A Two-stage Stochastic Programming Model for the Resource

Constrained Project Scheduling Problem under Uncertainty

Maria Elena Bruni, Luigi Di Puglia Pugliese, Patrizia Beraldi and Francesca Guerriero

Department of Mechanical, Energy and Management Engineering, Unical, Italy

Keywords:

RCPSP, Uncertainty, Integer L-shaped, Stochastic Programming.

Abstract:

Due to the increasing competitiveness of businesses, project planning and scheduling have become a chal-

lenging theme in the last years. In this paper, we propose a two-stage stochastic programming model for the

resource constrained project scheduling problem, taking into account the stochasticity of activity durations. In

this formulation, assuming that some activity duration scenarios are known, resource allocations are taken in

the ﬁrst stage, while scheduling decisions are postponed in the second stage. The resulting problem is a mixed

integer problem with recourse, where binary variables appear in the ﬁrst stage. In order to efﬁciently solve

the problem, a decomposition algorithm is developed, based on the well-known integer L-shaped method.

Detailed computational results are presented for a set of benchmark instances taken from the literature.

1 INTRODUCTION

The resource-constrained project scheduling problem

(RCPSP) deals with the sequencing of activities that

are usually related by precedence constraints and by

the simultaneous use of scarce resources. The RCPSP

has attracted the attention of scientists and practitio-

ners, given also its importance in many ﬁelds (Bruni

et al., 2011a,b), who have made considerable efforts

to enhance the efﬁciency of the solution process. In

the last decade, increasing ﬁnancial pressures have

put more emphasis on the project execution and, gi-

ven the uncertainty characterizing the project envi-

ronment, the role of deterministic static schedules has

been challenged. Due to machine breakdowns, em-

ployee absenteeism and delay in materials supply, bad

whether conditions and many other uncontrollable

factors, one or more project activities may experience

a delay, threatening the operational viability of the

planned schedule and its implementation. To address

these challenges, a ﬂourishing stream of literature has

focused on the RCPSP under uncertainty in which

activity durations are assumed to be random variables.

Two different approaches can be used depending on

the genuine interpretation of the RCPSP under uncer-

tainty and the way the uncertainty is tackled. Indeed,

the problem can be viewed as a stochastic dynamic

optimization problem, where decisions are made each

time new information becomes available, or as a pro-

blem where a tentative plan, which can be changed

during project execution, should be determined and

agreed before knowing the realization of uncertainty.

As far as the ﬁrst option is concerned, the literature

has focused on dynamic decision processes (called

policies) that deﬁne, at completion of some activities,

appropriate actions concerning the choice of a set of

activities that should be executed next. The vast ma-

jority of the research in this area is concerned only

with the expected makespan objective. M

ohring and

Stork (2000) following Igelmund and Radermacher

studied the so-called linear preselective policies, that

were exploited by Stork (2001) to develop a branch-

and-bound procedure, to efﬁciently compute an opti-

mal policy minimizing the expected makespan. Po-

licies have also been used for determining predictive

activity starting times, with the objective of minimi-

zing costs related to positive and negative deviations

of actual starting times, from the predicted ones, and

to penalties/bonuses associated with late/early project

completion. Deblaere et al. (2007) and Golenko et al.

(1997) presented solution procedures for the RCPSP

with random activities duration and the expected ma-

kespan as objective. Starting from the concept of cri-

tical chain introduced by Goldratt (Goldratt, 1997),

in Rabbani et al. (2007) a new heuristic was pre-

sented for minimizing the expected project duration

and its variance. Later, Ballest

ın (2007) developed

regret-based biased random sampling procedures then

embedded into a genetic algorithm, whereas Ballest

ın

and Leus (2009) examined multiple possible objective

194

Bruni, M., Pugliese, L., Beraldi, P. and Guerriero, F.

A Two-stage Stochastic Programming Model for the Resource Constrained Project Scheduling Problem under Uncertainty.

