The General Combinatorial Optimisation Problem: Towards Automated Algorithm Design

—This paper deﬁnes a new combinatorial optimisation problem, namely General Combinatorial Optimisation Problem (GCOP), whose decision variables are a set of parametric algorithmic components, i.e. algorithm design decisions. The solutions of GCOP, i.e. compositions of algorithmic components, thus represent different generic search algorithms. The objective of GCOP is to ﬁnd the optimal algorithmic compositions for solving the given optimisation problems. Solving the GCOP is thus equivalent to automatically designing the best algorithms for optimisation problems. Despite recent advances, the evolutionary computation and optimisation research communities are yet to embrace formal standards that underpin automated algorithm design. In this position paper, we establish GCOP as a new standard to deﬁne different search algorithms within one uniﬁed model. We demonstrate the new GCOP model to standardise various search algorithms as well as selection hyper-heuristics. A taxonomy is deﬁned to distinguish several widely used terminologies in automated algorithm design, namely automated algorithm composition, conﬁguration and selection. We would like to encourage a new line of exciting research directions addressing several challenging research issues including algorithm generality, algorithm reusability, and automated algorithm design.


I. INTRODUCTION
Along with advances in optimisation research, a rich set of Combinatorial Optimisation Problems (COPs) has been established.COPs represent a subset of operational research [1].They consist of, subject to given constraints, assigning discrete domain values to a finite set of decision variables, so as to optimise an objective function which evaluates the solutions.The well-established benchmark COPs (e.g. the OR Library [2] at http://people.brunel.ac.uk/∼mastjjb/jeb/info.html)have promoted the design of effective algorithms in evolutionary computation and computational intelligence.Problems addressed include job shop scheduling, knapsack problem, personnel Corresponding author: Rong Qu (rong.qu@nottingham.ac.uk) scheduling, timetabling and traveling salesman problem, as well as many others and their extensions with various real-world constraints and features.
In optimisation research, computational intelligence and evolutionary computation are two of the recent advances in automated algorithm design.That is, to automatically design search algorithms or solvers which are able to solve COPs or problem instances without extensive domain knowledge from human involvement.In the scientific literature, several terminologies have emerged in different contexts, sometimes being used interchangeably without a clear definition.
In this paper we formally define a taxonomy of automated algorithm design as follows.
• Automated algorithm configuration: to automatically configure the parameters of predefined target algorithm(s) upon a given set of training problem instances.• Automated algorithm selection: to automatically select from a portfolio of chosen algorithms with their associated parameters upon a set of training problem instances.• Automated algorithm composition: to automatically generate general algorithms by composing heuristics or components of some algorithms to solve problems.
Automated algorithm configuration has been well studied.The most studied algorithms include SAT solvers [3], [4], multi-objective ant colony optimisation [5], [6] and stochastic local search [7] for flow shop scheduling problems and traveling salesman problem, and mixed integer programming solvers for traveling salesman problem and vehicle routing problems with time windows [8].The automated configuration of parameters is usually conducted offline upon a set of training instances.
A number of frameworks have been built to search the configuration space of parameters for the target algorithms, achieving highly promising results which are superior to manually configured algorithms.These include ParamILS [3] which uses iterated local search, F-Race [5] which uses a racing mechanism, and the extended framework irace [9].
