUX

('ming00igjg0 + 'max

g00igjg0) = 1;

for i = 0; 8g00 2 G; (20)X(i;g;j;g0)2EUX

('ming00igjg0 + 'max

g00igjg0) = 1;

8g00 2 G; j = n+ 1; g0 = g00; (21)

'ming00igjg0 = 0; 8i 2 V; 8g; g00 2 G�G;

j = n+ 1; g0 6= g00; (22)

'maxg00igjg0 = 0; 8i 2 V; 8g; g00 2 G�G;

j = n+ 1; g0 6= g00; (23)


X(i;g;j;g0)2EUX

'ming00igjg0 + 'max

g00igjg0

=X

(j;g0;i;g)2EUX'ming00jg0ig + 'max

g00jg0ig;

8i 2 V n f0; n+ 1g ; 8g 2 G; 8g00 2 G; (24)

Tag � 0; 8g 2 G; (25)

'ming00igjg0 � 0; 8(i; g; j; g0) 2 EUX; 8g00 2 G; (26)

'maxg00igjg0 � 0; 8(i; g; j; g0) 2 EUX; 8g00 2 G; (27)

xigjg0 2 f0; 1g ; 8(i; j) 2 V � V ; 8g; g0 2 G�G;(28)

xigjg0 = 1; 8(i; g; j; g0) 2 E; (29)

aig 2 f0; 1g ; 8i 2 V; 8g 2 G; (30)

a0g = an+1g = 0; 8g 2 G; (31)

S0g = 0; 8g 2 G: (32)

In the second-stage model, the worst-case scenarioshould be found in such a way that the total weightedtardiness of the projects is maximized, as representedin Eq. (15). Eq. (16) shows how to obtain the projects'tardiness. In this equation, the �nish time of eachproject is obtained by the longest path (LPg) methodin the overall network of the projects, as demonstratedin Eq. (17). In the single project problem, the longestpath can be obtained by

P(i;j)2EUX

(pi:'ij), where pi is

the duration of activity i and 'ij is the ow transferredfrom activity i to activity j. The multiplication ofpi and 'ij leads to the nonlinearity of this formula.The binary variable, ai, is introduced to linearize theformula and is converted it to

P(i;j)2EUX

(pmini :'min

ij +

pmaxi :'max

ij ), in which pmini and pmax

i are the minimumand maximum values of the duration belonging toactivity i, respectively. For detailed information aboutcalculating the longest path of \single project" and howto linearize it, please refer to Artigues et al. [36].

In the multi-project scheduling problem, the EUXis the overall structure of all projects including theprimary precedence relationships between activities(E) and the extra precedence relationships caused byresource constraint (X). Therefore, in the studiedproblem, the projects are interrelated with each otherin this structure. Thus, for obtaining the longest pathsof the projects, a ow per project should be sent from0 activities to other activities in the overall structure,as demonstrated by 'g00igjg0 . It is worth mentioning

that the �rst index (g00) in the decision variable 'g00igjg0shows the project for which we want to calculate thelongest path.

In order to linearize the longest path formula,Constraints (18) and (19) are created in which thebinary variable, aig, takes the value 0 when theduration value of activity i of project g is minimumand, thus, 'max

g00igjg0 = 0. On the other hand, aig takesvalue 1, showing that the duration value of activity iof project g is maximum and, thus, 'min

g00igjg0 = 0.As mentioned before, the predecessor of one

activity can be the activity within the same projector from the other projects. Therefore, the longest pathof one project does not necessarily originate from theactivity 0 of that project, and it can also start fromthe 0 activity of other projects. According to Eq. (20),the summation of ows by calculating the longest pathof the project g00, originated from start nodes 0 of allprojects, to the overall structure should be equal to1. Besides, Eq. (21) implies that the ow calculatingthe longest path of project g00 should end in the noden+ 1 of project g00. Eqs. (22) and (23) ensure that the ow calculating the longest path of one project cannotenter the end node n + 1 of other projects. For each ow, the conservation law should be satis�ed, i.e., thesum of ows entering the activity i of project g shouldbe equal to the sum of ows exiting from the activity iof project g. This law is presented in Eq. (24).

Constraint (25) introduces the nonnegative vari-able of the projects' tardiness. The ows related to thelongest path calculations are represented in Eqs. (26)and (27). The binary variable, x, is de�ned in Eq. (28),while it should take value 1 for the activities withprecedence relationship between them, as stated inEq. (29). The binary variable, a, is described inEq. (30). Eq. (31) represents that, for all start nodes0, the variable a takes value 0. Finally, the start timeof the projects is set at time 0, as shown in Eq. (32).

