A Selective Scheduling Problem with Sequence-dependent Setup Times:

A Risk-averse Approach

Maria Elena Bruni

, Sara Khodaparasti

and Patrizia Beraldi

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

Department of Mathematics and Computer Science, Unical, Italy

Keywords:

Machine Scheduling, Sequence-dependent Setup Time, Distributionally Robust Optimization, Conditional

Value-at-Risk, Metaheuristic.

Abstract:

This paper addresses a scheduling problem with parallel identical machines and sequence-dependent setup

times in which the setup and the processing times are random parameters. The model aims at minimizing the

total completion time while the total revenue gained by the processed jobs satisﬁes the manufacturer’s thres-

hold. To handle the uncertainty of random parameters, we adopt a risk-averse distributionally robust approach

developed based on the Conditional Value-at-Risk measure hedging against the worst-case performance. The

proposed model is tested via extensive experimental results performed on a set of benchmark instances. We

also show the efﬁciency of the deterministic counterpart of our model, in comparison with the state-of-the-art

model proposed for a similar problem in a deterministic context.

1 INTRODUCTION

Machine scheduling (MS) is categorized as an out-

standing combinatorial problem whose applications

are far beyond the traditional problems arising in the

manufacturing sector and includes other ﬁelds such

as healthcare, logistics, computer science, and com-

munications (Blazewicz et al., 1991; Chang et al.,

2019). A typical parallel machine scheduling pro-

blem deals with the sequencing and scheduling of a

set of jobs to be processed on a set of identical ma-

chines where each job is processed by only one ma-

chine and each machine can process only one job at

a time. A processing time is assigned to each job.

Switching from one processed job to the next one in

the schedule might require setup times, which, gene-

rally, are sequence-dependent. The aim is, in gene-

ral, to ﬁnd a feasible scheduling decision optimizing

a time-related performance criterion such as the total

completion time. In classical scheduling contexts, it

is explicitly assumed that all existing jobs have to be

processed. In the last years, the MS literature focu-

sed on a different aspect, considering the contributi-

ons of researchers and practitioners who recognized

that there is a trade-off between the revenue gained

by processing a job and the increase in the total time

completion and that, in practice, it is impossible to

accept all customer orders (jobs) due to the ﬁrm limi-

tations in terms of resource and/or time. This led to

a new class of problems considering the order accep-

tance or rejection, called order acceptance and sche-

duling problems. (Oguz et al., 2010). Most contribu-

tions in the MS literature focus on deterministic mo-

dels in which the uncertainty is completely ignored or,

in the best case, is handled by replacing the random

parameter with an estimated value, such as the mat-

hematical expectation, as followed in the risk-neutral

approaches. On the contrary, in practice, the time-

dependent performance measures, such as the total

completion times, are affected by the ﬂuctuations in

the processing times and/or the sequence-dependent

setup times due to unexpected events such as the una-

vailability of raw materials and tools, machine fai-

lure, the variations in the worker skills or the chan-

ges in the working environment (Suwa and Sandoh,

2012; Bougeret et al., 2018). Clearly, the optimal

scheduling decisions obtained from the deterministic

models might be near-optimal or even not valid for

real-world applications with high uncertainty. This

motivated us to consider both the setup times and the

processing times uncertain. Since very often it is not

possible to have complete information on the distribu-

tion functions of these stochastic parameters, we as-

sume that only the ﬁrst and the second moments of

the probability distributions are known (Chang et al.,

2019). We apply a risk-averse approach based on the

idea of robust CVaR (RCVaR) to hedge against un-

certainty. Even in the deterministic context, the se-

lective structure makes the scheduling decisions chal-

lenging. If the decision-maker adopts a risk-averse

Bruni, M., Khodaparasti, S. and Beraldi, P.

A Selective Scheduling Problem with Sequence-dependent Setup Times: A Risk-averse Approach.