In automated algorithm selection, the portfolio of algorithms includes different parametric SAT solvers [10] and evolutionary algorithms [11], [12].One important research issue concerns the clustering of training instances according to their features, thus to select the best algorithms or solvers for unseen test instances of similar clusters [10].Frameworks developed include Population-based Algorithm Portfolios (PAP) [11] and Hydra [13] (based on extended ParamILS) for optimisation functions [11], Boolean satisfiability problem and traveling salesman problems [10].
In automated algorithm composition, a set of components or heuristics are automatically combined online to generate new generic algorithms.Some research concerns components for a type of target algorithms, e.g.evolutionary algorithms [14], [15].Another line of research on hyper-heuristics [16] decides at a higher level which low-level heuristics to apply [17].By searching the given low-level heuristics, multiple COPs can be solved online with the same or adaptive heuristic compositions.Frameworks developed include HyFlex [18] and EvoHyp [19], supporting automatic composition of general algorithms across multiple COPs [16].
The fundamental difference between the three lines of research in automated algorithm design is on the decision spaces.Automated algorithm configuration concerns a decision space of parameters within a template of target algorithm(s), rather than freely composing the algorithm components themselves.The resulting algorithms, which are likely variants of the same target algorithms, are best configured for solving the training and unseen testing instances offline.Automated algorithm selection explores a decision space with a family of given target algorithms, which are grouped against some features of training instances for solving similar testing instances offline.Automated algorithm composition explores a decision space of components or heuristics to flexibly compose or combine them.The resulting algorithms generated are new and generic for solving different unseen COPs.
The three lines of research are different ways automating algorithm design.Automated configuration and selection take a top-down approach to configure within given algorithm templates and select from a portfolio of target algorithms, respectively, resulting in variants of the same family of target algorithms.Automated algorithm composition takes a bottom-up approach, working with algorithm components, to flexibly compose and generate new algorithms.
In current automated algorithm design research, however, still requires some human expertise and empirical studies to manually select parameters, target algorithms/solvers, or heuristics/components.Challenges remain to gain the insights for effective algorithms to underpin automated algorithm design [20].Such advances require models and standards to conduct systematic investigations within coherent frameworks.
In this paper, a new model, namely General Combinatorial Optimisation Problem (GCOP), is defined to model the problem of algorithm design itself as a COP, solutions of which are new algorithms automatically generated to solve crossdomain COPs.By optimising the compositions of basic algorithmic components as decision variables in GCOP, new generic search algorithms can be automatically generated.
The major aim of GCOP is to establish a standard in algorithm design to model various search algorithms in one framework with the most basic algorithmic components.It is not our intention to model all existing algorithms with GCOP.Other common and user-defined algorithmic components could be added to GCOP to design new algorithms addressing COPs as well as other optimisation problems.To our knowledge, there is no existing standard in the literature which formally models the problem of designing search algorithms.Further studies can potentially provide additiuonal insights on how various algorithms work with the new standard, and thus underpin automated algorithm design.
In the rest of the paper, Section II presents the formal definition and search spaces of the new GCOP model.With the most basic algorithmic components, Section III demonstrates the application of GCOP as a standard to define various selection hyper-heuristics in the literature for solving two COPs.Section IV discusses research issues and future directions, followed by conclusions in Section V.