4.4. The two-stage exact approachThe scenario relaxation algorithm is an iterative op-timization algorithm that generates optimal robustdecisions with respect to the deviation and relativerobust objectives. The key insight of the scenariorelaxation algorithm is that, for a problem with alarge number of possible scenarios, only a small subsetof scenarios actually has to be explicitly examinedwhen searching for the deviations from the optimal(or relative) robust solution. For more informationabout the scenario relaxation algorithm, please referto [38].

In this paper, the objective function is to minimizethe maximum total weighted tardiness of the multi-project problem under uncertain activities' durations.A two-stage model is presented for the RRCMPSPin Sections 4.1 and 4.2. Based on the mentioned


modeling, the set of decision variables can be dividedinto two groups. The �rst-group variables are thoserelated to the su�cient selection decisions X 2 �. Thesecond group variables are related to the calculation ofthe longest paths in the structure obtained by the �rst-stage model. According to the Benders alphabet, the�rst-stage model corresponds to the \Master problem",and the second-stage model is similar in spirit to the\Sub problem".

According to Artigues et al. [36], a durationscenario p is extreme if pi = pmin

i or pi = pmaxi for all

i 2 �. They also proved that there is always an extremeduration scenario for which the maximum absoluteregret of an ES-policy X is reached. Therefore, in theworst case for the studied problem in this paper, thenumber of algorithm iterations can be jP j = 2�, where� is the number of activities belonging to all projects.

In this approach, the scenarios are graduallyadded to the problem structure in the sequentialiterations. First, one scenario of activity durationsis considered (any arbitrary number of scenarios canbe considered), and the �rst-stage model is solved.The aim is to obtain the structure E [ X, for whichthe total weighted di�erences between the projects'makespan and their due dates are minimized. Inother words, considering the existing scenario, wesearch for an optimized E [ X with minimum totalweighted tardiness of the projects. In the next step,the second-stage model is the worst-case scenario forthe obtained structure of the �rst-stage model suchthat the objective function (total weighted tardiness ofthe projects) will be maximized. Then, the mentionedscenario should be added to the scenario set of the�rst-stage model. This algorithm continues until theobjective functions of the both stages become equal.In other words, the algorithm terminates when theminimum weighted tardiness of the optimized structurefor the existing scenarios is equal to the maximumweighted tardiness of the worst-case scenario for theassigned structure.

The steps of the applied approach are describedin the following, where iter is the counter of algorithmiterations:

Step 1 (preliminary). The set P̂1 containing onlyone scenario p1 for the duration of all activities of theprojects is considered. In addition, iter = 1, LB = 0,and UB = +1 are assumed;

Step 2 (�rst-stage model). Models (1){(14)are solved in order to obtain LB = TTa�(P̂iter).In addition, the corresponding ES-policy, Xiter, isobtained;

Step 3 (second-stage model). Models (15){(32) are solved and the maximum TTamax(Xiter)for Xiter is obtained. The corresponding worst-

case scenario, piter+1, is obtained. In addition, theUB = TTamax(Xiter) is considered;

Step 4 (optimality investigation). If LB = UB,then stop the algorithm. If LB 6= UB, then iter =iter + 1, P̂iter = P̂iter�1 [ �piter and the algorithmshould continue from Step 2.

5. Numerical example

In this section, one simple example is presented toillustrate the application of the mentioned approachto the multi-project problem. Consider a multi-projectproblem that consists of three projects. Each projecthas only four activities (the start activities and endactivities are dummies), as shown in Figure 2. Thereis only one renewable resource with (b1 = 7). Therequired resource for performing each activity, thepossible durations of activities, the determined duedate of projects, and the importance weight of theprojects are all represented in Table 1. Both of the�rst-stage and second-stage models are coded in GAMSv24.1.2 and solved by the \CPLEX" solver.

The EUX1 is obtained after solving the �rst-stagemodel in the �rst iteration. According to this structure,the total weighted tardiness of the projects accordingto the �rst scenario will be minimized. In the �rstscenario, the durations of all activities are consideredat their minimum values (Table 2). To avoid untidinesscaused by too many arcs, the representation of thewhole EUXs is neglected in each iteration. The longestpaths of the projects according to the �rst scenario arecalculated and shown in Figure 3(a).

For the given EUX1 from the �rst-stage model,

Figure 2. Multi-project network.