DOI: 10.5220/0007578001950201

In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 195-201

ISBN: 978-989-758-352-0

195

approach, the trade-off between the job proﬁts, the

expected processing and setup times, and their va-

riations make the problem even more involved and

less tractable. Differently from the existing literature

which is more manufacturer-oriented and often max-

imizes the total revenue of the processed jobs, in the

present paper, we introduce a new selective schedu-

ling problem with proﬁts and sequence-dependent se-

tup times in the multi-machine environment with the

objective of minimizing the sum of the total comple-

tion times. The model prioritizes the customers sa-

tisfaction by minimizing a time-related performance

measure and accounts for the manufacturer’s interests

by enforcing the total revenue to be above a minimum

threshold. This study contributes to the machine sche-

duling context since, to the best of our knowledge,

this is the ﬁrst multi-machine scheduling model with

a selective structure in which the uncertainty of both

the setup and processing times are taken into account.

The new risk-averse model is very efﬁcient and al-

lows us to solve reasonable sized problem to optima-

lity. The rest of the paper is organized as follows.

Section 2 presents a review on the most related rese-

arch for machine scheduling problems under uncer-

tainty. In Section 2, the problem statement as well as

the proposed mathematical formulation are provided.

In Section 3, we present the extensive computational

results on a set of benchmark problems.

