Expert Systems With Applications 190 (2022) 116180
Contents lists available at ScienceDirect
Expert Systems With Applications
journal homepage: www.elsevier.com/locate/eswa
Review
Dynamic Programming algorithms and their applications in machine
scheduling: A review✩
Edson Antônio Gonçalves de Souza a ,1 , Marcelo Seido Nagano a ,∗,2 , Gustavo Alencar Rolim a,b ,3
a
Department of Production Engineering, São Carlos School of Engineering, University of São Paulo, Trabalhador São-Carlense 400, 13566-590, São Carlos,
SP, Brazil
b
Department of Industrial Engineering, Technology Center, Federal University of Ceará, Campus do Pici, 60455-900, Fortaleza, CE, Brazil
ARTICLE
INFO
Keywords:
Scheduling
Dynamic Programming
Survey
Exact methods
FPTAS
ABSTRACT
This paper aims at presenting a compilation of state-of art references in which dynamic programming (DP)
and its variants have been applied as a solution methods for the deterministic machine scheduling problems.
Overall, 183 articles have been gathered and their segmentation was carried out according to the machine
environment that characterized the problems addressed by the authors and ultimately, the objective functions
that were intended to be optimized. Additionally, we standardized the information provided by each article by
presenting the problems discussed by the authors, comparisons between previous works on the same problem
(if it was deemed necessary), the algorithms’ complexities and an extension of methods to computational
experiments (in case they have been stated). Finally, at the end of each section we furnish a discussion on
the main contributions of DP to the each environment and also suggest some further applications of DP to
machine scheduling problems, thus showing the potential resources that can be derived from the method in
terms of theoretical/practical approaches.
1. Introduction
Pinedo (2012) defines machine scheduling as the allocation of
machines to a set of jobs whose objective is to establish the ordering
that returns an optimal value for a given objective function. All these
functions are usually denoted as criteria or performance measures and
they are often described as time functions. Although this may seem
a theoretical description of the problem, one must acknowledge its
relevance, since it is related to several manufacturing scenarios and the
optimization of such problem and variations of it are one of the pillars
to place a company ahead in terms of competitiveness and efficiency
regarding the production process.
Fuchigami and Rangel (2018) describe thoroughly the applications
of machine scheduling in several segments of industry, such as chemical, printed circuits, clothing, iron and steel, pharmaceutical and furniture, among others. Their analysis is recent and therefore, it can be
accurate about the trends in machine scheduling over the last 20 years.
An interesting fact of their research is the discussion of the methods that
have been currently employed in the solution of the various problems
related to scheduling and from a general standpoint, one can notice
that 71.19% of the articles gathered cover heuristics and metaheuristics
as solution methods. This leads to a much smaller contribution of
exact methods and, even though the authors have only taken into
account MILP, the participation of other methods cannot be compared
to MILP, since it is one of the most common methods when contrasting
performances among algorithms.
Recently, a review presented by Tomazella and Nagano (2020) lists
articles in which Branch and Bound (B&B) is exploited as a solution
method for the flow shop environment with numerous evidences that
this method, only for this flow pattern has been repeatedly employed.
The same can be stated for less complex environments (e.g. single machine and parallel machines). However, when comparing the shares of
MILP and B&B with other exact methods, it is noticeable that the former
are recurrent options for the authors, even when being programmed in
order to verify the efficiency of some non-exact methods and therefore,
the use of other exact methods became less frequent over time, as it is
the case of Dynamic Programming (DP).
✩ This document is the result of the research project funded by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES, Brazil.
∗ Corresponding author.
E-mail addresses: edags90@usp.br (E.A.G.d. Souza), drnagano@sc.usp.br (M.S. Nagano), gustavo.rolim@usp.br (G.A. Rolim).
1
Researcher.
2
Co-ordinator.
3
Co-author.
https://doi.org/10.1016/j.eswa.2021.116180
Received 1 March 2021; Received in revised form 21 September 2021; Accepted 29 October 2021
Available online 12 November 2021
0957-4174/© 2021 Elsevier Ltd. All rights reserved.
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
The foundations of DP can be traced back to the fifties and the first
topics related to it are found in Bellman (1952) with the introduction
of a class of problems whose aim is to seek an optimal set of choices
which, in turn, would lead to an optimal solution. Such class had been
originally presented for an stochastic framework and the author focused
on outlining the Existence and Uniqueness Theorem for the problem
described and its relation to a functional equation. A subsequent work is
developed in Bellman et al. (1954), which extended the previous notion
of probabilistic nature to the deterministic case and then, consolidated
DP as an exact method used to solve optimization problems. Furthermore, the definition of DP is presented and according to the authors
it can be defined as a method that divides an original problem into
sub-problems of smaller dimensions by choosing, at every step, the set
of decisions that returns the best result in terms of a given objective
function. Such function is essentially modeled via a recursive relation
(functional equation) that will ultimately generate the optimal solution
for the original problem as previous solutions are aggregated.
Hereafter, the communication between DP and several classic problems of Operations Research has been established. Held and Karp
(1962) are probably one of the most revisited authors regarding this
connection and they show how to employ DP to a general scheduling
problem by developing a recursive relation that is dependent of time
and whose objective is to obtain the sequence that minimizes the time
the jobs remain on the system. Although the resources used in this
paper might be outdated, it is deemed a stepping stone for DP and
machine scheduling, once it is able to demonstrate its applicability to
such problem and originate derived methods of DP that have enabled
improvements for particular scenarios over time.
The premises for employing DP are completely compatible with the
structure of machine scheduling problems, once the problem of finding
an optimal sequence can be partitioned into sub-problems that stores
partial sequences until the original problem is solved and the value of
the objective function can be obtained through a specific functional
equation. Throughout time, DP has been diversified in scheduling and,
despite not being the most applied method regarding this problem
in Operations Research, its contribution can be considered substantial
to the development of the machine scheduling field. Some of the
contributions are: development of polynomial time algorithms (Moore,
1968; Potts & Van Wassenhove, 1982) that are still widely applied in
contemporary approaches for scheduling problems, methods to reduce
the state–space inherent to the DP recursive relations (Baker, 1977;
Lawler, 1977; Schrage & Baker, 1978), theoretical background for generation of a large class of fully polynomial time approximation schemes
(Woeginger, 2000), creation of methods derived from the original DP
features (Ozolins, 2019b; Tanaka & Fujikuma, 2012), among others.
This paper aims at compiling state-of-art references that relate Dynamic Programming and machine scheduling to motivate researchers
that are interested in studying the connection between these two topics
or those that intend to analyze the evolution of DP, its achievements
and potential capabilities. Despite DP being considered a method that
might be strongly associated with issues especially for NP-hard problems, as it is the case of most of scheduling literature, one of the
objectives of this paper is to show the applications of DP via derived
methods, which may include approximation schemes and other efficient
heuristics. With that said, we can demonstrate the branches of DP
in scheduling and their importance for future perspectives. We also
attempt to associate complexities regarding the problem and the algorithms whenever it is given, once those are indicators of improvements
in the tractability of combinatorial problems. Furthermore, the paper
includes the most recent applications of DP to develop either theoretical
analysis or practical algorithms. It also presents the last trends of DP
as well as the gaps in the current literature. Lastly, this paper suggests
promising methods that might be suitable for problems that have not
been widely explored or alternative solutions to the development of the
field.
We structure the paper as follows: Section 2 introduces formally the
concept of DP with the main elements that are intrinsic to it. Section 3
encompasses the general framework of the paper by presenting the
methodology applied for this research as well as the notations that
will be used throughout the review. From Section 4 to Section 8, the
papers regarding single machine, parallel machine, flow shop, jobshop and open shop environments are introduced in a concise manner,
respectively. In addition, the subsections are distributed according to
the objective functions that can be grouped due to their akin nature
with considerations being provided at the end of each section. Section 9
encloses concluding remarks and the direction of DP regarding further
studies in machine scheduling.
2. Dynamic programming
Whenever we conceive the idea of optimal solution, the first method
that often rises is the explicit enumeration or brute force method, which
can be thought of as counting over every possible solution a given
system can provide. A variety of problems in Operations Research are
formulated under a combinatorial nature (e.g. machine scheduling,
TSP) and therefore, each different ordering represents a distinct combination that can be evaluated as optimum according to the value of
the objective function. From either mathematical and computational
standpoints, this method might be infeasible because such approach
reaches factorial complexity (𝑂(𝑛!)) and consequently, it solves only
small-sized instances.
Dynamic Programming and Branch and Bound methods can be
classified as implicit enumeration methods and their structure is usually
based on finding the optimal solution by not analyzing every possible
combination and, for this reason, the overall complexity associated to a
given method is reduced. A common feature that can be found in both
methods is the partition of the original problem in sub-problems that
contain a smaller dimension. Once every subset reaches the number of
elements of the original problem, usually denoted by 𝑛, the problem is
solved and the optimal value is found for the objective function.
Before fully comprehending the concept of DP, it is relevant to
introduce some preliminary elements that are common to either simple
or complex formulations regarding such method. These concepts will
guide the reader throughout the next sections and they represent the
core of DP that should be unfolded into mathematical and theoretical
analyses. Hence, such terms can be detailed as follows:
• Stage (𝑙): Associated to the given level a sub-problem can be
found. Usually designates the dimension or cardinality of decision
making variables that have been allocated.
• Control/State (𝑢𝑙 ): A control is the decision making variable
associated with stage 𝑙, i.e. the variable that represents a decision
that will be allocated at a given stage 𝑙. The set of decisions at
stage 𝑙 are denoted by 𝑈𝑙 , such that 𝑈𝑙 = {𝑢1𝑙 , … , 𝑢𝑡𝑙 }.
• State variable (𝑥𝑙 ): Real-valued function that stores the value
of the objective function for a given state 𝑢𝑙 that is chosen to
aggregate the solution set at a given stage 𝑙.
• Policy (𝛼): Denotes the set of states that have been selected
at each stage, i.e. the set of feasible solutions that is allowed
at a given stage. Once optimal states are of interest here, one
can define an optimal policy as the set of optimal states that
are adopted at each stage and, consequently, return the optimal
solution.
Clearly, this set of concepts might be enlarged to adjust to a specific problem’s requirements, however, these four definitions will be
common to every problem involving DP and once they have been
established we are able to formally introduce a concept for DP. Despite Bellman et al. (1954) introducing a consistent mathematical formalism for DP’s definition, a simpler and formal characterization is
given in Bellman (1966) in which the author defines DP as a method
2
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 1
Processing times and description of function ℎ𝑗 (𝐶𝑗 ).
dedicated to the investigation of multistage decision processes that
demand a sequence of decisions over time. These decisions can be
accounted as sub-problems solutions and this is one of the premises
that confirm the viability of applying DP to a given problem.
Over the years more robust definitions have emerged in literature.
In the present days, a definition that seems to be conventional among
authors is the one seen in Bautista and Pereira (2009). The authors state
that DP is a technique that partitions the original problem into smaller
sub-problems, which are solved sequentially until the original problem
is solved through aggregations of solutions obtained in the previous
states. Additionally, it is also defined as a multistage graph 𝐺(𝑈𝑙 , 𝑇 ),
with 𝑇 being the transitions that occur from a stage to the next. Note
that 𝑈𝑙 corresponds to the set of vertices of the graph that can be stored
at the same level and 𝑇 is analogous to the edges of a graph.
The second premise that governs the proper use of DP to a problem
relies on the formulation of the problem via a recursive equation, which
is essentially a functional equation. Mathematically, a general model to
describe DP can be represented as follows
𝑓𝑙 (𝑢𝑙 , 𝑈𝑙 ) = min {𝑔𝑙 (𝑥𝑙 , 𝑈𝑙 ) + 𝑓𝑙−1 (𝑢𝑙−1 , 𝑈𝑙−1 )},
𝑢𝑙 ∈𝑈𝑙
𝑙 = 1, … , 𝑁
ℎ𝑗 (𝐶𝑗 )
4
3
6
𝐶1 + 𝐶12
3 + 𝐶23
8𝐶3
𝑖∈{𝑗⊕𝑆}
∀𝑡 ∈ 𝑆.
𝑘∈{𝑗⊕𝑆}
(3)
In order to find the minimum sum of this general function of
completion times, we need to find the best sequence that can yield this
sum. Let us consider the data in Table 1 for a 3-job problem:
We can calculate the allocation of the jobs at the first stage by using
the initial condition. Since each job is placed in the first position we
obtain
(1)
𝑓1 (1, ∅) = 4 + 16 = 20,
𝑓1 (2, ∅) = 3 + 27 = 30,
𝑓1 (3, ∅) = 8 × 6 = 48.
At the second stage we start calculating via the recursive equation
𝑓2 (2, {1}) = min{ℎ1 (𝑝1 + 𝑝2 ) + 𝑓1 (2, ∅); ℎ2 (𝑝1 + 𝑝2 )
+ 𝑓1 (1, ∅)} = min{56 + 30; 346 + 20} = 86
𝑓2 (3, {1}) = min{ℎ1 (𝑝1 + 𝑝3 ) + 𝑓1 (3, ∅); ℎ3 (𝑝1 + 𝑝3 ) + 𝑓1 (1, ∅)}
= min{110 + 48; 80 + 20} = 100
𝑓2 (3, {2}) = min{ℎ2 (𝑝2 + 𝑝3 ) + 𝑓1 (3, ∅); ℎ3 (𝑝2 + 𝑝3 ) + 𝑓1 (2, ∅)}
Theorem 1. An optimal policy has the property that whatever the initial
state and initial decisions are, the remaining decisions must constitute an
optimal policy with no regard to the state resulting from the first decisions.
= min{732 + 48; 72 + 30} = 102.
The last stage is calculated by assembling the jobs in one set. Therefore
𝑓3 (3, {1, 2}) = min {ℎ1 (𝑝1 + 𝑝2 + 𝑝3 ) + 𝑓2 (3, {2}); ℎ2 (𝑝1 + 𝑝2 + 𝑝3 )
Essentially, the principle correlates the optimization of the original
problem and the sub-problems. In other words, if the optimal policies
have been previously selected for the preceding stages and optimal
policies are assumed for the remaining ones, we guarantee that the
optimal solution will be found for the initial problem. Moreover, the
principle indicates that remaining states are dependent only on the
current stage rather than the initial ones.
Regarding the inner methods that are bound to the DP approach,
it is valid to mention that throughout literature references, extensive
records have been found involving forward and backward recursion.
The first type is characterized by placing the initial condition at stage
zero and gradually aggregating the states so as to obtain the objective
function at the final stage 𝑁. The second one works analogously,
however we consider an initial condition at the last stage and the
objective function is achieved at the first stage. Variations of those may
be also found, however their usage is not often employed and therefore,
will not be mentioned.
In order to apply the concepts introduced here, we present an example of dynamic programming applied to a single machine scheduling
problem aiming at minimizing a general function.
+ 𝑓2 (3, {1}); ℎ3 (𝑝1 + 𝑝2 + 𝑝3 ) + 𝑓2 (2, {1})}
= min{182 + 102; 2200 + 100; 104 + 86} = 190.
The example shown above is actually a generalization of functions
that deal with completion times (e.g. total tardiness, earliness, among
others). Despite displaying only the value of the minimum sum, it
is also necessary to be thorough in this analysis and present the
sequence that yields such value. As we can notice, this is a forward
DP mechanism because an initial condition is set at the first stage
rather than the last one and the sequence can be obtained by using
backtracking analysis derived from the optimal value. We may notice
that the minimum was attained by adding job 3 to 𝑓2 (2, {1}) and the
minimum in this set was originated from job 1 being placed after job
2. Therefore, the optimal sequence is given by 2-1-3.
After confirming the advantages of DP approach, the reader might
question the disadvantages of such method for several problems, mostly
those that involve an allocation process. Bellman and Lee (1978) explore the use of several functional equations for specific problems in
Operations Research and they state that as the number of state variables increases, so does the computational effort in terms of memory
requirements and CPU time and it can be justified due to the recursive
nature of the functional equations that compose the premises of DP.
Such effect is known as Curse of Dimensionality and it is clearly one of
the reasons related to the limited use of this approach to solve problems
in the optimization field.
Over the years, due to this disadvantage and the fact that most
scheduling problems are NP-hard, DP has not been a constant choice
of authors in the field as it can be seen for other exact methods and
Branch-and-Bound, Mixed Integer Linear Programming (MILP), among
others. Due to this disadvantage and the fact that most scheduling
Example 1. Let 𝑗 be a given job such that 𝑗 ∈ 𝐽 that must be processed
on a given machine. The interest of the scheduler is to minimize
the sum of a function ℎ𝑗 , such that 𝑓𝑗 = ℎ𝑗 (𝐶𝑗 ), with 𝐶𝑗 being the
completion time of job 𝑗 on the machine. By Graham’s notation, this
∑
problem can be denoted by 1∥ 𝑓𝑗 (𝐶𝑗 ). According to the definition
previously described we need a recursive relation for calculation the
objective function and an initial condition as an input for the recursion.
Therefore we define the initial condition as
∀𝑗 ∈ 𝐽
𝑝𝑗
1
2
3
and the recursive relation is given by
{ (
)
}
∑
𝑓𝑙 ({𝑗}, 𝑆) = min
ℎ𝑖
𝑝𝑘 + 𝑓𝑙−1 ({𝑡}, 𝑆 − {𝑡}) ,
Eq. (1) characterizes the functional relation that represents a recursive
mechanism related to DP. The function 𝑔𝑙 (𝑥𝑙 , 𝑢𝑙 ) indicates the cost
associated to the allocation of state 𝑢𝑙 at stage 𝑙 and the function
𝑓𝑙−1 (𝑢𝑙−1 , 𝑈𝑙−1 ) denotes the optimal objective function for the previous
stages and one can see that it is the main term related to the recursion
effect. As a consequence of this fact, it is worth mentioning that
Eq. (1) is only one of the several types of formulations designed for DP
functional equations, however all of them are derived from a common
factor known as Bellman’s Principle of Optimality, which is stated as
follows.
𝑓1 ({𝑗}, ∅) = ℎ𝑗 (𝑝𝑗 ),
𝑗
(2)
3
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
of articles that each subsection contains. For, example, as we shall
see on the next section, the number of tardy jobs criterion presents
a considerable amount of papers with regard to the single machine
environment and, therefore it is disjointed from the classic due date
related criteria subsection.
The two last determining conditions that have been set as a filter
were the relevance of scheduling, i.e. whether scheduling was the main
topic on that paper or just included as an intermediate process for
another part of the supply chain, and the nature of the scheduling
problem according to their deterministic or stochastic formulations.
Hence, for the first filter, the majority of papers that have been gathered
were verified through that analysis, except those that explored the
batching constraint, which have shown stronger features towards other
systems such as demand forecast and allocation and transportation
analysis. The second filter would include only deterministic cases in
our research because the number of stochastic cases would not be
significant in this analysis and also they would be scattered throughout
the environments.
More precisely, 219 papers linking DP and machine scheduling
have been initially found from 1952 to August 2020. According to our
methodology of inclusion/exclusion, 34 papers have not been considered, given that 13 of those corresponded to the stochastic approach of
the problem and 21 involved batching features that would not consider
scheduling as main topic. Hence, 183 articles have been included
and can be subdivided in 63 papers concerning the single machine
environment, 66 papers encompassing parallel machines, 32 dealing
with flow shop scheduling, 10 for the job shop environment and the
remaining 12 open shop scheduling. Furthermore, we have decided
to summarize the information concerning each article by introducing
the notations to describe the approached problems, a description of
how DP has been used, the algorithms’ complexity, in case they have
been provided, and lastly, results related to computational experiments,
when applicable.
problems are NP-hard, DP may not be a frequent choice in comparison
with other exact methods. However, a great advantage of DP is the
absence of the fixed formalism required by mathematical programming
approaches. This enables DP to not only rewrite these other methods
from its perspective but also to extend their possibilities.
Additionally, since it has been perceived that this is a useful trait of
DP for optimization problems in general, some authors in scheduling
have been concentrating their researches in developing hybrid algorithms that combine features of DP and other exact algorithms such
as B&B, Branch-and-Cut and Column Generation method. Also, due to
the mutability to heuristics, some authors have also been focused on
enhancing the quality of DP to already existing methods as well as the
generation of new methods that have gradually proven that the Curse
of Dimensionality is a disadvantage that can be overcome for larger
instances if additional properties of DP are explored.
3. Notations and methodology
This section is majorly part of the methodological comprehension
that entails the motives related to the production of this paper. It has
been divided into two subsections with the first one gathering the
topics that justify the collection of articles that have been included in
our research. The second subsection comprises the notations that are
commonly known to the machine scheduling experts with regard to the
environment, technological constraints and objective functions.
3.1. Methodology
The primary concern of this research was to define its purpose
based on the applications of DP to machine scheduling. After a series
of discussions we decided that this paper would represent a guide for
researchers in terms of the progress and the trends that have been
developed for DP in the machine scheduling and therefore, we defined
this paper as one of informative characteristics. Once the field is very
extensive, the filtering was set according to the classic environments
because by using them as keywords we would be able to cover a large
spectrum of papers concerning each environment and their variations
and also to gather information on correlated distributions of machines,
which has defined the first separation method.
The next step was to determine the most appropriate browsing
sources from which the papers would be extracted. Therefore we
adopted two types of sources: general and specific. The general source
was Google Scholar because it usually furnishes a larger number of
papers, even though some of them are not part of the previously defined
scope. The specific sources were Scopus and Web of Science because their
search method is more refined than that presented by Google Scholar,
even though some papers might not be included in their database.
Hence, the filtering process starts with selecting the papers in the
specific sources and, a posteriori, overlapping those with the papers
extracted from the general sources.
The search for articles depended on a standardized set of keywords
that can be seen in Table 2. These words were the main ones we
considered because their combination has generated distinct results and
the major portion of those have been included in our research. It is valid
to highlight that regarding the flow shop and job shop environments, it
was preferred to focus on the classic configuration of those because the
contributions of DP to the variations of such environments (e.g. flexible
flow shop, flexible job shop, non-permutation flow shop) have been
only related to references throughout the papers.
Sequentially, we have defined another segmentation within each
machine environment that is relative to the objective function. In order
to cover as many papers as possible and organize them with a reduced
number of subsections we divided the objective functions according to
their nature: classic due date related criteria, makespan and completion
time related criteria, multi-criteria and additional objective functions.
Clearly this configuration could be modified depending on the number
3.2. Notations
A problem in machine scheduling is defined as follows: Let 𝑘 ∈ 𝐾
be a given machine, with 𝐾 denoting the set of machines and 𝑗 ∈ 𝐽 be
a given job, which constitutes a set of jobs 𝐽 . The objective is to seek
the best ordering of jobs that will be processed on the machines, which
is denoted by 𝜋 ∈ 𝛱, such that it yields the optimal value depending
on the function the scheduler wishes to optimize. The time spent by
the job on a given machine is given by the processing time 𝑝𝑗𝑘 and
it is one of the fundamental parameters in machine scheduling. The
other parameters that are also representative in the field are the due
dates (𝑑𝑗 ), deadlines (𝑑 𝑗 ), common due dates (𝑑), due window (𝑑𝑤𝑗 )
and common due window (𝑑𝑤 ). All those have a relationship with the
moment in time the jobs should finish their processing and the first
three parameters are set for a given value, while the remaining ones
are set for a given interval.
The notation introduced in Graham, Lawler, Lenstra, and Kan
(1979) is still widely used in papers and books that are reference
in scheduling, even though recent ones have been developed. This
notation is based on three parameters as follows: 𝛼|𝛽|𝛾. Parameter 𝛼
is based on the pattern flow of the problem, which can be defined as
the manner the jobs flow through the production system or, in other
words, the environment that defines the problem. This parameter will
also be associated with the number of machines relative to the problem.
Parameter 𝛽 is related to the technological constraints in the problem.
In other words, there is a constraint added (e.g. blocking) that will
alter the configuration of the general problem. The last parameter,
𝛾, represents the criterion or criteria to be optimized. It is necessary
to understand and recognize the main elements that each parameter
may assume. A detailed description provided by Pinedo (2012) of these
elements is given as follows:
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E.A.G.d. Souza et al.
Table 2
Table concerning keywords related to the scope of the research.
Environment/Method
Keywords description
Single machine
‘‘single machine’’ and ‘‘dynamic programming’’;
‘‘single machine scheduling’’ and ‘‘dynamic programming’’;
‘‘single machine’’ and ‘‘DP’’
Parallel machines
‘‘parallel machine’’ and ‘‘dynamic programming’’;
‘‘parallel machine scheduling’’ and ‘‘dynamic programming’’;
‘‘parallel machine’’ and ‘‘DP’’
Flow-shop
‘‘flowshop scheduling’’ and ‘‘dynamic programming’’
‘‘flow-shop scheduling’’ and ‘‘dynamic programming’’;
‘‘flow shop scheduling’’ and ‘‘dynamic programming’’;
‘‘flowshop scheduling’’ and ‘‘DP’’
‘‘flow-shop scheduling’’ and ‘‘DP’’;
‘‘flow shop scheduling’’ and ‘‘DP’’
Job shop
‘‘job-shop
‘‘job shop
‘‘job-shop
‘‘job shop
Open shop
‘‘open shop scheduling’’ and ‘‘dynamic programming’’;
‘‘open shop scheduling’’ and ‘‘DP’’
Dynamic programming
‘‘dynamic
‘‘dynamic
‘‘dynamic
‘‘dynamic
‘‘dynamic
scheduling’’
scheduling’’
scheduling’’
scheduling’’
and
and
and
and
programming
programming
programming
programming
programming
‘‘dynamic programming’’;
‘‘dynamic programming’’;
‘‘DP’’;
‘‘DP’’
applied’’
applied’’
applied’’
applied’’
applied’’
to
to
to
to
to
the
the
the
the
the
‘‘single machine scheduling problem’’;
‘‘parallel machine scheduling problem’’;
‘‘flow shop scheduling problem’’;
‘‘job shop scheduling problem’’;
‘‘open shop scheduling problem’’
Table 3
Common notations for machine scheduling problems.
Parameter
Denomination
Notation
Additional comments
𝛼
Single machine
Identical parallel machine
Uniform parallel machine
Unrelated parallel machine
Flow shop
Job shop
Open shop
1
𝑃𝑚
𝑄𝑚
𝑅𝑚
𝐹𝑚
𝐽𝑚
𝑂𝑚
–
–
–
–
–
–
–
Release dates
Precedence
Preemption
Resumable jobs
Semi-resumable jobs
Non-resumable jobs
Sequence independent setup times
Sequence dependent setup times
Sequence independent setup family
Sequence dependent setup family
Controllable processing times
Learning effect
Aging effect
Blocking
Constrained rejection
Permutation
No-wait
𝑟𝑗
𝑝𝑟𝑒𝑐
𝑝𝑟𝑚𝑝
𝑟 − 𝑎(𝑀𝑘 )
𝑠𝑟 − 𝑎(𝑀𝑘 )
𝑛𝑟 − 𝑎(𝑀𝑘 )
𝑠𝑗 ∕𝑠𝑗𝑘
𝑠𝑖𝑗 ∕𝑠𝑖𝑗𝑘
𝑆𝐼𝑓
𝑆𝐷𝑓
𝑝(𝑦)
𝐿𝑒
𝐴𝑒
𝑏𝑙𝑜𝑐𝑘
∑ 𝑟𝑒𝑗
𝜃𝑗 ≤ 𝐾
𝑝𝑟𝑚𝑢
𝑛𝑤𝑡
–
–
–
–
–
–
–
–
–
–
Examples of 𝑦: 𝑟𝑗 , 𝑠𝑡𝑗 , 𝑥𝑗
–
–
–
𝐾 represents an upper bound
–
–
Makespan
𝐶max
𝐶max = max 𝐶[𝑛]𝑚
Maximum tardiness
Total tardiness
Total weighted tardiness
𝑇max
∑
𝑇
∑ 𝑗
𝑤𝑗 𝑇𝑗
𝑇𝑗 =
Maximum lateness
𝐿max
𝐿𝑗 = |𝐶𝑗 − 𝑑𝑗 |
Number of tardy jobs
Weighted number of tardy jobs
∑
𝑈
∑ 𝑗
𝑤𝑗 𝑈𝑗
𝑈𝑗 =
Total flow time
Total weighted flow time
∑
𝐹
∑ 𝑗
𝑤𝑗 𝐹𝑗
𝐹𝑗 = 𝐶𝑗 − 𝑟𝑗
Total earliness
Total weighted earliness
∑
𝐸
∑ 𝑗
𝑤𝑗 𝐸𝑗
𝐸𝑗 = max1≤𝑗≤𝑛 {0, 𝑑𝑗 − 𝐶𝑗 }
Total late work
∑
Total weighted late work
∑
𝛽
𝛾
5
𝑌𝑗
𝑤𝑗 𝑌𝑗
{
𝐶𝑗 − 𝑑𝑗 , if 𝐶𝑗 > 𝑑𝑗
0,
otherwise
{
1, if 𝐶𝑗 > 𝑑𝑗
0, otherwise
⎧0,
if 𝐶𝑗 ≤ 𝑑𝑗
⎪
𝑌𝑗 = ⎨𝐶𝑗 − 𝑑𝑗 , if 𝑑𝑗 < 𝐶𝑗 < 𝑑𝑗 + 𝑝𝑗
⎪
if 𝑑𝑗 + 𝑝𝑗 ≤ 𝐶𝑗
⎩𝑝𝑗 ,
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Clearly, other functions might appear but they are derived from
those that have been outlined in Table 3 or just have the converse
effect (e.g. early work criteria) and therefore they are not detailed in
this section. Nevertheless, whenever a new notation surfaces, we will
mention them in order to clarify it to the reader. The same can be stated
for the technological constraints. For further details we refer to Rolim
and Nagano (2020).
which are included to reduce the state space of feasible sets. It is
shown that decomposition algorithms outperformed precedence ones
with the most efficient one being able to solve 100 jobs problems within
reasonable time.
Abdul-Razaq, Potts, and Van Wassenhove (1990) compare two DP
formulations based on Lawler and Schrage–Baker works and four B&B
algorithms with lower bounds derived from Lagrangian, exponential,
∑
linear and state–space relaxations for the 1∥ 𝑤𝑗 𝑇𝑗 problem. The best
results have relied on the B&B algorithms, especially the one with linear
lower bound, which managed to solve up to 40 jobs. The DP algorithms
were limited by core storage as problem sizes grew larger, despite
yielding small time CPU times.