II. THE GENERAL COMBINATORIAL OPTIMISATION PROBLEM
A. Definition of the GCOP Let M be a mapping function The objective of GCOP is to search for the optimal c * ∈ C which produces the optimal s * ∈ S for p, so that F (c * ) is optimised, as defined in Objective (1).Without loss of generality, we assume p is a minimisation problem in this paper.
The following terminologies are defined in GCOP.
• Problem GCOP: a COP, whose decision variables take discrete values from a finite domain A of algorithmic components a, a ∈ A. In [20], it has been suggested that the optimisation research community should adopt a certain standard.The GCOP provides such a standard to define the design of search algorithms with a unified model.In the current research on automated algorithm composition, the search space is upon manually defined heuristics or components in a specific type of algorithm(s).For example, in hyperheuristics [16], the low-level heuristics are often manually determined, and fixed with pre-defined parameter values.GCOP significantly extends the search space to concern the most basic elementary algorithmic components, thus requires no human involvement.Methods finding the optimal composition of a for GCOP thus automatically design new generic search algorithms.

B. Decision Variables of GCOP
In Table I, we establish GCOP 1.0 with a domain A 1.0 of a set of most basic and elementary algorithmic components a.The underlying idea of the GCOP standard is to modularise the existing widely used basic components a ∈ A 1.0 which are grouped into two categories, namely operators A 1.0_o and acceptance criteria A 1.0_a , each with their associated heuristic and parametric settings.This categorisation has been widely used in the literature, although quite often manually "hardwired" into integrated or compound operators or heuristics.Note that acceptance criteria a ∈ A 1.0 _a are general for any problems, and are applicable with any a ∈ A 1.0_o .Different heuristic strategies and parameter settings can be associated with a ∈ A 1.0_o to define different components.
With the sets of basic generic components A 1.0 _o and A 1.0 _a, GCOP proposes a new standard which defines a large number of different algorithms in one common model.For example, Tabu Search variants can be defined by implementing o xchg (k,m,h1 t w ) associated with an acceptance criteria a tabu (n,l), where l (tabu length), n (number of neighbors sampled), and k, m (number of decision variables selected) could be set as fixed or variable values.Associated with a gd (n,t,r) and a oi (n), variants of Great Deluge and greedy search can instead be defined, respectively.Section III demonstrates that with the basic A 1.0 and some problem-specific components A p , most selection hyper-heuristics for two widely studied COPs in Improve and equal: a better or equal s is accepted.a late (q) Late acceptance: a s better than the last q visited s is accepted.a tabu (n,l) Tabu: the best s not in a tabu list of length l is accepted.a gd (n,t,r) Great Deluge: a worse s is accepted by a probability p = e −|f (s )−f (s)| .Better s is always accepted.amc(n,t,r) Monte Carlo: a worse s is accepted by a threshold t, t is decreased by r.
Better s is always accepted.asa(n,t,r) Simulated annealing: a worse s is accepted by p = e −|f (s )−f (s)|/t , t is decreased by r.Better s is always accepted.
the literature can be defined with the GCOP model.
With GCOP, the newly generated generic algorithms are likely to be highly different from those manually designed algorithms, which are likely subsets of GCOP solutions c ∈ C.These newly generated potentially introduce new coherent knowledge in algorithm design with the GCOP model.
The underlying idea of GCOP is to decompose algorithms into elementary algorithmic components, which can then be composed and optimised in a much more flexible way, and thus design new generic algorithms automatically.This is different from automated selection from a portfolio of target algorithms and automated configuration of target algorithm(s), where the resulting new algorithms usually belong to the same family or are variants of the target algorithm(s).The decomposition of algorithms into the most basic components allows the most flexibility and provides a much larger scope to design new generic algorithms.
The basic a when composed and configured in different ways can define either new or existing search algorithms.GCOP 1.0 can be seen as a problem instance of the GCOP model, with a small domain of only the basic a.It is not possible, also not intended, to model in GCOP exclusively all algorithmic components in the literature.We aim to establish the GCOP standard step by step to explore new effective algorithms that are automatically designed.
Other GCOP instances can be built with extended A consisting of more general or user-defined problemspecific A p to design new algorithms addressing new COPs.With extended A, the solution space of GCOP increases exponentially, leading to many more new potentially effective algorithms which have been designed automatically.Note that problem-specific features and solution structures are left with users who are familiar with p.The automated algorithm design is handled at a higher level by solving GCOP.
In [21], a unified mathematical formulation for hyper-heuristics is proposed as a high-level controller.Elements of heuristic design compete for resources within a shared repository workspace to configure better heuristics.Heuristics interoperate based on information shared from other heuristics.GCOP presents a more general and coherent model and standard with which a set of algorithmic components as the domain of a COP is formally defined and automatically composed.