2 LITERATURE REVIEW

There are only a few contributions on scheduling pro-

blems under uncertainty (Bruni et al., 2017; Bruni

et al., 2018a; Bruni et al., 2018b), and in particular

on selective problems in which the proﬁts and/or the

processing times are usually treated as random para-

meters (Emami et al., 2017; Xu et al., 2015). Emami

et al. (Emami et al., 2017) presented a robust appro-

ach for the order acceptance scheduling problem with

non-identical parallel machines, regarded as an exten-

sion of the model presented in (Oguz et al., 2010) to

the multi-machine case, in which the job proﬁts and

the processing times are random parameters speciﬁed

by interval data. They developed an equivalent deter-

ministic MIP model based on the work of Bertsimas

and Sim (Bertsimas and Sim, 2004) that was solved

using a Benders decomposition method enhanced by

valid cuts as well as a heuristic. In (Xu et al., 2015),

Xu et al. addressed the dynamic order acceptance

scheduling problem with sequence-dependent setup

times and uncertain demands where the arrival of or-

ders follows a Poisson distribution and the size and

the type of the product required by each order is not

known in advance. The objective is to maximize the

expected proﬁts of accepted orders respecting the cu-

stomer’s due date limitations. The authors develop an

order acceptance rule approach based on the system

capacity and opportunity cost where a heuristic is ap-

plied to estimate the capacity of system at the current

state. Almost all the aforementioned contributions on

selective scheduling problem share the same objective

of maximizing the total revenue gained, which is usu-

ally a decreasing function of tardiness while the de-

adline and the due date constraints are met. In (Ata-

kan et al., 2017), Atakan et al. proposed a risk-averse

approach to tackle the uncertainty of processing ti-

mes in a single-machine scheduling problem where a

probabilistic constraint is imposed on traditional per-

formance measures such as the total weighted com-

pletion time or the total weighted tardiness. The mo-

del is solved using a scenario decomposition appro-

ach providing promising results. Chang et al. (Chang

et al., 2017) adopted the distributionally robust appro-

ach to sequence a set of jobs with uncertain proces-

sing times on a single machine where no information

about the distribution function of the random parame-

ters is available, except their moments. The model

seeks to minimize the worst-case Conditional Value-

at-Risk (Robust CVaR) assigned to the total comple-

tion times. They formulate the problem as an inte-

ger second-order cone programming (I-SOCP) which

is solved using some approximation algorithms. In

(Niu et al., 2019), Niu et al. proposed a robust distri-

butionally approach to tackle the uncertainty of pro-

cessing times in the single-machine scheduling con-

text where the objective is expressed as the minimiza-

tion of the expected worst-case total tardiness. This

contribution is enhanced by a branch-and-bound al-

gorithm as well as a beam search heuristic to solve

large size instances. In (Pereira, 2016), Pereira pre-

sented a robust approach for a single-machine sche-

duling problem with uncertain processing times ca-

tegorized by closed intervals to minimize the worst

case total weighted completion time. They proposed

an exact branch-and-bound methodology to solve the

model. Following the risk-averse framework, Sarin et

al. presented a conditional value-at-risk (CVaR) ap-

proach to tackle the uncertainty of processing times in

a parallel multi-machine scheduling problem with the

objective function of the total weighted tardiness (Sa-

rin et al., 2014). They developed an integer L-shaped

algorithm as well as a dynamic programming-based

heuristic approach. Xu et al. proposed a min-max re-

gret robust approach to deal with the uncertainty of

the processing times for an identical parallel machine

scheduling where the uncertain parameters are cate-

gorized as closed intervals and the objective function

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

196

is expressed in terms of the makespan (the completion

time of the last processed job) (Xu et al., 2013). As

the solution approach, they designed some exact and

heuristic algorithms.

3 MATHEMATICAL

FORMULATION

Let G = (V, E) be a directed graph deﬁned over the

set of potential jobs in V = {0, 1,. . . , i, . . . , j, . . . , , n},

to be processed on k identical machines, and the set of

arcs E = {(i, j) ∈V ×

V |i 6= j} (

V = V −{0}) repre-

sents the precedence relationships among the jobs. To

each arc (i, j) ∈ E, we assign a setup time s

i j

which

is equivalent to the minimum required time to setup

the machine after processing job i and before starting

job j. Each potential job i requires a processing time

and beneﬁts a proﬁt value ψ

. Note that 0 ∈ V is

a dummy node representing the start of action over

each machine, obviously, p

= ψ

= 0. We need to

select a sub set of jobs in

V such that the total bene-

ﬁt gained by the chosen jobs, in percentage, is above

a minimum required level Γ determined by the ma-

nufacturer and the total completion time, including

the setup times and the processing times is minimi-

zed. To complete the notation, we deﬁne a level set

L = {1, . . . , r, . . . , N}, (N = |

V |−k + 1) where each

level r denotes the order of processing a job over each

machine. Since all machines are required to be used,

each machine can process up to N jobs. Considering

the above discussion, we deﬁne two binary decisions

variables as follows. Variable x

takes value 1 iff job

i is processed at level r and otherwise is set to 0; va-

riable y

i j

takes value 1 iff job j is processed at level

r after job i (which was processed at level r + 1) and

otherwise is set to 0.

min

∑

r=1

∑

j∈

r (s

0 j

+ p

0 j

∑

N−1

r=1

∑

i∈

∑

j∈

j6=i

r (s

i j

+ p

i j

(1)

∑

r=1

≤ 1 i ∈

V (2)

∑

r=1

∑

j∈

0 j

= k (3)

∑

i∈

= k (4)

∑

j∈

j6=i

i j

= x

r+1

i ∈

V , r = 1, 2, . . . , N −1 (5)

0 j

∑

i∈

i6= j

i j

= x

j ∈

V , r = 1, 2, . . . , N −1 (6)

0 j

= x

j ∈

V (7)

∑

i∈

∑

r=1

≥ Γ

∑

i∈

(8)

∈ {0, 1} i ∈

V , r = 1, 2, . . . , N (9)

i j

≥ 0 i ∈V, j ∈

V , r = 1, 2, . . . , N (10)

The objective function (1) minimizes the total com-

pletion times for all the processed jobs, including the

setup and the processing times. The set of constraints

(2) ensures that each each job is processed at most

once. Constraints (3) and (4) require the process of

exactly k jobs at the start and at the end of process,

respectively. The set of constraints in (5)-(7) are re-

lated to the connectivity of the problem. Constraints

(5) require that any job i processed at the upper level

r + 1 should be followed by exactly one task (let say

j) by traversing edge (i, j) at the lower level r. The

set of constraints in (6) impose that any job j proces-

sed at level r should be linked to exactly one recently

processed task (let say i) by traversing edge (i, j) or

linked directly to the dummy job by traversing edge

(0, j) at the same level. Constraints (7) guarantee that

the ﬁrst processed job, at the highest level N, should

be the processed just after the dummy task by traver-

sing edge (0, j). Constraint (8) states that the total

proﬁt gained from processing the selected jobs should

be above a predeﬁned threshold. Constraints (9)-(10)

deﬁne the binary nature of variables.