Sen and Borah (1991) develop a branching scheme based on order∑
ing theorems to compare it to a DP approach for solving the 1∥ 𝑇𝑗
problem. Even though the comparison was not made to measure efficiency, parameters such as dispersion of CPU times and generated state
space have been provided. The branching scheme presents a smaller
solution set, whereas a more scattered group in CPU times.
Tang, Xuan, and Liu (2007) aim at a different approach for the
∑
1|𝑝𝑟𝑒𝑐| 𝑤𝑗 𝑇𝑗 problem by proposing a Lagrangian relaxation of the
model and applying a hybrid backward and forward DP algorithm for
finding optimal solutions. This paper is considered a breakthrough in
the field due to its capacity of assessing all precedence relationships
and for also showing a faster convergence regarding the solutions for
small and large sized problems.
Tuong, Soukhal, and Billaut (2010) discuss general aspects from
the algorithm proposed by Lawler and Moore (1969) to solve the
∑
1|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑇𝑗 problem and some drawbacks such as the exclusion of
straddling jobs are pointed out. Therefore, they propose a DP algorithm
that takes into account that possibility, which is proven to have 𝑂(𝑛2 𝑑)
complexity.
Zhang, Lu, and Yuan (2010) explore the nature of several problems
that include the penalty for rejected jobs as a bounded constraint
(denoted here by 𝜃𝑗𝑟𝑒𝑗 ), which has been drawing considerable attention
lately since rejection might imply reduction in costs for jobs that have
larger processing times and do not alter significantly the composition of
∑
the process. The authors prove that 1| 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾|𝐿max is NP-complete
and they develop a DP formalism in which cases of acceptance and
∑
rejection are modeled recursively with 𝑂(𝑛𝐾 𝑝𝑗 ) complexity. In addition, a fully polynomial time approximation scheme (FPTAS) is derived
for this problem
Tanaka and Fujikuma (2012) develop a framework to solve several
problems by improving the successive sublimation DP algorithm seen
in Tanaka, Fujikuma, and Araki (2009). These improvements take into
account relaxations, reduction methods and shortened connections in
networks. The algorithm is able to solve optimally 80-job instances for
the 1|𝑟𝑗 |𝑤𝑗 𝑇𝑗 problem, thus outperforming previous methods.
∑
Tanaka and Sato (2013) take interest in studying the 1|𝑝𝑟𝑒𝑐| 𝑤𝑗 𝑇𝑗
by applying Lagrangian relaxation and reductions in paths generated
by the precedence networks that compose the proposed successive
sublimation DP algorithm. The algorithm is able to solve all 50-job
instances for the problem and most of the 100-job instances.
Rostami, Creemers, and Leus (2019) gather information on precedence theorems in order to develop a more efficient exact algorithm
∑
to solve the 1|𝑝𝑟𝑒𝑐| 𝑤𝑗 𝑇𝑗 problem. They construct a DP algorithm
with inbuilt formulations related to those theorems in order to diminish
the computational effort in generation of states and compare it to that
proposed by Tanaka and Sato (2013). The results were superior for the
most recent method, since it could solve some set of instances that had
not been solved by its predecessor and also larger ones.
Mor and Shapira (2020) investigate the use of theoretical formula∑
∑
tions in DP in order to solve the 1|𝑑𝑗 = 𝑑, 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾| 𝑇𝑗 , 1|𝑑𝑗 = 𝑑,
∑ 𝑟𝑒𝑗
∑
𝜃𝑗 ≤ 𝐾| 𝑤𝑗 𝑇𝑗 problems via computational tests and the complex∑
ities associated with them are 𝑂(𝑛𝐾 𝑝𝑗 ) and 𝑂(𝑛2 𝑑𝐾). The computational results show that good outcomes can be obtained for at most 150
jobs and a varying rate of rejected jobs within a short time for the first
problem and for at most 40 jobs regarding the second type of problem.
A summary of the papers in this subsection can be found in Table 4.
4. Single machine
Due to its simplicity when compared to other environments in
scheduling, the variations that come with this problem are abundant
regarding the constraints as well as the objective functions that present
potential interest in manufacturing. Therefore, experimenting on a
single machine and finding solutions to tackle variants related to it
may have positive impact on developing adapted methods for far
more complex systems. Among those solutions, dynamic programming
has shown substantial contribution when applied to single machine
problems, for both classic and contemporary approaches.
4.1. Classic due-dates related criteria
∑
∑
Lawler and Moore (1969) model the 1∥ 𝑤𝑗 𝐸𝑗 , 1|𝑑, 𝑑| 𝑇𝑗 and
∑
1|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑇𝑗 problems as knapsack-like formulations and, based on
that premise, a DP approach is obtained from a previous DP solution
for the knapsack problem. In addition, the formulation is combined
∑
with a non-increasing 𝑤𝑗 ∕𝑝𝑗 priority rule for 1∥ 𝑤𝑗 𝐸𝑗 , a SPT rule
∑
∑
for 1|𝑑, 𝑑| 𝑇𝑗 and a non-decreasing 𝑤𝑗 ∕𝑝𝑗 rule for 1|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑇𝑗 .
Complexities for the first and third DP approaches are given by 𝑂(𝑛𝑑)
and 𝑂(𝑛2 𝑑), respectively.
Srinivasan (1971) applies the theoretical background available on
precedence relations to a DP formulation ("hybrid" algorithm) to solve
∑
the 1∥ 𝑇𝑗 via the principle of optimality. The tests are performed in
data sets with up to 12 jobs and the CPU times results are far better
than those obtained by complete enumeration and full DP approach.
Lawler (1977) proposes a pseudo-polynomial algorithm endorsed
by properties of agreeable jobs and a DP formulation based on the
∑
principle of optimality to solve the 1∥ 𝑤𝑗 𝑇𝑗 problem. Although some
options have been offered and employed to reduce the algorithm’s
∑
complexity, the author proves it to be 𝑂(𝑛4 𝑝𝑗 ) or 𝑂(𝑛5 max{𝑝𝑗 }) in
a worst-case scenario.
Baker (1977) focuses on applying the dominance rules proposed
∑
by Emmons (1969) to find efficient solutions for the 1∥ 𝑇𝑗 . By coupling those rules with a DP formulation, the author manages to develop
a forward recursion DP chain algorithm. For a 15-job data set the
algorithm produces better results in terms of CPU times and storage
than conventional DP and the one proposed by Srinivasan (1971). The
comparison with a dual algorithm for a 20-job set generates better
results, however the storage requirements start approximating those in
conventional DP once cardinality of jobs is over 30 units.
Schrage and Baker (1978) consider an efficiency aspect of the DP
formulation by devising a labeling scheme in order to reduce the
computational effort when storing information from a previous stage
for problems with precedence constraints. The method is applied to
∑
∑
1|𝑝𝑟𝑒𝑐| 𝑇𝑗 and 1|𝑝𝑟𝑒𝑐| 𝑤𝑗 𝑇𝑗 problems and compared to a chain
algorithm, proving that DP outperforms the chain algorithm since the
former explores a smaller number of feasible sets.
Potts and Van Wassenhove (1982) combine features of the decomposition properties proposed by Lawler (1977) and labeling scheme
studied in Schrage and Baker (1978) in order to introduce a more
∑
powerful DP algorithm to solve 1∥ 𝑇𝑗 , which resorts to decomposition
if a given storage threshold is achieved. The experimental sets contain
data ranging from 50 to 100 jobs, which are solved within reasonable
time for instances up to 70 jobs and is proven to have 𝑂(𝑛4 ) complexity.
Potts and Van Wassenhove (1987) present a series of adapted DP
∑
algorithms to solve the 1∥ 𝑇𝑗 problem. The authors divide the algorithms according to their precedence or decomposition properties,
6
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 4
Due-date related problems for single machine.
Authors
Lawler and Moore (1969)
Problem notation
∑
𝑤𝑗 𝐸𝑗
∑
1|𝑑, 𝑑| 𝑇𝑗
∑
1|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑇𝑗
1∥
1∥
∑
Lawler (1977)
1∥
∑
Baker (1977)
1∥
∑
Srinivasan (1971)
Schrage and Baker (1978)
𝑝𝑗 ) or 𝑂(𝑛 max 𝑝𝑗 )
DP
–
DP
𝑤 𝑗 𝑇𝑗
–
DP
𝑇𝑗
𝑂(𝑛4 )
DP
𝑇𝑗
–
DP
𝑤𝑗 𝑇𝑗
–
DP and B&B
𝑇𝑗
–
1∥
1∥
∑
Sen and Borah (1991)
1∥
∑
Cheng and Ding (2000)
1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑑𝑗 |𝐿max
Tang et al. (2007)
1|𝑝𝑟𝑒𝑐|
∑
1|𝑑𝑗 = 𝑑|
𝜃𝑗𝑟𝑒𝑗
∑
1|𝑝𝑟𝑒𝑐|
Rostami et al. (2019)
1|𝑝𝑟𝑒𝑐|
∑
1|𝑑𝑗 =
1|𝑑𝑗 =
𝑂(𝑛 log 𝑛)
DP
𝑤 𝑗 𝑇𝑗
–
Hybrid DP
∑
𝑂(𝑛2 𝑑)
𝑤 𝑗 𝑇𝑗
𝑂(𝑛𝐾
∑
DP
𝑝𝑗 )
DP and FPTAS
–
SSDP
𝑤 𝑗 𝑇𝑗
–
SSDP
𝑤 𝑗 𝑇𝑗
–
𝑇𝑗
Tanaka and Sato (2013)
DP
6
≤ 𝐾|𝐿max
∑
Mor and Shapira (2020)
5
𝑇𝑗
∑
Abdul-Razaq et al. (1990)
1|𝑟𝑗 |
∑
DP
∑
Potts and Van Wassenhove (1987)
Tanaka and Fujikuma (2012)
DP
4
–
∑
1|
DP
𝑂(𝑛
1∥
Zhang et al. (2010)
DP
𝑂(𝑛2 𝑑)
𝑤𝑗 𝑇𝑗
1|𝑝𝑟𝑒𝑐|
∑
DP
–
𝑇𝑗
∑
Tuong et al. (2010)
Method
𝑂(𝑛𝑑)
–
𝑇𝑗
1|𝑝𝑟𝑒𝑐|
Potts and Van Wassenhove (1982)
Complexity
∑
𝑑, 𝜃𝑗𝑟𝑒𝑗
∑
𝑑, 𝜃𝑗𝑟𝑒𝑗
≤ 𝐾|
≤ 𝐾|
∑
∑
DP
∑
𝑝𝑗 )
𝑇𝑗
𝑂(𝑛𝐾
𝑤𝑗 𝑇𝑗
𝑂(𝑛2 𝑑𝐾)
DP
DP
(R). The most promising results are obtained from the R+BBDP and
R+DPLM algorithms, since they can be used for solving large problems
up to 1000 jobs.
Lawler (1990) incorporates aspects of a preemptive EDD priority
∑
rule to a DP formulation to solve the 1|𝑟𝑗 , 𝑝𝑟𝑚𝑝| 𝑤𝑗 𝑈𝑗 and
∑
1|𝑟𝑗 , 𝑝𝑟𝑚𝑝| 𝑈𝑗 . The DP algorithm is classified as pseudo-polynomial
with 𝑂(𝑛𝑘2 𝑊 2 ) time complexity and 𝑂(𝑘2 𝑊 ) space complexity for the
first problem and with time complexity 𝑂(𝑛3 𝑘2 ) for the second, where
∑
𝑘 is the number of distinct release dates and 𝑊 = 𝑤𝑗 . The algorithm
∑
is also extended to the special case 1|𝑟𝑗 | 𝑈𝑗 , producing the knapsack
DP formulation seen in Lawler and Moore (1969).
In Hariri and Potts (1994) a bounding scheme is presented by using
state–space relaxation and job penalties dependent on DP formulations
∑
to develop a B&B to solve 1|𝑑 𝑗 | 𝑤𝑗 𝑈𝑗 . The experiments range from
50 to 300 jobs and good results are obtained, provided that processing
∑
times are not too large since the algorithm is bounded by 𝑂(𝑛 𝑝𝑗 ) time
complexity.
Lee (1996) investigates a series of optimization problems involving
resumable and non-resumable availability constraints. Among them,
the author points that the algorithm in Moore (1968) with the inclusion of the unavailability period in the completion times solves the
∑
1|𝑟 − 𝑎| 𝑈𝑗 problem optimally.
∑
Dondeti and Mohanty (1998) model the 1|𝐹 𝑒, 𝐿𝑒| 𝑤𝑗 𝑈𝑗 as a 0–1
knapsack problem and combine a DP formulation with a maximumweighted network path algorithm in order to solve it. The validation
of such equivalence is established by lemmas based on EDD rule
and modifications on the recursive relation used for building the DP
algorithm.
∑
Baptiste (1999a) shows several properties for the 1|𝑟𝑗 , 𝑝𝑟𝑚𝑝| 𝑈𝑗
problem, which are the foundation for a DP algorithm devised for
solving such problem. The author is able to reduce time and space
complexities (compared do Lawler’s), respectively, from 𝑂(𝑛5 ) to 𝑂(𝑛4 )
and 𝑂(𝑛3 ) to 𝑂(𝑛2 ). Experiments to test the DP efficiency are designed
for 50 and 100 jobs and solved in reasonable CPU times.
4.2. Number of tardy jobs
∑
Moore (1968) presents a DP algorithm to solve the 1∥ 𝑈𝑗 problem,
by including SPT, LPT and earliest due date rules in its composition. It is
classified as a polynomial time method with 𝑂(𝑛 log 𝑛) time complexity.
A DP approach has also been found in Lawler and Moore (1969).
The authors use similar mechanisms seen in Classic due-dates related
criteria and resort to an earliest due date (EDD) priority rule to solve
∑
the 1∥ 𝑈𝑗 problem. Tardy jobs are identified whenever zero processing times are yielded from the DP formulation and its complexity is
∑
given by 𝑂(𝑛 min{ 𝑝𝑗 , max{𝑑𝑗 }}).
∑
Sahni (1976) uses its previous work concerning 1∥ 𝑤𝑗 𝑈𝑗 to formulate a DP algorithm that also employs EDD priority rule. In addition, an
elimination criterion is applied so as to reduce the state space by comparing tuples that include 𝑝𝑗 and 𝑤𝑗 . This generates a 𝑂(min{2𝑛 , 𝑛𝑀})
∑
∑
time complexity algorithm, where 𝑀 = min{ 𝑤𝑗 , 𝑝𝑗 , 𝑑𝑛 } + 1 tuples. The author also studies properties and presents an approximation
algorithm to solve the same problem.
∑
Kise, Ibaraki, and Mine (1978) aim at minimizing the 1|𝑟𝑗 | 𝑈𝑗
problem by imposing a non-decreasing restriction such that 𝑟𝑖 ≤ 𝑟𝑗
implies 𝑑𝑖 ≤ 𝑑𝑗 . The authors design an 𝑂(𝑛2 ) polynomial time DP
algorithm and validate its capacity to solve the problem.
Gens and Levner (1981) design an approximation scheme to min∑
imize the 1|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑈𝑗 problem by inserting a DP scheme with
a proposed modification to Sahni’s approximation algorithm so as to
eliminate redundant solutions, thus delivering a smaller set of feasible
solutions and 𝑂(𝑛2 log 𝑛 + 𝑛2 ∕𝜀) time complexity.
Potts and Van Wassenhove (1988) develop eight algorithms and
∑
analyze their performance in solving the 1∥ 𝑤𝑗 𝑈𝑗 problem. The algorithms are obtained by deriving lower bounds from Lawler and Moore
(1969) (DPLM), a LP relaxation (BBLP), a branch and bound method
(BBDP) and number of tardy jobs lower bound (BBNTJ). In addition,
all of them are also implemented by applying the Reduction Theorem
7
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 5
Number of tardy jobs related problems for single machine.
Authors
Problem notation
∑
𝑈𝑗
Moore (1968)
1∥
Lawler and Moore (1969)
1∥
∑
1∥
∑
Sahni (1976)
𝑤𝑗 𝑈𝑗
1|𝑟𝑗 |
1|𝑑𝑗 = 𝑑|
Potts and Van Wassenhove (1988)
1∥
Lawler (1990)
∑
1|𝑟𝑗 , 𝑝𝑟𝑚𝑝| 𝑤𝑗 𝑈𝑗
∑
1|𝑟𝑗 , 𝑝𝑟𝑚𝑝| 𝑈𝑗
Hariri and Potts (1994)
1|𝑑 𝑗 , 𝑑𝑗 |
𝑤𝑗 𝑈𝑗
𝑤𝑗 𝑈𝑗
∑
Lee (1996)
1|𝑟 − 𝑎|
1|𝐹 𝑒, 𝐿𝑒|
Baptiste (1999a)
∑
1|𝑟𝑗 , 𝑝𝑟𝑚𝑝|
2
∑
DP
2
𝑂(𝑛 ) - 𝑂(𝑛 )
1|𝑝𝑗 = 𝑝, 𝑟𝑗 | 𝑤𝑗 𝑈𝑗
∑
1|𝑝𝑟𝑚𝑝, 𝑝𝑗 = 𝑝, 𝑟𝑗 | 𝑤𝑗 𝑈𝑗
Woeginger (2000)
1∥
Zhang et al. (2010)
1|
7
–
∑
𝑤𝑗 𝑈𝑗
𝑂(𝑛𝐾
Baptiste (1999b) proposes DP-based polynomial time algorithms
∑
∑
to solve 1|𝑝𝑗 = 𝑝, 𝑟𝑗 | 𝑤𝑗 𝑈𝑗 and 1|𝑝𝑟𝑚𝑝, 𝑝𝑗 = 𝑝, 𝑟𝑗 | 𝑤𝑗 𝑈𝑗 problems,
respectively. By examining their properties, the author is able to prove
that the first algorithm has 𝑂(𝑛7 ) time complexity while the second is
𝑂(𝑛10 ).
Woeginger (2000) outlines a series of cases reasoning that, given
some properties, some scheduling problems could provide a FPTAS by
altering their DP formulations. Among them, the author suggests that,
employing properties seen in Gens and Levner (1981), Sahni (1976),
∑
the DP approaches generate a FPTAS for 1∥ 𝑤𝑗 𝑈𝑗 .
Zhang et al. (2010) analyze a DP formulation that depends on the
makespan of early jobs that have been accepted and a total rejection
∑
∑
penalty for the 1| 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾| 𝑤𝑗 𝑈𝑗 problem and it is bounded by
∑
𝑂(𝑛𝐾 𝑝𝑗 ). The problem is categorized as NP-hard.
Table 5 summarizes all the articles in this subsection.
DP
DP
DP
𝑂(𝑛 )
𝑂(𝑛10 )
𝑤𝑗 𝑈𝑗
𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾|
DP with weighted network path
4
𝑈𝑗
DP
DP
B&B with DP
𝑝𝑗 )
𝑂(𝑛 log 𝑛)
Baptiste (1999b)
∑
DP, B&B and LP relaxation
2
–
∑
∑
–
𝑂(𝑛
𝑤𝑗 𝑈𝑗
∑
DP
Approximation with DP
𝑂(𝑛𝑘 𝑊 ) - 𝑂(𝑘 𝑊 )
𝑂(𝑛3 𝑘2 )
𝑈𝑗
∑
𝑂(𝑛 )
𝑂(𝑛2 log 𝑛 + 𝑛2 ∕𝜀)
2
𝑤𝑗 𝑈𝑗
DP
DP
2
𝑈𝑗
Dondeti and Mohanty (1998)
𝑝𝑗 , max{𝑑𝑗 }})
𝑂(min{2 , 𝑛𝑀})
Kise et al. (1978)
∑
∑
𝑛
Gens and Levner (1981)
∑
Method
DP
𝑂(𝑛 min{
𝑈𝑗
∑
Complexity
𝑂(𝑛 log 𝑛)
FPTAS with DP
∑
𝑝𝑗 )
DP
Ji, He, and Cheng (2006) formulate a DP-based pseudo-polynomial
∑
time algorithm to deal with solutions for the 1|𝑛𝑟 − 𝑎, 𝑝𝑗 (𝑠𝑡𝑗 )| 𝐶𝑗
problem, with unavailability in the (𝑡1 , 𝑡2 ) interval. In addition, proof
on the NP-hardness of the problem is shown and a pseudo-polynomial
∏
DP algorithm with 𝑂(𝑛(𝑡1 − 𝑡0 )𝑡2 (1 + 𝑏𝑗 )) is formulated. Ultimately, a
heuristic procedure is also provided for the same problem.
Kacem, Chu, and Souissi (2008) propose a DP algorithm, an integer
programming formulation and a B&B for finding an optimal sched∑
ule regarding the 1|𝑟 − 𝑎| 𝑤𝑗 𝐶𝑗 problem. After tests being run and
comparisons being made in terms of CPU time and storage requirements among the methods, the authors conclude the DP algorithm
outperforms the other two, solving instances with up to 3000 jobs.
Fan, Li, Zhou, and Zhang (2011) contribute with theoretical analysis
∑
on the complexity of 1|ℎ(𝜇), 𝑟 − 𝑎, 𝑝𝑗 (𝑡)| 𝐶𝑗 , where 𝜇 corresponds to
the number of unavailable periods. Two DP algorithms are furnished to
solve a special case when 𝜇 = 1 with resumability with 𝑂(𝑛2 (𝑡2 − 𝑡1 )2 )
∏
and 𝑂(𝑛2 (𝑡2 − 𝑡1 )2 (1 + 𝑏𝑗 )) complexities, respectively. Furthermore, a
complexity assessment is given for both scenarios and for the general
case, no polynomial time approximation is likely to occur for constant
deteriorating rate.
Tanaka and Fujikuma (2012) also apply the modified successive
∑
sublimation DP method for 1|𝑟𝑗 | 𝑤𝑗 𝐶𝑗 and obtain optimal schedules
for problems containing up to 200 jobs.
Li and Fan (2012) intensify the characteristics of the problems
previously examined by Fan et al. (2011), however considering only
non-resumable scenarios and weighted completion time. For single
unavailable intervals, an 𝑂(𝑛𝑡1 ) pseudo-polynomial DP algorithm is
developed and a FPTAS is derived from it by using a state–space
trimming technique. In addition, the authors prove that the double
unavailable interval case in NP-hard and no polynomial time algorithm
can be devised unless 𝑃 = 𝑁𝑃 .
Gu, Lu, Gu, and Zhang (2016) study the time-dependent aging effect
with an associated processing speed (𝑣(𝑡)) and optional maintenance
(𝑚𝑎) constraints applied to the total completion time optimization
problem. A complexity analysis is made for makespan and, subsequently, total completion time criterion, with both being classified as
NP-complete. Moreover, two DP algorithms are presented to solve the
∑
∑
1|𝑣(𝑡), 𝑚𝑎| 𝐶𝑗 problem with 𝑂(2𝑛+1 ) and 𝑂(𝑛2 𝑝𝑗 ) time complexities,
respectively.
∑
Cheng, Kravchenko, and Lin (2020) analyze the 1|𝑝𝑗 (𝑠𝑡𝑗 )| 𝐶𝑗 problem, given a 𝑏𝑗 . The study of the complexity is evaluated and the
4.3. Completion time related criteria
∑
Lee (1996) proves that the 1|𝑟 − 𝑎| 𝑤𝑗 𝐶𝑗 problem is NP-hard
and presents a pseudo-polynomial DP algorithm to solve it based on
∑
weighted shortest processing time algorithm for 1∥ 𝑤𝑗 𝐶𝑗 as well as
a recursive equation for the objective function. The DP is bounded by
𝑂(𝑛(𝑡1 )𝑝max ), where 𝑡1 is the start time of maintenance.
Gélinas and Soumis (1997) present a thorough analysis regarding
∑
the 1|𝑟𝑗 | 𝐶𝑗 problem by studying a DP formulation coupled with a
labeling scheme and an elimination test in order to select reduced
and feasible sets of solution, a complexity investigation for each step
or modification in the formulation and also a pseudo-polynomial DP
algorithm that can solve problems up to 100 jobs out of 200-job
instances used in the experiments.
Bianco, Dell’Olmo, and Giordani (1999) address the 1|𝑟𝑗 , 𝑝𝑗 (𝜋𝑖𝑗 )|
∑
𝐶𝑗 problem with sequence dependent processing times. In order to
solve this problem, the authors establish its equivalence to the Cumulative TSP and propose a DP formulation, from which two lower bounds
are derived. Furthermore, two heuristics are defined and a comparative
performance analysis with the bounds is done for a realistic data set,
showing that heuristics are more efficient in such case.
Cheng and Ding (2000) examine the complexity analysis for
∑
1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑑𝑗 | 𝐶𝑗 , considering equal deteriorating rates 𝑏𝑗 = 𝑏. Such
problem is proved to be solved by a polynomial time DP algorithm of
𝑂(𝑛5 ) complexity.
8
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 6
Completion time related problems for single machine.
Authors
Lee (1996)
Gélinas and Soumis (1997)
Bianco et al. (1999)
Problem notation
∑
𝑤𝑗 𝐶𝑗
1|𝑟 − 𝑎|
1|𝑟𝑗 |
∑
𝐶𝑗
∑
1|𝑟𝑗 , 𝑝𝑗 (𝜋𝑖𝑗 )|
∑
Cheng and Ding (2000)
1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑑𝑗 |
1|𝑛𝑟 − 𝑎, 𝑝𝑗 (𝑠𝑡𝑗 )|
Kacem et al. (2008)
1|𝑟 − 𝑎|
Fan et al. (2011)
∑
1|𝑟 − 𝑎, 𝑝𝑗 (𝑡)| 𝐶𝑗
∑
1|𝑟 − 𝑎, 𝑝𝑗 (𝑡)| 𝐶𝑗
Tanaka and Fujikuma (2012)
1|𝑟𝑗 |
Li and Fan (2012)
1|𝑛𝑟 − 𝑎, 𝑝𝑗 (𝑡)|
∑
∑
𝑤𝑗 𝐶𝑗
Gu et al. (2016)
1|𝑣(𝑡), 𝑚𝑎|
Cheng et al. (2020)
1|𝑝𝑗 (𝑠𝑡𝑗 )|
Mor and Shapira (2020)
∑
∑
1| 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾| 𝐶𝑗
∑
∑
1| 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾| 𝑤𝑗 𝐶𝑗
∑
DP
DP and Heuristics
DP
𝑂(𝑛(𝑡1 − 𝑡0 )𝑡2
∏
(1 + 𝑏𝑗 ))
–
𝑤𝑗 𝐶𝑗
∑
–
𝑂(𝑛 )
𝐶𝑗
𝑤𝑗 𝐶𝑗
∑
DP
5
𝐶𝑗
∑
Method
–
𝐶𝑗
Ji et al. (2006)
Complexity
𝑂(𝑛(𝑡1 )𝑝max )
𝐶𝑗
𝐶𝑗
DP and Heuristic
DP, integer programming and B&B
2
2
𝑂(𝑛 (𝑡2 − 𝑡1 ) )
∏
𝑂(𝑛2 (𝑡2 − 𝑡1 )2 (1 + 𝑏𝑗 ))
DP
DP
–
SSDP
𝑂(𝑛𝑡1 )
FPTAS with DP
𝑂(2𝑛+1 )
∑
𝑂(𝑛2 𝑝𝑗 )
DP
DP
𝑂(𝑛3
∑
𝑎𝑗
∑
𝑂(𝑛𝐾)
∑
𝑂(𝑛𝐾 𝑝𝑗 )
(𝑎𝑗 + 𝑏𝑗 ))
DP
DP
DP
Jeng and Lin (2004) explore two optimal procedures to solve the
1|𝑝𝑗 (𝑑𝑗 , 𝑠𝑡𝑗 )|𝐶max problem, where the processing times are defined as
nonlinear step functions. The first approach is an 𝑂(𝑛(max 𝑎𝑗 + max 𝑑𝑗 ))
pseudo-polynomial DP algorithm and the second one relies on a B&B.
Each procedure is modified by the inclusion of dominance rules, however, as experiments show, the procedures reach memory capacity,
suggesting further research should focus on improvements.
Bosio and Righini (2009) evaluate the 1|𝑝𝑗 (𝑟𝑗 , 𝑠𝑡𝑗 ), 𝑟𝑗 |𝐶max problem.
A DP algorithm associated with construction of upper and lower bounds
is provided and an overall analysis regarding the range of 𝑟𝑗 and 𝑝𝑗 and
their effect upon the upper and lower bounds are described through
computational experiments.
Zhang et al. (2010) propose a DP formulation to solve the
∑
1|𝑟𝑗 , 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾|𝐶max problem taking into account a recursive relation
that is dependent on the makespan of the accepted jobs for each stage
∑
with 𝑂(𝑛(𝑟𝑛 + 𝑝𝑗 )) complexity. The problem is categorized as NP-hard,
due to the complexity analysis performed on the problem without the
release dates.
Davari, Ranjbar, De Causmaecker, and Leus (2020) address the
1|𝑟𝑗 , 𝑖𝑛𝑣|𝐶max , which implies that the inventory is now bounded by a
capacity constraint. The authors study the complexity and the problem
is proved to be strongly NP-hard even for an infinite capacity inventory.
In addition, two MIP formulations, a B&B method and a guess-andcheck DP algorithm are provided. Experiments are carried out on,
at most, 50 jobs and the guess-and-check DP outperforms the other
methods in terms of CPU times and number of solved problems.
Mor and Shapira (2020) address a DP formulation for solving the
∑
1|𝑟𝑗 , 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾|𝐶max problem. The theoretical approach is unfolded
into computational experiments that can solve instances with up to
2000 jobs and a varying set of rejection rate that can be extended to at
most 40% with 𝑂(𝑛𝐾) computational complexity.
Problems in which makespan is considered the main criterion are
summarized in Table 7.
problem is classified as binary NP-hard. The authors also work on
a pseudo-polynomial DP algorithm and show its complexity can be
∑ ∑
evaluated at 𝑂(𝑛3 𝑎𝑗 (𝑎𝑗 + 𝑏𝑗 )), where 𝑎𝑗 is the ordinary processing
time.