Table 1. The multi-project required data.

i g big1 pigDuedate

Projectweight

Pro

ject

1 1 1 0 f0g7 0.32 1 4 f4; 5; 6g

3 1 3 f2; 4g4 1 0 f0g

Pro

ject

2 1 2 0 f0g4 0.42 2 3 f3; 4g

3 2 5 f1; 2; 3g4 2 0 f0g

Pro

ject

3 1 3 0 f0g4 0.32 3 2 f2; 3; 5g

3 3 4 f4; 5; 6g4 3 0 f0g

Table 2. First scenario values for activities' durations.

ProjectActivity 1 2 3

1 0 0 02 4 3 23 2 1 44 0 0 0

the second-stage model should be solved. The maxi-mum total weighted tardiness for EUX1 is determinedby �nding the worst-case scenario, as presented inTable 3.

Figure 3(b) shows the longest paths of theprojects according to the worst-case scenario (thedemonstration of the longest paths based on the �rst

Table 3. The worst-case scenario obtained by thesecond-stage model in the �rst iteration.


1 0 0 02 6 3 23 2 3 64 0 0 0

Table 4. The worst-case scenario obtained by thesecond-stage model in the second iteration.


1 0 0 02 6 4 53 2 3 64 0 0 0

scenario is ignored) resulting from the second-stagemodel in the �rst iteration.

The �rst-stage model should be solved regardingtwo scenarios for the activities' durations in the seconditeration. The total weighted tardiness of the projectsshould be minimized with regard to these two scenarios.Therefore, the optimized EUX2 structure is obtained,which will be the input for the second-stage model.The longest paths of the projects only for the secondscenario are depicted in Figure 4(a).

After that, the second stage model is solvedwhile the objective is to maximize the total weightedtardiness of projects. In fact, for the given EUX2,the worst-case scenario should be achieved, which isdemonstrated in Table 4. The longest paths of the

Figure 3. (a) First-iteration/level 1. Obj: 1.3, Ta1: 2, Ta2: 1, Ta3: 1. (b) First-iteration/level 2. Obj: 5.9, Ta1: 8, Ta2:5, Ta3: 5, aig = 1 for f2:1; 3:2; 3:3g.


Figure 4. (a) Second-iteration/level 1 (second scenario): Obj: 5, Ta1: 8, Ta2: 5, Ta3: 2. (b) Second-iteration/level 2:Obj: 7.1, Ta1: 11, Ta2: 8, Ta3: 2, aig = 1 for f2:1; 2:2; 2:3; 3:2; 3:3g.

Figure 5. (a) Third-iteration/level 1 (third scenario): Obj: 5.9, Ta1: 8, Ta2: 5, Ta3: 5. (b) Third-iteration/level 2: Obj:5.9, Ta1: 8, Ta2: 5, Ta3: 5, aig = 1 for f2:1; 3:2; 3:3g.

projects based on only the worst-case scenario areshown in Figure 4(b).

In the third iteration, the �rst-stage model issolved regarding three scenarios. The longest pathsof projects only for the third scenario are shown inFigure 5(a).

By obtaining the EUX3, the second-stage modelcan be solved. The obtained worst-case scenario andthe longest paths of the projects are depicted in Table 5and Figure 5(b), respectively.

By comparing the objective functions of the �rst-stage and second-stage models in the third iteration, itis realized that the algorithm should stop when both ofthe objective functions have the same value, i.e. 5.9 inthis simple example. The results of each iteration arepresented in Table 6 in brief.

After three iterations, the optimized value of theobjective function of this example has been obtained.This value is found by the best structure according tothe resource-constraint and precedence relationships in


Table 5. The worst-case scenario obtained by thesecond-stage model in the third iteration.


1 0 0 02 6 3 23 2 3 64 0 0 0

which the total weighted tardiness of the projects isminimum. With regard to the existing uncertainty inthe activities' durations, the obtained result is robust.In fact, it ensures that if any scenarios occur forthe activities' durations (in this example, we have(34) � (22) = 324 possible scenarios occurrence), thetotal weighted tardiness of the projects will not begreater than 5.9. This is exactly the characteristicof the robust optimization method that immunizesthe problem against uncertainty and keeps the resultsfeasible and almost optimal.

6. Computational experiments

Both of the �rst-stage and second-stage models arecoded in GAMS v24.1.2 and solved with the CPLEXsolver. The experiments were run on a personalcomputer with an Intel(R) Xeon(R) CPU E7-8890 [email protected] GHz 2.19 GHz (2 processors) and 42 GB RAMunder Windows 10 operation system.

6.1. The test problemsIn this paper, in order to generate the test problems,the software RanGen [39] is applied to deterministicRCPSP. To adapt the test problems to RRCMPSP, therequired additional data are considered. In addition,the number of activities can be chosen. In this research,the number of activities, n = 30, is considered for eachproject in the multi-project problem. The applicationof this software provides us instances with di�erentvalues of the parameters related to the structure ofthe projects. The considered parameters include order

strength, resource factor, and resource constrainedness,which are explained brie y in the following:

� Order Strength (OS): The number of precedencerelations is divided by the theoretical maximumnumber of precedence relations in the network. Theminimum value of OS is 0 (in the parallel network),and the maximum value of OS is 1 (in the serialnetwork case). Therefore, it can take values from 0to 1. In this research, OS can be chosen from twovalues f0:4; 0:7g;

� Resource Factor (RF): How many di�erent re-sources used on average by the activities are de-termined by this factor. The minimum value ofRF is 0 (no resource requirements for executing theactivities), and the maximum value of RF is 1 (whenall the activities require all kinds of resources).Therefore, it can take values from 0 to 1. Inthis research, RF chooses a value from the setf0:25; 0:5; 0:75g;

� Resource Constrainedness (RC) (per re-source type): This factor can be obtained throughEq. (33) [40]:

RCk =DMNDk

Rk; for all k 2 R; (33)

where Rk is the capacity of resource type k, andDMNDk is the average quantity of resource type kdemanded when required by an activity and can becalculated by Eq. (34):

DMNDk=

PNrijkP

N

�1 if rijk � 00 if rijk = 0

� ; for all k 2 R;(34)

where rijk is the per-period requirement ofresource type k by activity j of project i, andN is the set of all activities to be scheduled.In this research, RC chooses a value from theset f0:3; 0:6g. For each combination of OS, RF,and RC, �ve instances of the RRCMPSP are

Table 6. The summary of the results obtained from iterations.

First iteration Second iteration Third iterationProjects Projects Projects

1 2 3 1 2 3 1 2 3

First stage model Projects tardiness 2 1 1 8 5 2 8 5 5The total weighted tardiness 1.3 5 5.9

Second stage modelProjects tardiness 8 5 5 11 8 2 8 5 5

The total weighted tardiness 5.9 7.1 5.9

Optimality check � � p


considered. Each multi-project problem is assumedto be containing three projects. Thus, 2(OS) �3(RF ) � 2(RC) � 5(per combination examples) �3(number of project in each multi project problem)= 180 instances are needed to be randomlygenerated by RanGen.

6.2. Computational experimentsThere are 12 classes of problems with respect todi�erent values of factors: OS, RF, and RC. Theaverage execution time per iteration for both of the�rst-stage and second-stage models is calculated. Inaddition, the average number of iterations is recorded.Table 7 represents the computational results of the testproblems.

Figures 6 to 8 illustrate the sensitivity analysis ofthe obtained results for di�erent levels of factors RC,RF, and OS, respectively. As is shown by these �gures,the behavior of the solution approach is strongly relatedto the instances and their characteristics.

There are 6 classes of problems based on thedi�erent values of OS and RF, in which the e�ect offactor RC should be examined. As shown in Figure 6,the most e�ective factor is RC, which strongly impactson the performance of the applied approach. It isnotable that the linear histogram is �tted just forshowing the e�ect of the factors' value on the resultsschematically. According to these 6 experiments,the computational time grows rapidly according tothe higher value of RC. In other words, when thevalue of RC increases, the instances become harderto solve and the approach needs requires time forexecution. Of note, this impact is mainly observed

Figure 6. The sensitivity analysis of ResourceConstrainedness (RC).

on the performance of the �rst-stage model, in whichextra precedence relationships (X) should be obtained(according to the resource constraint). Therefore, RC,

Table 7. The computational results.

Parameters Time

OS RF RCFirst-stage averagecomputational time

per iteration

Second-stage averagecomputational time

per iteration

Average totalcomputational

time periteration

Averagenumber ofiterations

1 0.4 0.25 0.3 23 min, 5 s 2 s 23 min, 7 s 3.22 0.6 54 min, 2 s 3 s 54 min, 5 s 5.63 0.5 0.3 28 min, 30 s 2 s 28 min, 32 s 44 0.6 1 h, 3 min, 13 s 2.5 s 1 h, 3 min, 15.5 s 4.85 0.75 0.3 45 min, 10 s 2 s 45 min, 12 s 4.26 0.6 1 h, 10 min, 28 s 3.2 s 1 h, 10 min, 31.2 s 67 0.7 0.25 0.3 17 min, 1.5 s 2 s 17 min, 3.5 s 3.28 0.6 53 min 3 s 53 min, 3 s 2.29 0.5 0.3 22 min, 28.3 s 2.7 s 22 min, 31 s 210 0.6 1 h, 1 min, 8 s 3 s 1 h, 1 min, 11 s 5.411 0.75 0.3 42 min, 2.5 s 3.5 s 42 min, 6 s 312 0.6 1 h, 3 min, 54.4 s 2.6 s 1 h, 3 min, 57 s 6.2


Figure 7. The sensitivity analysis of Resource Factor(RF).