3.1 Distributional Robust Formulation

In this Section, we ﬁrst present some preliminaries on

robust optimization (RO). The RO models do not re-

quire speciﬁcation of the exact distribution of the exo-

genous uncertainties of the model. This is the general

distinction between the approaches of robust optimi-

zation and stochastic programming toward modeling

problems with uncertainties. In the framework of ro-

bust optimization, uncertainties are usually modeled

as random variables with true distributions that are

unknown to the modeler, but are constrained to lie

within a known support. The uncertainty set can be

selected as a continuous interval or a ﬁnite set of dif-

ferent values. In this latter setting, the problem of

interest is in general the optimization of the perfor-

mance in the worst case scenario. Three criteria have

been introduced in the literature: absolute robustness,

robust deviation and relative robust deviation. Ab-

solute robustness considers minimizing the objective

value of the worst case directly. Robust deviation (or

absolute regret) minimizes the largest possible diffe-

rence between the observed objective value and the

optimal one, while relative robust deviation (or rela-

tive regret) deals with the ratio of the largest possible

observed value to the optimal value. In the continu-

ous case, the uncertainty sets are selected as conti-

nuous intervals. Under this assumption, the expected

solution performance is typically optimized. Howe-

ver, this criterion assumes that the decision maker is

risk-neutral and leads to solutions that may be ques-

A Selective Scheduling Problem with Sequence-dependent Setup Times: A Risk-averse Approach

197

tionable. In this case, the decision maker attitude to-

wards a risk should be taken into account. A criterion

called Conditional Value-at-Risk (CVaR), early app-

lied to a stochastic portfolio selection problem, can be

used. Using this criterion, the decision maker provi-

des a parameter α which reﬂects his attitude towards

a risk. When α = 0, then CVaR becomes the expec-

tation but for greater values, more attention is paid to

the worst outcomes, which ﬁts into the robust opti-

mization framework. In this Section, we consider the

worst-case CVaR in situation where the information

on the underlying probability distribution is not ex-

actly known. In fact, typically, the ﬁrst- and second-

order moments of the uncertain parameters may be

known, but it is unlikely to have complete informa-

tion about their distributions.

Let assume that the uncertain setup times ˜s

i j

and

processing times ˜p

are deﬁned by random vectors

s and p, respectively, We investigate a speciﬁc case

where the ambiguity set is determined by the mean

and covariance and the distributional set is a semi-

inﬁnite support set. Let P

and P

be the ambiguous

distribution of random vectors s and p, respectively,

which are described by their ﬁrst and second moments

as follows:

= {IP

|Sup( ˜s

i j

) = [0, ∞), ∀(i, j) ∈V ×

V ,

E( ˜s

i j

) = µ

˜s

i j

, Var( ˜s

i j

) = σ

˜s

i j

} (11)

= {IP

|Sup( ˜p

) = [0, ∞), ∀i ∈

V ,

E( ˜p

) = µ

˜p

, Var( ˜p

) = σ

˜p

} (12)

Following the risk-averse approach, we apply the

CVaR risk measure at a given conﬁdence level α ∈

(0, 1), denoted by CVaR

. This risk measure quanti-

ﬁes the expected loss of the random variable T in the

worst α% of cases described as follows:

CVaR

= E[T|T ≥ in f {t|P(T > t) ≤ 1 −α}] (13)

where T is a vector of random variables like s or p in

(11) or (12).

Therefore, considering the above deﬁnitions, we

deﬁne the robust risk measure CVaR

(T ), denoted by,

RCVaR

(T ), as follows:

RCVaR

(T ) = sup

∈P

CVaR

(T ) (14)

It is easy to see that the worst-case CVaR

(T ) is no-

thing but the robust RCVaR

(T ) which can be equi-

valently expressed as

min

(x,y)∈X

RCVaR

(T ) = min

(x,y)∈X

sup

∈P

CVaR

(T )

(15)

where X is the solution space describing the set of

constraints in (2)-(10).