Mor and Shapira (2020) investigate the use of theoretical for∑
∑
∑
mulations in DP in order to solve the 1| 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾| 𝐶𝑗 , 1| 𝜃𝑗𝑟𝑒𝑗
∑
≤ 𝐾| 𝑤𝑗 𝐶𝑗 problems via computational tests and the complexities
∑
associated with them are 𝑂(𝑛𝐾) and 𝑂(𝑛𝐾 𝑝𝑗 ). The computational
results show that excellent outcomes can be obtained for at most 2000
jobs and a varying rate of rejected jobs within a short time for the first
problem and for at most 60 jobs regarding the second type of problem.
Table 6 summarizes the content of this subsection.
4.4. Makespan
Kunnathur and Gupta (1990) propose several approaches to deal
with 1|𝑝𝑗 (𝑑𝑗 )|𝐶max problem, which are composed of two DP methods,
a B&B and five heuristics. Although, from a theoretical point of view,
DP is sustained by strong properties, its computational requirements
were considered infeasible. Therefore, the experimental analysis was
conducted from a comparison between the B&B and the best heuristic
out of the five that have been provided.
Kubiak and van de Velde (1998) apply theoretical and empirical
approaches to solve the 1|𝑝𝑗 (𝐷𝑏 , 𝑑𝑐 )|𝐶max , where 𝑑𝑐 is a common due
date at which jobs start deteriorating and 𝐷𝑏 is a bounded due date
at which jobs no further deteriorate. The theoretical content encloses
three pseudo-polynomial DP algorithms to solve the unbounded and
bounded job deteriorating cases while the empirical methodology encompasses a B&B algorithm and heuristics. The complexities of the DP
algorithms are described in Table 7.
Cheng and Ding (2000) prove that solving the 1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑑𝑗 |𝐶max
problem is equivalent to finding the optimal solution to the flow time
case. Therefore, the DP approach can be applied with no modifications
required. Ultimately, the DP algorithm presents itself as an alternative
solution for 1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑑𝑗 |𝐿max with 𝑂(𝑛6 log 𝑛) (see Table 4) time complexity. Furthermore, with the DP theoretical approach, the authors
manage to prove that both problems are NP-complete via a 3-partition
problem.
Woeginger (2000) studies the effect of the general transformation
scheme of DP algorithms in FPTAS for the 1|𝑝𝑗 (𝑑)|𝐶max problem as well.
Theoretical background is extracted from Kubiak and van de Velde
(1998) so as to perform the shifting procedure.
4.5. Multi-criteria and additional objective functions
Van Wassenhove and Gelders (1978) propose four approaches, two
∑
of which are DP, to solve the bicriteria 1∥ 𝑤𝑗 𝑇𝑗 + 𝜆𝑗 𝐶𝑗 problem. The
authors resort to dominance rules based on precedence constraints and
experimental tests are performed in a comparative manner for 25 jobs
maximum. The DP algorithms outperformed the other methods for
cases up to 20 jobs, however dual algorithms might be more efficient
as the number of jobs increases.
9
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 7
Makespan problems for single machine.
Authors
Problem notation
Complexity
Method
Kunnathur and Gupta (1990)
1|𝑝𝑗 (𝑠𝑡𝑗 )|𝐶max
–
DP, B&B and Heuristics
∑
Kubiak and van de Velde (1998)
1|𝑝𝑗 (𝑑𝑐 )|𝐶max
1|𝑝𝑗 (𝑑𝑐 , 𝐷𝑏 )|𝐶max
1|𝑝𝑗 (𝑑𝑐 , 𝐷𝑏 )|𝐶max
𝑂(𝑛𝑑𝑐 𝑝𝑗 ) − 𝑂(𝑛𝑑𝑐 )
∑
𝑂(𝑛2 𝑑𝑐 (𝐷𝑏 − 𝑑𝑐 ) 𝑝𝑗 ) − 𝑂(𝑛𝑑𝑐 (𝐷𝑏 − 𝑑𝑐 ))
∑ ∑
∑ ∑
𝑂(𝑛𝑑𝑐 𝑏𝑗 ( 𝑝𝑗 )2 ) − 𝑂(𝑛𝑑𝑐 𝑏𝑗 𝑝𝑗 )
DP and B&B
DP and Heuristics
DP and Heuristics
Cheng and Ding (2000)
1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑑𝑗 |𝐶max
–
DP
Woeginger (2000)
1|𝑝𝑗 (𝑑)|𝐶max
–
FPTAS with DP
Jeng and Lin (2004)
1|𝑝𝑗 (𝑑𝑗 , 𝑠𝑡𝑗 )|𝐶max
𝑂(𝑛(max 𝑎𝑗 + max 𝑑𝑗 ))
DP and B&B
Bosio and Righini (2009)
1|𝑝𝑗 (𝑟𝑗 , 𝑠𝑡𝑗 ), 𝑟𝑗 |𝐶max
–
DP
∑
𝜃𝑗𝑟𝑒𝑗
≤ 𝐾|𝐶max
Zhang et al. (2010)
1|𝑟𝑗 ,
Davari et al. (2020)
1|𝑟𝑗 , 𝑖𝑛𝑣|𝐶max
Mor and Shapira (2020)
1|𝑟𝑗 ,
∑
𝜃𝑗𝑟𝑒𝑗 |𝐶max
𝑂(𝑛(𝑟𝑛 +
∑
𝑝𝑗 ))
DP
–
Guess-and-Check DP
𝑂(𝑛𝐾)
DP
Hoogeveen and van de Velde (1991) explore the complexity of
∑
∑
1|𝑑𝑗 = 𝑑| 𝑤𝑗 |𝑑 − 𝐶𝑗 | when 𝑑 <
𝑝𝑗 is imposed. The authors prove
the problem is NP-hard and present a DP algorithm with time and space
complexities of 𝑂(𝑛2 𝑑) and 𝑂(𝑛𝑑), respectively. Moreover, some specific
cases within this scenario are presented as polynomially solvable.
Potts and Van Wassenhove (1992a) incorporate additional methods
∑
to the 1∥ 𝑌𝑗 problem by devising approximation algorithms derived
from DP formulations and B&B. The approximation methods have been
developed by aggregating rounding techniques into the DP so as to
produce higher quality and less complex (in the computational sense)
algorithms.
Potts and Van Wassenhove (1992b) analyze properties related to
∑
the 1∥ 𝑌𝑗 problem as well as complexity analysis by showing it
is NP-hard. Moreover, the authors develop a heuristic for setting an
initial sequencing and a pseudo-polynomial DP algorithm of 𝑂(𝑛𝑈 𝐵)
complexity, which is able to yield solution up to 10000-job instances
due to space–state reductions performed via optimality properties.
Carraway, Chambers, Morin, and Moskowitz (1992) apply general∑
ized DP to solve the nonlinear multi-criteria 1∥ (𝑒𝑘1 𝑤𝑗 𝑇𝑗 + 𝜆𝑒𝑘2 𝑤𝑗 𝑈𝑗 )
problem and compare the results with a DP-based heuristic and a DP
formulation to generate feasible sets. The heuristic and generalized DP
show similar results and both outperform the DP formulation, since the
last one may also admit sub-optimal schedules and fails rapidly due to
the curse of dimensionality. The authors suggest using the technique so
as to develop further studies regarding different nonlinear multi-criteria
problems.
De, Ghosh, and Wells (1992) devise a pseudo-polynomial DP algorithm by adding V-shaped properties to identify optimal solutions
∑
∑
for 1∥ (𝐶𝑗 − 𝐶 𝑗 ) with overall complexity of 𝑂(𝑛2 𝑝𝑗 ). Experiments
are carried out for at most 100 jobs and DP procedure outperforms a
B&B algorithm and an enumeration method for the same time limit.
Nevertheless the algorithm shows greater efficiency when processing
times are small, therefore an approximation algorithm is also developed. Additionally, the DP approach is extended to the 1∥(𝛼𝐶 𝑗 + (1 −
𝛼)(𝐶𝑗 − 𝐶 𝑗 )) bi-criteria problem, using V-shaped structure to construct
feasible sets that contain optimal solutions.
Bard, Venkatraman, and Feo (1993) combine the generation of state
spaces inherent to DP approaches with a B&B algorithm in order to
∑
reach optimality for the 1∥ (𝛼𝑗 𝐸𝑗 + 𝜆𝑗 𝐶𝑗 ) problem. A GRASP heuristic
is also programmed so as to furnish a tight upper bound and fortify the
fathoming in the B&B-DP. Optimality has been provided up to 30 jobs
regardless the composition of the data in less than 12 min and with
steady computational effort.
Ibaraki and Nakamura (1994) apply successive sublimation DP to
∑
solve the 1∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ) problem. The procedure includes heuristics
in order establish an initial upper bound and recursive lower bounds
are derived from modifiers, job penalties and state–space relaxation.
Barnes and Vanston (1981) present a DP method by coupling B&B
features (DPBB) to solve the problem of minimizing the sum of delay
penalties and setup costs. This method reduces the nodes explored in
the DP states, thus reducing the effects of the curse of dimensionality
inherent to a ‘‘pure’’ DP formulation. The computational experiments
are performed for 10, 15 and 20-job data sets and compared with
B&B and heuristics. The DPBB method has shown best performance
regarding solution quality and CPU times for the 15 and 20-job sets
whereas the B&B presented better outcomes for the 10-job set.
Chand and Schneeberger (1988) build a modified Smith heuristic
and a DP algorithm to minimize weighted earliness criterion subject
to no tardy jobs. The DP algorithm operates in backward recursion
and makes use of an efficient frontier, which stores optimal values of
weighted earliness and tardy jobs, to schedule jobs as further stages
are created. The authors acknowledged that DP was generated to serve
as benchmark for studying heuristics. In fact, the proposed heuristic
provides good quality solutions and better CPU times.
Abdul-Razaq and Potts (1988) derive six lower bounds from a DP
state–space relaxation to reduce the number of states generated by job
assignments. Each bound is built upon a combination of modifiers and
job penalties (similar to Lagrangian multipliers) so as to make them
tighter. In addition, three B&B algorithms are implemented with less
combinations than those designed for the DP formulation. Ultimately,
∑
the chosen problem to apply the methods was 1∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ) with
DP being suitable for small-sized instances whereas B&B performs
better for medium-sized instances with small processing times.
∑
Hall and Posner (1991) show that the 1|𝑑𝑗 = 𝑑| 𝑤𝑗 |𝑑 − 𝐶𝑗 | problem (𝑑 being unrestrictively late) is NP-complete in the ordinary sense
and presents a DP algorithm along with four special cases of the
problem, which can each be solved with polynomial time algorithms.
The second portion of this article is seen in Hall, Kubiak, and Sethi
∑
(1991) and investigates 1|𝑑𝑗 = 𝑑| |𝑑 − 𝐶𝑗 | (𝑑 placed early enough for
the allocation process) by proving it to be NP-complete in the ordinary
sense. A DP pseudo-polynomial algorithm is introduced and shows
promising results for being able to solve up to 1000 jobs.
De, Ghosh, and Wells (1991) focus on presenting properties related
∑
to V-shaped structures for 1|𝑑𝑗 = 𝑑| (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ) with asymmetrical
weights and state that this problem is NP-complete in the strong sense.
Furthermore, a DP algorithm is developed to ensure an exact approach
to the problem as well as an approximation algorithm. Ultimately some
particular cases are displayed.
Yano and Kim (1991) evaluate the use of a DP procedure for solving
∑
optimally the 1∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ) problem, assuming the sequence of
jobs is already given and focusing on timing as a decision making process. They also present theoretical and experimental B&B and heuristics
formulations to solve a specific case when penalties are proportional to
the jobs’ processing times.
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E.A.G.d. Souza et al.
The method is executed for data sets up to 35 jobs and compared with
an original DP formulation, obtaining superior outcomes both in CPU
times and number of generated states.
Chand, Chhajed, and Traub (1994) propose a DP algorithm to minimize the lead time (LT) and earliness costs considering fixed-delivery
intervals (𝑑𝑖 , 𝑑𝑓 ) by setting a first-come first-deliver priority rule. Reductions in the state space are provided through several dominance
rules and compared to a full DP approach in an experiment containing
up to 150 jobs. Results are considered better in terms of CPU times for
the former approach.
Liman and Ramaswamy (1994) describe properties related to restrictive and non-restrictive common due windows for the problem
∑
1|𝑑𝑤 | (𝛼𝑗 𝐸𝑗 + 𝑤𝑗 𝑈𝑗 ). In addition they show the NP-completeness for
both situations and develop a DP pseudo-polynomial algorithms for
∑
each case with overall complexity 𝑂(𝑛2 (𝑑𝑤 ) 𝑝𝑗 ).
Hariri, Potts, and Van Wassenhove (1995) bring a theoretical work
∑
on preemptive and non-preemptive 1∥ 𝑤𝑗 𝑌𝑗 by stating complexity
analysis for both problems. In addition, the authors offer a DP-based
B&B algorithm, in which DP, in conjunction with reduction properties,
is used for designing upper and lower bounds efficiently. Instances
ranging from 100 to 700 jobs show DP bounds efficiency by obtaining
a small number of nodes before reaching optimal solution.
Ventura and Weng (1995) improve the method proposed in Hall
et al. (1991) by providing a pseudo-polynomial DP algorithm that stems
from an elimination process of two subroutines in the original version.
This enhances the method significantly in computational requirements
∑
and yields an 𝑂(𝑛(𝑑 + 𝑝𝑗 )) overall complexity.
Weng and Ventura (1996b) propose a pseudo-polynomial DP al∑
gorithm and a heuristic to minimize 1|𝑑𝑗 = 𝑑, 𝑡𝑗 | (𝐸𝑗 + 𝑇𝑗 ). Some
properties involving LPT-SPT ordering are outlined and optimality is
ensured by DP if 𝑝𝑖 ≤ 𝑝𝑗 implies 𝑡𝑖 ≤ 𝑡𝑗 . Comparative tests are performed
for 𝑛 = 8, 9, 10 with both methods being quite efficient and DP reaching
near optimal solutions, indicating it could be used to ease the search
for optimal solutions in larger job sizes.
Weng and Ventura (1996a) describe properties related to the
∑
1|𝑑𝑗 = 𝑑| [(𝐶𝑗 − 𝑑)2 ]∕𝑛 problem by associating optimal scheduling
∑
with V-shaped structure. Furthermore, the authors design an 𝑂(𝑛 𝑝𝑗 )
pseudo-polynomial DP algorithm for the tightly restrictive case, for
which no strategy had yet been developed, and provide optimal solutions up to 100-job data sets in less than two seconds in CPU
time.
Lann and Mosheiov (1996) explore the minimization of
∑
1∥ (𝐷𝑗 + 𝑈𝑗 ), with 𝐷𝑗 denoting the number of early jobs, by providing algorithms for solving the job-independent and job-dependent
with symmetrical and asymmetrical costs scenarios (in Table 8, 𝑎𝑠𝑦
and 𝑠𝑦 stand for asymmetrical and symmetrical costs, respectively),
where DP formulations are given for the last two cases. They develop a polynomial DP algorithm of 𝑂(𝑛2 ) time complexity for the
symmetrical cost situation and a pseudo-polynomial DP method with
∑
𝑂(𝑛 max{ 𝑝𝑗 , 𝑑max }) complexity, which is the basis for two heuristics
used to solve the asymmetrical case.
∑
Cheng, Chen, and Li (1996) analyze the 1|𝑝𝑗 (𝑥𝑗 )|(𝑈𝑗 , 𝑥𝑗 ) bi-criteria
problem, where 𝑥𝑗 represents resources allocated to a given job and
𝑝𝑗 (𝑥𝑗 ) indicates controllable processing times dependent on this resources. Complexity theorems stating the NP-hardness are shown and
sequencing properties based on EDD rule are given. In addition pseudopolynomial DP algorithms are provided for a linear non-decreasing
function and a general non-decreasing function of 𝑥𝑗 .
Cheng, Janiak, and Kovalyov (1998) examine several bi-criteria
minimization problems with resource dependent processing times, considering a non-negative time compressing rate 𝑞𝑗 and initial processing
∑
∑
time 𝑎𝑗 . The DP algorithms are applied to the 1∥( 𝑣𝑗 𝑥𝑗 ≤ 𝐾, 𝑤𝑗 𝑈𝑗 )
∑
∑
∑
and 1∥( 𝑣𝑗 𝑥𝑗 , 𝑤𝑗 𝑈𝑗 ≤ 𝐾) problems with 𝑂(𝐾𝑛𝑈 𝐵 𝜏𝑗 ), where 𝜏𝑗 ≤
𝑎𝑗 ∕𝑞𝑗 . In addition, the DP algorithms can be altered into approximation
algorithms.
Woeginger (2000) also applies the properties delineated by Hariri
et al. (1995), Potts and Van Wassenhove (1992a) to show the develop∑
ment of a FPTAS is likely to occur when the DP formulations for 1∥ 𝑌𝑗
∑
and 1∥ 𝑤𝑗 𝑌𝑗 are modified accordingly.
Klamroth and Wiecek (2001) develop a DP algorithm in order to
find the optimal schedule for a general multiple criteria optimization
problem, whose objective functions are dependent on time and also
bounded by a resource availability constraint. The authors model the
problem as a multiple criteria knapsack problem and develop the DP
algorithm based on previous references of DP for the related knapsack
formulation.
Yeung, Oğuz, and Cheng (2001a) introduce optimal properties for
∑
the 1|𝑑𝑤 | (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 ) problem and prove the NP-completeness as∑
sociated to it. They also develop an 𝑂(𝑛2 𝑝max (1 + 𝑝𝑗 ∕2)) pseudopolynomial DP algorithm, which is unfolded in two types for solving
problems with either presence or absence of straddling jobs. Computational tests are performed and the algorithm is limited to solving
problems with 100 jobs within reasonable time.
Yeung, Oguz, and Cheng (2001b) discuss optimal schedule proper∑
ties related to the multi-criteria 1|𝑑𝑤 | (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 + 𝜌𝑗 𝐷𝑗 + 𝜃𝑗 𝑈𝑗 ) +
∑
𝐿(𝑑𝑤 ) and 1|𝑑𝑤 | (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 + 𝑤𝑗 𝐶𝑗 ) + 𝐿(𝑑𝑤 ) problems bounded by
a given due window. Two pseudo-polynomial DP algorithms, one for
each problem, are designed with time complexity given by 𝑂(𝑛3 (𝐾 +
∑
∑
1)𝑝2max ( 𝑝𝑗 + 1)( 𝑝𝑗 − 𝐾 + 1)2 ). Additional polynomially solvable cases
are presented.
Ventura, Kim, and Garriga (2002) aim at finding the optimal sched∑
ule as well as the release dates that minimize the 1|𝑟𝑗 | (𝑓 (𝑟𝑗 ) + 𝛼𝑗 𝐸𝑗
+ 𝛽𝑗 𝑇𝑗 ) problem, in which 𝑓 is a non-increasing function of release
dates. A DP algorithm imbued with V-shaped ordering properties is
conceived and heuristics are also provided. Despite small CPU times,
the DP approach does not seem to be more advantageous than the
heuristics since memory overreaching occurs for instances larger than
40 jobs while heuristics achieve high quality solutions by demanding
less computational effort.
Hendel and Sourd (2005) apply the DP concepts to generate a
∑
formulation to solve the 1∥ (𝐸𝑗 + 𝑇𝑗 ). Despite being the main element
for a pairwise interchange approach, the DP formulation, which is of
𝑂(𝑛2 ) complexity, is seen as an auxiliary method that can be coupled
within a neighborhood search algorithm to finally solve the problem.
Kedad-Sidhoum, Solis, and Sourd (2008) evaluate the performance
of several lower bounds and a DP-based heuristic to solve the
∑
1∥ (𝐸𝑗 + 𝑇𝑗 ). The 𝑂(𝑛2 ) polynomial DP formulation is included in the
process in order to reduce the memory usage when storing previous
information regarding the heuristic solutions.
∑
Tanaka et al. (2009) investigate the 1|𝑛𝑜 − 𝑖𝑑𝑙𝑒| 𝑓𝑗 (𝐶𝑗 ) problem. A
series of theorems are outlined and applied in order to reduce the computational space requirements to store the DP states in the successive
sublimation DP algorithm proposed to solve the scheduling problem.
These modifications lead to an efficient algorithm with capacity to
optimally solve up to 300 jobs within reasonable time.
Cheng and Sun (2009) develop several DP algorithms and FPTAS for
∑
∑
∑
the 1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑟𝑗 , 𝑟𝑒𝑗| 𝑤𝑗 𝐶𝑗 + 𝜃𝑗𝑟𝑒𝑗 , 1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑟𝑗 , 𝑟𝑒𝑗|𝐶max + 𝜃𝑗𝑟𝑒𝑗 and
∑ 𝑟𝑒𝑗
1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑟𝑗 , 𝑟𝑒𝑗|(𝑇max ∕𝐿max ) + 𝜃𝑗 problems. A detailed assessment of
each algorithm’s complexity is given (see Table 8) and an 𝑂(𝑛2 ) DP al∑
gorithm is provided for 1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑟𝑗 , 𝑟𝑒𝑗|𝐶max + 𝜃𝑗𝑟𝑒𝑗 with deteriorating
rates 𝑏𝑗 = 𝑏.
Zhang, Lu, and Yuan (2009) analyze the complexity formulations
∑
the 1|𝑟𝑒𝑗, 𝑟𝑗 |𝐶max + 𝜃𝑗𝑟𝑒𝑗 and categorize the problem as NP-hard. In
addition, the authors provide polynomial and pseudo-polynomial DP
algorithms as well as approximation schemes (see Table 8) for the
general problem and specific cases.
Zhao and Tang (2011) supplement the work done in Fan et al.
(2011) by extending the completion time model to two other optimiza∑
∑
tion problems: 1|ℎ(𝜇), 𝑟 − 𝑎, 𝑝𝑗 (𝑡)|𝛼 𝐶𝑗 + 𝛽 𝑇𝑗 and 1|ℎ(𝜇), 𝑟 − 𝑎, 𝑝𝑗 (𝑡)|
∑
∑
𝛼 𝐶𝑗 + 𝛽 𝐸𝑗 . Each problem is solved from a DP perspective and
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E.A.G.d. Souza et al.
Li and Yuan (2020) focus on a variety of problems related to a multiagent scheduling with the intent of minimizing the total weighted late
work criterion. They introduce some notations in order to separate the
various forms of minimization, classify some problems regarding their
complexity and develop an exact pseudo-polynomial DP algorithm with
∏
𝑂(𝑛2 𝑈 𝐵𝑥 ) and an approximation method for a specific case in which
Pareto-optimal solutions are sought with a fixed number 𝑥 of agents.
Mosheiov and Oron (2021) develop a pseudo-polynomial DP al∑
gorithm in order to solve the 1|𝑅𝑀𝐴| 𝑌𝑗 , where RMA denotes a
rate modifying activity associated with a maintenance the machine
undergoes, which means that the processing time is subject to a reduction after the maintenance process, for a given time interval 𝑇 . The
algorithm is based on cases in which EDD, RMA and delayed jobs are
taken into account. For each case, an appropriate recursive relation
is formulated and the sequence is obtained in a backward manner.
∑
In the end, the algorithm’s complexity is estimated at 𝑂(𝑛( 𝑝𝑗 + 𝑇 )).
Computational experiments are performed for instances with up to 200
jobs and DP proves to be efficient in solving them.
the authors examine the complexity of the algorithms and classify both
problems as NP-hard.
Tanaka and Fujikuma (2012) extend the successive sublimation al∑
gorithm DP for the following two cases: 1∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ) and
∑
1|𝑟𝑗 | (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ). The SSDP has been able to find optimal solution
for instances with up to 200 jobs.
Tanaka and Sato (2013) also apply the successive sublimation al∑
gorithm to the 1|𝑛𝑜 − 𝑖𝑑𝑙𝑒, 𝑝𝑟𝑒𝑐| (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ) problem and the results
are quite promising, given that solution has been found for all instances
in the 50-job data set.
Yin, Liu, Cheng, Wu, and Cheng (2013) bring a contemporary
approach to a class of problems that include common due date as
an endogenous element in the decision making process, considering
positional learning effect and past sequence-dependent delivery times
(psd). Four different objective functions are considered and also some
specific cases are outlined. An 𝑂(𝑛3 ) time complexity DP algorithm is
∑
∑
developed to minimize the 1|𝐿𝑒, 𝑞𝑝𝑠𝑑 |𝛼 𝐸𝑗 + 𝑤𝑗 𝑈𝑗 + 𝛾𝑑. Moreover,
another 𝑂(𝑛3 ) time complexity DP approach is offered as solution to the
∑
∑
1|𝐿𝑒, 𝑞𝑝𝑠𝑑 |𝛾 𝑑 + 𝑤𝑗 problem.
Zhao, Hsu, Cheng, Yin, and Wu (2014) evaluate solutions to min∑
imize the 1|𝑝𝑗 (𝑠𝑡𝑗 )| 𝑤𝑗 𝑈𝑗 + 𝛾𝑑, where 𝑑 is also to be determined by
the optimization process. The authors prove the problem is NP-hard and
∑
design two pseudo-polynomial DP algorithms, which possess 𝑂(𝑛 𝑤𝑗 )
∏
and 𝑂(𝑎𝑗 (1 + 𝑏𝑗 )) complexities, and an approximation scheme to deal
with the optimization.
Xingong and Yong (2015) analyze two problems with positional
learning effect, which main objective is to minimize 𝐹 (𝑑, 𝜋) = 𝜙(𝑑) +
∑
∑
𝛼 𝐸𝑗 + 𝛽𝑗 , where 𝜙(𝑑) is a variable that represents the assigned
due dates and 𝛽𝑗 denotes a penalty associated with a discarded job.
Common (CON) and slack due date (SLK) rules are applied for the due
date assignment. 𝑂(𝑛4 ) and 𝑂(𝑛3 ) polynomial time DP algorithms are
employed as solution methods for each problem, respectively.
Ben-Yehoshua and Mosheiov (2016) investigate the complexity be∑
havior related to the 1∥ 𝑉𝑗 problem, where 𝑉𝑗 represents the early
work criteria, and present proofs for its NP-hardness in the ordinary
∑
sense. Furthermore, a pseudo-polynomial DP algorithm with 𝑂(𝑛 𝑝𝑗 )
complexity is proposed and its results show that it can solve instances
with up to 200 jobs efficiently.
∑
Yin, Xu et al. (2016) study the 1|𝑚𝑎| 𝑌𝑗 problem, given that a fixed
interval (𝑡1 , 𝑡2 ) is defined for the maintenance to occur. The authors
propose two pseudo-polynomial DP algorithms, whose complexities are
given by 𝑂(𝑛𝑡1 (𝑡1 +𝑈 )) and 𝑂(𝑛𝑈 (𝑡1 +𝑈 )) (with 𝑈 being a threshold), respectively. Additionally, a FPTAS is also provided and experiments are
performed on each algorithm. The second DP algorithm outperforms
the first in terms of CPU time and optimal solutions have been found for
at most 45 jobs for either. The FPTAS reduces the state space generated
by the DP algorithms and is able to find solution with up to 85 jobs
depending on the approximation parameter.
Wang, Kang, Shiau, Wu, and Hsu (2017) address the two-agent
∑
problem denoted by 1∥ 𝑌𝑗 ∶ 𝐿max ≤ 𝐾. The article focuses on exploring exact methods, which are comprised of two pseudo-polynomial
DP algorithms and a B&B, and a Tabu Search (TS) in order to analyze
their performances. Although the DP approaches for this problem are
deemed fairly good methods, their practical usage is limited to smallsized instances and therefore, the comparison is estimated for the B&B
and TS. For the complexity description seen in Table 8 we consider two
∑
∑
parameters, which are 𝑆𝑃 = ( 𝑝𝑗 )𝐴 + ( 𝑝𝑗 )𝐵 and 𝑠𝑝 = max{(𝑝𝑗 )𝐴 +
(𝑑𝑗 )𝐴 − 1}.
Bülbül, Kedad-Sidhoum, and Şen (2019) analyze solvability for the
∑
∑
following problems 1|ℎ(𝜇), 𝑟 − 𝑎| (𝐸𝑗 + 𝑇𝑗 ), 1|ℎ(𝜇), 𝑠𝑟 − 𝑎| (𝐸𝑗 + 𝑇𝑗 )
∑
and 1|ℎ(𝜇), 𝑛𝑟 − 𝑎| (𝐸𝑗 + 𝑇𝑗 ). The authors prove that the nonresumable and semi-resumable cases about a common non-restrictive
common due date is NP-hard in the strong sense and for special cases
involving the three scenarios, in which the problem is NP-hard in the
ordinary sense, pseudo-polynomial DP algorithms are developed for
optimal scheduling.
4.6. Discussion on single machine problems
One of the enriching discussions of this environment is that DP
enabled the development and application of several approaches, and
as one can notice, no pattern has been dominant over the years. This
feature is a confirmation that the single machine environment has a
large potential in the innovation field in machine scheduling due to
a variety of theoretical and practical background it has to offer. For
starters, we can cite the paper developed by Lawler and Moore (1969),
which correlates the single machine environment with the knapsack
problem for a handful of objective functions. As a result of that,
the DP formalism applied to the knapsack problem was, with proper
∑
∑
∑
modifications, used as a resource to model 𝑤𝑗 𝐸𝑗 , 𝑇𝑗 , 𝑤𝑗 𝑇𝑗 and
∑
𝑈𝑗 functions and minimize them via recursion.
Baker (1977) and Lawler (1977) have also shown relevant contribution to the machine scheduling environment for tardiness-related
functions by developing decomposition properties and labeling scheme,
respectively so as to reduce the state space associated with the storage
of DP recursive methods. Potts and Van Wassenhove (1982) have made
use of both approaches and combined them, being able to reach practical results for 70 jobs with a polynomial time complexity algorithm. At
a posterior work (Potts & Van Wassenhove, 1987), they adapt a series of
algorithms by including decomposition and precedence properties and
prove that decomposition algorithms outperform the precedence ones
by reaching optimal solution for 100 jobs at most within reasonable
∑
time for
𝑇𝑗 function. Ultimately, this can be seen as a remarkable
work because it was a leap for DP algorithms due to the Curse of
Dimensionality in terms of efficiency relating core storage and CPU
time.
Authors have also been concerned with combining DP with other
methods to either reinforce the DP efficiency by reducing its state
space or to generate a more powerful algorithm on the other end.
This can be observed in Potts and Van Wassenhove (1988) to minimize
∑
𝑤𝑗 𝑈𝑗 by designing lower bounds from the knapsack-like DP approach
seen in Lawler and Moore (1969) and a combination of B&B and DP
that are improved with reduction theorems and can solve problems as
large as 1000 jobs. Hariri and Potts (1994) also aim at minimizing
the weighted version of the same function with deadlines by using
DP as a subroutine for job penalties, showing promising results for at
most 300 jobs. Additionally, the organization and theorems proposed
by Woeginger (2000) has enabled the existence of several FPTAS that
depend on DP not only in single machine scheduling but also in other
environments, as we will mention later.
SSDP is one of the modified structures of pure DP that has also
appeared recurrently and is usually introduced in Tanaka’s papers to
solve tardiness related problems (Tanaka & Fujikuma, 2012; Tanaka
& Sato, 2013) and makespan problems (Tanaka & Fujikuma, 2012).
12
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E.A.G.d. Souza et al.
Table 8
Multicriteria and additional functions problems for single machine.
Authors
Van Wassenhove and Gelders (1978)
Barnes and Vanston (1981)
Chand and Schneeberger (1988)
Abdul-Razaq and Potts (1988)
Hall and Posner (1991)
Hall et al. (1991)
De et al. (1991)
Yano and Kim (1991)
Hoogeveen and van de Velde (1991)
Potts and Van Wassenhove (1992b)
Potts and Van Wassenhove (1992a)
Carraway et al. (1992)
De et al. (1992)
Bard et al. (1993)
Ibaraki and Nakamura (1994)
Chand et al. (1994)
Liman and Ramaswamy (1994)
Hariri et al. (1995)
Ventura and Weng (1995)
Weng and Ventura (1996b)
Weng and Ventura (1996a)
Lann and Mosheiov (1996)
Cheng et al. (1996)
Cheng et al. (1998)
Woeginger (2000)
Klamroth and Wiecek (2001)
Yeung et al. (2001a)
Yeung et al. (2001b)
Ventura et al. (2002)
Hendel and Sourd (2005)
Kedad-Sidhoum et al. (2008)
Tanaka et al. (2009)
Cheng and Sun (2009)
Zhang et al. (2009)
Zhao and Tang (2011)
Tanaka and Fujikuma (2012)
Tanaka and Sato (2013)
Yin et al. (2013)
Zhao et al. (2014)
Xingong and Yong (2015)
Ben-Yehoshua and Mosheiov (2016)
Yin, Xu et al. (2016)
Wang et al. (2017)
Bülbül et al. (2019)
Li and Yuan (2020)
Mosheiov and Oron (2021)
Problem notation
∑
1∥ 𝑤𝑗 𝑇𝑗 + 𝜆𝑗 𝐶𝑗
∑
1∥ 𝐹 (𝑠𝑖𝑗 ) + 𝐺(𝑤𝑗 𝑈𝑗 )
∑
∑
1∥( 𝑤𝑗 𝐸𝑗 , 𝑈𝑗 = 0)
∑
1∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 )
∑
1|𝑑𝑗 = 𝑑| 𝑤𝑗 |𝑑 − 𝐶𝑗 |
∑
1|𝑑𝑗 = 𝑑| |𝑑 − 𝐶𝑗 |
∑
1|𝑑𝑗 = 𝑑| (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 )
∑
1∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 )
∑
1|𝑑𝑗 = 𝑑| 𝑤𝑗 |𝑑 − 𝐶𝑗 |
∑
1∥ 𝑌𝑗
∑
1∥ 𝑌𝑗
1∥(𝑒𝑘1 𝑤𝑗 𝑇𝑗 + 𝜆𝑒𝑘2 𝑤𝑗 𝑈𝑗 )
∑
1∥ (𝐶𝑗 − 𝐶 𝑗 )
1∥(𝛼𝐶 𝑗 + (1 − 𝛼)(𝐶𝑗 − 𝐶 𝑗 ))
∑
1∥ (𝛼𝑗 𝐸𝑗 + 𝜆𝑗 𝐶𝑗 )
∑
1∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 )
∑
1|(𝑑𝑖 , 𝑑𝑓 )| 𝐸𝑗 + 𝜎𝑗 𝐿𝑇
∑
1|𝑑𝑤 | (𝛼𝑗 𝐸𝑗 + 𝑤𝑗 𝑈𝑗 )
∑
1∥ 𝑤𝑗 𝑌𝑗
∑
1|𝑑𝑗 = 𝑑| |𝑑 − 𝐶𝑗 |
∑
1|𝑑𝑗 = 𝑑, 𝑡𝑗 | (𝐸𝑗 + 𝑇𝑗 )
∑
1|𝑑𝑗 = 𝑑| [(𝐶𝑗 − 𝑑)2 ]∕𝑛
∑
1∥ (𝐷𝑗 + 𝑈𝑗 )
∑
1∥ (𝐷𝑗 𝑠𝑦 + 𝑈𝑗 𝑠𝑦 )
∑
1∥ (𝐷𝑗 𝑎𝑠𝑦 + 𝑈𝑗 𝑎𝑠𝑦 )
∑
1|𝑝𝑗 (𝑥𝑗 )|(𝑈𝑗 , 𝑥𝑗 )
∑
∑
1∥( 𝑣𝑗 𝑥𝑗 ≤ 𝐾, 𝑤𝑗 𝑈𝑗 )
∑
∑
1∥( 𝑣𝑗 𝑥𝑗 , 𝑤𝑗 𝑈𝑗 ≤ 𝐾)
∑
1∥ 𝑌𝑗
∑
1∥ 𝑤𝑗 𝑌𝑗
∑
1∥(𝑓𝑖 (𝑡)), 𝑥𝑗
∑
1|𝑑𝑤 | (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )
∑
1|𝑑𝑤 | (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 + 𝜌𝑗 𝐷𝑗 + 𝜃𝑗 𝑈𝑗 ) + 𝐿(𝑑𝑤 )
∑
1|𝑑𝑤 | (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 + 𝑤𝑗 𝐶𝑗 ) + 𝐿(𝑑𝑤 )
∑
1|𝑟𝑗 | (𝑓 (𝑟𝑗 ) + 𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 )
∑
1∥ (𝐸𝑗 + 𝑇𝑗 )
∑
1∥ (𝐸𝑗 + 𝑇𝑗 )
∑
1|𝑛𝑜 − 𝑖𝑑𝑙𝑒| 𝑓𝑗 (𝐶𝑗 )
∑
∑
1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑟𝑗 , 𝑟𝑒𝑗| 𝑤𝑗 𝐶𝑗 + 𝜃𝑗𝑟𝑒𝑗
∑
1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑟𝑗 , 𝑟𝑒𝑗|𝐶max + 𝜃𝑗𝑟𝑒𝑗
∑
1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑟𝑗 , 𝑟𝑒𝑗|(𝑇max ∕𝐿max ) + 𝜃𝑗𝑟𝑒𝑗
∑ 𝑟𝑒𝑗
1|𝑝𝑗 (𝑠𝑡𝑗 ), 𝑏𝑗 = 𝑏, 𝑟𝑗 , 𝑟𝑒𝑗|𝐶max + 𝜃𝑗
∑
1|𝑟𝑒𝑗, 𝑟𝑗 |𝐶max + 𝜃𝑗𝑟𝑒𝑗
∑
1|𝑟𝑒𝑗, 𝑟𝑗 |𝐶max + 𝜃𝑗𝑟𝑒𝑗
∑
1|𝑟𝑒𝑗, 𝑟𝑗 , 𝜃𝑗𝑟𝑒𝑗 = 𝜃 𝑟𝑒𝑗 |𝐶max + 𝜃𝑗𝑟𝑒𝑗
∑ 𝑟𝑒𝑗
1|𝑟𝑒𝑗, 𝑟𝑗 , 𝑝𝑗 = 𝑝|𝐶max + 𝜃𝑗
∑
∑
1|ℎ(𝜇), 𝑟 − 𝑎, 𝑝𝑗 (𝑡)|𝛼 𝐶𝑗 + 𝛽 𝑇𝑗
∑
∑
1|ℎ(𝜇), 𝑟 − 𝑎, 𝑝𝑗 (𝑡)|𝛼 𝐶𝑗 + 𝛽 𝐸𝑗
∑
1∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 )
∑
1|𝑟𝑗 | ((𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ))
∑
1|𝑛𝑜 − 𝑖𝑑𝑙𝑒, 𝑝𝑟𝑒𝑐| (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 )
∑
∑
1|𝐿𝑒, 𝑞𝑝𝑠𝑑 |𝛼 𝐸𝑗 + 𝑤𝑗 𝑈𝑗 + 𝛾𝑑
∑
∑
1|𝐿𝑒, 𝑞𝑝𝑠𝑑 |𝛾 𝑑 + 𝑤𝑗
∑
1|𝑝𝑗 (𝑠𝑡𝑗 )| 𝑤𝑗 𝑈𝑗 + 𝛾𝑑
∑
1|𝑝𝑗 (𝑠𝑡𝑗 )| 𝑤𝑗 𝑈𝑗 + 𝛾𝑑
∑
∑
1|𝐿𝑒, 𝑆𝐿𝐾|𝜙(𝑑) + 𝛼 𝐸𝑗 + 𝛽𝑗
∑
∑
1|𝐿𝑒, 𝐶𝑂𝑁|𝜙(𝑑) + 𝛼 𝐸𝑗 + 𝛽𝑗
∑
1∥ 𝑉𝑗
∑
1|𝑚𝑎| 𝑌𝑗
∑
1|𝑚𝑎| 𝑌𝑗
∑
1∥ 𝑌𝑗 ∶ 𝐿max ≤ 𝐾
∑
1∥ 𝑌𝑗 ∶ 𝐿max ≤ 𝐾
∑
1|ℎ(𝜇), 𝑟 − 𝑎| (𝐸𝑗 + 𝑇𝑗 )
∑
1|ℎ(𝜇), 𝑠𝑟 − 𝑎| (𝐸𝑗 + 𝑇𝑗 )
∑
1|ℎ(𝜇), 𝑛𝑟 − 𝑎| (𝐸𝑗 + 𝑇𝑗 )
∑
∑
1∥𝑃 ( 𝑤𝑗 𝑌𝑗(1) , … , 𝑤𝑗 𝑌𝑗(𝑥) )
∑
1|𝑅𝑀𝐴| 𝑌𝑗
Guess-and-Check DP is also a derived method proposed by Davari et al.
Complexity
Method
–
–
–
–
∑
𝑂(𝑛 𝑝𝑗 )
∑
𝑂(𝑛 𝑝𝑗 )
–
–
𝑂(𝑛2 𝑑) - 𝑂(𝑛𝑑)
𝑂(𝑛𝑈 𝐵)
–
–
∑
𝑂(𝑛2 𝑝𝑗 )
–
–
–
–
∑
𝑂(𝑛2 (𝑑𝑤 ) 𝑝𝑗 )
–
∑
𝑂(𝑛(𝑑 + 𝑝𝑗 ))
–
∑
𝑂(𝑛 𝑝𝑗 )
–
𝑂(𝑛2 )
∑
𝑂(𝑛 max{ 𝑝𝑗 , 𝑑max })
–
∑
𝑂(𝐾𝑛𝑈 𝐵 𝜏𝑗 )
∑
𝑂(𝐾𝑛𝑈 𝐵 𝜏𝑗 )
–
–
–
∑
𝑂(𝑛2 𝑝max (1 + 𝑝𝑗 ∕2))
∑
∑
𝑂(𝑛3 (𝐾 + 1)𝑝2max ( 𝑝𝑗 + 1)( 𝑝𝑗 − 𝐾 + 1)2 )
∑
∑
𝑂(𝑛3 (𝐾 + 1)𝑝2max ( 𝑝𝑗 + 1)( 𝑝𝑗 − 𝐾 + 1)2 )
–
𝑂(𝑛2 )
𝑂(𝑛2 )
–
∑
𝑂(𝑛 𝜃𝑗𝑟𝑒𝑗 )
∑
𝑂(𝑛 𝜃𝑗𝑟𝑒𝑗 )
∏
∑
𝑂(𝑛 (1 + 𝑏𝑗 ) 𝜃𝑗𝑟𝑒𝑗 )
2
𝑂(𝑛 )
∑
𝑂(𝑛 𝜃𝑗𝑟𝑒𝑗 )
∑
𝑂(𝑛(𝑟max + 𝑝𝑗 ))
𝑂(𝑛2 )
𝑂(𝑛3 )
∏
∏
𝑂(𝑛𝑡1 ( 𝑎𝑏 ( (1 + 𝑏𝑝𝑗 ) − 1) + 𝑡2 ( (1 + 𝑏𝑝𝑗 ) − 1)))
∏
∏
𝑂(𝑛𝑡1 ( 𝑎𝑏 ( (1 + 𝑏𝑝𝑗 ) − 1) + 𝑡2 ( (1 + 𝑏𝑝𝑗 ) − 1)))
–
–
–
𝑂(𝑛3 )
𝑂(𝑛3 )
∑
𝑂(𝑛 𝑤𝑗 )
∏
𝑂(𝑎𝑗 (1 + 𝑏𝑗 ))
𝑂(𝑛4 )
𝑂(𝑛3 )
∑
𝑂(𝑛 𝑝𝑗 )
𝑂(𝑛𝑡1 (𝑡1 + 𝑈 ))
𝑂(𝑛𝑈 (𝑡1 + 𝑈 ))
∑
𝑂(𝑛𝐴 𝑛𝐵 min{(𝑆𝑃 , 𝑠𝑝, ( 𝑑𝑗 )𝐵 }))
∑
𝑂(𝑛𝐴 𝑛𝐵 ( 𝑝𝑗 )𝐴 )
–
–
–
∏
𝑂(𝑛2 𝑈 𝐵𝑥 )
∑
𝑂(𝑛( 𝑝𝑗 + 𝑇 ))
DP and Dual algorithm
DPBB
DP and Heuristic
DP and B&B
DP and FPTAS
DP
DP and Approximation algorithm
DP
DP
DP
Approximation with DP and B&B
Heuristic with DP and B&B
DP
DP
B&B with DP and Heuristic
SSDP
DP
DP
B&B with DP
DP
DP and Heuristic
DP
DP and Heuristics
DP
DP
DP
DP and Approximation algorithm
DP and Approximation algorithm
FPTAS with DP
FPTAS with DP
DP
DP
DP
DP
DP
Heuristic with DP
Heuristic with DP and LBs
SSDP
DP and FPTAS
DP and FPTAS
DP and FPTAS
DP
DP and Approximation algorithm
DP and Approximation algorithm
DP
DP
DP
DP
SSDP
SSDP
SSDP
DP
DP
DP
DP
DP
DP
DP
DP and FPTAS
DP and FPTAS
DP, B&B and TS
DP, B&B and TS
DP
DP
DP
DP and Approximation algorithm
DP
Regarding completion time related criteria, one can notice that most
of them are concentrated on maintenance constraints and that the
majority resorts to pure DP in order to solve the various problems
involving completion times with different types of constraints. The
papers encompass pure DP formulations, comparisons with heuristics,
integer programming and B&B and also the development of FPTAS with
(2020) for the makespan analysis. Note that this last one is a recent
paper and therefore, a novel method that could be a potential asset in
further research involving DP, given that for the 1|𝑟𝑗 , 𝑖𝑛𝑣|𝐶max (which
is strongly NP-hard) it has outperformed a B&B method and a MIP.
13
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Fig. 1. Number of papers, separated by journal, that includes DP as solution method for single machine scheduling problems.
Fig. 2. Frequency of DP in single machine scheduling problems per journal.
DP as main subroutine. Makespan, which is analyzed separately in this
section, entails majorly problems with controllable processing times,
which is one of the forms that makespan becomes a function of interest
in the optimization for the single machine environment. Pure DP is
presented in conjunction with other methods or studied as one of the
methods that could be applied in order to solve the problem.
The last subsection encompasses multi-criteria and additional functions that are not considered regular according to the definition in the
previous section. The main point that must be discussed regarding it
is that since last decade, many objective functions have been leaning
towards the use of resources constrained by a threshold, penalties on
rejected jobs and multi-agent functions. The reason for that might be
explained due to the fact that many of the manufacturing companies operate on shortage of resources and, while the process is ongoing, there
might exist the need to prioritize some of the components in the system.
The classic objective functions in literature, despite being crucial for
the development of scheduling, may often incorporate assumptions that
could be unrealistic in the industrial scenario. Therefore, many authors
have been more interested in investigating situations that might appear
more frequently in the shop floor.
Despite all the papers that have been produced over these years
with respect to single machine scheduling there are some gaps that can
be pointed out in literature. The first one is related to the tardinessrelated criteria and according to our research, precedence constraints
and unconstrained are often investigated using DP as solution method.
Therefore, problems considering other constraints such as release dates,
common due dates, preemption, among others should be looked at.
Even though the Curse of Dimensionality may represent a major drawback, with new methods derived from DP, its effects can be mitigated when including such constraints. This also leads to the second
point that should be the formulation of new dominance rules, labeling
schemes and decomposition properties in order to reduce the state
space. By doing so, the storage problems are also diminished and the
efficiency of DP can also be improved. In addition, SSDP, proposed
in Tanaka’s papers, has shown promising results for several classic
objective functions and it may indicate that it is a strong candidate to
be applied to other models involving makespan, earliness and tardiness
and also problems that entail the more recent functions (e.g. job
rejection). Lastly, one can notice that a large portion of the articles focuses on the mathematical development of formulations and complexity
14
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 9
Due date related problems for parallel machines.
Authors
Lawler and Moore (1969)
Baptiste (2000)
Hall, Lesaoana, and Potts (2001)
Lopes and de Carvalho (2007)
Tuong et al. (2010)
Pessoa, Uchoa, De Aragão, and Rodrigues (2010)
Yoo and Lee (2016)
Tadumadze, Emde, and Diefenbach (2020)
Problem notation
∑
𝑃 𝑚∥ 𝑤𝑗 𝐸𝑗
∑
𝑄𝑚∥ 𝑤𝑗 𝐸𝑗
∑
𝑃 𝑚|𝑑, 𝑑| 𝑇𝑗
∑
𝑄𝑚|𝑑, 𝑑| 𝑇𝑗
∑
𝑃 𝑚|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑇𝑗
∑
𝑄𝑚|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑇𝑗
∑
𝑃 𝑚|𝑝𝑗 = 𝑝, 𝑟𝑗 | 𝑇𝑗
𝑃 𝑚|𝑑𝑠 |𝐿max
𝑃 𝑚|𝑑𝑠 |𝐿max
∑
𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠| 𝑇𝑗
∑
𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠| 𝑤𝑗 𝑇𝑗
∑
𝑅𝑚|𝑠𝑖𝑗𝑘 , 𝑟𝑗 , 𝑎𝑘 | 𝑤𝑗 𝑇𝑗
∑
𝑃 𝑚|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑇𝑗
∑
𝑄𝑚|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑇𝑗
∑
𝑃 𝑚∥ 𝑤𝑗 𝑇𝑗
𝑃 𝑚|𝑖𝑛𝑑|𝐿max
∑
𝑃 𝑚|𝑖𝑛𝑑| 𝐿𝑗
𝑃 𝑚|𝑑𝑒𝑝|𝐿max
∑
𝑃 𝑚|𝑑𝑒𝑝| 𝐿𝑗
𝑃 𝑚|𝑑𝑒𝑝, 𝑝𝑗 = 𝑝|𝐿max
∑
𝑃 𝑚|𝑑𝑒𝑝, 𝑝𝑗 = 𝑝| 𝐿𝑗
𝑅𝑚|𝑟𝑗 , 𝑑 𝑗 |𝑤𝑗 𝐿max
analysis rather than computational experiments. Logically, it is relevant to demonstrate the former, however showing the computational
performance might be as important as the mathematical analysis so as
to understand the limitations a given model can present and also to
encourage new researchers to tackle these challenges.
The last analysis that we have made regarding the information
we could extract is a qualitative analysis concerning the journals that
have published the articles in which DP has been presented as solution
method for single machine scheduling problems. From Fig. 1, the journal that has presented the most prominent contribution is the European
Journal of Operational Research, which is accountable for 22.6% of
the amount of papers that have been gathered in this research, as it
is shown in Fig. 2. Other journals have also contributed significantly in
publishing the DP-related articles in the single machine environment,
as it can be seen by the percentages attributed to them (Operations
Research, Computers and Operations Research (8.3% each), Operations
Research Letters (7.1%) and Journal of Scheduling (6%)). Note that
when comparing Figs. 1 and 2 more journals have been categorized
in the first one. This can be explained due to the fact that the first
picture considers all papers involved in our research and the second one
only those that have been the most recurrent in publishing DP in single
machine problems. Those that have shown less than 2% of contribution
were placed in the ‘‘Others’’ category.
Complexity
Method
–
–
–
–
–
–
𝑂(𝑛2𝑚+2 )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚−1 )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚𝑠−1 )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚𝑠−1 )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚𝑠−1 )
–
∑
𝑂(𝑛𝑚+1 𝑑 2𝑚 ( 𝑝𝑗 )𝑚−1 )
∑
𝑂(𝑚𝑛𝑚+1 𝑑 2𝑚 ( 𝑝𝑗 )𝑚−1 )
–
𝑂(𝑛𝑚(𝑅1 + 𝑅2 )𝑚 )
𝑂(𝑛𝑚(𝑅1 + 𝑅2 )𝑚 )
𝑂(𝑛𝑚𝑅𝑚3 𝑅𝑚4 𝑇 𝑚 )
𝑂(𝑛𝑚𝑅𝑚3 𝑅𝑚4 𝑇 𝑚 )
𝑂((𝑛 + 1)3𝑚 )
𝑂((𝑛 + 1)3𝑚 )
–
DP
DP
DP
DP
DP
DP
DP and Decomposition algorithm
DP
DP
DP
DP
B&P with DP
DP
DP
BCP with DP
DP
DP
DP
DP
DP
DP
MIP, BB&C with BDP and Heuristics
formulation is derived from it. Due to that, it is possible to develop
∑
a polynomial solution of 𝑂(𝑛2𝑚+2 ) complexity for 𝑃 𝑚|𝑝𝑗 = 𝑝, 𝑟𝑗 | 𝑇𝑗 .
Hall et al. (2001) explore a series of problems concerning fixed
delivery dates, which will be denoted as 𝑑𝑠 with 𝑠 indicating the number of fixed deliveries. The authors formulate two DP algorithms with
∑
∑
𝑂(𝑛( 𝑝𝑗 )𝑚−1 ) and 𝑂(𝑛( 𝑝𝑗 )𝑚𝑠−1 ) complexities that provide optimal
solutions for the 𝑃 𝑚|𝑑𝑠 |𝐿max and one of them is able to yield optimal
∑
∑
solution for the 𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠| 𝑇𝑗 and 𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠| 𝑤𝑗 𝑇𝑗 .
Lopes and de Carvalho (2007) analyze a branch-and-price (B&P)
∑
algorithm to minimize 𝑅𝑚|𝑠𝑖𝑗𝑘 , 𝑟𝑗 , 𝑎𝑘 | 𝑤𝑗 𝑇𝑗 . Due to the problem’s
robustness, the algorithm relies on a series of improvements and one
of those is directed to the DP formulations that compose the pricing
algorithm embedded in the B&P, which is named primal box. The
method is applied in instances with up to 150 jobs and 50 machines
and promising results are attained.
Tuong et al. (2010) extend their single-machine DP formulation to
the identical and uniform machines under modifications. The assumptions of their DP are based on the existence of straddling jobs and
WSPT priority rule for the fully tardy jobs. These properties generate two algorithms, each designated to its associated problem, with
∑
∑
𝑂(𝑛𝑚+1 𝑑 2𝑚 ( 𝑝𝑗 )𝑚−1 ) and 𝑂(𝑚𝑛𝑚+1 𝑑 2𝑚 ( 𝑝𝑗 )).
Pessoa et al. (2010) present a Branch-Cut-and-Price (BCP) algorithm
∑
in order to solve the 𝑃 𝑚∥ 𝑤𝑗 𝑇𝑗 . This algorithm relies on a forward
and backward DP recursion to fix variables related to Lagrangian
bounds that optimize the pricing algorithm, which is one of the essential elements in the internal structure of the BCP. Experiments are
placed for at most 100 jobs and 4 machines and results are promising
for the scenario with the authors emphasizing the need of sophisticated
methods to large-sized arc-time indexed problems such as the one
presented.
Yoo and Lee (2016) investigate a series of problems concerning the
dependent and independent cases of maintenance on parallel machines.
∑
Their analysis include classification of 𝑃 𝑚|𝑖𝑛𝑑|𝐿max , 𝑃 𝑚|𝑖𝑛𝑑| 𝐿𝑗 ,
∑
𝑃 𝑚|𝑑𝑒𝑝|𝐿max and 𝑃 𝑚|𝑑𝑒𝑝| 𝐿𝑗 as NP-hard in the strong sense and DP
formulations are proposed in order to solve them. Their complexities
are functions of some elements relative to the maintenance activity
∑
and those are defined as 𝑅1 = min{𝑇 − (𝑡2 − 𝑡1 ), ( 𝑝𝑗 + (𝑚 + 1)(𝑡2 −
∑
𝑡1 ) − 𝑝max )∕𝑚 + 𝑝max }, 𝑅2 = ( 𝑝𝑗 − 𝑝max )∕𝑚 + (𝑡2 − 𝑡1 ) + 𝑝max , 𝑅3 =
∑
min{𝑇 − (𝑚 − 𝑘 + 1), ( 𝑝𝑗 + (𝑚 − 1)(𝑡2 − 𝑡1 ) + (𝑚 − 2)𝑝max )∕𝑚 + 𝑝max } and
∑
𝑅4 = ( 𝑝𝑗 + (𝑚 − 1)𝑝max )∕𝑚 + (𝑡2 − 𝑡1 ) + 𝑝max . Furthermore, the authors
extend the DP approaches for problems with identical processing times
for all the jobs. Detailed information about their complexities is shown
in Table 9.
5. Parallel machines
Dynamic Programming has also contributed for an extensive list of
solutions regarding the parallel machine environment. Several mathematical properties have been developed for this multi-machine scenario
and this fact has enabled sole or combined methods relying on dynamic
programming resources to be designed.
5.1. Classic due date related criteria
Lawler and Moore (1969) extend the functional DP formulation
for the same problems studied in the single machine environment to
parallel machines. Since loss functions are part of the formulation,
they might be equal or not. This feature, combined to the behavior of processing times, generates solutions for identical and uniform
machines.
Baptiste (2000) proves that, since pairwise monotonicity and nondecreasing behavior can be observed for some classes of functions
in scheduling, a decomposition algorithm can be applied and a DP
15
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 10
Number of tardy jobs related problems for parallel machines.
Authors
Lawler and Moore (1969)
Li (1995)
Chen and Powell (1999b)
Hall et al. (2001)
Chen and Powell (2003)
Problem notation
∑
𝑃 𝑚∥ 𝑈𝑗
∑
𝑄𝑚∥ 𝑈𝑗
∑
𝑃 𝑚∥ 𝑈𝑗
∑
𝑃 𝑚∥ 𝑤𝑗 𝑈𝑗
∑
𝑄𝑚∥ 𝑤𝑗 𝑈𝑗
∑
𝑅𝑚∥ 𝑤𝑗 𝑈𝑗
∑
𝑃 𝑚|𝑑𝑠 | 𝑈𝑗
∑
𝑃 𝑚|𝑑𝑠 | 𝑤𝑗 𝑈𝑗
∑
𝑃 𝑚|𝑆𝐼𝑓 | 𝑤𝑗 𝑈𝑗
∑
𝑃 𝑚|𝑆𝐷𝑓 | 𝑤𝑗 𝑈𝑗
Tadumadze et al. (2020) develop a MIP, a Benders-andCut algorithm and a heuristic in order to solve the 𝑅𝑚|𝑟𝑗 , 𝑑 𝑗 |𝑤𝑗 𝐿max .
The second algorithm is in fact a decomposition of two problems, which
are the master problem and the slave problem. The second one is solved
by using a Bounded Dynamic Programming (BDP) approach, which is a
branched method from DP that includes lower bounds and dominance
rules. The authors state that the Benders-and-Cut algorithm is able to
outperform the other methods for instances with up to 60 jobs and 13
machines.
Complexity
Method
–
–
–
∑
𝑂(𝑛2 𝑝𝑗𝑘 )
∑
𝑂(𝑛2 𝑝𝑗𝑘 )
∑
𝑂(𝑛2 𝑝𝑗𝑘 )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚 )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚 )
2
𝑂(𝑛 max{𝑡2 𝑗 − 𝑡2 𝑗 + 1})
𝑂(𝑛2 max{𝑡2 𝑗 − 𝑡2 𝑗 + 1})
DP
DP
DP and Heuristic
B&B with DP
B&B with DP
B&B with DP
DP
DP
DP
DP
Dessouky, Lageweg, Lenstra, and van de Velde (1990) present algorithms of different types to solve several problems in uniform machines.
Among those, a DP formulation is designed to compute recursively the
∑
completion times in the 𝑄𝑚|𝑟𝑗 , 𝑝𝑖𝑗 = 1| 𝐶𝑗 problem. The algorithm is
bounded by an 𝑂(𝑚𝑛2𝑚+1 ) time complexity.
∑
Lee and Liman (1993) prove that the 𝑃 2|𝑚2 ≤ 𝐾| 𝐶𝑗 problem is
NP-complete, where 𝐾 represents a capacity constraint imposed on the
second machine, and develop a pseudo-polynomial DP algorithm to
compute the completion time recursively at each stage and a heuristic,
which is proven to have a 50% error bound assuming a worst-case
scenario.
∑
Alidaee (1993) establishes similarity patterns between 𝑃 2∥ 𝑤𝑗 𝐹𝑗
and the single machine problem investigated by Hall and Posner
(1991). The author incorporates properties previously known such
as V-shaped structure and proposes an adapted DP formulation with
∑
𝑂(𝑛 𝑤𝑗 ) complexity for solving the identical parallel machines problem.
Webster (1994) describes the DP formulation proposed by Alidaee
(1993) as a specific case of the one proposed by Rothkopf (1966) for an
arbitrary number of machines and a more general function, in which
discounts and dependencies between jobs might be allowed.
Cheng and Diamond (1995) study the problem for minimizing the
∑
𝑃 𝑚|𝐽𝑓 | 𝐶𝑗 , where 𝐽𝑓 denotes job families and no job in the second
family is allowed to precede jobs in the first one. The authors investigate a two-job family case and a general DP recursive formulation
with 𝑂(𝑛2 𝑚) overall complexity is devised by adding the contribution
of each scheduled job in a family to the objective function separately.
Lee (1996) addresses two problems involving unavailability for the
∑
∑
second machine on 𝑃 2|𝑟 − 𝑎| 𝑤𝑗 𝐶𝑗 and 𝑃 2|𝑛𝑟 − 𝑎| 𝑤𝑗 𝐶𝑗 . Initially, a
re-indexation of jobs is made through the WSPT priority rule and a DP
formulation is provided to compute the objective function in each case,
∑
∑
yielding 𝑂(𝑛 𝑝𝑗𝑘 𝑠2 𝑝max ) and 𝑂(𝑛 𝑝𝑗𝑘 𝑠2 ) complexities.
Alon, Azar, Woeginger, and Yadid (1998) present an ILP and a
DP approach as alternative methods to constitute the formulation of
∑
a PTAS to solve the 𝑃 𝑚∥ 𝑓 (𝐶𝑗 ). Although the DP formulation is estimated by a pseudo-polynomial bound, the ILP displays more effective
results by producing a polynomial time complexity.
van Den Akker, Hoogeveen, and van de Velde (1999) describe a
pricing algorithm as a forward DP formulation to solve a column generation approach to a linear programming relaxation. A B&B algorithm
is also provided and the authors prove that, when combined with the
linear programming relaxation, it outperforms the previous algorithms
∑
used for solving 𝑃 𝑚∥ 𝑤𝑗 𝐶𝑗 regarding number of jobs and CPU time.
Chen and Powell (1999b) devise DP formulations, which are modifications from those used for solving the weighted number of tardy jobs,
to minimize the weighted completion time for the same environments.
These formulations solve each single machine sub-problem with the
∑
same 𝑂(𝑛2 𝑝𝑗𝑘 ) complexity.
Baptiste (2000) also verifies that the premises to apply the decom∑
position algorithm hold for the 𝑃 𝑚|𝑝𝑗 = 𝑝, 𝑟𝑗 | 𝑤𝑗 𝐶𝑗 problem. Since
the algorithm has no modification from the total tardiness version, the
complexity is also computed at 𝑂(𝑛2𝑚+2 ) bound.
Lee and Chen (2000) analyze the NP-hardness related to the
∑
∑
𝑃 𝑚|𝑛𝑟 − 𝑎, 𝑖𝑛𝑑| 𝑤𝑗 𝐶𝑗 and 𝑃 𝑚|𝑛𝑟 − 𝑎, 𝑑𝑒𝑝| 𝑤𝑗 𝐶𝑗 problem, where 𝑑𝑒𝑝
5.2. Number of tardy jobs
Lawler and Moore (1969) also extend the properties of the functional DP formulation for uniform and identical machines. Despite
being presented as a natural extension this analysis performs calculations with exponential growth even for cases where machines are
identical.
∑
Li (1995) proposes a heuristic method as solution for the 𝑃 𝑚∥ 𝑈𝑗 ,
considering agreeable due dates in the process and in order to compare
its results, the author develops a bounding scheme derived from a DP
formulation. The experiments are applied in job sets from 50 to 200
jobs and machine sets from 2 to 5 machines.
Chen and Powell (1999b) explore a branch and bound algorithm
by deriving bounds from a decomposition method in order to solve
∑
∑
∑
the 𝑃 𝑚∥ 𝑤𝑗 𝑈𝑗 , 𝑄𝑚∥ 𝑤𝑗 𝑈𝑗 and 𝑅𝑚∥ 𝑤𝑗 𝑈𝑗 . This decomposition
method is solved by a column generation algorithm which, in turn, is
subdivided into two single machine sub-problems, being each solved
∑
by a DP formulation with 𝑂(𝑛2 𝑝𝑗𝑘 ) complexity.
Hall et al. (2001) investigate the application of the DP formulations
∑
∑
on 𝑃 𝑚|𝑑𝑠 | 𝑈𝑗 and 𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠| 𝑤𝑗 𝑈𝑗 with complexity bounded by
∑ 𝑚
𝑂(𝑛( 𝑝𝑗 ) ).
Chen and Powell (2003) resort to a B&B method to solve the
∑
∑
𝑃 𝑚|𝑆𝐼𝑓 | 𝑤𝑗 𝑈𝑗 and 𝑃 𝑚|𝑆𝐷𝑓 | 𝑤𝑗 𝑈𝑗 when a given time window
(𝑡1𝑗 , 𝑡2𝑗 ) is considered. The branching process is partitioned via decomposition and column generation algorithms until the 𝑚-machine
problem can be converted into single machines sub-problems, which
are solved by a DP recursion of 𝑂(𝑛2 max{𝑡2𝑗 − 𝑡2𝑗 + 1}) complexity (see
Table 10).
5.3. Completion time related criteria
Baker and Merten (1973) report a variety of results concerning the
mean weighted flow time minimization, which are divided in practical
and theoretical approaches. The latter states, among a series of findings,
that the DP formulation proposed by Held and Karp can be used to
reduce the state space for optimal solution in identical and uniform
machines, however it is only fit for small and medium sized sets.
Horowitz and Sahni (1976) provide algorithms to solve the
∑
𝑄2∥ 𝑤𝑗 𝐹𝑗 problem and, among those, a DP algorithm is presented.
Such algorithm is used to determine the partition that produces the optimal value for the objective function and subsequently, it is extended
to the 𝑚-machine case with 𝑂(min{𝑛𝑈 𝐵, 2𝑛 }) complexity. Approximation algorithms are also presented as solution methods.
16
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
and 𝑖𝑛𝑑 define dependent and independent availability, respectively. A
column generation method is also proposed in order to provide optimal
solutions for these problems working similarly to that seen in Chen
and Powell (1999b). The inbuilt DP formulations are defined with
𝑂(𝑛2 𝑅2 ) and 𝑂(𝑛2 𝑅3 + 𝑛𝑅4 𝑡2 ) for the independent and dependent case,
∑
respectively. In this notation 𝑅2 = [ 𝑝𝑗 − 𝑝max ]∕𝑚 + (𝑡2 − 𝑡1 ) + 𝑝max ,
∑
∑
𝑅3 = [ 𝑝𝑗 + (𝑚 − 1)(𝑡2 − 𝑡1 ) + (𝑚 − 2)𝑝max ]∕𝑚 + 𝑝max and 𝑅4 = [ 𝑝𝑗 + (𝑚 −
1)𝑝max ]∕𝑚 + (𝑡2 − 𝑡1 ) + 𝑝max . Moreover, computational experiments are
performed in order to test the algorithm’s efficiency for medium sized
jobs (up to 60) and 𝑚 ranging from 2 to 8 machines.
Webster and Azizoglu (2001) propose a forward and a backward
∑
DP algorithm to solve the 𝑃 𝑚|𝑆𝐼𝑓 | 𝑤𝑗 𝐶𝑗 problem, where 𝑆𝐼𝑓 denotes family independent setup times. The problem is classified as
NP-hard and the algorithms are pseudo-polynomial with 𝑂(𝑚𝑓 𝑚+1−𝑓 𝑛𝑓
∑
∑
( 𝑤 )𝑚−1 ) complexity for the backward DP and 𝑂(𝑚𝑓 𝑚+2−𝑓 𝑛𝑓 ( 𝑤𝑗 +
∑ 𝑗𝑚
𝑝𝑗 ) ) complexity for the forward DP.
Hall et al. (2001) also present the DP algorithms in the last two
∑
sections to furnish optimal solutions for the 𝑃 𝑚|𝑑𝑠 | 𝐶𝑗 , 𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠|
∑
∑
𝐶𝑗 and 𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠| 𝑤𝑗 𝐶𝑗 . The algorithm used to solve the first
∑
problem produces an optimal schedule within a 𝑂(𝑛( 𝑝𝑗 )𝑚−1 ) complexity while the other one, which solves the remaining problems, shows
∑
an 𝑂(𝑛( 𝑝𝑗 )𝑚𝑠−1 ) complexity.
Chen and Powell (2003) use a network-like structure for 𝑃 𝑚
∑
∑
|𝑆𝐼𝑓 | 𝑤𝑗 𝐶𝑗 and 𝑃 𝑚|𝑆𝐷𝑓 | 𝑤𝑗 𝐶𝑗 problems in association with decomposition and column generation algorithms, reaching singlemachine sub-problems, which are solved by a DP formulation bounded
by 𝑂(𝑛2 max{𝑡2𝑗 − 𝑡1𝑗 + 1}) complexity.
Ramachandra and Elmaghraby (2006) offer a BIP method and a
∑
DP algorithm to find optimal solutions for 𝑃 2|𝑝𝑟𝑒𝑐| 𝑤𝑗 𝐶𝑗 . The DP
presents better results for a larger number of jobs and processing times
range, even though the DP can generate a smaller number of solutions
for reduced number of jobs and processing times. A genetic algorithm
(GA) is also proposed for larger number of jobs and the 2 and 3-machine
cases and experimental results are registered.
Mellouli, Sadfi, Chu, and Kacem (2009) evaluate the performance of
∑
exact algorithms based on MILP, B&B and DP for the 𝑃 𝑚|ℎ(𝜇𝑘 )| 𝐶𝑗 .
Experiments are performed on job sets with up to 200 jobs and machine
sets ranging from 2 to 4 machines. DP shows the best results for 𝑚 = 2,
being able to solve 200-job instances and reasonable results for 𝑚 ≥ 2,
however, for this case, its efficiency is compromised as jobs as job sets
become higher than 25.
Kim, Sung, and Lee (2009) investigate the minimization of
∑
𝑃 𝑚|𝑠 − 𝑝𝑟𝑒𝑐| 𝐶𝑗 , where 𝑠 − 𝑝𝑟𝑒𝑐 indicates that 𝑠 precedence relationships are allowed into the scenario. The authors propose a LP∑
based heuristic, whose core solution is given by an 𝑂(𝑛 𝑝𝑗 ) pseudopolynomial DP approach to a separation problem. Experiments are
performed for at most 300 jobs and 10 machines and the heuristic is
compared to a SPT-based heuristic, proving the former obtains better
results in terms of (𝑛∕𝑚) ratio and CPU times.
Zhao, Tang, and Cheng (2009) investigate a special case of maintenance, which is the rate-modifying (denoted by 𝛼𝑗 ) activity, for two
∑
identical parallel machines and
𝑤𝑗 𝐶𝑗 criterion. The authors also
develop a DP algorithm for cases where jobs are agreeable accord∑
ing to the WSPT-ratio and 𝛼𝑗 -WSPT-ratio and it presents 𝑂(𝑛( max
6
{𝑝𝑗 , 𝛼𝑗 𝑝𝑗 }) ) complexity and show that the approach can be extended
∑
to a 𝑚-machine scenario with 𝑂(𝑛( max{𝑝𝑗 , 𝛼𝑗 𝑝𝑗 })3𝑚 ).
Tang and Zhang (2011) devise a Lagrangian relaxation algorithm
∑
in order to minimize the 𝑅𝑚|𝑟𝑗 | 𝐶𝑗 problem. Such algorithm can
be divided in two parts, which are connected. A relaxation decomposition is performed and it originates machine-level sub-problems,
whose solvability is given by a combination of a forward DP recursion
and WSPT priority rule. Experiments are performed in order to verify
the efficiency of the method and it furnishes high quality solutions in
reasonable CPU time for instances with up to 100 jobs and 10 machines.
Zhao and Tang (2014) develop a DP approach for the
∑
𝑃 𝑚|𝑛𝑟 − 𝑎, 𝑝𝑗 (𝑠𝑡𝑗 )| 𝑤𝑗 𝐶𝑗 problem. This algorithm utilizes the premises
of the Principle of Optimality and indexation of jobs follows the
weighted shortest deteriorating rate (WSDR) priority rule with com∏
∏
∏
plexity bounded by 𝑂(𝑛 (𝑡1𝑘 − 𝑡0 ) (𝑡2𝑘 ( 1 + 𝑏𝑗 )𝑚 ).
∑
Yoo and Lee (2016) also derive DP algorithms for the 𝑃 𝑚|𝑖𝑛𝑑| 𝐶𝑗 ,
∑
𝑃 𝑚|𝑖𝑛𝑑| 𝑤𝑗 𝐶𝑗 problems, their dependent cases counterparts and the
dependent cases with equal processing times for all jobs and their
complexities are equivalent to those shown for the total lateness and
maximum lateness.
Tadumadze et al. (2020) analyze the 𝑅𝑚|𝑟𝑗 , 𝑑 𝑗 |𝑤𝑗 𝐹max problem via
the same methods used for the 𝑤𝑗 𝐿max counterpart. The results also
indicate that the BB&C algorithm outperforms the other ones tested for
small and medium-sized instances with up to 60 jobs and 13 machines,
however the heuristic works more efficiently when problems with 100
jobs and 25 machines are considered (see Table 11).
5.4. Makespan
Horowitz and Sahni (1976) also investigate solution methods to
minimize makespan in parallel machines. In order to do so, a DP algorithm is devised for the 𝑅2∥𝐶max problem, in which the optimal schedule can be found using an SPT rule and the partition of jobs can be determined
through
the
DP.
This
algorithm
presents
∑
𝑂(min{𝑛 min{ 𝐶𝑖𝑘 }, 2𝑛 }) complexity and, can be naturally extended for
∑
the 𝑄𝑚∥𝐶max problem with 𝑂(min{𝑛 min{ 𝐶𝑖𝑘 }, 𝑚𝑛 }) complexity.
Jansen and Porkolab (2001) develop an approximation algorithm
to solve the 𝑅𝑚∥𝐶max problem. This method contains several steps and
among them, a DP algorithm is devised in order to find an optimal
schedule for jobs with long processing times, which are grouped according similarities tested against a given criterion. Such algorithm is bound
with 𝑂(𝑙𝑚(𝑙𝑚∕𝛿)𝑚 ), with 𝑙 being the number of jobs grouped accordingly
and 𝛿 a partition parameter.
Hall et al. (2001) applies their DP algorithms to in order to analyze the behavior for makespan. Both of them are suitable for such
∑
criterion and the complexities associated are also 𝑂(𝑛( 𝑝𝑗 )𝑚−1 ) and
∑ 𝑚𝑠−1
𝑂(𝑛( 𝑝𝑗 )
).
Ghirardi and Potts (2005) propose a beam search algorithm to
solve the 𝑅𝑚∥𝐶max problem. Such algorithm presents similarities to a
B&B algorithm, however a DP-based pruning criterion is used in order
to select which vertices should compose the solution. The theoretical
results are computationally tested and compared to an approximation
algorithm for jobs ranging from 100 to 1000 and machines ranging
from 10 to 100. Results show that for a majority of cases the beam
search shows superior results in terms of CPU times and solution
quality.
Tang and Luo (2006) develop a powerful iterated local search (ILS)
algorithm for solving instances with up to 1000 and 40 machines
regarding the 𝑃 𝑚∥𝐶max problem. An approximately DP algorithm is
integrated to its constitution to furnish the minimum cost to a cyclic
exchange in an associated graph so as to design an initial solution for
the algorithm.
Glass and Kellerer (2007) consider a job assignment restriction
for the 𝑃 2∥𝐶max , which means that there might occur impositions on
which machines jobs are allocated to. They propose a FPTAS and an
𝑂(𝑛𝑝max ) complexity DP formulation to tackle the problem, with the
latter being a knapsack adapted formulation through a Subset Sum
Problem analysis.
Kellerer and Strusevich (2008) evaluate solutions to solve the job assignment restriction problem denoted by 𝑃 2|𝑝𝑗 (𝑥𝑗 ), 𝑥𝑗 = 1, 𝑟𝑛𝑤, 𝐵𝑖|𝐶max ,
where 𝑟𝑛𝑤 represents a renewable resource and 𝐵𝑖 reinforces that
consumption occurs while this resource is being used as a ‘‘speed-up’’
mechanism for the processing times. A DP algorithm is proposed to
solve the problem, which is partitioned into two knapsack problems,
however the DP does not contribute efficiently to this objective and
a DP-based FPTAS is provided instead. The complexity of the DP algo∑
rithm is bounded by 𝑂(𝑛𝑅), where 𝑅 = max{𝐴, 𝐵}, 𝐴 = 𝑗∈𝑁1 (𝑎𝑗1 − 𝑏𝑗1 )
∑
and 𝐵 = 𝑗∈𝑁2 (𝑎𝑗2 − 𝑏𝑗2 ).
17
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 11
Completion time related problems for parallel machines.
Authors
Baker and Merten (1973)
Horowitz and Sahni (1976)
Dessouky et al. (1990)
Lee and Liman (1993)
Alidaee (1993)
Webster (1994)
Cheng and Diamond (1995)
Lee (1996)
Alon et al. (1998)
van Den Akker et al. (1999)
Chen and Powell (1999b)
Baptiste (2000)
Lee and Chen (2000)
Webster and Azizoglu (2001)
Hall et al. (2001)
Chen and Powell (2003)
Ramachandra and Elmaghraby (2006)
Mellouli et al. (2009)
Kim et al. (2009)
Zhao et al. (2009)
Tang and Zhang (2011)
Zhao and Tang (2014)
Yoo and Lee (2016)
Tadumadze et al. (2020)
Problem notation
∑
𝑃 𝑚∥ 𝑤𝑗 𝐹𝑗
∑
𝑄𝑚∥ 𝑤𝑗 𝐹𝑗
∑
𝑄𝑚∥ 𝑤𝑗 𝐹𝑗
∑
𝑄𝑚|𝑟𝑗 , 𝑝𝑖𝑗 = 1| 𝐶𝑗
∑
𝑃 2|𝑚2 ≤ 𝐾| 𝐶𝑗
∑
𝑃 2∥ 𝑤𝑗 𝐹𝑗
∑
𝑄𝑚∥ 𝑤𝑗 𝐹𝑗
∑
𝑃 𝑚|𝐽𝑓 | 𝐶𝑗
∑
𝑃 2|𝑟 − 𝑎| 𝑤𝑗 𝐶𝑗
∑
𝑃 2|𝑛𝑟 − 𝑎| 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚∥ 𝑓 (𝐶𝑗 )
∑
𝑃 𝑚∥ 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚∥ 𝑤𝑗 𝐶𝑗
∑
𝑄𝑚∥ 𝑤𝑗 𝐶𝑗
∑
𝑅𝑚∥ 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑝𝑗 = 𝑝, 𝑟𝑗 | 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑛𝑟 − 𝑎, 𝑖𝑛𝑑| 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑛𝑟 − 𝑎, 𝑑𝑒𝑝| 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑆𝐼𝑓 | 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑆𝐼𝑓 | 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑑𝑠 | 𝐶𝑗
∑
𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠| 𝐶𝑗
∑
𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠| 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑆𝐼𝑓 | 𝑤𝑗 𝑈𝑗
∑
𝑃 𝑚|𝑆𝐷𝑓 | 𝑤𝑗 𝑈𝑗
∑
𝑃 2|𝑝𝑟𝑒𝑐| 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|ℎ(𝜇𝑘 )| 𝐶𝑗
∑
𝑃 𝑚|𝑠 − 𝑝𝑟𝑒𝑐| 𝐶𝑗
∑
𝑃 2|𝑝𝑗 (𝛼𝑗 )| 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑝𝑗 (𝛼𝑗 )| 𝑤𝑗 𝐶𝑗
∑
𝑅𝑚|𝑟𝑗 | 𝐶𝑗
∑
𝑃 𝑚|𝑛𝑟 − 𝑎, 𝑝𝑗 (𝑠𝑡𝑗 )| 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑖𝑛𝑑| 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑖𝑛𝑑| 𝐶𝑗
∑
𝑃 𝑚|𝑑𝑒𝑝| 𝐶𝑗
∑
𝑃 𝑚|𝑑𝑒𝑝| 𝑤𝑗 𝐶𝑗
∑
𝑃 𝑚|𝑑𝑒𝑝, 𝑝𝑗 = 𝑝| 𝐶𝑗
∑
𝑃 𝑚|𝑑𝑒𝑝, 𝑝𝑗 = 𝑝| 𝑤𝑗 𝐶𝑗
𝑅𝑚|𝑟𝑗 , 𝑑 𝑗 |𝑤𝑗 𝐹max
Complexity
Method
–
–
𝑂(min{𝑛𝑈 𝐵, 2𝑛 })
𝑂(𝑚𝑛2𝑚+1 )
–
∑
𝑂(𝑛 𝑤𝑗 )
–
𝑂(𝑛2 𝑚)
∑
𝑂(𝑛 𝑝𝑗𝑘 𝑠2 𝑝max )
∑
𝑂(𝑛 𝑝𝑗𝑘 𝑠2 )
–
∑
𝑂(𝑛 𝑝𝑗 )
2∑
𝑂(𝑛
𝑝 )
∑ 𝑗𝑘
𝑂(𝑛2 𝑝𝑗𝑘 )
∑
2
𝑂(𝑛
𝑝𝑗𝑘 )
𝑂(𝑛2𝑚+2 )
𝑂(𝑛2 𝑅2 )
𝑂(𝑛2 𝑅3 + 𝑛𝑅4 𝑡2 )
∑
𝑂(𝑚𝑓 𝑚+1−𝑓 𝑛𝑓 ( 𝑤𝑗 )𝑚−1 )
∑
𝑚+2−𝑓 𝑓 ∑
𝑂(𝑚𝑓
𝑛 ( 𝑤 𝑗 + 𝑝𝑗 ) 𝑚 )
∑ 𝑚−1
𝑂(𝑛( 𝑝𝑗 ) )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚𝑠−1 )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚𝑠−1 )
𝑂(𝑛2 max{𝑡2 𝑗 − 𝑡1 𝑗 + 1})
𝑂(𝑛2 max{𝑡2 𝑗 − 𝑡1 𝑗 + 1})
–
–
∑
𝑂(𝑛 𝑝𝑗 )
∑
𝑂(𝑛( max{𝑝𝑗 , 𝛼𝑗 𝑝𝑗 })6 )
∑
𝑂(𝑛( max{𝑝𝑗 , 𝛼𝑗 𝑝𝑗 })3𝑚 )
∑
𝑂(𝑛 𝑝𝑗𝑘 )
∏
∏
∏
𝑂(𝑛 (𝑡1𝑘 − 𝑡0 ) (𝑡2𝑘 ( 1 + 𝑏𝑗 )𝑚 ))
𝑂(𝑛𝑚(𝑅1 + 𝑅2 )𝑚 )
𝑂(𝑛𝑅2𝑚
𝑟𝑚 )
1 2
𝑂(𝑛𝑚𝑅𝑚3 𝑅𝑚4 𝑇 𝑚 )
𝑂(𝑛𝑚𝑅𝑚3 𝑅𝑚4 𝑇 𝑚 )
𝑂((𝑛 + 1)3𝑚 )
𝑂((𝑛 + 1)3𝑚 )
–
DP and Heuristics
DP
DP and Approximation algorithm
DP
DP and Heuristic
DP
DP
DP
DP
DP
DP, ILP and PTAS
Linear Relaxation with DP
B&B with DP
B&B with DP
B&B with DP
DP and Decomposition algorithm
B&B with DP
B&B with DP
DP
DP
DP
DP
DP
DP
DP
DP, BIP and GA
DP, B&B and MILP
LP-based Heuristic with DP
DP
DP
Lagrangian relaxation with DP
DP
DP
DP
DP
DP
DP
DP
MIP, BB&C with BDP and Heuristic
Li and Wang (2010) conduct a research regarding the 𝑃 𝑚|𝑟𝑗 |𝐶max
considering the job assignment restriction constraint with a DP-based
PTAS as solution method for an arbitrary number of machines and a
DP-based FPTAS for a fixed number of machines. The algorithms are
bounded by 𝑂(𝑚|𝑋𝑚 ∥𝑌𝑚 |) (with |𝑋𝑚 |, |𝑌𝑚 | being assignment subsets for
the last machine) and 𝑂(𝑛𝑚+1 (1∕𝜀)𝑚 ), respectively.
Haned, Soukhal, Boudhar, and Tuong (2012) consider the problem
of minimizing the makespan taking into account delay transportation
and preemption constraints, which is denoted by 𝑃 2|𝑑𝑒𝑙𝑎𝑦, 𝑝𝑟𝑚𝑝|𝐶max .
Despite the paper focusing on a FPTAS, the authors also present a DP
∑
algorithm with 𝑂(𝑛2 𝑝𝑗 ) complexity to solve the problem, which is
derived from a DP formulation proposed for the 𝑃 2∥𝐶max in this paper.
Yoo and Lee (2016) develop DP procedures, which have been also
applied to lateness and total completion times functions, and convey
them to solve the 𝑃 𝑚|𝑖𝑛𝑑|𝐶max , 𝑃 𝑚|𝑑𝑒𝑝|𝐶max and 𝑃 𝑚|𝑑𝑒𝑝, 𝑝𝑗 = 𝑝|𝐶max
problems with the same complexity achieved by the other criteria.
Rudek (2017) discusses the influence on learning or aging effects regarding arbitrary functions of processing times to minimize
the makespan. The author includes preliminary topics on continuous
functions for the single machine environment and devises a pseudopolynomial DP algorithm for integer-valued functions, which is
∑
∑
bounded by 𝑂(𝑛( 𝑝𝑗 )𝑚 ( 𝑝𝑗 𝑓 (𝑗))𝑚 ) as well as the real-valued.
Ghalami and Grosu (2019) attempt a more sophisticated approach
to finding solutions for the 𝑃 𝑚∥𝐶max problem, in which they resort to a
parallel DP method to construct a PTAS. First a DP approach is designed
in order to select all the machines where 𝐶max does not surpass a given
threshold and then the DP formulation is associated to a graph-like
structure in order to determine how to proceed with the adaptation
∑
2
to the parallel DP version with 𝑂((𝑛∕𝜀)(1∕𝜀 ) ∕ 𝑝𝑗 ).
Tadumadze et al. (2020) investigate the 𝑅𝑚|𝑟𝑗 , 𝑑 𝑗 |𝐶max problem via
the same methods used for the 𝑤𝑗 𝐿max counterpart. The BB&C method
shows an outstanding performance even for larger problems with 100
jobs and 25 machines (see Table 12).
5.5. Multi-criteria and additional objective functions
Rothkopf (1966) describes a DP formulation to solve two problems
related to the minimization of a linear waiting cost function in 𝑚
uniform machines and a non-linear waiting cost function in 𝑚 identical machines. This method is able to create 𝑛 − 1 independent state
variables, however the calculations reach exponential order.
Gupta and Maykut (1973) develop a DP formulation for three
waiting cost functions. For the linear and exponential cases, the authors
prove that the allocation problem is reduced to a solving one sequencing problem per stage, however for non-decreasing monotonic general
functions the same cannot be stated and exponential growth of 𝑂(2𝑛 )
order is reached.
So (1990) derives three heuristic procedures to solve the
∑
𝑃 𝑚|𝑆𝑖𝑓 | 𝑅𝑒𝑤𝑗 problem in order to maximize the reward by scheduling composed by 𝐼 job families and 𝐽 job types. The first and second
heuristics employ DP formulations with by finding an optimal schedule
for each machine as a single machine problem (Sequential Heuristic)
and assigning each machine to a set of jobs (Decomposition Heuristic). The algorithms present 𝑂(𝐽 𝑚𝐾 2 ) and 𝑂(𝐽 𝑚2 𝐾 2 + 𝐽 𝑚2 𝐾 ln (𝑚𝐾)),
respectively.
Krämer and Lee (1994) study optimality properties and complexity
∑
analysis for the 𝑃 2|𝑑𝑤 | (𝐸𝑗 + 𝑇𝑗 ) problem and proves it to be NP-hard.
The optimality properties are applied to the proposed algorithm that
18
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 12
Makespan problems for parallel machines.
Authors
Problem notation
Complexity
Method
Horowitz and Sahni (1976)
𝑅2∥𝐶max
𝑄𝑚∥𝐶max
𝑅𝑚∥𝐶max
𝑃 𝑚|𝑑𝑠 |𝐶max
𝑃 𝑚|𝑑𝑠 , 𝑠 = 𝑠|𝐶max
𝑅𝑚∥𝐶max
𝑃 𝑚∥𝐶max
𝑃 2∥𝐶max
𝑃 2|𝑝𝑗 (𝑥𝑗 ), 𝑥𝑗 = 1, 𝑟𝑛𝑤, 𝐵𝑖|𝐶max
𝑃 𝑚|𝑟𝑗 |𝐶max
𝑃 𝑚|𝑟𝑗 |𝐶max
𝑃 2|𝑑𝑒𝑙𝑎𝑦, 𝑝𝑟𝑚𝑝|𝐶max
𝑃 𝑚|𝑖𝑛𝑑|𝐶max
𝑃 𝑚|𝑑𝑒𝑝|𝐶max
𝑃 𝑚|𝑑𝑒𝑝, 𝑝𝑗 = 𝑝|𝐶max
𝑃 𝑚|𝐴𝑒|𝐶max
𝑃 𝑚|𝐿𝑒|𝐶max
𝑃 𝑚∥𝐶max
𝑅𝑚|𝑟𝑗 , 𝑑 𝑗 |𝐶max
∑
𝑂(min{𝑛 min{ 𝐶𝑖𝑘 }, 2𝑛 })
∑
𝑂(min{𝑛 min{ 𝐶𝑖𝑘 }, 𝑚𝑛 })
𝑚
𝑂(𝑙𝑚(𝑙𝑚∕𝛿) )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚−1 )
∑
𝑂(𝑛( 𝑝𝑗 )𝑚𝑠−1 )
–
–
𝑂(𝑛𝑝max )
𝑂(𝑛𝑅)
𝑂(𝑚|𝑋𝑚 ∥𝑌𝑚 |)
𝑂(𝑛𝑚+1 (1∕𝜀)𝑚 )
∑
𝑂(𝑛2 𝑝𝑗 )
𝑂(𝑛𝑅2𝑚
𝑟𝑚 )
1 2
𝑂(𝑛𝑚𝑅𝑚3 𝑅𝑚4 𝑇 𝑚 )
𝑂((𝑛 + 1)3𝑚 )
∑
∑
𝑂(𝑛( 𝑝𝑗 )𝑚 ( 𝑝𝑗 𝑓 (𝑗))𝑚 )
∑ 𝑚 ∑
𝑂(𝑛( 𝑝𝑗 ) ( 𝑝𝑗 𝑓 (𝑗))𝑚 )
∑
2
𝑂((𝑛∕𝜀)(1∕𝜀 ) ∕ 𝑝𝑗 )
–
DP and Approximation algorithm
DP and Approximation algorithm
Approximation algorithm with DP
DP
DP
Beam Search with DP
Heuristic with DP
DP and FPTAS
FPTAS with DP
PTAS with DP
FPTAS with DP
DP and FPTAS
DP
DP
DP
DP
DP
PTAS with Parallel DP
MIP, BB&C with BDP and Heuristic
Jansen and Porkolab (2001)
Hall et al. (2001)
Ghirardi and Potts (2005)
Tang and Luo (2006)
Glass and Kellerer (2007)
Kellerer and Strusevich (2008)
Li and Wang (2010)
Haned et al. (2012)
Yoo and Lee (2016)
Rudek (2017)
Ghalami and Grosu (2019)
Tadumadze et al. (2020)
is partitioned into an 𝑂(𝑛2 𝑑𝑤 ) complexity DP algorithm to schedule
the jobs that fit in the due window interval and an absolute error
bound heuristic method, which reduces the overall complexity of the
algorithm.
De, Ghosh, and Wells (1994) outline the complexity analysis for
∑
∑
the 𝑃 𝑚|𝑑𝑗 = 𝑑| (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ) and 𝑃 𝑚|𝑑𝑗 = 𝑑| 𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 + 𝛾𝑗 𝐿(𝑑)
problems and prove their NP-hardness in the strong sense and their particular cases involving two machines are NP-hard. A pseudo-polynomial
DP algorithm is provided for each problem based on the optimality
properties and computational experiments are performed, indicating
that the proposed DP formulation yields better results in terms of CPU
∑
times for the 𝑃 𝑚|𝑑𝑗 = 𝑑| (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ) problem.
Chen and Powell (1999a) develop a complex scheme to analyze
∑
the 𝑃 𝑚∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ). A branch and bound algorithm is proposed
with a decomposition process via column generation process until each
node of the tree can be examined as two single machines sub-problems
∑
which, in turn, rely on two DP algorithms of 𝑂(𝑛2 𝑝𝑗 ) complexity to
be solved optimally.
Bartal, Leonardi, Marchetti-Spaccamela, Sgall, and Stougie (2000)
apply the rounding DP method seen in Horowitz and Sahni (1976)
∑
to design a FPTAS to solve the 𝑃 𝑚∥𝐶max + 𝜃𝑗𝑟𝑒𝑗 problem. The DP
formulation denotes optimal solutions for each instance and it is used
to verify the approximation ratio of the FPTAS.
Sun and Wang (2003) elaborate a DP algorithm based on the
LPT priority rule to establish a initial indexation of jobs and define a recursive relation to compute the objective function for the
∑
𝑃 𝑚|𝑤𝑗 = 𝑎𝑗 𝑝𝑗 , 𝑑𝑗 = 𝑑| 𝑤𝑗 |𝐶𝑗 − 𝑑| problem. Since the DP is classified
∑
as a pseudo-polynomial algorithm with 𝑂(𝑛𝑚𝑑 𝑚 ( 𝑝𝑗 )𝑚−1 ) complexity,
the problem is NP-hard in the ordinary sense. Furthermore, an additional study on heuristics is shown with calculation of error bounds.
Chen (2004) elaborates a similar algorithmic structure to previous
works on B&B solved via partition, decomposition,column generation
and single machine problems. The last portion of this division generates
two DP algorithms, one of which is employed on finding the solution
∑
∑
for the 𝑃 𝑚|𝑝(𝑥𝑗 )| 𝑤𝑗 𝑈𝑗 + 𝐺(𝑥𝑗 ) problem with 𝑂(𝑛2 𝑡max max{𝑏𝑗 − 𝑎𝑗 })
complexity. A second DP algorithm is applied to the single machine
problem generated via decomposition to find the solution for the
∑
∑
∑
𝑃 𝑚|𝑝(𝑥𝑗 )| 𝑤𝑗 𝐶𝑗 + 𝐺(𝑥𝑗 ) problem with 𝑂(𝑛2 𝑝𝑗 ) complexity and
based on the principle of optimality.
Jansen and Mastrolilli (2004) incorporate PTASs to solve three
problems related to makespan, in which three bi-criteria involving a
cost function with respect to resource allocation. Among these, one of
the PTASs, which is applied to the 𝑃 𝑚|𝑝𝑗 (𝑥𝑗 )|(𝐺(𝑥𝑗 ), 𝐶max ≤ 𝐾) uses a
DP algorithm to reduce to a polynomial number the state space of the
problem
Sung and Vlach (2005) investigate the minimization the
∑
𝑅𝑚∥ (𝐷𝑗 + 𝑈𝑗 ) problem by analyzing its complexity and classifying it
as NP-hard in the strong sense. In addition to this analysis, the authors
design an 𝑂(𝑚𝑛𝑚+1 ) DP polynomial-time algorithm that consists of three
steps and incorporates V-bounded structure properties.
Li, Shen, Ghenniwa, and Wang (2005) work with a criterion that
computes the weighted square sum of the difference between the real
length of time on a given machine (𝑦𝑘 ) and the predefined length
of time on a given machine (𝑧𝑘 ). The problem is modeled as a 0–1
quadratic model and a DP formulation is provided in order to solve
it.
Rios-Solis and Sourd (2008) evaluate theoretical and practical aspects of an exponential search neighborhood algorithm for the
∑
𝑃 𝑚|𝑑𝑗 = 𝑑| 𝐸𝑗 + 𝑇𝑗 problem. Optimality properties are presented and
a DP approach, which is based on interchange operations of early
and tardy jobs in the schedule, is given. This formulation is the main
element to optimize the search of feasible sequences and, therefore,
the algorithm’s complexity is mostly defined by the DP’s, which is
dependent on number of allocations. Experiments are performed in a
set of jobs with up to 150 and applied in 𝑚 = {4, 8} with satisfactory
results given the CPU times and the gap between upper and lower
bounds.
Kedad-Sidhoum et al. (2008) adapt their own approach of the
single-machine problem with earliness-tardiness criterion and devise
∑
bounds for the 𝑃 𝑚∥ (𝐸𝑗 + 𝑇𝑗 ) problem. Experimental results are provided for at most 90 jobs and 6 machines. The DP-based heuristic
and the Lagrangian bound show the best results when compared to
the remaining bounds and the heuristic yields the smallest gap with
regard to CPU times that are close to those registered for the Lagrangian
bound.
Agnetis, Alfieri, and Nicosia (2009) resort to a bounding scheme to
assess the quality of given heuristic to solve general problems whose
functions are defined as max{𝑓 (𝐶𝑗 )}. Therefore, a bounding scheme
is developed in order to provide a comparison with the proposed
heuristics. Such scheme relies on the column generation technique,
which, in turn, is divided to the level of a pricing algorithm that might
be solved by a DP formulation or a DP-based heuristic. Experimental
results are presented for 𝑇max and 𝐶max but we omit them since they
are not part of the scope of our research.
Leyvand, Shabtay, Steiner, and Yedidsion (2010) analyze a variety
of problems involving resource consumption and earliness-tardiness
criteria with controllable processing times. They develop a 𝑂(𝑚𝑛𝑚+1 )
∑
∑
time DP algorithm to solve the 𝑅𝑚|𝑝𝑗 (𝑥𝑗 )| (𝐸𝑗 + 𝑇𝑗 ) + 𝐺(𝑥𝑗 ) and a
FPTAS is derived from a DP formulation for the remaining problems.
Li and Yuan (2010) design a FPTAS and a polynomial time DP
∑
∑
∑
algorithm to solve 𝑃 𝑚|𝑟𝑒𝑗, 𝑝𝑗 (𝑡)| 𝑤𝑗 𝐶𝑗 + 𝜃𝑗𝑟𝑒𝑗 and 𝑃 𝑚|𝑟𝑒𝑗, 𝑝𝑗 (𝑡)| 𝐶𝑗 +
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Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
∑
𝜃𝑗𝑟𝑒𝑗 problems, respectively. The FPTAS depends on a DP recursion,
∑
which can be computed in 𝑂(𝑛( 𝑛 log (1 + 𝑏𝑗 )∕𝜀)𝑚 ) complexity. The
polynomial DP algorithm is an extension of the algorithm devised
by Cheng and Sun (2009) for single machine environment and yields
𝑂(𝑛2 ) complexity.
Zhao and Lu (2013) analyze two distinct models involving two𝐴 ∶
agent scheduling criterion. The first model is related to the 𝑃 𝑚∥𝐶max
∑
𝐵
𝐶max
≤ 𝐾 and a DP algorithm is developed with 𝑂(𝑚𝑛𝐴 𝑛𝐵 ( 𝑝𝑗 )𝑚
𝐴
∑
( 𝑝𝑗 )𝑚
) as well as an FPTAS derived from it. An akin procedure is
𝐵
∑ 𝐴
𝐵
devised for the 𝑃 𝑚∥ 𝐶𝑗 ∶ 𝐶max
≤ 𝐾 with the previously mentioned
complexity.
Dong (2013) aims at minimizing functions that are related to machine setup costs, job processing and energy consumption considering
that the maintenance intervals are set within the problem rather than
being prespecified. A B&B algorithm is proposed and its inner mechanism falls under the set partition scenario, which is ultimately solved
by a DP algorithm in conjunction with a column generation method.
The algorithm is tested via computational experiments, which proves
the method is efficient for medium sized problems.
Rebai, Kacem, and Adjallah (2013) evaluate solutions for the
∑
∑
𝑃 𝑚|ℎ(𝜇)| 𝑤𝑗 𝐶𝑗 + 𝐻(𝑡1 , 𝑡2 ) where 𝐻(𝑡1 , 𝑡2 ) represents the preventive
maintenance cost. The solutions proposed rely on a B&B algorithm and
a GA, which can be reinforced by applying a DP formulation in order
to obtain better initial solutions for the heuristic. The authors show
that for problems with up to 300 jobs and 5 machines, both algorithms
produce high quality solutions within reasonable time. In addition, the
use of DP enhances the GA performance regarding solution quality.
Zhang and Lu (2016) design a theoretical analysis on the
∑ 𝑟𝑒𝑗
𝑃 𝑚|𝑟𝑒𝑗, 𝑟𝑗 |𝐶max +
𝜃𝑗 problem. The authors propose a pseudopolynomial DP approach in conjunction with earliest release date (ERD)
priority rule for indexation and also apply it to a special case denoted
∑
by 𝑃 𝑚|𝑟𝑒𝑗, 𝑟𝑗 , 𝑝𝑗 = 𝑝|𝐶max + 𝜃𝑗𝑟𝑒𝑗 . Each problem presents, respectively,
∑ 𝑚
𝑂(𝑚𝑛(𝑟max + 𝑝𝑗 ) ) and 𝑂(𝑚𝑛2𝑚+1 ) complexities. Furthermore, a FPTAS
is also offered as solution method for the first problem.
Yin, Cheng et al. (2016) present a theoretical assessment for design∑
ing algorithms to solve two-competing agents 𝑅𝑚∥( (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )𝐴 ,
∑
∑
min (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )𝐵 ) and 𝑅𝑚∥( (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )𝐴 , (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )𝐵 ) problems. The authors develop a pseudo-polynomial DP algorithm for
each case, including elimination properties to reduce the solutions
state space and their complexities are bounded by 𝑂(𝑚𝑛𝑚+1
𝑛𝐵 ) and
𝐴
∑
∑
𝑂(𝑚2 𝑛𝑚+1 min{ (𝛼𝑗 + 𝛽𝑗 )𝐴 , (𝛼𝑗 + 𝛽𝑗 )𝐵 }), respectively. In addition, and
FPTAS is derived from the DP approach for the last problem.
Chen, Sterna, Han, and Blazewicz (2016) evaluate online and of∑
fline scheduling problems stemming from the 𝑃 2|𝑑𝑗 = 𝑑| 𝑌𝑗 . The
authors prove the given problem is NP-hard, however a simple pseudopolynomial DP algorithm with 𝑂(𝑛𝑑 2 ) complexity is provided. In addition, they also state how close this problem is to a knapsack problem/bin packing problem as well as how it can relate to the 𝑃 2∥𝐶max
problem.
Sterna and Czerniachowska (2017) develop a PTAS in order to solve
∑
the 𝑃 𝑚∥ 𝑉𝑗 . Its construction derives from adaptations made in a series
of list algorithms and the DP formulation designed by Chen et al.
(2016). Dominance properties are concocted from relative improvements performed by observing the algorithms’ behavior and the PTAS
is generated upon them so as to work in an optimal manner.
Yin, Chen, Qin, and Wang (2019) devise a B&P algorithm to de∑
∑
termine solutions for the two-agent 𝑃 𝑚∥ 𝐶𝑗 𝐴 ∶
𝑤𝑗 𝑈𝑗 𝐵 problem.
The solution method is composed of a three-step algorithm and an
∑
𝑂(𝑛𝐴 𝑛𝐵 max{𝑛𝐴 , 𝑛𝐵 } 𝑝𝑗 ) DP forward recursion is used in order to
provide solutions for the pricing sub-problem generated from a column
generation method in each node. Furthermore, a dominance rule is also
coupled to the DP in order to reduce the state space via fathoming.
Experiments are performed for at most 80 jobs and 10 machines.
Chen, Liang, Sterna, Wang, and Błażewicz (2020) implement a DP
and a FPTAS for solving the 𝑃 𝑚|𝑑𝑗 = 𝑑| max 𝐷𝑗 problem. A theoretical
analysis is carried out in order to deliver a complexity analysis and the
results are tested computationally. Although DP might not be the best
choice for solving this particular problem, it is relevant to state that a
DP formulation is also integrated in the FPTAS in order to generate a
modified instance that will be used as input for the algorithm, which
have been experimented on at most 65 jobs and three machines.
T’kindt, Shang, and Della Croce (2020) generate a theoretical al∑
∑
gorithm for the 𝑃 𝑚|𝑑𝑗 = 𝑑 ≥ 𝑝𝑗 | 𝑤𝑗 (𝐸𝑗 + 𝑇𝑗 ) problem based on the
DP across subsets and machines and Sort & Search algorithms in order
to place the jobs. This ordering also relies on the use of an WSPT rule
for the initial set of solutions and produces an 𝑂(3𝑛 ) time complexity
DP formulation in the worst case. Additionally, this analysis can be
extended to the uniform and unrelated parallel machines case, however
the recursive relations needs to be adapted to such environments with
no additional cost in terms of complexity (see Table 13).
5.6. Discussion on parallel machines problems
Parallel machines problems that used DP as main or secondary
methods did not follow a pattern either, as it was the case for single
machine problems. One fact that can be noticed is that some of the
methods that have been applied to single machine problems have been
extended to parallel machines (e.g. Lawler & Moore, 1969). Furthermore, improvements have been made in order to consider a more
realistic scenario, which can be seen, for example, in Tuong et al.
(2010) by incorporating straddling jobs in the Lawler and Moore (1969)
∑
modeling of the problems involving 𝑤𝑗 𝑇𝑗 .
Problems involving more complex cases of parallel machines such
as unrelated parallel machines and uniform parallel machines have
also been investigated and despite their theoretical analysis, a series
of open problems could be classified according to their computational
complexity, as it is shown in Dessouky et al. (1990), Sung and Vlach
(2005) and Tuong et al. (2010). Yet, the numbers for the identical
parallel machines problems are far larger and even recent papers have
been drawn to these types of machines, once they are less complex
systems and therefore, the adherence of technological constraints or
objective functions becomes more compatible.
In terms of segmentation regarding objective functions, completion time-related criteria have dominated the topics of interest of the
authors. Initially these problems concentrated on the unconstrained
versions of flow time and completion times and on the early nineties,
the inclusion of constraints started to appear, however a pattern could
not be observed since they have varied from time to time. Nevertheless,
some occurrences are indeed more frequent than others, as it is the
case of dependent and independent cases of maintenance and also
controllable processing times.
Some authors have also made use of DP quite recurrently and
consequently gained notoriety by studying adaptations of their initial
approaches. Two that can be cited here are Baptiste, whose main
interest was to apply a decomposition algorithm and derive a DP formulation from it in order to solve tardiness problems and those related to
completion times (e.g. Baptiste, 2000). This author has also participated
in works in single machine environments and flow shop scheduling and
usually his papers discuss equal processing times. Chen’s papers have
also received attention due to the decomposition method and column
generation approach that would unfold into single machine problems
that could be solved via DP (e.g. Chen & Powell, 1999a, 1999b).
The recent trends in parallel machines are similar to those seen for
single machine scheduling and they include more realistic objective
functions and constraints, such as rejected jobs functions and constraints, multi-agent and hierarchical functions, just-in-time functions,
resource allocation and costs, deteriorating jobs and maintenance. It
is important to mention that the analysis for these problems remain
theoretical for most cases when using DP due to the storage limitations
associated to the Curse of Dimensionality.
From this analysis, some conclusions can be drawn about the gaps
that the studies in such environment have generated. Firstly, despite
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E.A.G.d. Souza et al.
Table 13
Multicriteria and additional functions problems for parallel machines.
Authors
Rothkopf (1966)
Gupta and Maykut (1973)
So (1990)
Krämer and Lee (1994)
De et al. (1994)
Chen and Powell (1999a)
Bartal et al. (2000)
Sun and Wang (2003)
Chen (2004)
Sung and Vlach (2005)
Li et al. (2005)
Rios-Solis and Sourd (2008)
Kedad-Sidhoum et al. (2008)
Agnetis et al. (2009)
Leyvand et al. (2010)
Li and Yuan (2010)
Zhao and Lu (2013)
Dong (2013)
Rebai et al. (2013)
Zhang and Lu (2016)
Yin, Cheng et al. (2016)
Chen et al. (2016)
Sterna and Czerniachowska (2017)
Yin et al. (2019)
Chen et al. (2020)
T’kindt et al. (2020)
Problem notation
∑
𝑃 𝑚∥ 𝑐𝑗 𝑡
∑
𝑄𝑚∥ 𝑐𝑗 (1 − 𝑒−𝑟𝑡 )
∑
𝑃 𝑚∥ 𝑐𝑗 𝑡
∑
𝑃 𝑚∥ 𝑐𝑗 (1 − 𝑒−𝑟𝑡 )
∑
𝑃 𝑚∥ 𝐺(𝑡)
∑
𝑃 𝑚|𝑆𝑖𝑓 | 𝑅𝑒𝑤𝑗
∑
𝑃 𝑚|𝑆𝑖𝑓 | 𝑅𝑒𝑤𝑗
∑
𝑃 2|𝑑𝑤 | 𝐸𝑗 + 𝑇𝑗
∑
𝑃 𝑚|𝑑𝑗 = 𝑑| 𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗
∑
𝑃 𝑚|𝑑𝑗 = 𝑑| 𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 + 𝛾𝑗 𝐿(𝑑)
∑
𝑃 𝑚∥ 𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗
∑
𝑃 𝑚∥𝐶max + 𝜃𝑗𝑟𝑒𝑗
∑
𝑃 𝑚|𝑤𝑗 = 𝑎𝑗 𝑝𝑗 , 𝑑𝑗 = 𝑑| 𝑤𝑗 |𝐶𝑗 − 𝑑|
∑
∑
𝑃 𝑚|𝑝(𝑥𝑗 )| 𝑤𝑗 𝑈𝑗 + 𝐺(𝑥𝑗 )
∑
∑
𝑃 𝑚|𝑝(𝑥𝑗 )| 𝑤𝑗 𝐶𝑗 + 𝐺(𝑥𝑗 )
∑
𝑅𝑚∥ 𝐷𝑗 + 𝑈𝑗
∑
𝑅𝑚∥ (𝑦𝑘 − 𝑧𝑘 )2
∑
𝑃 𝑚|𝑑𝑗 = 𝑑| 𝐸𝑗 + 𝑇𝑗
∑
𝑃 𝑚∥ (𝐸𝑗 + 𝑇𝑗 )
𝑃 𝑚∥ max{𝑓 (𝐶𝑗 )}
∑
∑
𝑅𝑚|𝑝𝑗 (𝑥𝑗 )| (𝐸𝑗 + 𝑇𝑗 ) + 𝐺(𝑥𝑗 )
∑
∑
𝑃 𝑚|𝑟𝑒𝑗, 𝑝𝑗 (𝑡)| 𝑤𝑗 𝐶𝑗 + 𝜃𝑗𝑟𝑒𝑗
∑
∑
𝑃 𝑚|𝑟𝑒𝑗, 𝑝𝑗 (𝑡)| 𝐶𝑗 + 𝜃𝑗𝑟𝑒𝑗
𝐴 ∶ 𝐶𝐵 ≤ 𝐾
𝑃 𝑚∥𝐶max
max
∑
𝐵 ≤𝐾
𝑃 𝑚∥ 𝐶𝑗𝐴 ∶ 𝐶max
∑
∑
∑
𝑃 𝑚|ℎ(𝜇)| 𝑆𝐶(𝑡) + 𝐸𝐶(𝑡) + 𝐽 𝑃 𝐶(𝑡)
∑
∑
𝑃 𝑚|ℎ(𝜇)| 𝑤𝑗 𝐶𝑗 + 𝐻(𝑡1 , 𝑡2 )
∑
𝑃 𝑚|𝑟𝑒𝑗, 𝑟𝑗 |𝐶max + 𝜃𝑗𝑟𝑒𝑗
∑
𝑃 𝑚|𝑟𝑒𝑗, 𝑟𝑗 , 𝑝𝑗 = 𝑝|𝐶max + 𝜃𝑗𝑟𝑒𝑗
∑
𝑅𝑚∥( (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )𝐴 , min (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )𝐵 )
∑
∑
𝑅𝑚∥( (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )𝐴 , (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )𝐵 )
∑
𝑃 2|𝑑𝑗 = 𝑑| 𝑌𝑗
∑
𝑃 𝑚|𝑑𝑗 = 𝑑| 𝑉𝑗
∑
∑
𝑃 𝑚∥ 𝐶𝑗 𝐴 ∶ 𝑤𝑗 𝑈𝑗 𝐵
𝑃 𝑚|𝑑𝑗 = 𝑑| max 𝐷𝑗
∑
∑
𝑃 𝑚|𝑑𝑗 = 𝑑 ≥ 𝑝𝑗 | 𝑤𝑗 (𝐸𝑗 + 𝑇𝑗 )
∑
∑
𝑄𝑚|𝑑𝑗 = 𝑑 ≥ 𝑝𝑗 | 𝑤𝑗 (𝐸𝑗 + 𝑇𝑗 )
∑
∑
𝑅𝑚|𝑑𝑗 = 𝑑 ≥ 𝑝𝑗 | 𝑤𝑗 (𝐸𝑗 + 𝑇𝑗 )
Complexity
Method
–
–
–
–
𝑂(2𝑛 )
𝑂(𝐽 𝑚𝐾 2 )
𝑂(𝐽 𝑚2 𝐾 2 + 𝐽 𝑚2 𝐾 ln (𝑚𝐾))
𝑂(𝑛2 𝑑𝑤 )
𝑂(𝑛𝑚(2𝑝1 )𝑚 )
𝑂(𝑛𝑚[([𝑛∕𝑚] + 1)𝑝1 ]2𝑚 )
∑
𝑂(𝑛2 𝑝𝑗 )
–
∑
𝑂(𝑛𝑚𝑑 𝑚 ( 𝑝𝑗 )𝑚−1 )
𝑂(𝑛2 𝑡max max{𝑏𝑗 − 𝑎𝑗 })
∑
𝑂(𝑛2 𝑝𝑗 )
𝑂(𝑚𝑛𝑚+1 )
–
–
𝑂(𝑛2 )
–
𝑂(𝑚𝑛𝑚+1 )
∑
𝑂(𝑛( 𝑛 log (1 + 𝑏𝑗 )∕𝜀)𝑚 )
2
𝑂(𝑛 )
∑
∑
𝑂(𝑚𝑛𝐴 𝑛𝐵 ( 𝑝𝑗 )𝑚𝐴 ( 𝑝𝑗 )𝑚𝐵 )
∑
∑
𝑂(𝑚𝑛𝐴 𝑛𝐵 ( 𝑝𝑗 )𝑚𝐴 ( 𝑝𝑗 )𝑚𝐵 )
–
–
∑
𝑂(𝑚𝑛(𝑟max + 𝑝𝑗 )𝑚 )
2𝑚+1
𝑂(𝑚𝑛
)
𝑂(𝑚𝑛𝑚+1
𝑛𝐵 )
𝐴
∑
∑
𝑂(𝑚2 𝑛𝑚+1 min{ (𝛼𝑗 + 𝛽𝑗 )𝐴 , (𝛼𝑗 + 𝛽𝑗 )𝐵 })
𝑂(𝑛𝑑 2 )
–
∑
𝑂(𝑛𝐴 𝑛𝐵 max{𝑛𝐴 , 𝑛𝐵 } 𝑝𝑗 )
𝑂(𝑛𝑑 𝑚 )
𝑂(3𝑛 )
𝑂(3𝑛 )
𝑂(3𝑛 )
DP
DP
DP
DP
DP
Heuristic with DP
Heuristic with DP
DP and Heuristic
DP
DP
B&B with DP
FPTAS with DP
DP and Heuristic
B&B with DP
B&B with DP
DP
DP
DP and Heuristic
DP with Heuristic and LBs
Bounding scheme with DP
DP and FPTAS
FPTAS with DP
DP
DP and FPTAS
DP and FPTAS
B&B with DP
GA with DP and B&B
DP and FPTAS
DP
DP
DP and FPTAS
DP
PTAS with DP
B&P with DP
DP and FPTAS
DP
DP
DP
Fig. 3. Number of papers, separated by journal, that includes DP as solution method for parallel machine scheduling problems.
the existence of DP applications in unrelated and uniform machines,
the researches have been concentrating in the identical machines environment. Clearly, challenges may be found along the study due to
the exponential nature that DP tends to incorporate, however, with
the analysis developed by Woeginger (2000), FPTAS derived from DP
could be designed. Furthermore, methods that have been promising
for a given objective function could be extended with modifications
for others, as it is the case of BCP method shown in Pessoa et al.
(2010) and B&P algorithm seen in Lopes and de Carvalho (2007) for
tardiness-related functions. As one can notice, no author has developed
21
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Fig. 4. Frequency of DP in parallel scheduling problems per journal.
works in which DP was used as a subroutine of BCP or B&P for completion time-related functions and makespan and this could actually
generate good results from both theoretical and practical standpoints.
Additionally, decomposition methods have taken place regarding the
parallel machine problems in several cases, which indicates that they
could be exploited in recent topics that have been of interest, such as
contemporary objective functions.
As we can also notice, the computational experiments are also less
frequent than the theoretical developments related to all types of parallel machine scheduling problems. Once again, the DP approach has
given rise to a series of hybrid algorithms and also to new methods that
are branched through alterations, being able to create more efficient
resources in terms of CPU time and storage. Therefore, one of the main
goals here would be to verify to which extent these methods can be
applied and also to confirm their experimental limitations.
We also highlight that a qualitative analysis concerning the journals that have published the articles in which DP has been presented
as solution method for parallel machine scheduling problems might
be needed. From Fig. 3, the journal that has presented a significant
contribution regarding the characteristics stated above is the European
Journal of Operational Research, which is accountable for 14.3% of the
papers that described in this section, as it is shown in Fig. 4. Other
journals have also contributed significantly in publishing the DP-related
articles in the single machine environment, as it can be seen by the
percentages attributed to them (Naval Research Logistics, Computers
and Operations Research (7.9% each), Journal of Scheduling (6.3%)
and Discrete Applied Mathematics and Computers and Industrial Engineering (4.8%)). Those that have shown less than 2% of contribution
were placed in the ‘‘Others’’ category. In addition, one can tell that,
compared to the single machine environment, there are less papers
concerning DP, even though it is still a substantial amount.
that, in this section, some of the criteria have been grouped in the same
subsection due to a decrease in the number of articles involving DP and
a specific criterion.
6.1. Classic due dates-related criteria
∑
Józefowska, Jurisch, and Kubiak (1994) address the 𝐹 2|𝑑𝑗 = 𝑑| 𝑈𝑗
problem among a series of shop environments and develop a study on
the complexity proving it is ordinary NP-hard by association with a partition problem. In addition a 𝑂(𝑛𝑑 2 ) pseudo-polynomial DP formalism
is introduced as solution for the given problem.
Sonmez and Baykasoglu (1998) incorporate the theoretical analysis
∑
of the 𝐹 𝑚|𝑠𝑖𝑗𝑘 | 𝑤𝑗 𝑇𝑗 problem into a study case in a pipe manufacturing industry. The first part of the article is dedicated to introducing
a DP formulation for the problem, which seems to be quite simple
compared to some robust exact methods found in literature, although
no complexity analysis is displayed. Sequentially, the authors employ
the method to the pipe scenario and, due to the shop floor structure,
they are able to convert the problem into a single machine and apply
the DP formulation to solve the problem efficiently.
Nishi and Hiranaka (2013) explore features of a Lagrangian relax∑
ation to solve the 𝐹 𝑚|𝑠𝑖𝑗𝑘 | 𝑤𝑗 𝑇𝑗 problem. The algorithm undergoes a
decomposition via machine capacity relaxation and a DP subroutine is
proposed for its solution. Furthermore, the authors improve the lower
bound by applying a cut generation method that also resorts to a DP
subroutine. Experiments show that the algorithm is quite efficient for
medium sized instances and they suggest the investigation of other
methods to carry out more accurate cuts.
