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1 Introduction
AMulti-Agent Problem (MAP) is usually identified with a class of different problems of in Artificial Intelligence (see: [28]) involved in a multiple of interacting and intelligent agents. Despite of some particular differences between problems from this class, at least one of the two following properties may be asssociated to them. It is A) an inventing of the action sequence in order to perform a goal or B) associating actions to agents that could be performed by them due to their skills.
Although a conceptual provenance of different problems under the banner MAP should be found in a class of different optimizations problems, such as: the so-called Nurse Job Scheduling Problem (NJSP)fn – intensively explored in such works of Nottingham’s school as: [3, 1, 5, 23, 21] – MAP remains close-related to Constraints Satisfaction Problems (CSP). One can venture to fomulate the thesis that MAP forms a unique correlate of a simplified temporal variant of CSP – the so-called Simple Temporal Problem (STP) – introduced and broadly discussed together with its extensions (STPU and STPPfn) in such works of Dechter-Khatib’s school as: [4, 15, 26, 18, 16, 29, 24, 11].
In practice, different combinations of MAP with CSPs have been elaborated under the banner of ’Multi-Agent Approaches to CSP’ – as described in [19, 22, 20, 30]. These (usually sophisticated) synergic combinations have either a nature of asychronous solvers, such as: Asynchronous Backtracking (ABT) or Asynchronous Weak-Commitment Search (AWCS) – due to [30] or a character of distributed local search methods, such as: Distributed Brekout Algorithm (see: [30]) or Environment, Reactive rules and Agents (ERA) approach (see: [19]). It appears that MAP in the context of CST forces some new special requirements imposed both on agents activity and for their mutual interaction. For example, in ABT – each agent performs a distributed version of of the backtracking in asynchronous way and a total order among agents is required. In contrast – in ERA, agents move synchronously and cannot operate in the domain of other agents, etc. (see: [19].) Finally, an idea of the combination of MAP with CSP found its reflection in a synergic definition of Distributed Constraint Satisfaction Problem (DisCSP) as an algebraic foundation of the Multi-Agent CSP approach.
MAP sucessfully finds its programming-wise reflection in such languages of a declarative paradigm as PROLOG, Answer Set Programming (ASP) or – recently – in Bousi-PROLOG, which integrates fuzzy logic and logic programming for a use of dealing with approximate reasoning – due to ideas from [7, 8].
Meanwhile, an inherent nature of NJSP – as a conceptual foundation of MAS – sometimes required a piece of fuzziness because of a need to grasp vaguenees of information (for example, about the hospital objectives or personal preferences). Thus, a kind of a fuzzified MAP (FMAP) seems to be expected in order to provide a high quality scheduling tasks. A new step towards a fuzzification of NJSP involved in fuzzy goal programming (GP) models has recently been proposed in [2, 25].
Anyhow, it seems that fuzziness may be introduced in different manner to the optimization problems. In the context of MAS, it may be naturally introduced by different (sometimes unknown) degrees of ability of agents to perform their tasks. A similar comprehension of FMAP is just proposed in this paper.
1.1 The Paper Motivation
Independently of a visible success of the Multi-Agent Approaches to CSP and of a still increasing effectiveness of different specified algorithms, solvers and variants of dynamic programming, some difficultes of the proposed approches may be easily detected.
-
A Nurse Scheduling Problem – as a basis of Multi-Agent Problem should be rather seen a basis reservoir of possible, more specified formulations and it still waits for a more advanced fuzzy complementation. In particular:
— Admittedly, Ernst’s approach in [5] is mathematically general, but it refers to some simplified situations.
— In contrast, Lepegue’s approach from [17] is pretty detailed, but formalization of possible temporal constraints in it seems to be too excessive and it is counterintuitive.
— Finally, a fuzzification from [25] more predicts some results than puts forward them in detail.
B It is not completely clear how to relate MAP to the simplified temporal CSP in terms of STP and its different extensions – as defined in [4, 15, 29, 24].