As proposed in (Chang et al., 2017), it can be pro-

ved that the solution of the robust selective parallel

scheduling model (1)- (10) can be found by applying

the following Theorem.

Theorem 1. For any random variable T ∈ R

, with

a distribution function IP

belonging to the distri-

butional set P

= {IP

|Sup(

T ) = [0, ∞), E(T ) =

, Var(T ) = σ

}, the RCVaR

(T ) is calculated as

follows:







1−α

, if 0 ≤ α ≤

+µ

1−α

, if

+µ

≤ α ≤ 1

Proof: See (Chang et al., 2017). 

Theorem 1 provides a baseline to present an equi-

valent mixed integer mathematical model for the ro-

bust model that we are going to present. In fact,

for any feasible solution (x, y) ∈ X described by

the set of constraints in (2)-(10) with the distri-

butional loss function Z

, Z

subject to a distribu-

tion in P

= {IP

|Sup(Z

(x,y)

) = [0, ∞), E(Z

(x,y)

) =

(x, y), Var(Z

(x,y)

) = σ

(x, y)}, the following result

holds (Chang et al., 2017).

min

(x,y)

RCVaR

(x,y)

) = min

(x,y)∈X

∗

, z

∗

), (16)

where z

∗

is the optimal objective function value of the

following integer linear problem

min

1−α

(

∑

r=1

∑

j∈

r [µ( ˜s

0 j

) + µ( ˜p

)]y

0 j

∑

r=1

∑

i∈

∑

j∈

j6=i

r [µ( ˜s

i j

) + µ( ˜p

)]y

i j

)

s.t.(2)−(10) (17)

and z

∗

is the optimal objective function value of the

following nonlinear integer problem

min(

∑

r=1

∑

j∈

r [µ( ˜s

0 j

) + µ( ˜p

)]y

0 j

∑

r=1

∑

i∈

∑

j∈

j6=i

r [µ( ˜s

i j

) + µ( ˜p

)]y

) +

1 −α

√

(18)

s.t. (2) −(10)

where

b =

∑

r=1

∑

j∈

[σ

( ˜s

0 j

) + σ

( ˜p

)]y

∑

r=1

∑

i∈

∑

j∈

V j6=i

[σ

( ˜s

i j

) + σ

( ˜p

)]y

(19)

hence, the optimal solution of the distributionally ro-

bust model is obtained by solving one linear and one

non-linear mixed integer mathematical model. Then,

we should take the minimum between the optimal va-

lues of z

∗

and z

∗

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

198

4 COMPUTATIONAL

EXPERIMENTS

To test the model, we have chosen some instances

from the benchmark on order acceptance in single

scheduling context (Oguz et al., 2010). The se-

tup times, processing times, and the proﬁts are the

same as those reported in the benchmark and the se-

tup time variance σ

( ˜s

i j

) as well as the processing

time variances σ

( ˜s

) were set to σ

(

i j

) =





and

( ˜s

) =





where ζ

and ζ

are random numbers

uniformly distributed in intervals [1,

( min

i∈V, j∈

µ(

i j

max

i∈V, j∈

µ(

i j

))] and [1,

(min

i∈

µ( ˜s

) + max

i∈

µ( ˜s

))], re-

spectively. The value of Γ is set to 0.6 and the number

of machines (k) varies from one to six depending on

the size of instance. The proposed model was imple-

mented in C++. The experiments were executed on

an Intel



Core

i7 2.90 GHz, with 8.0 GB of RAM

memory.

To show the efﬁciency of the proposed model, we

perform a set of experiments on 20 instances selected

from the benchmark and compare the deterministic

counterpart of our model with the order acceptance

scheduling model presented in (Oguz et al., 2010),

which shares the same idea of job acceptance or re-

jection with our model. For more information on the

deterministic order acceptance scheduling problems,

see for example (Geramipour et al., 2017; Nguyen,

2016).

