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Scheduling in production, supply chain and Industry 4.0
systems by optimal control: fundamentals,
state-of-the-art and applications
Alexandre Dolgui, Dmitry Ivanov, Suresh Sethi, Boris Sokolov
To cite this version:
Alexandre Dolgui, Dmitry Ivanov, Suresh Sethi, Boris Sokolov. Scheduling in production, supply
chain and Industry 4.0 systems by optimal control: fundamentals, state-of-the-art and applica-
tions. International Journal of Production Research, Taylor & Francis, 2019, 57 (2), pp.411-432.
�10.1080/00207543.2018.1442948�. �hal-01998181�
Scheduling in production, supply chain and Industry 4.0 systems by optimal control: fundamen-
tals, state-of-the-art, and applications
Alexandre Dolgui
1
, Dmitry Ivanov
2*
, Suresh Sethi
3
, Boris Sokolov
4,5
1
Automation, Production and Computer Sciences Dept.
IMT Atlantique LS2N - CNRS UMR 6004 La Chantrerie
email: alexandre.dolgui@imt-atlantique.fr
2*
Berlin School of Economics and Law
Department of Business Administration; Professor for Supply Chain Management
10825 Berlin, Germany
Phone: +49 30 85789155
E-Mail: divanov@hwr-berlin.de
3
Eugene McDermott Chair
Center for Intelligent Supply Networks (C4iSN)
Naveen Jindal School of Management, Mail Station SM30
The University of Texas at Dallas
800 W. Campbell Rd., Richardson, Texas 75080-3021
E-mail: sethi@utdallas.edu
4
Saint Petersburg Institute for Informatics and Automation of the RAS (SPIIRAS)
V.O. 14 line, 39 199178 St. Petersburg, Russia
E-Mail: sokol@iias.spb.su
5
ITMO University, St. Petersburg, Russia
E-Mail: sokol@iias.spb.su
* Corresponding author
Dmitry Ivanov
Abstract
Specific scheduling problems with complex hybrid logical and terminal constraints, non-stationarity in
process execution as well as complex interrelations between dynamics in process design, capacity utili-
zation, and machine setups require further investigation and the application of a broad range of method-
ical approaches. One of these approaches is optimal control. The objectives of this survey are twofold.
The first objective is to derive major contributions, application areas, limitations, as well as research and
application recommendations for the future regarding optimal control applications to scheduling. The
second objective is to explain control engineering models in terms of industrial engineering and produc-
tion management. In this paper, we provided a survey on the applications of optimal control to schedul-
ing in production, supply chain, and Industry 4.0 systems with a focus on the deterministic maximum
principle. Optimal control approaches take a different perspective as mathematical programming meth-
ods which represent schedules as trajectories. We consider optimal control models, performance analy-
sis qualitative methods, and computational methods for optimal control. We provide a brief historic
overview and clarify major mathematical fundamentals whereby the control engineering terms are
brought into correspondence with industrial engineering and management. The survey allows the group-
ing of models with only terminal constraints with application to master production scheduling, models
with hybrid terminal-logical constraints with applications to short term job and flow shop scheduling,
and hybrid structural-terminal-logical constraints with applications to customized assembly systems
such as Industry 4.0. Computational algorithms in state, control, and adjoint variable spaces are dis-
cussed. Finally, we derive major contributions, application areas of different control methods, and their
limitations. This paper also provides recommendations for future research and applications.
Keywords: optimal program control, deterministic control, maximum principle, scheduling, attainable
sets, algorithms
1. Introduction
Short-term scheduling belongs to the fundamentals of scheduling theory. It considers jobs containing
operation chains with equal (i.e., flow shop) or different (i.e., job shop) machine sequences and different
processing times. The operations need to be scheduled for machines with different processing power
subject to some criteria such as makespan, lead time, or due dates (Blazewicz et al. 2001, Pinedo 2008,
Dolgui and Proth 2010, Werner and Sotskov 2014).
Over the last decades, mathematical optimization applications to scheduling problems have been studied
from different perspectives whereby significant progress can be observed in the development of rigor-
ous theoretical models and efficient solution techniques. Lauff and Werner (2004), Jungwattanakit et al.
(2009), Sotskov et al. (2013), Choi et al. (2013), Harjunkoski et al. (2014), Bożek and Wysocki (2015),
Ivanov et al. (2016a,c) have pointed out that specific scheduling problems with complex hybrid logical
and terminal constraints, non-stationarity in process execution as well as complex interrelations between
dynamics in process design, capacity utilization, and machine setups require further investigation and
the application of a broad range of methodical approaches.
Optimal control approaches take a different perspective as mathematical programming methods which
represent schedules as trajectories. Optimal control applications to scheduling problems are encountered
in production systems with single machines (Giglio 2015), job sequencing in two-stage production sys-
tems (Lou and Van Ryzin 1989, Sethi and Zhou 1996) and multi-stage machine structures with alterna-
tives in job assignments and intensity-dependent processing rates such as flexible manufacturing sys-
tems (Sharifnia et al. 1991, Maimon et al. 1998, Yang et al. 1999, Ivanov and Sokolov 2013a, Pinha et
al. 2015), supply chains as multi-stage networks (Ivanov and Sokolov 2012, Ivanov et al. 2013), and
Industry 4.0 systems based on data interchange between the product and stations, flexible stations dedi-
cated to various technological operations, and real-time capacity utilization control (Ivanov et al.
2016a).
This survey considers research on optimal control applications to production scheduling with analysis of
model parameters and computational algorithms published in the last 55 years. The objectives of this
survey are twofold. The first objective is to derive major contributions, application areas, limitations, as
well as research and application recommendations for the future regarding optimal control applications
to scheduling. The second objective is to explain control engineering models in terms of industrial engi-
neering and production management. We provide a survey on the applications of optimal control to
scheduling in production, supply chain, and Industry 4.0 systems whereby we restrict ourselves to de-
terministic maximum principle-based approaches and omit detailed analysis of stochastic optimal con-
trol approaches as well as dynamic programming algorithms. Regarding the related topics which are not
covered in this paper, we refer to the surveys by Sethi (1978, 1984) for applications of the maximum
principle to production-inventory problems and to works (Lou et al. 1994, Sethi and Zhang 1994,
Presman et al. 1995, Samaratunga et al. 1997, Presman et al. 1997, 2000, Feng and Yan 2000, Sethi et
al. 2002, Khmelnitsky et al. 2011) which extend this survey to stochastic scheduling problems.
The survey follows the structure “optimal control models – performance analysis qualitative methods –
computational methods”. In Section 2, a brief historic overview and clarification of major mathematical
fundamentals are provided. The control engineering terms are brought into correspondence with indus-
trial engineering and management. In Section 3, optimal control models for scheduling in production,
supply chain, and Industry 4.0 systems are presented and classified in terms of their analytical contents
and application areas. Section 4 deals with attainable (reachable) sets as a method of qualitative analysis
of optimal control performance. Section 5 is devoted to computational algorithms regarding state, con-
trol, and adjoint variable spaces. In Section 6, we derive major contributions, application areas of differ-
ent methods, and their limitations. This section also provides recommendations for future research and
applications. The paper concludes in Section 7 by summarizing the insights from this survey.
2. Fundamentals of optimal control models with applications to scheduling
2.1. Historical development
Optimal control approaches represent schedules as trajectories. The first studies in this area were devot-
ed to inventory control. One of these studies (Eilon 1961) was published in the first volume of the Inter-
national Journal of Production Research (IJPR). Later, Hwang et al. (1967, 1969) were among the first
to apply optimal control and the maximum principle to multi-level and multi-period master production
scheduling which determined the optimal control (i.e., production) with the corresponding state (i.e.,
inventory) trajectory. Albright and Collins (1977) developed a Bayesian approach to the optimal control
of continuous industrial processes. Bedini and Toni (1980) developed a dynamic model for the planning
of a manufacturing system. The maximum principle has been used to formulate the problem and obtain
a solution. A large research area of flexible manufacturing systems and their dynamics has been exam-
ined in numerous studies (e.g., Stecke and Solberg 1981). The stream of production scheduling was
continued by Kimemia and Gershwin (1985), Kogan and Khmelnitsky (1996), and Khmelnitsky et al.
(1997), who applied the maximum principle in discrete form to the planning of continuous-time flows in
flexible manufacturing systems
The origins of control scheduling techniques can be found in network planning, dynamic programming
and waiting line theory (Moiseev 1974, Sotskov et al. 2013). Consider the graph in Fig. 1.
х
1
х
2
х
3
х
5
х
4
х
6
х
п–2
х
п–1
х
п
Fig. 1. Network planning graph
In the classical network planning theory, the state
i
x
of the i-job can be determined subject to Eq. (1):
i
i
ii
Q
Tx
, (1)
where T
i
is the task time to process the i-job, Q
i
is the processing volume of the i-job, and
i
is the in-
tensity (i.e., processing rate) of processing the i-job, whereby the jobs i= 1,...,n are interconnected with
each other in terms of precedence constraints “or”/“and”. It is notable that even if elements of dynamics
can be observed in the aforementioned system regarding process deployment in time, job execution
dynamics itself have not been considered explicitly. In other words, operations execution has been treat-
ed in a static manner since task times nwere assumed to be fixed. In reality, task times may change in
job execution dynamics. As such, dynamics of job execution requires an explicit description of job pro-
cess execution, distribution (allocation) of resources required for job execution, and changes in the job
states and the respective control inputs. The aforementioned dynamic interpretation of job execution
was extensively developed in the 1970s (Zimin and Ivanilov 1971, Moiseev 1974) within the framework
of optimal control theory.
Optimal control theory is devoted to determining some functions known as controls that lead to optimi-
zation(minimization or maximization) of an objective (Pontryagin et al. 1964, Athaus and Falb 1966,
Lee and Markus 1967, Moiseev 1974, Bryson and Ho 1975, Hartl and Sethi 1984, Soner 1986, Gersh-
win 1994, Sethi and Thompson 2000). This theory evolved over the centuries based on calculus varia-
tion principles developed by Fermat, Lagrange, Bernulli, Newton, Hamilton and Jacobi. In the 20
th
cen-
tury, two computational fundamentals of optimal control theory, the maximum principle (Pontryagin et
al., 1964) and the dynamic programming method (Bellmann 1972), were developed. These methods
extend the classical calculus variation theory which is based on control variations of a continuous trajec-
tory and observing the performance impact of these variations at the trajectory end. Since control sys-
tems in the middle of the 20
th
century were increasingly characterized by piecewise continuous func-
tions (such as 0-1 switch automats), the development of the maximum principle and the dynamic pro-
gramming was needed for solving problems with complex constraints on state and control variables.
This section aims to clarify the notions of state, control, and performance at the optimal control model
level, bridging these notions to industrial management and engineering. Moreover, the computational
level will be considered and the maximum principle, adjoint equation system, and transversality condi-
tions will be clarified.
2.2. Major elements of an optimal control model for scheduling
Consider the evolution of a quantifiable object (e.g., inventory or production quantity) in time
),u),(x,(f)(x ttt
f
ttt
dt
d
t
0
,
x
)(x
where
m
R
u
is the control,
n
Rx
is the terminal system state vector,
n
Rf
is a given function,
n
R
is
Euclid space of dimensionality
0
,tn
is the initial point in time and
f
t
is final point in time. For exam-
ple, in a job shop,
)(x t
can describe a buffer such as production quantity or inventory, and control
variable
)(tu
can describe processed flow volume of a job at a machine. The system state vector is de-
termined by the evolution of state variables
)(x t
that characterize the system at each point in time. State
evolution in dynamics is determined by control variables
)(tu
that correspond to the decisions of a per-
son or an algorithm governing the system dynamics.
In real practice of control engineering, control variables are typically considered bounded piecewise
continuous functions. Examples of controls in operational systems include processing rates ofmachines
in manufacturing or shipment rates in transportation. In production scheduling, binary control variables
are used to describe the assignment of a job to a machine. Consider an example (Fig. 2).
(t)=1
1
2
3
4
5
6
7
6
5
4
3
2
0
t
8
9
1
)(tx
10
11
12
13
(t)=1
(t)=1
1)(;)(
21
tuctu
)(
tx