C It seems that a fuzzy and multi-valued logic-based approach to combining CSP with other ’entities’ of fuzzy temporal-reasoning, such as preferences – due to [13, 12, 11, 10, 14] – does not indicate any appropriate method to grasp MAP in their contexts.
D It also seems – due to author’s best knowledge – that no explicit approach to fuzziness has been proposed in the contaxt of MAP on the conceptual level.
E Finally, an expressive power of PROLOG and its solvers has not yet been completely explored with respect to any fuzzified version of MAP – independently of a variety of modern alternative tools (such as GP, ADOPT) used for a fuzzified NJP or Multi-Agent Constraints Satisfaction (see:[25, 19, 2].)
Meanwhile, the choice of the PROLOG-solvers in the context of FMAP requires a brief justification in the light of the fact that Bousi-PROLOG is just addressed to model vague knowledge and approximate reasoning. The solution is dictated by two facts. At first, development of Bousi-PROLOG still remains in progress, so it is relatively difficult to evaluate its expressive power. Secondly, exploting of PROLOG itself allows us to deeply recognize an expressive power and some adequacy of PROLOG for representation of FMAP and its workable subcases. Investigations of the paper partially stem from research presented in [9].
1.2 The Objectives of the Paper and its Organization
According to the requirements and needs, just formulated, and slightly against the tendency of the technology-oriented research on Multi-Agent Constraints Satisfaction – the paper forms a unique return to the conceptual foundation of the considered issues. In fact, this paper will be devoted to a new formulation of Multi-Agent Problem (MAP) and its fuzzy extension (FMAP). The newly introduced FMAP and DisCSP will be also referred to STPU as their conceptual correlates in order to detect some important similarities and discrepancies between them. Finally, some workable subcases of FMAP will be solved by means of PROLOG machinary. In addition, complexity of these solutions will be briefly discussed.
Novelty of the paper with respect to earlier approaches consists in:
N1 proposing a fuzzy modification of MAP – both in a practical and in a slightly genral depiction,
N2 describing both MAP and FMAP as a synergy syntesis of planning and scheduling components in a referrence to both STPU and DisCSP,
N3 detecting both similarities and discrepancies between FMAP and STPU and DisCSP.
N3 considering the PROLOG-solvers for some workable fuzzy cubcases of FMAP.
The rest of the paper is organized as follows.Section 2 introduces a terminological background of the paper analysis. Section 3 contains both the (more) practical depiction of MAP, its generalization and FMAP. Finally, a brief taxonomy of different constraints imposed on these problems is also put forward here. Section 4 is devoted to the programming-wise aspects of FMAP. Section 5 contains closing remarks.
2 Terminological Background
Before we move to the main body of the paper, we preface our analysis by introducing the concepts of STP and STPU as further extension of STP. The terminological framework determined by these concepts allows us to better grasp a conceptual specificity of both MAP and FMAP, introduced in detail in Section 3.
Simple Temporal Problemfn forms such a simple subproblem of TCSP that all
constraints are specified by a single interval. More formally, STP constitutes the
kind of the Constraints Satisfaction Problem, where a constraint between time-points
Solving STP – due to [24] – means to find an assignment of values to variables such that all temporal constraints are satisfied. It appears that STPs can be solved in polynomial time, whereas the complexity of a general TCSP belongs to a class of NP-problemsfn.
In [29] STP was extended by considering contingent events, whose occurrence is somehow controlled by exogenous factors sometimes referred to ’Nature’. In this way we introduce Simple Temporal Problem under Uncertainty (STPU) as a further specification of STP, which still preserves some important point of the STP-based conceptualization (such as a controllability of start points of an agent activity).
Nevertheless, the end times (end time points) are divided into two classes: required (requirements) (’free’ in [29]) and the contingent ones. It immediately introduces a distinction between STPU-variables for the executable, required and contingent time points. The first ones – are associated to performing agents, the second ones – are assigned by ’external’ worlds (factors).