With
GCOP, new algorithms generated automatically may cater for multiple p with improved level of generality.The newly evolved algorithms c may also reflect a certain level of reusability for other p, saving algorithm development costs.
In GCOP, Objective (1) defines the performance measure F (c) on c for solving p.When addressing multiple p, GCOP can be seen as a multi-objective optimisation problem where c * is automatically composed to simultaneously optimise the objective function f i for each p i , i = 1, ..., I, I is the number of p under consideration.Note that the same c * is used to solve all p, rather than configured individually or manually to solve each p, respectively, i.e. each p acts as a specific problem instance for c * .Objective (2) can be defined to measure performance F m (c) of c for solving multiple p.
In Objective (1) and Objective (2), F could be the same as f , which reflects direct evaluation of s ∈ S for p. F (c) could also be different measures of c producing the corresponding s.For example, in hyper-heuristics, rewards or aggregated scores have been used to assess the performance of lowlevel heuristics during the search.Such rewards, rather than direct solution evaluation f (s), can be used in F (c) to provide informed search for GCOP.
In addition to solution quality as the evaluation F m (c) in Objective (2), further extensions and variants could be defined to evaluate different aspects of the automatically designed c, details discussed in Section IV on future research directions.

D. Search Spaces of GCOP
The search space C for GCOP presents some interesting and unique characteristics compared to that of S for p, see Table II.They can be defined based on the following three factors: • Solution encoding: represents all solutions based on some finite alphabet for the decision variables.The encoding of c ∈ C is highly different from that of s ∈ S, leading to different upper bounds of C for GCOP and S for p.In [22], the concept of two search spaces, namely high-level heuristic space and low-level solution space in hyper-heuristics, has been introduced into the scientific literature of search algorithms.Fitness landscape analysis on local optimal solutions in the heuristic space revealed interesting findings [22], [23].The search space of selection constructive hyper-heuristics has also been analysed in [24] to reveal common landscape features of this type of algorithms.

III. EXAMPLE METHODS FOR GCOP
With the GCOP model, a large number of algorithms could be defined with a subset of a ∈ A 1.0 in Table I [25] in Section III-A, and Nurse Rostering Problems (NRP) [26] in Section III-B are selected as the p in GCOP.The aim is not to provide an exclusive review on all existing selection hyper-heuristics, but to demonstrate the modelling of various search algorithms in one GCOP standard towards automated algorithm design.

A. GCOP Methods for Vehicle Routing Problems
As one of the most studied COPs, VRP and its variants have been used to model a range of real-world applications, e.g.transport logistics in supply chain with different constraints.The basic VRP involves constructing a set of closed routes from and to a depot, each route delivering the required demands to an ordered list of customers by a vehicle of limited capacity.The objective is to minimise the total costs (e.g.distance and/or vehicles), whilst satisfying the capacity constraints.Evolutionary algorithms and computational intelligence techniques have been extensively studied for VRP variants with complex constraints [25], [27], [28], [29].
In GCOP, c upon the configured a operates on s for VRP.In the most commonly used solution encoding in the VRP literature, customers or tasks are modeled as nodes (i.e.decision variables s i,j for p) in a directed routing network.A large number of operators in VRP algorithms can be modeled using the basic generic a ∈ A 1.0 , as shown in Table III Some frameworks, e.g.HyFlex [33], POEMS [34] and ALNS [35], have been adopted to develop the hyper-heuristics, i.e.GCOP composition methods.

Table IV in the extended GCOP.
In Table IV, the GCOP composition methods in selection hyper-heuristics for VRP range from various search methods to classifiers trained using machine learning.Of particular interest is the research on c for different COPs developed in the HyFlex framework with a set of built-in low-level heuristics.With the consistent GCOP standard, further investigations on c composed more flexibly with the automatically selected a ∈ A 1.0 could gain useful insights on the effectiveness of complicated as well as simple c for various COPs.
In Table IV, in addition to using the direct evaluation on s for p, i.e. f (s), F (c) for GCOP has also used various assessment metrics to evaluate the effectiveness of a in c.Instead of assessing the final solution, such measurements reflect the short-term performance of a in c during problem solving, thus providing more informed decision making composing a into the generated algorithm c.IV can be seen as compound operators, integrating more than one a in Table I.For example, o xchg can be seen as applying o asg twice, and o rr can be seen as applying o rm followed by o asg .A large number of heuristics h1 and h2 have been used in the literature.For example, in [36], seven different removal criteria have been used to remove requests (nodes) from the routes using different measures (e.g.time-oriented, history-based, cluster, worst, and related removals).Two insertion criteria (i.e.greedy and regret) are also used to re-assign requests back to routes.