∑
Koulamas (2020) analyzes solutions for the 𝐹 𝑚|𝑝𝑗𝑘 = 𝑝𝑗 | 𝑇𝑗 problem by developing a DP formalism, which is actually an extension from
the one presented by Lawler (1977) by expanding the recursion method
∑
to 𝑚 machines. The author proves the algorithm runs in 𝑂(𝑛4 𝑝𝑗 ) and
performs experiments for problems with up to 50 jobs and 20 machines
(see Table 14).
6. Flow shop
Flow shop scheduling has been an environment that, despite being
far more complex compared to the ones previously outlined in this
research, has received nearly undivided attention from most authors
whose scheduling is a core theme and this interest can be traced back
nearly to 70 years. Nevertheless, when seeking information on the
contribution of dynamic programming in this environment, one can
notice that only few authors have actually employed it as a primary or
secondary method to solve several problems that compose the literature
in flow shop scheduling and those are presented below. One may notice
6.2. Makespan and completion time-related criteria
Corwin and Esogbue (1974) address the 𝐹 2|𝑠𝑖𝑗 |𝐶max problem by
dividing it into two different problems that are characterized by the
absence of sequence dependent setup times. When assuming the first
one is allowed to present setup times, a backward DP formulation is
proposed. Conversely, the authors offer a forward DP formulation for
the problem where setup times are allowed on the second machine.
22
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 14
Due date related problems for flow shop.
Authors
Józefowska et al. (1994)
Sonmez and Baykasoglu (1998)
Nishi and Hiranaka (2013)
Koulamas (2020)
Problem notation
∑
𝐹 2|𝑑𝑗 = 𝑑| 𝑈𝑗
∑
𝐹 𝑚|𝑠𝑖𝑗𝑘 | 𝑇𝑗
∑
𝐹 𝑚|𝑠𝑖𝑗𝑘 | 𝑤𝑗 𝑇𝑗
∑
𝐹 𝑚|𝑝𝑗𝑘 = 𝑝𝑗 | 𝑇𝑗
Complexity
Method
𝑂(𝑛𝑑 2 )
–
–
∑
𝑂(𝑛4 𝑝𝑗 )
DP
DP
Lagrangian relaxation with DP
DP and FPTAS
problems of main interest was the 𝐹 2|ℎ(𝜇)|𝐶max problem and the authors develop a theoretical analysis on its complexity, which has been
categorized as binary NP-hard with the manufacturing of a DP formulation that delivers the makespan in 𝑂(𝑛𝑀 4 ), with 𝑀 being the number
of generated state variables dependent on the maintenance interval.
Furthermore, a FPTAS can be derived from this approach to solve the
same problem using the DP as a subroutine.
Bautista, Cano, Companys, and Ribas (2012) delineate properties
related to a DP-based heuristic, which is commonly known as Bounded
Dynamic Programming (BDP), for solving the 𝐹 𝑚|𝑏𝑙𝑜𝑐𝑘|𝐶max problem.
It considers features of both DP (transition through states of given stage
and dominance properties) and B&B (bounding scheme) to generate solutions and given some conditions, the method can actually be viewed
as an exact method. The method is applied the Taillard benchmark (up
to 500 jobs) and it actually improves some of the instances for the given
problem.
Shang, Lenté, Liedloff, and T’Kindt (2018) apply a DP algorithm to
the 𝐹 3∥𝐶max in order to generate a Pareto frontier whose composition
are vectors of non-dominated solutions based on the completion times
of permutations 𝜋 and 𝜋 ′ . By employing these dominance rules, the DP
is bounded by an 𝑂(3𝑛 ) complexity.
Ozolins (2019b) improves the technique developed by Bautista
et al. (2012) in order to fully guarantee the optimality of the BDP
approach. The author devises three adapted lower bounds and uses a
recursive procedure in order to consider as many solutions as possible
without memory overreaching. Optimality is proved for 20 jobs and 10
machines in reasonable time and some cases with 20 machines.
Mor, Mosheiov, and Shapira (2019) investigate three proportionate
flow shop problems on two machines taking into account a threshold
for the penalty incurred on rejected jobs and learning effect, aiming
∑
∑
at minimizing 𝐶max , 𝐶𝑗 and 𝐶max functions. Three DP algorithms
are designed and, despite differing according to the objective function,
their functioning mechanism is fairly similar since the main recursive
equation determines which jobs are selected or rejected. Furthermore,
the three problems are classified as NP-hard and the algorithms are
bounded by an 𝑂(𝑛2 𝐾) complexity. The authors also report their computational results for different learning factors and problems containing
at most 50 jobs with efficient outcomes for medium-sized instances.
Mor and Mosheiov (2021) devise a theoretical analysis on two
∑
problems, which are makespan-related (𝐹 𝑚|𝑝(𝑏𝑗 ), 𝜃𝑗𝑟𝑒𝑗 ≤ 𝑈 𝐵|𝐶max and
∑ 𝑟𝑒𝑗
∑
𝐹 𝑚|𝑝(𝑏𝑗 ), 𝜃𝑗 ≤ 𝑈 𝐵| 𝐶max ) and in order to carry it out two DP algorithms are generated. For the first problem, the solution is guaranteed
once that all states are included, the process of rejecting or accepting
a given job is verified and the processed jobs are ordered according
to a dispatching rule involving the deteorating rate. Additionally the
complexity generated by this algorithm is of 𝑂(𝑛𝑈 𝐵) order. The second
DP algorithm is similar ot the first one, however there are some slight
changes regarding the recursive relation for the objective function (see
Table 15).
Furthermore, in order to reduce storage in each case, some dominance
rules are also included in both cases and the authors are able to find
optimal solutions for 15 job-instances.
Dutta and Cunningham (1975) develop a theoretical study on mathematical formulations to describe the 𝐹 2|𝑏𝑗,𝑗+1 |𝐶max with 𝑏𝑗,𝑗+1 denoting a limited buffer scenario. First, the authors introduce a DP
formulation in order to find optimal solutions for the problem and,
although it might be considered a fairly realistic model, it is an expensive one from the computational requirements standpoint. Therefore,
a reduction in its complexity is performed and a sub-optimal DP is
presented.
van de Velde (1990) devise a B&B algorithm that essentially converges to a Lagrangian Relaxation which, in turn, results in a linear
ordering problem. In order to create a more efficient bound, a DP
dominance rule in association with two other based on Johnson’s rule
is applied to reduce the search tree.
Kovalyov and Werner (1997) analyze theoretical aspects of the
𝐹 2|𝑟𝑗 |𝐶max problem and devise a PTAS to compare with preliminary
studies. The approach is based on a DP formulation that bear resemblance to those proposed by Lawler and Moore (1969), Rothkopf (1966)
and has 𝑂(𝑛𝑖(2𝑅)𝑖 𝑅2𝑖−2 ) complexity, being 𝑖 a subset of 𝐽 and 𝑅 =
∑
∑
max{ 𝑝𝑗1 , 𝑝𝑗2 }.
Lee (1997) evaluates the 𝐹 2|𝑟 − 𝑎(𝑀1 )|𝐶max and 𝐹 2|𝑟 − 𝑎(𝑀2 )|𝐶max
problems and a complexity analysis proves both problems are NP-hard.
Furthermore, the author design a pseudo-polynomial DP algorithm for
the problem with unavailability placed on machine 1. Although the
method is deemed fairly strong, its complexity renders it not viable and
Johnson-based heuristics are offered as alternative solutions.
Lee (1999) also extends the analysis in Lee (1997) for the semiresumable case problem. A complexity analysis is stated about the
NP-hardness of the problem, since it is considered a generalized problem of the non-resumable scenario. In addition, the results are the same
found in its previous work regarding the DP formulation.
Hou and Hoogeveen (2003) investigate the 𝐹 3|𝑝𝑗𝑘 = 𝑝𝑗 ∕𝑠𝑘 |𝐶max
problems under the assumption that all machines operate under different speeds and machine 2 is the slowest one. An 𝑂(𝑛𝑠1 𝑠22 𝑠3 max{𝑝𝑗 }2 ) DP
algorithm is proposed by incorporating V-shaped optimality properties
as well as a recursive function that computes the makespan for subsequences. Furthermore, a combination of optimality properties and a
series of lemmas prove that the problem can be classified as ordinary
NP-hard.
Bouquard, Billaut, Kubzin, and Strusevich (2005) develop a diversified research on problems involving number of regular and nowait jobs and methods are provided so as to solve them. Among
them, the 𝐹 2|𝑟𝑒𝑔 + 𝑛𝑤𝑡, 𝑝𝑚𝑟𝑢, 𝑛𝑛𝑤𝑡 = 𝑞|𝐶max can be solved via a pseudopolynomial time DP algorithm and a subsequent FPTAS (by converting
the former approach). The complexity associated with the DP is of
̃ 3𝑞+1 ), where 𝑀
̃ represents the maximum value among
the form 𝑂(𝑛(𝑀)
states generated when scheduling a given job.
Allaoui, Artiba, Elmaghraby, and Riane (2006) investigate the effects of heuristics derived from Johnson’s algorithm and a DP model
applied to the 𝐹 2|𝑟 − 𝑎(𝑀1 )|𝐶max and 𝐹 2|𝑛𝑟 − 𝑎(𝑀1 )|𝐶max problems. A
complexity analysis is carried out for the pseudo-polynomial DP model
and, despite containing an exponential component and the number
of subsets in 𝐽 , it performs significantly better than the formulation
proposed by Lee (1997), since it does not depend on processing times.
Kubzin and Strusevich (2006) investigate some shop problems considering unavailability for either one or two machines. One of the
6.3. Multi-criteria and additional objective functions
Lawler and Moore (1969) also take interest in applying their DP
knapsack functional equation to problems considering two machines
with the intent of minimizing a general loss time function. This application of the recursion takes into account a given common due date
and the formulation is bounded by a 𝑂(𝑛𝑑 2 ) complexity.
23
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 15
Makespan and completion time related problems for flow shop.
Authors
Problem notation
Complexity
Method
Corwin and Esogbue (1974)
Dutta and Cunningham (1975)
van de Velde (1990)
Kovalyov and Werner (1997)
Lee (1997)
Lee (1999)
Hou and Hoogeveen (2003)
Bouquard et al. (2005)
Allaoui et al. (2006)
Kubzin and Strusevich (2006)
Bautista et al. (2012)
Shang et al. (2018)
Ozolins (2019b)
Mor et al. (2019)
𝐹 2|𝑠𝑖𝑗 |𝐶max
𝐹 2|𝑏𝑗,𝑗+1 |𝐶max
∑
𝐹 2∥ 𝐶𝑗
𝐹 2|𝑟𝑗 |𝐶max
𝐹 2|𝑟 − 𝑎(𝑀1 )|𝐶max
𝐹 2|𝑠𝑟 − 𝑎(𝑀1 )|𝐶max
𝐹 3|𝑝𝑖𝑘 = 𝑝𝑗 ∕𝑠𝑘 |𝐶max
𝐹 2|𝑟𝑒𝑔 + 𝑛𝑤𝑡, 𝑝𝑚𝑟𝑢, 𝑛𝑛𝑤𝑡 = 𝑞|𝐶max
𝐹 2|𝑟 − 𝑎(𝑀1 )|𝐶max
𝐹 2|ℎ(𝜇)|𝐶max
𝐹 𝑚|𝑏𝑙𝑜𝑐𝑘|𝐶max
𝐹 3∥𝐶max
𝐹 𝑚|𝑏𝑙𝑜𝑐𝑘|𝐶max
𝐹 2|𝐿𝑒, 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾|𝐶max
∑
𝐹 2|𝐿𝑒, 𝜃𝑗𝑟𝑒𝑗 ≤ 𝐾| 𝐶𝑗
∑
𝑟𝑒𝑗
𝐹 2|𝐿𝑒, 𝜃𝑗 ≤ 𝐾| 𝐶max
∑
𝐹 𝑚|𝑝(𝑏𝑗 ), 𝜃𝑗𝑟𝑒𝑗 ≤ 𝑈 𝐵|𝐶max
∑
∑
𝐹 𝑚|𝑝(𝑏𝑗 ), 𝜃𝑗𝑟𝑒𝑗 ≤ 𝑈 𝐵| 𝐶max
–
–
–
𝑂(𝑛𝑖(2𝑅)𝑖 𝑅2𝑖−2 )
∑
𝑂(𝑛𝑡11 ( 𝑝𝑗𝑘 )2 𝑝max 1 )
∑
𝑂(𝑛𝑡11 ( 𝑝𝑗𝑘 )2 𝑝max 1 )
𝑂(𝑛𝑠1 𝑠22 𝑠3 max{𝑝𝑗 }2 )
̃ 3𝑞+1 )
𝑂(𝑛(𝑀)
𝑂(2𝑛1 𝑛 log 𝑛)
𝑂(𝑛𝑀 4 )
–
𝑂(3𝑛 )
–
𝑂(𝑛2 𝐾)
𝑂(𝑛2 𝐾)
𝑂(𝑛2 𝐾)
𝑂(𝑛𝑈 𝐵)
𝑂(𝑛𝑈 𝐵)
DP
DP and Sub-optimal DP
DP
PTAS with DP
DP and Heuristic
DP and Heuristic
DP
DP and FPTAS
DP and Heuristics
DP and FPTAS
BDP
DP
BDP
DP
DP
DP
DP
DP
Mor and Mosheiov (2021)
presented. Both algorithms are pseudo-polynomial with 𝑂(𝑛2 (𝐶(𝑆))4 𝐾 2 )
∑
and 𝑂(𝑛2 (𝐶(𝑆))4 𝐾 2 𝑝𝑗𝑘 ) complexities, respectively.
Luo, Chen, and Zhang (2012) outline a theoretical approach to the
𝐵
𝐴
𝐴 + 𝜃𝐶 𝐵
𝐹 2∥𝐶max
max and 𝐹 2∥𝐶max ∶ 𝐶max ≤ 𝐾 problems, respectively.
Approximation algorithms and a DP that is suitable for both situations
are also presented and, due to the latter, the problems can be classified
as weakly NP-hard. It might be valid to comment that the complexity
of the DP approach is relies on variables denoted here by 𝑞𝑗𝐴 and 𝑞𝑗𝐵 ,
which are valued according to allocation in analogy with the partition
problem.
Shabtay and Bensoussan (2012) develop a serial analysis on shop
problems for the earliness-tardiness minimization. One of the problems
∑
of interest is the 𝐹 2∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ). Its solution relies on an algorithm whose DP is mainly used for updating the recursive function
and partitioning the jobs in groups of early and tardy based on a
∑
dominance rule and bounded by 𝑂(𝑛2 𝑝𝑗1 ) complexity. The authors
also provide an FPTAS with the DP applied as a fundamental element
for its functioning.
𝐴 +𝛼𝐶 𝐵 problem, taking
Fan and Cheng (2016) consider the 𝐹 2∥𝐶max
max
into account a pseudo-polynomial DP algorithm that generates a partition of subsets depending on their start time to update the objective
function, which is a four-dimension vector containing partial sequences
and start times of both agents. The complexity of the problem is proven
to be ordinary NP-hard and the algorithm’s complexity is given by
∑
∑
∑
𝑂(𝑛 𝑝𝑗1 ( 𝑝𝑗1 + 𝑝𝑗2 )2 ). Ultimately an approximation algorithm based
on Johnson’s rule is also presented.
Shang et al. (2018) extend their approach to the case with general
∑
functions 𝑓max and
𝑓𝑗 , which can display similar behavior to the
makespan criterion. For the first class of problems, the dominance rule
adds the objective function as an element to discard partial sequences
and the complexity is associated with an exponential form of total
∗
feasible solutions of the Pareto frontier (𝑂(4|𝑆 | )). The second group
of functions is solved analogously and is bounded by 𝑂(5𝑛 ).
Wang, Zhu, Fang, Chu, and Chu (2018) investigate a MILP model
∗ |𝑇 𝐸𝐶 problem, which is a problem that aims
for the 𝐹 2, 𝑇 𝑜𝑈 |𝐶max
at minimizing the total electricity cost function without altering the
∗
makespan value (hence the 𝐶max
being adopted as a constraint) and
considering variable time-of-use tariffs to the manufacturing problem.
A DP-based heuristic is developed as well for solving the problem with
a fixed sequence by modeling the energy cost function as a recursive
equation in terms of completion time. Such algorithm produces an
optimal scheduling with 𝑂(𝑛𝑇 4 ) time complexity, where 𝑇 represents
a given horizon. An ILS algorithm, a Johnson’s Rule-based heuristic
and an ILSDP (which combines features of DP in the end of the
solution given by) are also provided. Comparisons are established via
computational experiments for at most 50 jobs and the results show that
ILSDP can outperform the other methods in terms of solution quality
T’kindt, Gupta, and Billaut (2003) furnish a variety of solution
methods in order to solve the lexicographic bi-criteria problem denoted
∑
by 𝐹 2∥𝐿𝑒𝑥(𝐶max , 𝐶𝑗 ), which are divided into mathematical programming formulations, exact methods and heuristic. Among, those a DP
formulation is created and its complexity is bounded by 𝑂(𝑈 𝐵𝑛2𝑛 ).
Although the method is mathematically structured, the computational
requirements are high enough to make it the worst option among the
others, being able to solve optimally only problems with up to 17 jobs.
Yeung, Oğuz, and Cheng (2004) propose a B&B and a heuristic to
∑
solve the 𝐹 2|𝑑𝑤 | (𝐸𝑗 + 𝑇𝑗 ) problem. The B&B is structure is connected
′
to an 𝑂(𝑛 (𝐶(𝑆)+𝑝𝑛′ 2 −𝑠min )) DP formulation via a bounding scheme and
computational experiments are performed in order to verify efficiency.
Although optimal solutions are found for 15-job instances in about
5 min, the heuristic is able to find near-optimal solutions for instances
as ten times as large in 20 s.
Błażewicz, Pesch, Sterna, and Werner (2005b) outline the proof
∑
for the NP-hardness associated to the 𝐹 2|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑌𝑗 problem by
transforming the partition problem into their scheduling problem. They
also develop a general DP formulation with 𝑂(𝑛2 𝑑 4 ) time complexity,
which is able to prove the problem can actually be classified as binary
NP-hard. Furthermore, a comparison among the DP, an enumeration
method and list algorithms can be seen in Błażewicz, Pesch, Sterna, and
Werner (2005a) and the authors claim that, even though DP cannot be
discarded because it was used for classifying the problem, it might not
be the best method compared to the other ones in terms of efficiency.
Dawande, Gavirneni, and Rachamadugu (2006) compile a variety
of algorithms with the intent of maximizing a reward function under a makespan constraint for a two-machine scenario. Similarities
between the knapsack DP functional approach seen in Lawler and
Moore (1969) and the proposed DP algorithm are established, being the
latter an 𝑂(𝑛2 𝐾 2 log 𝑛) pseudo-polynomial time algorithm. Moreover
comparisons are made among the DP, MIP and heuristics and due to its
complexity, the DP algorithm is the one with the worst performance.
T’kindt, Croce, and Bouquard (2007) address the problem of minimizing the number of tardy jobs and finding the best common due
date in order to optimize it according to a Pareto-optimum curve.
The authors design a 𝜀-exact method and an ILP in order to verify
their efficiency according to the experimental results. Furthermore, the
problem is defined as ordinary NP-hard and a pseudo-polynomial DP of
𝑂(𝑛𝑑 2 ) complexity is also formulated by merging Johnson’s rule with
the method proposed by Józefowska et al. (1994).
Yeung, Oğuz, and Cheng (2009) branch their research on the
JIT framework and develop two DP algorithms to solve 𝐹 2|𝑑𝑗 =
∑
∑
𝑑| (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 ) and 𝐹 2|𝑑𝑗 = 𝑑| (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )+𝐹 (𝑑) problems. First
they prove both problems are NP-hard through a partition problem
analogy and the optimality properties regarding the jobs positioning are
24
Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 16
Multicriteria and additional functions problems for flow shop.
Authors
Lawler and Moore (1969)
T’kindt et al. (2003)
Yeung et al. (2004)
Błażewicz et al. (2005b)
Błażewicz et al. (2005a)
Dawande et al. (2006)
T’kindt et al. (2007)
Yeung et al. (2009)
Luo et al. (2012)
Shabtay and Bensoussan (2012)
Fan and Cheng (2016)
Shang et al. (2018)
Wang et al. (2018)
Kovalev et al. (2019)
Koulamas and Kyparisis (2021)
Problem notation
∑
𝐹 2|𝑑𝑗 = 𝑑| 𝐹 (𝑡)
∑
𝐹 2∥𝐿𝑒𝑥(𝐶max , 𝐶𝑗 )
∑
𝐹 2|𝑑𝑤 | (𝐸𝑗 + 𝑇𝑗 )
∑
𝐹 2|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑌𝑗
∑
𝐹 2|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑌𝑗
∑
𝐹 2∥( 𝑅𝑒𝑤𝑗 , 𝐶max ≤ 𝐾)
∑
𝐹 2|𝑑𝑗 = 𝑑|𝑃 ( 𝑈𝑗 , 𝑑)
∑
𝐹 2|𝑑𝑗 = 𝑑| (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 )
∑
𝐹 2|𝑑𝑗 = 𝑑| (𝛼𝑗 𝐷𝑗 + 𝛽𝑗 𝑈𝑗 ) + 𝐹 (𝑑)
𝐴
𝐵
𝐹 2∥𝐶max + 𝜃𝐶max
𝐴 ∶ 𝐶𝐵 ≤ 𝐾
𝐹 2∥𝐶max
max
∑
𝐹 2∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 )
𝐴
𝐵
𝐹 2∥𝐶max
∶ 𝛼𝐶max
𝐹 3∥𝑓max
∑
𝐹 3∥ 𝑓𝑗
∗ |𝑇 𝐸𝐶
𝐹 2, 𝑇 𝑜𝑈 |𝐶max
∑
𝐹 2|𝑛𝑤𝑡, 𝑝𝑗𝑘 = 𝑝𝑗 | |𝐶𝑗 − 𝐶𝑖 |
∑
𝐹 𝑚|𝑛𝑤𝑡, 𝑠𝑜𝑟𝑗|𝐶max + 𝜃𝑗𝑟𝑒𝑗
∑
∑
𝐹 𝑚|𝑛𝑤𝑡, 𝑠𝑜𝑟𝑗| 𝐶𝑗 + 𝜃𝑗𝑟𝑒𝑗
∑
∑
𝐹 𝑚|𝑛𝑜 − 𝑖𝑑𝑙𝑒, 𝑠𝑜𝑟𝑗| 𝐶𝑗 + 𝜃𝑗𝑟𝑒𝑗
Complexity
Method
𝑂(𝑛𝑑 2 )
𝑂(𝑛𝑈 𝐵2𝑛 )
𝑂(𝑛′ (𝐶(𝑆) + 𝑝𝑛′ 2 − 𝑠𝑡min ))
𝑂(𝑛2 𝑑 4 )
𝑂(𝑛2 𝑑 4 )
𝑂(𝑛2 𝐾 2 log 𝑛)
𝑂(𝑛𝑑 2 )
𝑂(𝑛2 (𝐶(𝑆))4 𝐾 2 )
∑
𝑂(𝑛2 (𝐶(𝑆))4 𝐾 2 𝑝𝑗𝑘 )
∑ 𝐴
∑
𝑂((𝑛𝐴 + 𝑛𝐵 )( (𝑝𝑗 + 𝑞𝑗𝐴 ) + (𝑝𝐴
+ 𝑞𝑗𝐴 ))4 )
∑
∑ 𝑗
𝑂((𝑛𝐴 + 𝑛𝐵 )( (𝑝𝐴
+ 𝑞𝑗𝐴 ) + (𝑝𝐴
+ 𝑞𝑗𝐴 ))4 )
𝑗
𝑗
∑
𝑂(𝑛2 𝑝𝑗1 )
∑
∑
∑
𝑂(𝑛 𝑝𝑗1 ( 𝑝𝑗1 + 𝑝𝑗2 )2 )
∗
𝑂(4|𝑆 | )
𝑂(5𝑛 )
𝑂(𝑛𝑇 4 )
𝑂(𝑛3 )
𝑂(𝑛3 )
𝑂(𝑛3 )
𝑂(𝑛3 )
DP
DP, MIP and Heuristic
B&B with DP and Heuristic
DP
DP, Enumeration method and Heuristic
DP, MIP and Heuristics
DP and Enumeration Methods
DP
DP
DP and Approximation Algorithm
DP and FPTAS
DP and FPTAS
DP and Approximation algorithm
DP
DP
DP-based heuristics and Heuristics
DP
DP
DP
DP
and, even though its average time might reach ten times more the value
obtained by ILS, it is still a powerful algorithm and the trade-off might
be acceptable.
Kovalev, Kovalyov, Mosheiov, and Gerstl (2019) design a polynomial time DP algorithm of 𝑂(𝑛3 ) complexity to minimize the total
absolute deviation of completion times for a no-wait proportionate flow
∑
shop, also known as TADC, denoted by 𝐹 2|𝑛𝑤𝑡, 𝑝𝑗𝑘 = 𝑝𝑗 | |𝐶𝑗 − 𝐶𝑖 |.
The DP is based on computing the objective function for the partial
sequences and find an optimal schedule based on a 𝑉 -shaped structure
and in case it cannot be found, the algorithm shifts into a heuristic.
Experiments are performed for at most 300 jobs and the algorithm is
able to find 𝑉 -shaped schedules within few seconds on average.
Koulamas and Kyparisis (2021) develop a theoretical analysis on
flow shop problems involving the minimization of completion timesrelated functions associated with rejection penalties. The investigation
∑ 𝑟𝑒𝑗
is able to model the problem 𝐹 𝑚|𝑛𝑤𝑡|𝐶max +
𝜃𝑗 as a TSP and
subsequently a DP polynomial time algorithm of 𝑂(𝑛3 ) complexity
is devised to solve a special case, which is derived by studying the
∑
𝐹 𝑚|𝑠𝑜𝑟𝑗, 𝑛𝑤𝑡|𝐶max + 𝜃𝑗𝑟𝑒𝑗 problem, where 𝑠𝑜𝑟𝑗 denotes semi-ordered
jobs via a pyramidal sequencing (SPT-LPT). A similar algorithm is
∑
∑
generated for the 𝐹 𝑚|𝑠𝑜𝑟𝑗, 𝑛𝑤𝑡| 𝐶𝑗 + 𝜃𝑗𝑟𝑒𝑗 problem with the same
time complexity and as an additional gain, the authors prove that this
∑
∑
algorithm also solves optimally the 𝐹 𝑚|𝑠𝑜𝑟𝑗, 𝑛𝑜 − 𝑖𝑑𝑙𝑒| 𝐶𝑗 + 𝜃𝑗𝑟𝑒𝑗
problem (see Table 16).
Regarding the problems associated with the Curse of Dimensionality, it might be recommended to develop new dominance rules and
some trimming-based algorithms that are able to reduce the state
space of feasible solutions. It may be also relevant to devise DPbased heuristics (see Bautista et al. (2012)) or combination with other
exact algorithms so as to establish a more efficient approach. Although
heuristics do not guarantee optimality, recent applications of those
have proven to yield high quality solutions and depend on exact methods as subroutines to generate initial solutions or to enhance intrinsic
mechanisms of the method.
Another suggestion is related to the number of machines analyzed
by the authors. It is seen that most two-machine flow shop problems
have been widely studied when DP is involved while formulations for
the 𝑚-machine scenario have not been accounted as often. Logically,
due to the difficulties associated with storage and also the behavior
of such systems, their modeling might also become harder to conceive. However, some adaptations could be used and mathematical
formulations in terms of recursive equations and complexity could
be devised. Eve though these may wind up being not practical for
computational experiments, they might serve as background for future
studies regarding the 𝑚-machine problems.
Finally, since not many computational experiments have been provided in the state-of-art references, it might be necessary to investigate
more of that in future research. Once again, since DP adherence to
other methods (exact and non-exact) is commonly resorted to, it may
be interesting to evaluate the application and response of such methods
in the flow shop environment as well as to compare their performance
to other ones that are already present.
Due to the smaller share of articles that compose this environment,
it is expected that the number of papers also undergoes a reduction
and it is in fact what occurs according to Fig. 5. Only 16 journals
have aggregated material regarding the use of DP in the flow shop
environment, whereas the previous environments have presented over
25 journals each. In addition, Operations Research Letters is accountable for the largest contribution, however, the European Journal of
Operational Research still represent a substantial share in the total
amount of papers, which totaled 29. These statements can be confirmed
by analyzing also Fig. 6, which shows the percentage related to the
occurrence of DP in the flow shop environment on journals up until
2020.
6.4. Discussion on flow shop scheduling problems
Flow shop is actually considered the simplest version of the shop
scheduling problems, however it is much more complex when compared to the models in the previous sections. Therefore, one might
expect that the number of papers that are dedicated to this environment
is considerably smaller, and this is, in fact, what happens.
The first observation that one might be able to conclude is that the
largest share of articles are concentrated in the multi-criteria and additional functions and this indicates that several gaps are yet to be filled
regarding classic criteria when DP is taken into account as solution
method. This generates a large scenario of possibilities for exploring DP
features in the flow shop environment in terms of objective functions
as well as technological constraints. Therefore, authors should focus on
developing DP new or additional formulations for problems involving,
for instance, release dates, no-idle, no wait, blocking, sequence dependent setup times, among others. Analogously, completion time-related
functions and due date-related functions have accounted for a small
number when compare to makespan and other functions, which means
that this may also be a lead for future studies in the area with DP.
7. Job shop
Job shop scheduling has also been an environment which has been
a target for researchers due to the establishment of different routes
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E.A.G.d. Souza et al.
Fig. 5. Number of papers, separated by journal, that includes DP as solution method for flow shop scheduling problems.
Fig. 6. Frequency of DP in flow shop scheduling problems per journal.
on distinct machines. Some studies have been developed since the
sixties but contribution has grown quite moderately only in the eighties.