Definition 1. (STPU.) A Simple Temporal Problem
with Uncertainty (STPU) is such a 4-tuple N =
Example.
Consider the following scenario connected to the following two
activities:
On a base of a general definition of CSP, the following definition of DisCSP has been elaborated in [30].
Definition 2. (DisCSP forms a 5-tuple
This generic formulation that combines a multi-agent problem with CSP may be specified towards some additional requirements imposed on agent interaction and capabilities.
3 Multi-Agent Problem and Fuzzy Multi-Agent Problem
In this chapter, our initial depiction of MAP will be put forward, but its formulation is motivated by Nurse Rostering Problem. This solution allows us to grasp many intuitions that find their reflection in a definition of FMAP and in the appplication-based part of the paper. In order to make it, let us note that each well-founded depiction of MAP must satisfy the following general criteria.
C1 A finite (non-empty) of agants schould be given,
C2 Agents should be involved in some activities in some time periods and subperiods (for examples: days and shifts),
C3 There are some hard constraints imposed on agant activities thatm must be absolutely satisfied to perform the task,
C4 There are some soft constraints imposed on agent activities that may be satisfied.
Taking into account these general criteria, one can formulate the following generic Multi-Agent Problem as follows:
Multi-Agent Problem (MAP). Consider a factory with
HC1 The charm of the shift organization should be fair: each agent must to have equally: 2 day-shifts and 2 night-shifts.
HC2 Each agent can be associated to at most one shift,
HC3 Some shifts are prohibited for agents,
HC4 Length of the shifts sequences associated to each agent is restricted,
HC5 Quantity of the shifts in a scheduling period is restricted,
HC6 Quantity of the shifts per a day is restricted.
The MAP consists in a construction of a scheduling diagram, which respects all these constraints.
The constraints HC1-HC6 form an exemplification of the so-called hard constraints as these constraints, which should be satisfied in a scheduling task. They ensure a feasibility of the scheduling task. The soft constraints may not be satisfied, but a degree of their satisfaction is a measure how good is a performed plan. In order to make the requirements with respect to the hard constraints more liberal, the so-called relaxation, i.e. a weakening of the strong constraints will be used. We often use this solution, when satisfaction of all hard constraints leads to an inconsistency.
A further relaxaction of requirements or expectations allows us to consider other entities, such as preferences or fuzzy requirements. Only the last ones will concern us is this paper. In general, the fuzzy constraints might be the entities of a nature, which introduces a kind of fuzzines to MAP.
The fuzziness may be determined by uncertainty, partial observability, a lack of a complete knowledge, etc. In this paper, a meaning of ’fuzziness’ is restricted to an existence of different (usually unknown) degrees of agent ability to perform their tasks in the context of MAP. All these degrees determines a [0,1]-continuum with 0 as a complete lack of such a disposition and with 1 as a full disposition to perform the appropriate tasks.
In such a conceptual framework, Fuzzy Multi-Agent Problem (FMAP) may be viewed as Multi-Agent Problem equipped with additional fuzzy requirements imposed on the task performing. An exemplary, paradigmatic formulation of FMAP is given as follows.
Fuzzy Multi-Agent Problem (FMAP). Consider a factory with
n-agents working in a rhythm of the day-night shifts:
HC1 The charm of the shift organization should be fair: each agent must to have equally: 2 day-shifts and 2 night-shifts.
HC2 Each agent can be associated to at most one shift,
HC3 Some shifts are prohibited for agents,
HC4 Length of the shifts sequences associated to each agent is restricted,
HC5 Quantity of the shifts in a scheduling period is restricted,
HC6 Quantity of the shifts per a day is restricted.
Assuming also an agent
SC7 A quantity of shifts in a scheduling period is established,
SC8 A scheduling charm’s covering by shifts in a scheduling period is established,
SC9 A length of the shifts sequence associated to an agent is fixed,
Fuzz1 A number of degrees of agent ability to perform actions is a natural number from the set
Fuzz2 An agent
The FMAP consists in a construction of a scheduling diagram, which respects all these hard and soft constraints and the fuzzy requirements.
Although the list of possible fuzzy requirements might be enlarged, let us underline that both Fuzz1 and Fuzz2 rather constitute a scheme or a class name for different particular fuzzy constraints: Fuzz2 – for the constraints introduced by preferences, but Fuzz1 – for fuzzy constraints defined without them. Thus, both classes have a whiff of a paradigmatic generality.
3.1 Types of Constraints of FMAS and their Arithmetic Representation
Different approaches to a representation of both hard and soft constraints and preferences has been proposed. Majority of them grasps an arithmetic represnetation based on a calculus of characteristic functions – due: [17, 2, 25], but a multitude of different additional indexed parameters (c.a. 20) in their arithmetic depiction often makes an idea of representation only partially elusive.
Thus, a restricted set of parameters is introduced for a mathematical representation of temporal constraints imposed on (F)MAP. Instead of agent skills we will consider agent roles (contracts)fn:
—
—
—
—
—
It enables of representing MAP by its formal instances in the form of the triple
where
where
This notation allows us to elaborate the following representation of hard and
soft constraints. Since their list is not exhaustivefn, it might be relatively naturally extended. A
formal representation of each constrain is based on a characteristic function
HC 1: The charm of the shift organization should be fair: each agent must
to have equally 2–day shifts and 2–night shifts. Assume that
HC 2: Each agent can be associated to at most one shift.
HC 3: Some shifts are prohibited for an agent
n. If
HC4: Length of the shifts sequence associated to an agent. This
constraint defines a restriction for a sequence of shifts associated to an agent
n. If we denote the minimal and the maximal number of
shifts for an agent
HC 5: Quantity of shifts in a scheduling period. It defines the
minimal and the maximal quantity of shifts during a given scheduling period
(day) – associated to a single agent
where
Obviously, all the constraints listed above are not exhaustive and their list may be naturally enlarged. However, we interrupt their presentation in this point for a cost of a new presentation of FMAP in a more general depiction.
4 General Formulation of a Multi-Agent Problem
In this section a kind of a generalization of MAP and FMAP is put forward in order to provide a formal distinction criterion between MAP and FMAP. The conceptual framework of this generalization is determined by a set of agents, each of them possessing specific skills, a set of some temporal tasks to be completed and associated to agents.
Each agent can accept only tasks consistent with its skillsfn. Below, a generic, simple formalization of this problem is proposed.
Consider a set
For practical reasons, it is convenient to represent this function in a tabular (matrix) form as follows:
where
Similarly, consider a set
Again, for practical reasons it is convenient to represent this function in a tabular (matrix) form as follows:
where
For simplicity, it is assumed that a single task can be executed by a single agent,
one task at a time. Task
and the execution can start whenever the agent is free; this holds for all
In this depiction, a distinction between MAP and FMAP reflects itself in another way
of defining of functions
satisfies the condition:
5 FMAP versus STPU versus DisCSP
Fuzzy Multi-Agent Problem has already been introduced not only in a practical NJSP-based depiction, but also – in a more general formulation. This two-side presentation allows us to propose an outline of a comparative analysis between FMAP,STPU and DisCSP.
5.1 FMAP versus STPU
It is not difficult to detect both a couple of similarities between FMAP and STPU, and some discrepancies. On one hand, constraint satisfaction plays an important role in both classes of problems. In addition, a similar taxonomy of constraints – divided (at least) into two groups: hand and soft constraints – is usualy considered in both cases. Nevertheless, the role of the constraints and their satisfaction seem to be a little bit different. In fact, the problem forms a consitutive components of STPU, its real conceptual ’core’.
This conviction may be easily verified in the light of STPU-definition as an extension of STP that forms a simplified TCSP – due to [29] repeated in Section 2. Meanwhile, different types of constraints constitute only an additional component of FMAP – as imposed on performing the appropriate scheduling tasks. Conversely – agents, their mutual iteraction and their capabilities constitute FMAP as such a one, whereas these componants may be radically restricted or even omitted in the generic formulation of STPU.