Some of the components in Table
In the literature, some decisions of algorithm design (e.g. the acceptance criteria A 1.0a , or the number of n neighbors explored) are not always provided or take some default settings.This leads to some ambiguity in reproducing the published algorithms.With A 1.0 defined with consistent parametric and heuristic settings within the GCOP standard, these details can be clearly defined with a, thus supporting consistent research in the literature.

B. GCOP Methods for Nurse Rostering Problems
The NRP has received extensive research attention in the last five decades [26] due to the high demands of quality healthcare, limited resources, and various legislation around the world.The problem consists of assigning a set of nurses of different skills to a set of different shifts on each day of a scheduling period.A set of hard constraints must be satisfied, including the legislation (e.g.maximum consecutive shifts) and coverage (i.e.all demands must be covered).The objective is to minimise the violations of soft constraints, e.g.personal preferences, free weekend, and preferred shift patterns.Algorithms investigated include exact methods, evolutionary algorithms and hyper-heuristics, and many more ruin and recreate: use h1 b to reassign all k shifts of a set of nurses chosen by h1w.[26].
With the GCOP standard, most of the low-level heuristics in hyper-heuristics and components in other algorithms in the NRP literature can be modelled as a ∈ A 1.0 in Table V.They all operate on shifts assigned to selected nurses working on particular days (i.e.decision variables in p).For example, o bw xchg (k, k, h1 w ) swaps k shifts of two nurses selected by h1 w .If shifts of one of the nurses are empty (no shift assigned), it defines the move shift operator in the NRP literature.
The GCOP composition methods in Table VI employed various local search algorithms and different techniques for NRP.As for VRP, F in GCOP for NRP measures either the quality of the resulting s or the performance of a. Compared to VRP, there are not many problem-specific a in NRP, i.e. most of the a in Table V are in A 1.0 .Also, various acceptance criteria have been studied for NRP, which is not the case for VRP.Note that most widely used acceptance criteria in A 1.0 are not problem-specific, and can be used across different COPs.
Only three benchmark datasets (i.e. the INRC2010 competition [56], Nottingham and UK [55] datasets) have been widely tested in NRP, the first two with extensive NRP variants.An interesting study on INRC2010 [52] focuses on choosing a compact set of low-level heuristics.The methods developed showed to be highly effective on the benchmark as well as two real-world scheduling problems.Such analysis can also be conducted automatically with the GCOP model.In [22], it is found that simple configuration methods work as effective as complicated algorithms on effective low-level heuristics.Such studies provide useful insights on a ∈ A 1.0 for further research in GCOP.
In this paper, we define the most basic a ∈ A 1.0 across COPs, aiming to establish the fundamentals of the GCOP standard.More advanced research tensor analysis [54].
NRP a in GCOP methods o upon pre-defined Bayesian network [55] learns to shift patterns for select a set of good shift patterns specific p for p. p: UK dataset will be conducted as discussed in Section IV, and also strongly encouraged from the research communities, to further enhance the GCOP standard towards automated algorithm design.Following the recommended good practice in OR [20], updates of extensions and resources on GCOP will be provided at a dedicated GCOP web site at https://sites.google.com/view/general-cop.