To make the comparison in a fair way, we replace

those constraints related to the tardiness and deadlines

in the order acceptance model with the service level

constraint (8) and set the objective function equal to

the total completion time. Since the order acceptance

model in (Oguz et al., 2010) is, indeed, designed for

the single machine case, we set k = 1 in our proposed

model. Both models were solved by CPLEX conside-

ring a time limit of 1000 seconds. Table 1 summarizes

the obtained results where Column 1 and 2 represent

the instance name and the number of jobs, respecti-

vely; Columns 3 and 4 exhibit, the best objective va-

lue and the computational time (in seconds) for the

proposed model followed by its relative gap (in per-

centage) with respect to the CPLEX linear relaxation

lower bound. In a similar way, Columns 6-8 present

the same information exhibited in Columns 3-5 for

the other model. Column 9 shows the optimality gap

for the order acceptance model and, ﬁnally, Column

10 indicates the speed up in solution time calculated

as ∆ =

CPU

Proposed Model

CPU

Order acceptance Model

×100. In terms of the solu-

tion quality, the proposed model was able to ﬁnd the

optimal solution, veriﬁed by the zero gap values in

Table 1: Comparing the results of two models.

Instance #Jobs Proposed Model Traditional Model

Obj Val.CPU(s)Gap

(%)Obj Val.CPU(s)Gap

(%)Gap

Opt

(%)∆(%)

10Tao-R1-1 10 131 0.15 0 131 4.1 0 0 3.66

10Tao-R3-1 10 150 0.05 0 150 9.36 0 0 0.53

10Tao-R5-1 10 106 0.05 0 106 1.46 0 0 3.42

10Tao-R7-1 10 155 0.09 0 155 5.94 0 0 1.52

10Tao-R9-1 10 192 0.09 0 192 11.85 0 0 0.76

15Tao-R1-1 15 175 0.1 0 175 1000 29.7 0 0.01

15Tao-R3-1 15 287 0.4 0 287 1000 44.9 0 0.04

15Tao-R5-1 15 311 0.24 0 311 1000 46.3 0 0.02

15Tao-R7-1 15 244 0.23 0 282 1000 74 13.48 0.01

15Tao-R9-1 15 269 0.13 0 269 1000 40 0 0.01

25Tao-R1-1 25 646 2.24 0 648 1000 79.3 0.31 0.21

25Tao-R3-1 25 566 1.56 0 569 1000 77.9 0.53 0.16

25Tao-R5-1 25 772 2.24 0 843 1000 82.2 8.42 0.22

25Tao-R7-1 25 555 1.43 0 557 1000 78.9 0.36 0.14

25Tao-R9-1 25 620 2.43 0 649 1000 77.2 4.47 0.24

50Tao-R1-1 50 1501 43.7 0 1720 1000 91.2 12.73 4.34

50Tao-R3-1 50 1977 24.24 0 - 1000 ∞ ∞ 2.33

50Tao-R5-1 50 2300 131.24 0 - 1000 ∞ ∞ 13.12

50Tao-R7-1 50 1854 23.01 0 - 1000 ∞ ∞ 2.3

50Tao-R9-1 50 2216 49.39 0 - 1000 ∞ ∞ 4.94

Avg 14.15 1.9

Column 5, in a solution time limited to 132 seconds.