Figure 8. The sensitivity analysis of Order Strength(OS).

which is mainly related to the resource constraint, hasstrong in uence on the performance of the �rst-stagemodel, and the second-stage model is not in uencedsigni�cantly by this factor.

The next e�ective factor is RF that is also related

to the resource constraint and has an observable in- uence on the �rst-stage model. When RF increases,the �rst-stage model becomes harder to solve and,consequently, consumes more time. Therefore, RF isthe second e�ective factor that in uences the obtainedresults. As demonstrated by Figure 7, a sensibleincrease in the computational time occurs by increasingthe RF.

The OS factor has the least in uence on thecomputational time of the obtained results, as depictedin Figure 8. In most cases, there is a decreasein the computational time when the value of factorOS changes (in some cases, there are not signi�cantchanges). With respect to the de�nition of OS factor,when OS increases, the structure of the projects movesfrom the parallel structure to the serial structure.Therefore, the problem becomes easier according tothe resource constraints, and it is expected that therunning time of the algorithm decreases signi�cantly.However, why does it not happen? The reason isrelated to the extra precedence relationship X that isadded to the E set to remove the resource con ict. Inother words, the E set (based on the OS factor) is notthe only parameter that a�ects the physical structureof the project network, and EUX is the �nal structureof the network.

7. Conclusion and further research

The Robust Resource-Constrained Multi-ProjectScheduling Problem (RRCMPSP) was studied inthis paper, in which the objective function was tominimize the maximum total weighted tardiness of theprojects. The durations of the activities belonging tothe projects were uncertain and de�ned with discretevalues called scenarios. The resource sharing policywas applied in this study for resource allocation inthe multi-project problem. Moreover, there was adeadline for each project determined by the globalproject manager, and each project had its own weightof importance, which dictates which project shouldreceive more consideration. To ensure exact results,a scenario-relaxation algorithm was applied andimplemented for the proposed robust multi-projectscheduling problem. Then, the computational resultswere discussed. It was found obviously that the factorRC had more in uence on the behavior of the solutionapproach, which is an important factor, especially inthe multi-project problems.

Some extensions of this research as a future studymight be of interest. While the limitation of thisstudy is that the presented exact solution method isnot able to solve large-size problems in a reasonableamount of time, developing the heuristic and meta-heuristic algorithms is suggested to solve the large-sized RRCMPSP. As another extension, considering


uncertainty in the resources availability and its e�ectin managing the multi-project problems would be ofinterest. In addition, some constraints can be added tothis model such as the multi-mode activities, nonrenew-able resources, multi-skill resources, etc. while the otherobjective functions such as minimum cost or maximumquality are considered.

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Biographies

Elham Nabipoor Afruzi is currently a PhD Candi-date in Industrial Engineering in K.N. Toosi Universityof Technology (KNTU), Tehran, Iran. She obtained BSand MS degrees in Industrial Engineering both fromKNTU in 2010 and 2012, respectively. Her researchinterests include project scheduling and management,uncertainty modeling, and exact and meta-heuristicalgorithms.

Abdollah Aghaie is a Professor of Industrial En-gineering at K.N. Toosi University of Technology inTehran, Iran. He received his BSc from Sharif Uni-versity of Technology in Tehran, Iran, MSc from NewSouth Wales University in Sydney in Australia, andPhD from Loughborough University in the U.K. Hismain research interests are in modeling and simula-tion, queuing system, quality management and control,knowledge management, and stochastic process.

Amir Abbas Naja� is currently an Associate Profes-sor of Industrial Engineering at K.N. Toosi Universityof Technology. He received his BS degree in IndustrialEngineering from Isfahan University of TechnologyIsfahan, Iran in 1996 and his MS and PhD degreesin Industrial Engineering from Sharif University ofTechnology Tehran, Iran in 1998 and 2005, respectively.His research interests include project scheduling andmanagement, portfolio selection models, and appliedoperations research.

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