IV. DISCUSSIONS AND FUTURE DIRECTIONS
As a fast emerging topic in computational intelligence, evolutionary computation and optimisation research, automated algorithm design has recently attracted increasing research attention.In this paper, GCOP is formally established as a new standard to define various search algorithms in one model, providing the fundamentals and opening a number of potential new research directions in automated algorithm design.
New knowledge towards automated algorithm design: The new GCOP model provides a standard for systematic analysis on the basic a of different behaviors in the optimised c.Some studies in hyper-heuristics identified a compact subset of effective low-level heuristics and revealed synergy among them, enabling effective methods to be built [53].In [57], a runtime analysis on a selection hyper-heuristic shows that online reinforcement learning for configuring operators may perform poorer than a fixed distribution of operators.These analyses could also be conducted within the consistent GCOP model.The balanced intensification and diversification may be modeled in c considering synergy among a.New findings on new effective algorithms with different categories of algorithmic components may also lead to new knowledge and deeper understanding in algorithm design and introduce new effective algorithms to the literature.
Generality and reusability of algorithms: In GCOP, the automatically designed new algorithms evolve to perform well for solving different p, thus may cater for similar types of new p with a certain level of generality and reusability.Recent research has made some progress on the generality of algorithms.However, the reusability of algorithms remains under-explored.With the GCOP standard, the optimised components may be analysed to derive new knowledge potentially transferable to solve unseen p. GCOP may contribute to addressing the challenging research issue of generality and reusability of algorithms.
• Generality: Recent hyper-heuristics where shown to be able to address cross-domain COPs [58].There is to our knowledge, however, not yet a formal definition of algorithm generality in the literature.In [59], a new assessment method has been proposed to evaluate hyper-heuristics against four levels of generality, in terms of solving different problem domains, problems, problem instances and benchmarks, respectively.The automatically generated new algorithms with GCOP can be evaluated against these four levels of generality for solving different p.Note that the assessment of algorithm optimality is different from that of algorithm generality.The latter may also measure the robustness and speed, in addition to solution quality for multiple problems/domains.• Reusability: Recent research has made some progress on reusing algorithms, although the main research focus may not be exactly on reusability.For example, the automatically selected algorithms on training instances [10], [11] could be reused to solve testing instances of certain similar features.In generation hyper-heuristics [60], new heuristics can be automatically generated by using genetic programming based on problem state features [61], [62], thus could be potentially reusable for problems of similar features.However, the problem of code bloat may lead to the issues of readability and interpretability [63].
Fundamentals of GCOP: Advanced theoretical investigations are needed to underpin the fundamentals of the new GCOP model in operational research.
• Evaluation of GCOP: In solving GCOP, the objective function can be extended with multiple objectives including generality, reusability and computational time.The new performance measure in [59] can be adopted in the objective function to measure different levels of generality.In GCOP, instead of designing algorithms using human expertise, as happens in most of the research, the time is spent on automatically searching for, or composing the optimal c * for p.The trade-off between solving a number of p and the increased computational time presents another interesting research issue.The c for each p can be further evaluated in F to assess its convergence, the number of operations and number of fitness evaluations used, using different statistical measures as shown in [64].With the new GCOP model, we define a set A 1.0 of the mostly used basic elementary algorithmic components for widely studied combinatorial optimisation problems in the literature.This set can be extended to include more common components, as well as userdefined problem-specific components A p for different optimisation problems.With the new standard, we also demonstrate the application of the GCOP model with A 1.0 and A p to formally define a large number of selection hyperheuristics for solving two benchmark combinatorial optimisation problems, namely vehicle routing and nurse rostering problems.This can be seen as the implementation of the GCOP standard, demonstrating its effectiveness for modeling a large number of existing algorithms.To our knowledge, this is the first standard in the literature defining a large number of search algorithms in one common model.
The established new GCOP opens a new line of interesting research directions in optimisation research.Further studies will investigate theoretical issues including landscape analysis on the search spaces of GCOP.The objective function can be extended to measure the generality, reusability and computational time of the newly generated algorithms.The new automatically designed algorithms introduced to the literature brings new knowledge which can be used to design new effective algorithms, and reused for solving other optimisation problems.In addition to combinatorial optimisation problems, continuous optimisation problems and multiobjective optimisation problems could also be addressed with extended GCOP models.
With the new GCOP standard, this position paper calls for further investigations on the emerging topic of automated algorithm design to stimulate more advances in evolutionary computation and optimisation research.We strongly encourage future research in the research communities to adopt and extend the GCOP standard.Resources and latest developments will be continuously updated at the GCOP web site.
Objective function f for p: f (s) → R evaluates solutions s for p, s ∈ S are obtained by using algorithmic compositions c ∈ C.
* for p, i.e. c * → s * .•• Solution space S of p: consists of solutions s for p.•