Hence, when comparing the use of dynamic programming to other
methods, there might not be numerous articles tying this method to
such environment. In this section we outline these papers and, due to a
shortage in the number of references, all the criteria will be summarized
in Table 18.
windows (TSPTW) and a heuristic to compose a B&B algorithm. The authors collaborate with a DP formulation to solve the TSPTW relaxations
and they compare the effectiveness of such method to propagation
methods that have also been provided. They develop experiments for
small, medium and large sized problems and notice that it could
perform well for small instances and suggest that such method can be
improved for larger sets.
Balas, Simonetti, and Vazacopoulos (2008) also develop an adapted
Bottleneck Heuristic for the problem 𝐽 𝑚|𝑠𝑖𝑗𝑘 , 𝑑 𝑗 |𝐶max by reformulating
it as an asymmetric TSPTW problem in order to solve a general problem
with fixed deadlines and as consequence, this problem, with addition
of precedence constraints, can be solved via a DP subroutine combined
with feasibility check properties in order to reduce the state space for
accepted solutions. An overall comparison indicates the approach is
promising even for harder instances, hence it might be useful in further
studies.
Gromicho, Van Hoorn, Saldanha-da-Gama, and Timmer (2012) furnish a DP algorithm to solve the 𝐽 𝑚∥𝐶max problem since no other exact
method had been provided as solution for it. The paper focuses on
deriving a dominance rule in order to supplement the algorithm with
the adequate formulation via Bellman’s principle and the authors, in
7.1. Classic due dates-related criteria
Józefowska et al. (1994) describe a DP formulation for the
∑
𝐽 2|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑈𝑗 in which the computation of the tardy jobs depends
on a conditional formalism that groups the jobs according to their
selection for being early or tardy. It is a backward recursive relation
bounded by an 𝑂(𝑛𝑑 3 ) complexity.
7.2. Makespan and completion time-related criteria
Artigues and Feillet (2008) investigate the 𝐽 𝑚|𝑠𝑖𝑗𝑘 |𝐶max by incorporating varied concepts such as graph disjunction and TSP with time
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E.A.G.d. Souza et al.
be interesting to extend the method for number of tardy jobs or to
develop adapted methods with backward and/or forward recursion for
the tardiness-related objective functions.
For the completion time-related criteria, one can notice that TSP
with time windows has been a recurrent artifice in order to solve the
unconstrained version for the 𝑚-machine makespan problem. Apparently, it has been considered a successful approach, as it can be seen
in Ozolins (2018a). Therefore, this approach could be extended to other
completion time functions by resorting to TSPTW. Another approach
that could also be used is to adapt the BDP version proposed by Bautista
et al. (2012) by including TSPTW. Despite being considered a DP-based
heuristic, it has produced significant results for the flow shop case and
could increase the number of job and machine instances for the job
shop case, even if it does not reach the optimum for all the data in the
benchmark.
Regarding the approaches, it is noticed that most of them rely on the
classic objective functions, even for the non-regular criteria (which tend
to analyze the earliness-tardiness function and its variants). Therefore
some considerations concerning more contemporary objective functions
might be needed (e.g. multi-agent and job rejection). These attempt
may be challenging, once job shop itself is clearly a less studied
environment with scarce background in concentrated topics but this
could also lead to more theoretical sources that, a posteriori, should be
unfolded onto empirical analysis.
The job shop environment, due to a reduced number of articles,
has been represented only by seven journals, given that most titles
have been published Annals of Operations Research, Computers and
Operations Research and Journal of Scheduling as it is shown in Figs. 7
and 8. This reinforces the need for more publications that involve the
job shop scheduling as well as the use of DP as solution method to
several types of segments within this environment that could be of
interest either in classic or contemporary approaches.
order to optimize the DP due to the curse of dimensionality, consider
optimality conditions that avert a full set of feasible solutions. Its
𝑛
complexity is bounded by 𝑂(𝑝2𝑛
max (𝑚 + 1) ) complexity and empirical
results show that the proposed algorithm performs exponentially better
than brute force.
Ozolins (2018a) proposes a BDP algorithm, which can be considered
an improvement of the DP algorithm presented by Gromicho et al.
(2012), however including sequence dependent setup times. A tight
bound is provided and also the selection of feasible solutions is based
on the solution for the TSPTW seen in Artigues and Feillet (2008).
Computational results show that the BDP solves to optimality and
even achieve better solutions that the previous best known solutions
in several benchmarks with at most 20 jobs and 10 machines.
Ozolins (2018b) conducts a robust study on the 𝐽 𝑚|𝑛𝑤𝑡|𝐶max problem. Two remarks must be made concerning the DP structure of the
solution method. The first one is that de DP algorithm presents a
dominance rule that is part of the dynamic programming graph-like
definition. Additionally, a recursive formalism is also presented as well
as a prominent bound. The results are proven to achieve optimality for
moderate size of jobs and machines (10 × 10) from several benchmarks
in seconds.
7.3. Multi-criteria and additional objective functions
∑
Wang, Luh, Zhao, and Wang (1997) examine the 𝐽 𝑚∥ (𝛼𝑗 𝐸𝑗2 +
𝛽𝑗 𝑇𝑗2 ) problem considering that some machines can be grouped into
a set 𝐻𝑖𝑗 according to their capabilities and a given horizon 𝐿. The
authors design a Lagrangian relaxation method to solve the problem
and it is split into sub-problems that are relative to each part of the
objective function. A backward DP formulation is proposed to furnish
∑
a solution to the weighted tardiness and yields an 𝑂(𝐿 |𝐻𝑖𝑗 |).
Luh et al. (1998) also extend the complexity of the problem analyzed in Wang et al. (1997) with the inclusion of limited buffer, family
dependent setup times and a long horizon. Their solution also relies
on decomposing the problem and the relaxation occurs on the family
constraints and machine capacity. In addition, a backward DP formulation is presented to solve the decomposed part of the Lagrangian
relaxation and its concept can be extended to the constraints without
further adjustments. Since similarities can be shared with Wang’s DP
∑
formalism, the complexity remains 𝑂(𝐿 |𝐻𝑖𝑗 |).
Błażewicz, Pesch, Sterna, and Werner (2007) design a DP approach
∑
to find the optimal schedule for the 𝐽 2|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑌𝑗 problem in which
the recursive equation is based on the positioning of jobs throughout
the sequence depending on separation of subgroups of early and tardy
jobs. The overall complexity is given by 𝑂(𝑛3 𝑑 11 ) and the problem
is classified as binary NP-hard due to this construction of optimal
solutions based on DP.
Baptiste, Flamini, and Sourd (2008) bounds are based on precedence constraints relaxation and resource constraint relaxation for the
∑
𝐽 𝑚∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 ). The second type, which relies on a time-indexed
formulation of the problem with release dates on a single machine, can
be solved by a DP approach and when experiments are performed, it
can present advantage over the precedence type even for large sets of
machines. The complexity for this relaxation is given by 𝑂(𝑛𝑖 (max{𝑑𝑗 +
∑
𝑝𝑗𝑘 })), where 𝑛𝑖 is a subset of 𝐽 considering jobs performing a given
number of operations (see Table 17).
8. Open shop
The randomness of the open shop environment has also turned
this problem into one of the most interesting settings in machine
scheduling, since its behavior can be considered an ordeal. Due to this
fact, the use of exact methods in solving such method has been scarce
or just theoretically conceived. Nevertheless, when verifying the application of dynamic programming in this scenario one can notice that
an unexpected amount of papers concerning this topic might appear,
which is surprising considering the storage issues readily associated
with the DP algorithms. These articles have been organized as follows
and, as in the previous sections, metaheuristics have not been included
nor have been articles involving the dynamic and flexible scenarios.
8.1. Classic due dates-related criteria
Brucker, Jurisch, Jurisch et al. (1993) address problems in which
unit processing times are considered in the open shop environment.
First several problems are proved to be solved under the assumption
that some of the open shop problems can be transformed into identical
parallel machine problems. Then, further results are presented, in
which an 𝑂(𝑛2 ) DP formulation is developed to construct an optimal
schedule of all early jobs that can be assembled in block structures to
∑
solve the 𝑂2|𝑝𝑗𝑘 = 1| 𝑤𝑗 𝑈𝑗 .
Brucker, Jurisch, Tautenhahn et al. (1993) investigate the
∑
𝑂𝑚|𝑝𝑗𝑘 = 1| 𝑤𝑗 𝑈𝑗 problem by incorporating the findings regarding
the open shop being transformed in parallel machine seen in Brucker,
Jurisch, Jurisch et al. (1993). This needs to be done in order to
apply the DP formulation, which is of 𝑂(𝑛2 𝑚𝑚+1 ) complexity, for the
construction of an optimal scheduling according to the sequencing of
early jobs in optimal manner and tardy jobs arbitrarily.
7.4. Discussions on job shop scheduling problems
Similar to the observations made for the flow shop scheduling
problems, due to the shortage of state-of-art references, it is expected
that the possibilities of problems that can modeled via DP are larger
and they could be implemented to a handful of classes of objective
functions and technological constraints. For instance, only one article
has been found for the due date-related criteria with an specific function, which is weighted number of tardy jobs. In this sense, it may
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E.A.G.d. Souza et al.
Table 17
Problems in the job-shop environment.
Authors
Józefowska et al. (1994)
Wang et al. (1997)
Luh et al. (1998)
Błażewicz et al. (2007)
Artigues and Feillet (2008)
Balas et al. (2008)
Baptiste et al. (2008)
Gromicho et al. (2012)
Ozolins (2018a)
Ozolins (2018b)
Problem notation
∑
𝐽 2|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑈𝑗
∑
𝐽 𝑚∥ (𝛼𝑗 𝐸𝑗2 + 𝛽𝑗 𝑇𝑗2 )
∑
𝐽 𝑚|𝑆𝐷𝑓 , 𝑏𝑗,𝑗+1 | (𝛼𝑗 𝐸𝑗2 + 𝛽𝑗 𝑇𝑗2 )
∑
𝐽 2|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑌𝑗
𝐽 𝑚|𝑠𝑖𝑗𝑘 |𝐶max
𝐽 𝑚|𝑠𝑖𝑗𝑘 , 𝑑 𝑗 |𝐶max
∑
𝐽 𝑚∥ (𝛼𝑗 𝐸𝑗 + 𝛽𝑗 𝑇𝑗 )
𝐽 𝑚∥𝐶max
𝐽 𝑚|𝑠𝑖𝑗𝑘 |𝐶max
𝐽 𝑚|𝑛𝑤𝑡|𝐶max
Complexity
Method
𝑂(𝑛𝑑 3 )
∑
𝑂(𝐿 |𝐻𝑖𝑗 |)
∑
𝑂(𝐿 |𝐻𝑖𝑗 |)
𝑂(𝑛3 𝑑 11 )
–
–
∑
𝑂(𝑛𝑖 (max{𝑑𝑗 + 𝑝𝑗𝑘 }))
2𝑛
𝑛
𝑂(𝑝max (𝑚 + 1) )
–
–
DP
Lagrangian relaxation with DP
Lagrangian relaxation with DP
DP
B&B with DP
Heuristic with DP
Lagrangian relaxation with DP
DP
BDP
DP
Fig. 7. Number of papers, separated by journal, that includes DP as solution method for job shop scheduling problems.
Fig. 8. Frequency of DP in job shop scheduling problems per journal.
previous stage or calculate the completion time of a block of jobs on the
∑
second machine. Moreover, its complexity is given by 𝑂(𝑛 𝑝𝑗2 ) and a
FPTAS is also derived from it.
∑
Tautenhahn and Woeginger (1997) study the 𝑂𝑚|𝑝𝑗𝑘 = 1, 𝑟𝑗 | 𝐶𝑗
2
by designing an 𝑂(𝑛 ) DP algorithm to solve it. First, some optimality
properties regarding the ordering of jobs is presented according to a
increasing monotonic behavior of completion times and a recursive
8.2. Makespan and completion times-related criteria
Strusevich and Hall (1997) consider the minimization of makespan
setting one of the machines as non-bottleneck (𝑁𝐵). The problem
is denoted by 𝑂2|𝑁𝐵|𝐶max and the authors prove it to be ordinary
NP-hard by developing a forward DP formulation that computes the
makespan at each stage, which may either keep the value of the
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Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
relation is introduced with real and bounded values for cases where the
sequence follows the optimality conditions and it is infinite, otherwise.
Shafransky and Strusevich (1998) analyze the problem of minimizing the makespan by imposing that the jobs have to be scheduled
according a given sequence on the last machine. According to Graham’s
notation, the problem is denoted by 𝑂2|𝐺𝑆(1)|𝐶max . Cases of preemption are considered, however, the DP algorithm is developed for the
non-preemptive case. The algorithm allows partitions and deletion of
some of these partitions 𝑎𝜆 that might correspond to partial infeasible
∑
sequences. The algorithm’s complexity is given by 𝑂(𝑛 𝑝𝑗1 𝑎𝜆 ) and a
FPTAS is provided considering a DP subroutine based on the previous
formulation.
Guéret and Prins (1999) develop a lower bound for the problem
𝑂𝑚∥𝐶max in which one of the steps that consist the B&B algorithm
is to calculate values that work as lower and upper bounds for the
completion times. By doing that, the authors create an analogy with the
subset-sum problem and apply a DP algorithm, which solves it quickly,
and therefore, the DP is viewed as a subroutine.
Li (2011) considers a non-bottleneck machine under jobs deterioration regime. The author delineates optimality properties based on
preliminary sources for obtaining an optimal schedule and apply those
in order to devise a DP algorithm that is unfolded into two formulations
depending on the route for scheduling a given job. Furthermore, the
algorithm can be transformed in an FPTAS by reusing some optimality
concepts.
Zhang and Bai (2014) construct a lower bound by using Lagrangian
∑
relaxation for the 𝑂𝑚∥ 𝐶𝑗2 problem. The problem is actually decomposed in two parts, being the first one relative to sequencing the
operations and the second one to calculating the Lagrangian multipliers
(referred to as assignment costs) via a forward DP formulation. The authors prove that the DP complexity is estimated as 𝑂(𝑚 max{𝑚, 𝑛}𝑝max )
and it is appropriate for small-sized instances.
Ozolins (2019a) apply a DP algorithm by relying on dominance
rules and a lower bound to find optimal solutions for the 𝑂𝑚∥𝐶max
problem. These features are incorporated in order to reduce the state
space compared to an enumeration method, which considers all possible schedules. In addition, the algorithm has its performance tested for
problems with up to 7 jobs and 7 machines and, despite showing good
results only for small-sized instances, the dominance rule can be useful
in further research and it can assist on reducing a variety of solutions
that can be disregarded.
that include cases for idle and no-idle machines in order to apply the
recursive relation and also a longest alternate processing times (LAPT)
∑
∑
rule, producing an overall 𝑂(𝑛( 𝑝𝑗1 + 𝑝𝑗2 )3 ) complexity. Moreover,
since a DP pseudo-polynomial algorithm can be constructed for such
problem, it is categorized as ordinary NP-hard (see Figs. 9 and 10).
8.4. Discussion on open shop scheduling problems
The open shop scheduling problems solved via DP also represent a
small compared to simpler environments. The papers gathered in this
section shows us that the problems studied are diversified and this
might represent potential for DP to be applied to this environment.
Problems related with makespan and completion times-related criteria
are those that account for the majority of research, which leads to a
substantial background to develop further studies with these objective
functions. On the other hand, the other objective functions cannot be
accounted for a significant contribution and this might mean research
in this area needs to be developed.
Regarding the methods used, one can notice that they vary frequently. Also, combinations have been used in order to improve the
efficiency of a given method or DP’s itself, as it can be seen in Guéret
and Prins (1999) and Zhang and Bai (2014). It might be interesting
to investigate some properties related to dominance rules and also the
results of combining DP with heuristics in order to reduce the effects
of storage intrinsic to DP. In addition, some of these may facilitate the
use of DP in practical studies, which have not been developed widely
and, those that have been, are only limited to very small instances.
The qualitative analysis for the open shop environment shows that
it has been represented by ten journals, and 25% of them have been
published in the Operations Research Letters, as it is shown in Figs. 7
and 8. This, as it has been stated previously for the job shop discussion,
reinforces the need for more publications that involve the open shop
scheduling as well as the use of DP as solution method to several types
of segments within this environment that could be of interest either in
classic or contemporary approaches.
9. Concluding remarks
DP is a method that, over the years, has been reinvented by the
researchers in machine scheduling. Clearly it had to happen because of
the drawbacks brought by the Curse of Dimensionality. These artifices
used are usually adaptations made in order to make the original method
more efficient and also in order to verify the adherence of DP to other
methods, whether it is used as a primary tool or a secondary one.
The studies developed for the single machine environment have
shown that DP has undergone several transitions. We can notice that
the first class of solutions relying on DP would focus on precedence
rules and labeling schemes that have been introduced in due daterelated criteria problems (e.g. total tardiness and weighted total tardiness) by Baker (1977) and Schrage and Baker (1978). Even though
these methods are reported to solve instances with a limited number
of jobs, they inspired more efficient approaches such as the hybrid
algorithms presented by Potts and Van Wassenhove (1982, 1987) that
could solve problems at least three times larger in comparison to
their predecessors. A second type of transition is seen with the adoption of different strategies to overcome the drawbacks inherent to DP
(e.g. Carraway et al., 1992; Hariri & Potts, 1994; Potts & Van Wassenhove, 1992b), Woeginger (2000) represents a landmark on shifting
procedures to alter DP formulations into FPTAS, which represents an
important advance in algorithmic analysis.
In summary, the last 20 years of research were focused on the
development of new techniques to improve efficiency and on new
heuristics derived from DP. This progress offered a background for
more complex problems which are recurrent in industrial scenarios. It
is worth noting that problems related to makespan, completion-times
related criteria and non-regular functions have grown considerably.
8.3. Multi-criteria and additional objective functions
Błażewicz, Pesch, Sterna, and Werner (2004) present some problems characterized by minimization of late work criterion and variations, among which a DP formulation is provided as solution for
∑
the 𝑂2|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑌𝑗 problem. The algorithm is able to calculate the
minimum total weighted late work based on parts of jobs that are
processed earlier according to a common due date. Its complexity
is bounded by 𝑂(𝑛𝑑 2 min{𝑝max , 𝑑}) and due to the pseudo-polynomial
nature of the algorithm, the problem is proven to be binary NP-hard.
∑
Zhang, Lu, and Yuan (2016) investigate the 𝑂2|𝑟𝑒𝑗|𝐶max + 𝜃𝑗𝑟𝑒𝑗
problem by studying its complexity and proposing algorithms for solving it. Firstly, the problem is actually proved to be NP-hard even
for unit processing times and equal rejection penalties. Then a DP
algorithm is presented and its computation of the makespan depends
on a recursive formulation that separates rejected from accepted jobs.
Therefore, the problem and its aforementioned special cases can be
classified as ordinary NP-hard due to this algorithm. Also, the complex∑
∑
ity of the algorithm is estimated as 𝑂(𝑛2 ( 𝑝𝑗1 )( 𝑝𝑗2 )) and, thereafter
the approximation scheme is provided.
Jiang, Zhang, Bai, and Wu (2018) develop a scheme similar to the
one presented in Fan and Cheng (2016) so as to provide solutions for
𝐴 + 𝛼𝐶 𝐵
the 𝑂2∥𝐶max
max problem. A DP algorithm and an approximation
algorithm are devised and the first one relies on optimal properties
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Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Table 18
Problems in the open-shop environment.
Authors
Brucker, Jurisch, Jurisch et al. (1993)
Brucker, Jurisch, Tautenhahn et al. (1993)
Strusevich and Hall (1997)
Tautenhahn and Woeginger (1997)
Shafransky and Strusevich (1998)
Guéret and Prins (1999)
Błażewicz et al. (2004)
Li (2011)
Zhang and Bai (2014)
Zhang et al. (2016)
Jiang et al. (2018)
Ozolins (2019a)
Problem notation
∑
𝑂2|𝑝𝑗𝑘 = 1| 𝑤𝑗 𝑈𝑗
∑
𝑂2|𝑝𝑗𝑘 = 1| 𝑤𝑗 𝑈𝑗
𝑂2|𝑁𝐵|𝐶max
∑
𝑂𝑚|𝑝𝑗𝑘 = 1, 𝑟𝑗 | 𝐶𝑗
𝑂2|𝐺𝑆(1)|𝐶max
𝑂𝑚∥𝐶max
∑
𝑂2|𝑑𝑗 = 𝑑| 𝑤𝑗 𝑌𝑗
𝑂2|𝑁𝐵, 𝑝𝑗𝑘 (𝑡)|𝐶max
∑
𝑂𝑚∥ 𝐶𝑗2
∑
𝑂2|𝑟𝑒𝑗|𝐶max + 𝜃𝑗𝑟𝑒𝑗
𝐴
𝐵
𝑂2∥𝐶max
+ 𝛼𝐶max
𝑂𝑚∥𝐶max
Complexity
Method
𝑂(𝑛2 )
𝑂(𝑛2 𝑚𝑚+1 )
∑
𝑂(𝑛 𝑝𝑗2 )
𝑂(𝑛2 )
∑
𝑂(𝑛 𝑝𝑗1 𝑎𝜆 )
–
𝑂(𝑛𝑑 2 min{𝑝max , 𝑑})
–
𝑂(𝑚 max{𝑚, 𝑛}𝑝max )
∑
∑
𝑂(𝑛2 ( 𝑝𝑗1 )( 𝑝𝑗2 ))
∑
∑
𝑂(𝑛( 𝑝𝑗1 + 𝑝𝑗2 )3 )
–
DP
DP
DP and FPTAS
DP
DP and FPTAS
B&B with DP
DP
DP
Lagrangian relaxation with DP
DP and Approximation algorithm
DP and Approximation algorithm
DP
Fig. 9. Number of papers, separated by journal, that includes DP as solution method for open shop scheduling problems.
Special considerations such as controllable processing times, resource
constraints, and job deterioration seem to be the core application of DP
in recent years.
DP-based algorithms such as SSDP and Guess-and-Check DP have
also shown some promising results since the optimality conditions are
maintained while inner drawbacks seem to exert a smaller impact when
compared to the pure DP analysis. Additionally, theoretical algorithms
and analysis were derived so as to include and classify problems
related to contemporary functions and constraints (e.g. rejection, multiobjective functions as a combination of due date-related criteria, among
others).
Regarding parallel machines, the contributions of DP have been numerous, and patterns can be observed more closely than those present
in the single machine environment. Some techniques have been extensively employed, which is the case of decomposition algorithms, Branch
and Price methods, Branch and Cut methods, among others (e.g. Chen &
Powell, 1999b; Lopes & de Carvalho, 2007; Pessoa et al., 2010). For this
machine environment, most papers are based on a theoretical approach.
This type of contribution permits the identification of optimality conditions as well as structural properties that can be incorporated into
heuristics to reduce their search space.
The most recent topics that entangle DP and parallel machines have
been those related to constraints such as bounded resources, job rejection, job deterioration and controllable processing times. Regarding the
objective functions, most of them is associated with the contemporary
analysis brought by the industrial scenario, such as earliness-tardiness,
number of early and tardy jobs, job rejection and multi-agent functions.
One can also notice that those that are developed with a practical
purpose are a junction of new DP formulations applied to routines that
had already been presented in previous works (e.g. Yin et al., 2019).
The largest share of papers involving flow shop environment have
also addressed a more theoretical background rather than a practical
one. Despite this fact, the problems that have been solved via DP
can be divided into two classes: pure DP and DP-based algorithms.
Applying pure DP to flow shop problems can, in fact, yield optimal
solutions (e.g. T’kindt et al., 2003; Yeung et al., 2004), however the
computational burden is still too high even for medium-sized instances
in most cases. For problems with identical processing times, DP appears
to present better results and find optimal solutions for instances with
up to 50 jobs and 20 machines (e.g. Koulamas, 2020; Mor et al., 2019).
On the other hand, DP-based algorithms seem to have adhered better to
some flow shop problems such as BDP (e.g. Bautista et al., 2012), which
has been able to find solutions of blocking flow shop for instances with
up to 500 jobs.
The theoretical approach of DP for flow shop problems follows the
same trends of its predecessors by generating a series of FPTAS and
complexity analysis associated with the algorithmic development. Also,
it is notorious that for some cases, Johnson’s algorithm is modified to be
applied as initial solution before developing some algorithms into full
DP formulations. Additionally, some cases are also focused in devising
dominance rules so as to reduce the state space created by DP.
The considerations regarding job shop and open shop are minimum
because DP has not been widely explored for these two environments.
Some papers focused on solving the problem by using experimental
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Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
Fig. 10. Frequency of DP in open shop scheduling problems per journal.
tests and yielding solutions for up to 20 jobs in the job shop environment and 7 jobs in the open shop, leaving most observations
to the theoretical analysis, which were mostly about the complexity
associated either with the algorithm or the problem.
Overall, these observations describe how DP has been applied over
the years to the machine scheduling, however they also can help us list
some gaps that have not been addressed in the papers concerning these
two topics and how to suggest some improvements for future research
themes. From our perspective, all the gaps and issues related to DP
in machine scheduling can be summarized in three classes: Curse of
Dimensionality, complexity of machine scheduling environments and
theoretical versus practical analysis.
The major drawback attributed to DP is the Curse of Dimensionality and when analyzing its effects on machine scheduling one can
notice that they might occur due to the complex recursive relations
generated when modeling the given machine scheduling problem. This
can be explained because these relations often create algorithms with
exponential complexity in terms of storage and tend to fail rapidly
when solutions for large and even medium-sized problems are required. Again, to overcome this issue, it is necessary to appeal to
hybrid algorithms where DP is commonly associated with Lagrangian
relaxation, bounding schemes and dominance rules. Therefore, more
research should be conducted to improve DP performance. For example, bounding schemes are versatile structures and therefore, being
able to couple an already existing bound with a new bound to the
DP formulation might yield better results than working with pure DP
approach. It is also worth noting that it would be interesting to see more
papers that explore dominance properties since they are fundamental
aspects for improving the performance of DP algorithms.
We are aware that most of the machine scheduling problems fall
under the NP-hard category. Single machine and parallel machines are
accountable for approximately 70% of the papers we gathered and
the remaining 30% are relative to other shop floor environments. The
main reason for that is that finding recursive are more challenging in
relation to the structures of their formulations that need to be both
mathematically and computationally feasible. To tackle this problem,
we have seen that some algorithms have been branched from pure DP
(e.g., BDP, SSDP, Guess-and-Check DP, Parallel DP) and, in some cases,
they have shown better results when compared to pure DP, B&B, and
MILP. When analyzing the structure of BDP and SSDP, for example, one
can notice that they construct a codependency on bounding schemes,
window widths and dominance rules, thus reducing their need to
completely depend on a recursive relation to solve the problems. Hence,
since these derived methods have not been yet explored sufficiently on
some complex environments (mostly job shop and open shop).
For instance, when we refer to Tanaka et al. (2009), the authors
managed not only to adapt the SSDP algorithm in Ibaraki and Nakamura (1994) but also improve its performance by including a powerful
lower bound whose dominance was applied upon two adjacent jobs,
a dominance for four adjacent jobs, a sub-gradient optimization for
selecting the best Lagrangian multipliers and also a sophisticated upper
bound that is coupled in the state space modifiers computation. All
these elements combined with the relaxation property continuously
reduce the gaps between upper and lower bounds, generating an exact
method, however with a more efficient performance of a pure DP
approach. Nevertheless, the same adaptations that the authors have designed are also the main challenge because the formulation of efficient
bounding structures and modifiers tend to increase in complexity as
environments and objective functions change. The same can be stated
for the BDP algorithm, which has been frequently employed due to its
dominance rules and bounding schemes. Although Ozolins (2019b) has
been able to enhance the quality of the algorithm proposed by Bautista
et al. (2012) in terms of exactness, one of the main ordeals for this
is that formulating Bellman’s equation for the 𝐹 𝑚|𝑏𝑙𝑜𝑐𝑘|𝐶max has an
significant computational cost. However, the state space cost associated
can be reduced, as it is shown in Ma and Stachurski (2021) by altering
the Bellman equation via practical sets of transformations, which maintain the optimality conditions but establish preferences according to
a desired level of robustness or recursive characteristics. Additionally,
some references in DP whose kernel are other problems in Operations
Research might be useful for developing formulations for scheduling
problems, as it is the case in Clautiaux, Detienne, and Guillot (2021)
when resorting to SSDP to solve efficiently the knapsack problem but
also confirming that various techniques used on their paper could
be further adapted for other applications of SSDP (e.g. problems in
scheduling involving precedence constraints).
Lastly, as it can be noticed, most papers tend to address theoretical solutions to the scheduling problems and, therefore, the practical
analysis of the method seems to be often neglected. Although this may
strongly suggest that DP is not as practical, there are some points
that should be considered: DP has largely contributed for unfolding
the complexity of several problems in machine scheduling. This is
relevant to comprehend how practical a given algorithm is and to
guide researchers in search for improvements. Moreover, investigating
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Expert Systems With Applications 190 (2022) 116180
E.A.G.d. Souza et al.
theoretical analysis in DP is relevant because it is associated with
new mathematical formulations that might be incorporated to the
development of other practical algorithms that are unrelated to DP as
well as future applications to DP and DP-based algorithms.
In comparison with other exact methods, DP may not be a well
explored field to tackle machine scheduling problems. However, it is
important to mention that several advances in scheduling literature
were possible due to the theoretical findings of studies that employ such
technique. Moreover, future research trends include the hybridization
of DP with heuristics to diminish the effects of the curse of dimensionality or the combination with other exact methods. In fact, the
recent progresses in computer science, especially the ones related with
memory and processing capacities may help dealing with the common
drawbacks associated with this type of algorithm.
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CRediT authorship contribution statement
Edson Antônio Gonçalves de Souza: Conceptualization of this
study, Text elaboration, Methodology, Review and Discussion. Marcelo
Seido Nagano: Conceptualization of the study, Orientation, Text review. Gustavo Alencar Rolim: Search for additional articles, Text
elaboration, Text review.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgments
This research was supported by Coordenação de Aperfeiçoamento
de Pessoal de Nível Superior (CAPES) – Brazil, under grant number
88882.379101/2019-01. The research of the authors is also partially
supported by the grants numbers 306075/2017-2 and 430137/2018-4
from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.
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