Some further observable discrepancy between STPU and FMAP stems from a semantic discrepancy between fuzziness and uncertainty as their founding concepts. Although fuzziness (in particular – in a sense given by fuzzy logic) may be exploited to represent and model uncertainty, the fuzziness seems to rather refer to a vague, imprecise nature of different modeled entities or states(such as as: ’small’ disposition of an agent to work). In contrast – uncertainty seems to rather refer to some epistemic unclarity in acting and reasoning (such as: partial observability, a lack of some premises in reasoning). This fact may be rendered in alternative way as follows: uncertainty refers more to epistemic capabilities or skills of agents or an external observer, meanwhile fuzziness grasps more an ontological side of the systems/problems.
Finally, some interesting aspects of STPU and FMAP are ellucidated by the category theory-based concepts of graphs and monoids. MAS may be viewed as the organizing thoughs about agants acting on objects.
This property due to [27], p. 115 – may be algebraically rendered by monoids. In contrast, STPU materializes an idea of information flow (between nodes), that – due to Spivak’s conviction from [27], p. 115 – may be naturally represented by systems of nodes and arrows (vertices). It explained why STPU is naturally associated to graphs and different methods of the graph search may find an application in STPU-solutions.
5.2 FMAP versus DisCSP
Let us recall that FMAP is defined as the tuple
6 Programming-wise Aspects of Fuzzy Multi-Agent Problem
In this section, we intend to face the Multi-Agent Problem in order to illustrate how to solve some of its workable subcases by means of SWI-PROLOG-solvers. Earlier arrangements from Section 3 and Section 4 allow us to formulate a couple of general guidelines of the solver construction. In particular, the proposed representation method for hard and soft constraints in Section 3 suggests a choice of their arithmetic, summeric depiction in PROLOG.
Meanwhile, the proposed general depiction of FMAP from Section 4 suggests to associate agents to their skills or degrees of their abilities (to work).
6.1 A More Detailed Specification of Requirements
In order to illustrate this method of the solver construction, let us assume that
a non-empty set
In such a framework, the PROLOG-solver task is to give a schedule respecting the temporal constrains imposed on a task performing and the corresponding agent activity. The achieved solutions will be represented by lists of the form:
where
In general, two following types of situations may be considered:
crisp-type situations, when
fuzzyfn-type situations, when
6.2 General Specification of the PROLOG Code
SWI-PROLOG solvers will be exploited to find so-lutions for scheduling tasks of FMAP. As mentioned above, each admissible solution will have a form of the appropriate list of natural numbers.
The required constraints imposed on the given scheduling tasks will define the
appropriate 2- argument predicate
a general specification of each agent as a list of its working days in a scheduling period, for example,
an association the admissible degrees of ability to agents – introduced by the code of the form:
an arithmetic restrictions on agent during a scheduling period, for example,
a logical restriction on the agents interaction, for example:
The expected solutions in a form of lists (9)-(10) have just been specified.
6.3 Crisp Cases
We will consider 3 subcases of MAP dependently on the problem complexity. In all
the cases: 1 – in solution-lists – represents a state when an agent
As the first one, a case of 2 agents working 5 days in a week (with a week restriction for each of them equal to 4) will be considered. It will be shortly denoted as MAP(2, 5, 4). It is assumed that a maximal sum of working days of each of them does not exceed 5.
Case 1 – MAP(2, 5, 4). In this case, we consider the following situation:
— 2 agents operating during a 5-day scheduling period,
— only two degrees of ability to work are associated to the agents,
— a restriction imposed on number of activities (shifts) is smaller than 4,
— each day (1-5), a working activity of the first agent excludes a working activity of the second agent and conversaly.
The requirements may be represented by the following code (The sense of lines of the PROLOG-code is explained on the right side):
In result, the following list of solutions is obtained:
It is not difficult to observe that each pair of the achieved solutions respect
all the requirements (1 for
Case 2 – MAP(2, 6, 4). In this situation, all the earlier requirements remain without changes with exception of the length of a working period for both agents (6 instead of 5) and the admitted number of acitivities during a week (we exchange an upper bound equal to 5 for 4). In this framework, the appropriate PROLOG-schedule is given by the code clauses:
In result, we obtain 19 pairs of lists for
6.4 Fuzzy Cases
In order to exchange the crisp cases for the fuzzy ones it is enough to admit some new degrees of agent dispositions In the current cases, the following list of values is admitted:
0 – to represent the fact that an agent A is absent (on a shift),
1 – to represent a physical absence of the agent A, but a real disposition to be present.
2 – to represent a physical presence of A, which is only in a partial disposition to work.
3 – to represent a full disposition of A to workfn.
Let us begin with an exemplary case of two agents: A1 and A2 working 5 days in a week and having four degrees of disposition denoted by 0,1,2 and 3 as above.
MAP(2,5,4)Fuzz. This situation is defined by the following requirements:
— 2 agents: A1, A2 operating during a 5-day scheduling period,
— 4 degrees of ability to work are associated to the agents: 0, 1, 2, 3.
— a summaric number of admissible degrees of A1-disposition is smaller than 12,
— a summaric number of admissible degrees of A2-disposition is smaller than 9,
— a summaric restriction on the admissible number of the disposition degrees for both A1 and A2 (together) in the first day is smaller than 4, for A1 (alone) is greather than 1, what excludes a degree of A2-activity greather than 2,
— a summaric restriction on the admissible number of the disposition degrees for both A1 and A2 (together) in the second, the third, the 4th and in the 5th day (D1, D2, D3, D2, D5) is smaller than 4, for A1 (alone) – is greather than 2, what excludes a degree of A2-activity greather than 2,
— a summaric activity of A1 in the day triples: (D1, D2, D3) and (D3,D4,D5) is smaller than 6,
— a summaric activity of A2 in the day triples: (D1, D2, D3) and (D3,D4,D5) is smaller than 7,
— a summaric activity of A1 in a day triple (D2, D2, D4) is smaller than 7,
— a summaric activity of A2 in a day triple (D2, D3, D4) is smaller than 5.
This situation may be reflected in the following PROLOG-program (As earlier, the sense of lines of the PROLOG-code is explained on the right side of the programm):
In this case, the PROLOG-solver returns us the following solution-lists:
MAP(2,5,5)
instead of 0,1,2,3 we consider five values: 0,1,2,3,4 as admissible fuzzy disposition degrees of A1 and A2,
restriction on A2-activity during a week is relaxed – instead of
Taking into account these requirements, one achieves the following program:
In this situation, the PROLOG-solver returns us the following (slightly longer) list of solutions:
6.5 A Brief Discussion of the Results: Diagnosis and Prediction
It easy to see that a number of solutions
It may be an intriguing fact that exchanging a number of admissible degrees of
agent ability (4 for 5) only slightly exchanges the number of solutions – (2 for
3). In fact, we only get 2 solutions (pairs) for
Simoultaneously, if we admit 6 degrees instead of 5 and exchange 4 for 5 in each
place of the program for MAP
Further relaxation of temporal constrains usually changes a combinatorial
explosion of the algorithm relatively quickly. For example, if exchange also a
requirement
7 Conclusion
It has already been shown how FMAP may be put forward in two depictions. The first depiction enables an arithmetic representation of different constraints imposed on FMAP, whereas the second general depiction ellucidates a disctinction criterion between FMAP and the original MAP. Finally, some workable subsaces of FMAP were solved by means of SWI-PROLOG solvers.
Nevertheless, it appears that PROLOG only approximates a kind of fuzziness in the context of MAP. In fact, considering different degrees of agant ability introduces a kind of a multi-valency to different workable subcases of FMAP. In this perspective, Bousi-PROLOG could lay claim to the role of the more adequate language to grasp different FMAP.