DOI: 10.5220/0006612601940200

In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems (ICORES 2018), pages 194-200

ISBN: 978-989-758-285-1

functions for project scheduling with stochastic acti-

vity durations. For a special case involving only one

renewable resource (the budget) a two-stage integer

linear stochastic program has been proposed in Zhu

et al. (2007), whereas Bruni et al. (2011a), proposed

a chance-constrained based heuristic aiming at buil-

ding a baseline schedule which is protected against

possible disruptions.

Contrary to the stochastic project scheduling

(Bruni et al. 2015), robust project scheduling assumes

that the distribution of the uncertain activity duration

is not known or only partially known. Robustness can

be referred to either the project makespan (in this case

it is referred as quality-robustness) or to possible devi-

ations between the planned and realized starting times

of the projected schedule (we call this solution robus-

tness). The literature on robust project scheduling has

mainly dealt with the development of effective and

efﬁcient proactive and reactive scheduling procedu-

res. Proactive scheduling aims at generating robust

baseline schedules, that incorporate some protection

against possible disruptions, whereas reactive schedu-

ling procedures can be invoked during the execution

of the project, to repair the baseline schedule by de-

viating as little as possible from the original baseline

schedule. For an extensive review of research in this

ﬁeld, the reader is referred to Demeulemeester and

Herroelen (2011).

Robust optimization approaches have been re-

cently proposed for the RCPSP, under general poly-

hedral uncertainty (Bruni et al., 2017a,b). Assuming

that scenarios represent different realizations of the

activity durations, a min-max absolute-regret problem

is proposed by Artigues et al. (2011), to minimize the

maximum absolute difference between the makespan

obtained by the robust solution and the scenario de-

pendent optimal solutions.

In this paper, we study a stochastic programming

optimization approach for the RCPSP, assuming that

the random variables are discretely distributed. We

observe that such an assumption is quite general,

since discrete distributions arise either naturally in

many real-world applications, or as empirical approx-

imations of the continuous ones derived, for example,

by taking a Monte Carlo sample from a general distri-

bution.

The two-stage stochastic RCPSP (TSRCPSP) is

investigated by assuming that scenarios represent dif-

ferent realizations for the activity durations. The de-

cision variables are divided into wait-and-see varia-

bles, that must be determined before the realization

of the uncertain parameters, and here-and-now varia-

bles, that can adjust to the uncertain data when they

become known. The objective is to ﬁnd a schedule

that minimizes the expected makespan over all sce-

narios. To solve the problem, we have designed and

implemented an integer L-shaped decomposition ap-

proach, both in the single cut and the multi-cut versi-

ons.

The remainder of the paper is organized as fol-

lows: in Section 2, a formal deﬁnition of the TSR-

CPSP is given. Section 3 presents a detailed descrip-

tion of the proposed exact algorithm. Section 4 dis-

cusses the computational results obtained on a set of

benchmark problems. Finally, some conclusions are

drawn in Section 5.

2 PROBLEM FORMULATION

Project activity duration is typically unknown when

scheduling decisions need to be taken; under uncer-

tainty, specialized models able to cope with this un-

certainty should be developed. Speciﬁcally, these

models should allow to make some decisions about

the schedule before the actual activities duration is

known. Then, after the uncertainty is disclosed, a

recourse action can be implemented to compensate

deﬁciencies in the previously made schedules. Sto-

chastic programming formulations extend and adapt

deterministic models to allow this kind of schedule

modiﬁcations. In these models, some decisions must

be made in the ﬁrst-stage under uncertainty. Then, in

the second-stage, a recourse action can be made after

observing the actual values of the random variables.

In real cases, it is a very common practice to de-

cide upon the resource allocation well in advance,

since often resources (e.g. expert staff) cannot be ea-

sily transferred between activities at short notice, (for

instance in a multi-project environment), nor alloca-

ted without sufﬁcient time lapse. In our model, we

assume that the resource allocation is a static here-

and-now decision, that can be made in advance, un-

der uncertainty about the actual duration of activities.

The starting times, on the contrary, can be decided la-

ter on.

Let deﬁne a project as a set of activities V =

{0, . . . , n +1} (where 0 and n +1 are two dummy acti-

vities representing the project start and end, respecti-

vely) that have to be scheduled. A set of renewable

resources K = {1, . . . , m}, each with ﬁnite capacity

for all k belonging to K is used during the project

execution. Precedence constraints between different

activities within the project are modeled by a set of

project arcs E such that (i, j) ∈ E means that activity

j has to start after the completion of activity i. Each

activity i ∈ V requires a non-negative amount r

and

for the activities 0 and n + 1 r

= r

n+1 k

= R

for all

A Two-stage Stochastic Programming Model for the Resource Constrained Project Scheduling Problem under Uncertainty

195

k ∈ K.

The TSRCPSP can be deﬁned on the basis of the

ﬂow network model, originally proposed by Artigues

and Roubellat (2000). Different sets of variables are

used. Variables f

i jk

denote the number of units of re-

source k directly transferred from an activity i to an

activity j. Binary variables y

i j

assume the value 1 if

activity j starts after the completion of activity i and

0 otherwise and deﬁne an extended set of precedence

relations with the aim of resolving possible resource

conﬂicts. Finally variables S

deﬁne the starting times

of each activity i.

We assume that the random vector, represen-

ting stochastic activity durations, has a ﬁnite sup-

port. Henceforth, we deﬁne Σ as the set of its

possible realizations (scenarios) and we denote by

{0, d

, . . . , d

, 0}, σ ∈ Σ the durations of the activi-

ties under scenario σ occurring with probability p

The TSRCPSP may be formulated as follows.

min

Q(y) (1)

i j

= 1 ∀(i, j) ∈ E, (2)

i j

+ y

≤ 1 ∀i, j ∈ V, i < j, (3)

≥ y

i j

+ y

− 1 ∀i, j, l ∈ V, i 6= j 6= l, (4)

i jk

− min(r

, r

i j

≤ 0 ∀i, j ∈ V, i 6= j,

i 6= n + 1, j 6= 0, ∀k ∈ K, (5)

∑

j∈V\{i,0}

i jk

= r

i ∈ V \ {n + 1}, ∀k ∈ K, (6)

∑

i∈V \{ j,n+1}

i jk

= r

j ∈ V \ {0}, ∀k ∈ K, (7)

i jk

≥ 0 ∀i, j ∈ V, i 6= j, i 6= n + 1, j 6= 0,

∀k ∈ K, (8)

i j

∈ {0, 1} ∀i, j ∈ V, i 6= j. (9)

It should be noted that the constraints (3) and (4)

are valid inequalities preventing cycles into the ex-

tended graph, and constraints (5), (6) and (7) are re-

source ﬂow-conservation constraints, ensuring feasi-

ble resource allocations. Preprocessing constraints

(2) impose the preexisting precedence constraints.

Hence, the solution of this ﬁrst stage problem, gi-

ves a resource allocation, i.e., the sequence of the acti-

vities and the resource ﬂow among them. It should

be clear that it is often possible to make different re-

source allocation decisions, each represented by a dif-

ferent resource ﬂow, for the same schedule, having se-

rious impacts on the expected makespan of the sche-

dule. For this reason, the TSRCPSP minimizes in

the second stage the expected value of the makespan,

Q(y) =

∑

σ∈Σ

Q(y, σ), resulting from the resource

allocation decisions made in the ﬁrst stage. For a gi-

ven realization σ, Q(y, σ) is the optimal solution of a

scenario subproblem that can be obtained by solving

the following model:

Q(y, σ) = minS

n+1

(10)

− S

− My

i j

≥ d

− M ∀i, j ∈ V

∀σ ∈ Σ (11)

= 0 (12)

here M is large enough constant used to model the

disjunctive precedence constraints (11).

For a ﬁxed y = ˆy, the subproblem (10)–(12) is

equivalent (for strong duality) to the following model.

Q( ˆy, σ) = max

∑

(i, j)|ˆy

i j

∑

j∈V\n+1

∑

j∈V\0

i j

= 0 ∀i ∈ V \ {0, n + 1}

∑

i∈V \n+1

in+1

= 1

∑

i∈V \0

= 1

i j

≤ ˆy

i j

∀i, j ∈ V

i j

≥ 0 ∀i, j ∈ V

3 SOLUTION APPROACH

Problem (1)–(12) is a two-stage stochastic mixed

integer model with continuous recourse, where the

ﬁrst-stage problem includes binary decision variables,

whereas the recourse problem is a longest path pro-

blem. In general, solving two-stage stochastic pro-

grams is computationally demanding and decomposi-

tion techniques are usually used to solve the problem

efﬁciently. Speciﬁcally, for the continuous recourse

case, the L-shaped algorithm can be used to solve the

two-stage stochastic programming problem based on

Benders decomposition, a well-known technique for

solving large scale MIP problems that exhibit a spe-

cial structure. This approach decomposes the original

problem into an integer master problem and one or

more linear subproblems. In particular, after solving

the master problem, the current optimal master solu-

tion is fed into the recourse problems. Subsequently,

a series of cuts is generated based on the subproblems

solutions and new constraints are added to the master

problem which is then solved again. The above pro-

cess is executed iteratively until an optimal solution

is found. The L-shaped method was ﬁrst proposed in

Van Slyke and Wets (1969) for solving two-stage sto-

chastic linear problems. Its main idea is to approxi-

mate the expected recourse function Q(y) (whose eva-

luation requires the solution of all the second-stage

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

196

recourse problems) in the objective function of the

two-stage stochastic problem by adding cuts into the

master problem. The cuts are classiﬁed into two types

known as optimality cuts and feasibility cuts. The for-

mer drive the master solution toward optimality, while

the latter ensure the feasibility of the subproblems. In

the case of complete recourse (which is also our case)

it is not longer required to take care of feasibility is-

sues.

The general scheme of the L-shaped approach, in

this case, is the following.

• Set L B := −∞; UB := +∞ t = 0.

• while UB > LB do

– Solve the master problem obtaining y

∗

, f

∗

– Assign to LB the optimal objective function va-

lue of the master problem.

– Solve the subproblems for each scenario σ ∈ Σ

and let Q(y

∗

)

be the optimal expected make-

span.

– Update UB = min(UB, Q(y

∗

)

– Add an optimality cut and update t := t + 1.

– end while

• Return L B

By deﬁning η as an additional variable, the master

problem in our case assumes the following form:

minη (13)

s.t.(2) − (9) (14)

3.1 Optimality Cuts

At a given iteration of the algorithm, once the master

problem has been solved and the corresponding opti-

mal solution obtained, we should solve all the scena-

rio related subproblems (10)–(12). Let Q(y∗, σ)

, for

all σ ∈ Σ the optimal second stage objective function,

representing the makespan under scenario σ. Let π

∗t

the optimal longest path for the scenario σ. Let denote

with π

∗t

= ∪

σ∈Σ

∗t

the subset of arcs belonging to the

union of the optimal longest paths for each scenario.

Given a ﬁnite global lower bound L of the TSRCPSP,

and deﬁning Q = (Q(y∗)

− L), the following valid

cut can be added to the master problem formulation:

η ≥ Q

∑

(i, j)|(i, j)∈π

∗t

i j

− [Q (|π

∗t

| − 1) − L] (15)

Since there is a ﬁnite number of such cuts the al-

gorithm converges to an optimal solution to the TSR-

CPSP.

Proposition 1. The cut (15) is a valid optimality cut.

Proof. The quantity

∑

(i, j)∈π

∗t

i j

is always less than or

equal to |π∗|, taking exactly the value |π

∗t

| when y

i j

1, ∀(i, j) ∈ π

∗t

. In this case, the right-hand side takes

the value Q(y∗)

, otherwise the right-hand side takes

a value less than or equal to L. Since the ﬁrst stage

decision variables y are binary, there is only a ﬁnite

number of feasible ﬁrst stage solutions and therefore,

the number of cuts that can be added to the master is

ﬁnite.

We have also implemented a multicut L-shaped al-

gorithm (Birge and Louveaux, 1988). In general, the

multicut L-shaped algorithm has less major iterations

than the L-shaped algorithm. However, solving the

master problem requires more computation time. In

particular, by deﬁning an additional set of free vari-

ables η

and denoting with Q

= Q(y∗, σ)

− L, the

following optimality cut can be added for each σ ∈ Σ:

≥ Q

∑

(i, j)|(i, j)∈π

∗t

i j

− [Q

(|π

∗t

| − 1) − L] (16)

Optimality cuts (16) ensure that the value of each va-

riable η

is larger than or equal to the optimal value

of its corresponding second-stage problem for each

scenario.

Proposition 2. The cut (16) is a valid optimality cut.

Proof. The proof is omitted since trivial.

In this case, the master problem objective function

is min

∑

σ∈Σ

4 COMPUTATIONAL

EXPERIMENTS

This Section reports on the computational experi-

ments carried out to assess and compare the perfor-

mance of the proposed approaches. The algorithms

have been coded in Java and run on a PC with 16

GB RAM and 2.50 GHz Intel Core i7 - 4710 HQ

CPU. The numerical results have been collected on

120 instances generated from the benchmark determi-

nistic instances of the PSPLIB (Kolisch and Sprecher,

1997) (available at http://www.om-db.wi.tum.de/psplib)

with 30 activities. The instances differ essentially

for three main parameters, namely the network com-

plexity (NC ∈ {1.50, 1.80, 2.10}) , the resource fac-

tor (RF ∈ {0.25, 0.50}) and the resource strength

(RS ∈ {0.70, 1.00}). The ﬁrst parameter measures

the average number of non-redundant arcs per nodes

including the dummy activities, the second reﬂects

the average percentage of different resource types for

which a non-dummy activity has a non-zero resource

A Two-stage Stochastic Programming Model for the Resource Constrained Project Scheduling Problem under Uncertainty

197

demand and the last one concerns the strength of the

resource constraints.

A time limit of 20 minutes has been imposed for

both algorithms. We have assumed that each activity

either may have a normal duration or it can be de-

layed. In this case the duration assumes a peak value.

In practical settings, it is unlikely that all the activities

exhibit longer durations than expected. Henceforth,

we have considered single disruption scenarios, that

is in each scenario only one activity can be disrupted.

Consequently, the number of scenarios is 30. The nu-

merical results are collected in Tables 1–4.

Columns NC, RF, and RS correspond to the cha-

racteristics of the problem instance. The total compu-

tational times and the time spent for solving the sub-

problems, in seconds, are indicated in the table with

headers time and timeSP, respectively. The average

number of iterations-calculated over the instances sol-

ved within the time limit (column #solved)-is repor-

ted in column iter. The number of cuts is reported in

columns #cuts.

The behavior of the Benders approach is strongly

inﬂuenced by the characteristics of the instances, as

it happens also in the deterministic case. In the next

Sections,we present a detailed analysis of the compu-

tational performance of the proposed approaches de-

pending on the network characteristics.

4.1 Sensitivity to the Network

Characteristics

For the sake of comprehension, let us consider the pa-

rameters NC, RF, and RS separately.

Parameter NC. Tables 1 and 2 show the average

results over all solved instances varying NC, for the

single cut and the multi-cut algorithm, respectively.

Table 1: Average computational results over all the instan-

ces solved within the time limit varying NC, single cut.

NC time iter #cuts timeSP #solv

1.5 197.74 53.60 52.60 0.011 30

1.8 227.52 73.48 72.48 0.016 27

2.1 140.25 66.66 65.66 0.017 32

From the analysis of the results, we can observe

that, on average, 75% of the instances are solved to

optimality within the time limit. As can be observed,

the total computational time is similar for both algo-

rithms, but the number of cuts that should be added

into the master problem is higher for single cut than

for the multi-cut version. In general, the subproblems

do not represent a computational challenge since they

Table 2: Average computational results over all the instan-

ces solved within the time limit varying NC, multi-cut.

NC time iter #cuts timeSP #solv

1.5 192.90 24.90 23.90 0.005 25

1.8 229.68 38.65 37.65 0.014 27

2.1 145.13 35.70 34.70 0.044 26

can be solved in polynomial time. This behaviour is

observed for all values of NC.

Parameter RF. Referring to the values of RF, Ta-

bles 3 and 4 highlight the strong relation between the

average portion of the resources used and consumed

and the behaviour of the proposed solution approach.

Indeed, the lower the value of RF, the higher the num-

ber of solved instances to optimality.

Table 3: Average computational results over the instances

solved within the time limit, single cut.

RF time iter #cuts timeSP #solv

0.25 63.78 26.44 25.44 0.01 59

0.5 426.68 138.83 137.83 0.03 30

Table 4: Average computational results over the instances

solved within the time limit, multi-cut.

RF time iter #cuts timeSP #solv

0.25 128.63 14.92 13.92 0.00 47

0.5 288.78 64.09 63.09 0.05 31

The single cut version is more efﬁcient on pro-

blem with a small RF, whereas when the RF incre-

ases the computational time increases as well. The

multi-cut version of the algorithm is able to solve one

more instance with RF = 0.5. The higher is the va-

lue of the RF, the harder are the instances to be sol-

ved, since when very few resource types are required

by the activities (low values for RF) it is easy to take

good resource allocation decisions.

The last part of the Section is devoted to the analy-

sis of the impact of the resources strength. The higher

the value of RS, the higher the number of instances

solved to optimality, revealing the difﬁculty of the in-

stances with low RS values. The analysis of the re-

sults shows that the most time consuming instances

for the single-cut algorithm are those with RS = 1,

whereas for the multi-cut version is the opposite. On

the other hand, the multi-cut version is able to solve

the instances with RS = 1 more efﬁciently than the

single-cut algorithm, with an average time of around

148 seconds.

This might be due to the fact that problems with

RS = 1 have less resource conﬂicts than problems

with lower values of RS. We notice that this has a di-

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

198

Table 5: Average computational results over the instances

solved within the time limit, single-cut.

RS time iter #cuts timeSP #solv

0.7 131.26 52.68 51.68 0.01 34

1 220.01 71.53 70.53 0.02 55

Table 6: Average computational results over the instances

solved within the time limit, multi-cut.

RS time iter #cuts timeSP #solv

0.7 252.13 33.44 32.44 0.04 27

1 148.62 33.09 32.09 0.01 51

rect impact on the master problem, where a low num-

ber of extra precedences need to be added in order to

eventually solve resource conﬂicts. Since the master

it is easy to be solved, the multi-cut version turns out

to be more efﬁcient than the single cut on average.

For the instances not solved to optimality within the

time limit, the gap is below 1% for both the single cut

and the multi-cut L-shaped algorithm.

5 CONCLUSIONS

In this paper, we have studied the RCPSP under un-

certainty. In summary, the main contributions of this

paper are the following. First, we have proposed a

two stage stochastic programming approach, where

resource allocation is a ﬁrst stage decision, whereas

the starting times are second stage decisions. An

integer L-shaped algorithm has been tested on a set

of benchmark instances generated from deterministic

benchmark instances.

The analysis of the results shows that the single-

cut version provides better results than the multi-cut

one. This consideration would suggest the use of dif-

ferent cut aggregation strategies. The design of these

strategies together with the deﬁnition of tailored ap-

proaches for the master problem solution is the sub-

ject of ongoing research.

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