For the order acceptance model, CPLEX found the

optimal solutions only in 9 instances, including 4

cases for which the optimality did not proved (ve-

riﬁed by the non-zero gaps in Column 8). In 7

cases, CPLEX provided only near-optimal solutions

with different optimality gap varying from 0.31% to

12.73% and for the 4 last instances with the largest

size, CPLEX did not ﬁnd any feasible solution (spe-

ciﬁed by ”-”). Also, with respect to the solution time,

the time limit for all cases but those with 10 jobs

was reached. Apart from that, the considerable dif-

ference between Gap

Opt

and Gap

in Columns 8

and 9 is an informative insight showing that the lower

bound resulted from the linear relaxation of the order

acceptance model is not tight at all even for mode-

rate instances with 25 jobs. In summary, the proposed

model outperforms the other model in terms of both

the solution quality and the computational time. In

terms of the solution quality, the proposed model was

able to ﬁnd the optimal solution, veriﬁed by the zero

gap values in Column 5, in a solution time limited to

132 seconds. Regarding the order acceptance model,

CPLEX was able to ﬁnd the optimal solutions only

in 9 cases including 5 cases for which the optimality

was not veriﬁed. For 7 cases, CPLEX provided only

near-optimal solutions with different optimality gaps

varying from 0.31% to 12.73% and for the 4 last in-

stances with the largest size, CPLEX did not ﬁnd any

feasible solution (those speciﬁed by ”-”). Also, with

respect to the solution time, only for the smallest in-

stances with size 10 the time limit did not reached.

We should mention that the considerable difference

between Gap

Opt

and Gap

is an informative insight

showing the weak performance of lower bounds pro-

vided by the linear relaxation of the order acceptance

model even for moderate instances with 25 jobs. In

summary, the proposed model outperforms the order

A Selective Scheduling Problem with Sequence-dependent Setup Times: A Risk-averse Approach

199

Table 2: CPU time for 10 jobs, one machine.

α = 0.1 α = 0.5 α = 0.9

Instance Avg. CPU(s) Avg. CPU(s) Avg. CPU(s)

Tao1R1 7.75 7.52 7.98

Tao1R3 8.33 8.13 9.01

Tao1R5 8.82 8.18 9.05

Tao1R7 8.15 7.91 8.76

Tao1R9 8.54 8.56 9.33

Tao3R1 8.52 7.51 8.04

Tao3R3 8.43 7.7 7.98

Tao3R5 8.97 8.55 9.4

Tao3R7 8.55 7.86 8.38

Tao3R9 8.98 8.17 8.93

Tao5R1 7.79 7.39 8.29

Tao5R3 8.16 8.74 9.52

Tao5R5 8.61 8.52 8.59

Tao5R7 8.07 7.61 8.6

Tao5R9 7.16 7.47 8.01

Tao7R1 8.18 8.32 9.53

Tao7R3 7.64 7.12 7.5

Tao7R5 8.04 7.55 8.52

Tao7R7 8.09 8.11 8.9

Tao7R9 8.19 8.26 9.2

Tao9R1 8.65 7.96 8.75

Tao9R3 8.39 6.96 7.8

Tao9R5 9.05 8.6 9.28

Tao9R7 9.75 9.09 9.89

Tao9R9 8.77 8.5 8.9

Avg. 8.38 8.01 8.73

acceptance model in terms of both the solution quality

and the computational time.

The second set of the experiments was devoted to

the assessment of the computational tractability of the

proposed distributionally robust model. To this aim,

the non-linear MIP model was solved using the open

source SCIP library, released 3.2.0.

Tables 2 and 3 show the average CPU time in se-

conds, provided by SCIP for each class of instances.

For all the instances with 10 and 15 nodes and one

machine, SCIP was able to ﬁnd the optimal solution

within short computational times. The CPU time does

not show a signiﬁcant variation with the increase of α.

For the biggest instances with 25 nodes (Tables 4 and

5 for one and two machines, respectively), with the in-

crease of α from 0.5 to 0.9, the CPU time drastically

increases (64% and 104%).

5 CONCLUSION

In this paper, we introduced the selective job sche-

duling problem with sequence dependent setup times

in a multi-machine context where the setup times as

well as the job processing times are uncertain. The

problem seeks the minimization of the total comple-

Table 3: CPU time for 15 jobs, one machine.

α = 0.1 α = 0.5 α = 0.9

Instance Avg. CPU(s) Avg. CPU(s) Avg. CPU(s)

Tao1R1 36.69 38.6 40.06

Tao1R3 37.17 39.66 42.05

Tao1R5 39.36 40.99 41.71

Tao1R7 39.76 46.09 43.54

Tao1R9 38.66 40.31 41.53

Tao3R1 37.79 40.41 40.29

Tao3R3 38.23 39.38 43.55

Tao3R5 39.94 39.82 43.27

Tao3R7 41.68 46.59 45.33

Tao3R9 36 40.66 40.62

Tao5R1 39.2 42.34 43.64

Tao5R3 40.57 40.18 39.94

Tao5R5 37.67 41.1 43.3

Tao5R7 38.64 41.65 43.48

Tao5R9 34.46 37.74 38.01

Tao7R1 39.28 40.21 41.84

Tao7R3 35.88 39.11 41.88

Tao7R5 41.32 41.91 45.64

Tao7R7 35.99 36.73 39.23

Tao7R9 39.61 41.68 43.52

Tao9R1 38.93 42.09 43.04

Tao9R3 34.23 35.46 36.17

Tao9R5 36.87 39.39 42.31

Tao9R7 39.67 42.86 43.41

Tao9R9 39.18 41.33 41.53

Avg. 38.27 40.65 41.96

Table 4: CPU time for 25 jobs, one machine.

α = 0.1 α = 0.5 α = 0.9

Instance Avg. CPU(s) Avg. CPU(s) Avg. CPU(s)

Tao1R1 295.95 380.85 696.31

Tao1R3 284.4 330.59 538.97

Tao1R5 383.52 449.52 621.13

Tao1R7 292.87 395.81 622.81

Tao1R9 329.85 492.69 948.22

Tao3R1 303.51 457.73 779.48

Tao3R3 324.32 367.85 630.11

Tao3R5 388.36 452.19 558.76

Tao3R7 395.02 411.51 714.42

Tao3R9 349.44 440.82 748.21

Tao5R1 340.54 424.23 810.03

Tao5R3 296.48 323.62 499.65

Tao5R5 272.08 350.51 614.55

Tao5R7 288.31 318.23 626.53

Tao5R9 344.98 377.1 775.88

Tao7R1 312.19 342.35 572.17

Tao7R3 292.1 339.05 518.48

Tao7R5 270.79 335.57 587.55

Tao7R7 275.4 354.04 454.84

Tao7R9 287.97 345.61 563.71

Tao9R1 229.7 370.06 417.83

Tao9R3 259.06 340.6 529.14

Tao9R5 280.73 345 455.18

Tao9R7 233.08 281.73 565.33

Tao9R9 259.3 292.25 430.72

Avg. 303.6 372.78 611.2

tion time such that a minimum service level in terms

of the proﬁts of the selected jobs is met. We adopted

a distributionally robust approach and formulated the

problem as a non-linear MIP risk-averse model. We

tested the efﬁciency of the proposed model on a large

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

200

Table 5: CPU time for 25 jobs, two machines.

α = 0.1 α = 0.5 α = 0.9

Instance Avg. CPU(s) Avg. CPU(s) Avg. CPU(s)

Tao1R1 130.69 89.78 122.66

Tao1R3 120.01 146.64 148.64

Tao1R5 94.19 123.81 259.42

Tao1R7 114.16 174.75 568.06

Tao1R9 134.41 172.25 461.29

Tao3R1 112.58 160.54 358.55

Tao3R3 97.51 142.8 492.49

Tao3R5 141.19 163.15 330.12

Tao3R7 117.34 160.33 189.83

Tao3R9 100.38 68.21 78.84

Tao5R1 89.53 88.82 281.69

Tao5R3 93.56 86.06 259.5

Tao5R5 70.34 112.47 158.87

Tao5R7 89.2 89.77 188.48

Tao5R9 125.8 115.44 440.91

Tao7R1 73.28 91.76 159.64

Tao7R3 80.9 85.39 322.09

Tao7R5 116.16 123.9 243.05

Tao7R7 110.08 193.51 322.81

Tao7R9 88.09 158.97 281.59

Tao9R1 101.87 138.27 255.47

Tao9R3 147.58 185.53 255.07

Tao9R5 112.69 88.8 74.32

Tao9R7 100.9 135.96 279.13

Tao9R9 77.43 148.42 84.86

Avg. 105.59 129.81 264.7

set of scheduling benchmark instances providing the

optimal solution within short computational time for

the set of small and moderate sized instances. For the

biggest instances the computational effort increases,

calling for the development of a tailored heuristic ap-

proach, that could be a promising avenue for future

research.

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