Table I
DOMAIN a ∈ A 1.0 OF DECISION VARIABLES FOR GCOP 1.0 b .o in ins /o bw ins : s i,j are from the same/different i. orr(k,h1w,h1 b ) Use h1 t b or h1 r b to reassign values of k decision variables selected by h1 t w or h1 r w .oxo(k,m,h2 b ) Swap k and m randomly chosen decision variables between two solutions in P chosen by h2 t b /h2 r b .Domain a with parameters n, t, r. neighbor A 1.0_a s of s is produced by o ∈ A 1.0_o .n: number neighbors sampled.a all Accept all, Naïve accept: s is always accepted, i.e. random strategy.a oi (n) Only improving: better s is accepted.a ie (n)

•
Successor operator: modifies values of the decision variables thus defines connections of the encoded solutions in the search space.The successor operators in C and S operate upon c and s of different encodings.
• Objective function: evaluates the solutions.In GCOP, F (c) assesses the performance of c, which produces s ∈ S. F (c) thus depends on f (s), however, may potentially be different from f (s), see the examples in Section III.In solving GCOP, assume c for the decision variables are encoded as one-dimensional strings of a ∈ A in TableI.With the simple encoding c for GCOP, it is possible and highly useful to analyse the landscape of C, whose spatial structure

Table II CHARACTERISTICS
OF SEARCH SPACES: GCOP VS. p It is shown to be very difficult, if not impossible, to analyse the landscape of S for many complex p with d-dimensional solution encodings, d ≥ 2, see the example COPs in Section III.
. Furthermore, GCOP provides a standard towards automated algorithm design.With a ∈ A, different high-level methods, e.g.local search, tools, rules or models could be used to compose a flexibly and design new algorithms automatically in a bottom-up way.Hyper-heuristics can be seen as one type of GCOP implementation which combine low-level heuristics, which are defined and integrated based on a subset of basic a ∈ A 1.0 in TableI, to design new algorithms automatically.That is, the low-level heuristics are "hard-wired" with the basic a thus requiring a certain level of human knowledge.

Table III OPERATORS
FOR VRP MODELED AS a ∈ A 1.0 IN GCOP

Table V
LOW-LEVEL HEURISTICS IN HYPER-HEURISTICS FOR NRP MODELED AS BASICa ∈ A 1.0 IN GCOP [67]d on more findings in evolutionary computation, A 1.0 could be extended with more effective common a.Those a used in the literature (TableIIIfor VRP and TableVfor NRP) represent only a subset of A 1.0 in TableI.The library of general and basic a can be easily extended with problemspecific A p and is portable to solve a wider range of p.Such efforts are highly valuable and strongly encouraged to promote future advances of automated algorithm design.Resources will be updated at the GCOP web site.•Otheroptimisationproblems:Recentresearchhas developed effective selection hyperheuristics for continuous optimisation problems[66],[67].The GCOP model can also be extended to solve other optimisation problems in addition to COPs.V. CONCLUSIONS This position paper introduces the General Combinatorial Optimisation Problem (GCOP) as a new standard for algorithm design.The objective of GCOP is to optimise the compositions of basic algorithmic components, i.e. the decision variables, to automatically design new algorithms for solving different optimisation problems.A taxonomy of automated algorithm design, i.e. automated algorithm configuration, automated algorithm selection and automated algorithm composition, has been formally defined.GCOP which standardises various search algorithms within one model underpins the fundamentals of automated algorithm design.
• Extensions of A: