Fuzzy Continuous Resource Allocation Mechanisms in Workflow Management Systems

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1 Fuzzy Continuous Resource Allocation Mechanisms in Workflow Management Systems Joslaine Cristina Jeske, Stéphane Julia, Robert Valette Faculdade de Computação, Universidade Federal de Uberlândia, Campus Sta. Mônica, Av. João Naves de Ávila, 2160, P.O. Box 593, , Uberlândia-M.G.-Brazil LAAS-CNRS, 7 Avenue du Colonel-Roche, Toulouse Cedex France joslainejeske@hotmail.com stephane@facom.ufu.br robert@laas.fr Abstract In this paper, an approach based on a fuzzy hybrid Petri net model is proposed to solve the resource allocation problem of Workflow Management Systems. Initially, an ordinary Petri net model is used to show the main activities of the system the different routings of the Workflow Process. Hybrid resource allocation mechanisms are modeled by hybrid Petri net with discrete transitions where discrete resources represent equipment continuous resources represent employees availability. With the intent of expressing in a more realistic way the resource allocation mechanisms when human behavior is considered, fuzzy sets delimited by possibility distributions of the triangular form are associated with the marking of the places which represent human availability. New firing rules the fuzzy marking evolution of the new model are defined. Such a fuzzy resource allocation model is applied to an example of resource allocation mechanism of a Hle Complaint Process. I. INTRODUCTION The purpose of Workflow Management Systems is to execute Workflow Processes. Workflow Processes represent the sequence of activities that have to be executed within an organization to treat specific cases to reach a welldefined goal. Of all notations used for the modeling of Workflow Processes, Petri nets are very suitable to represent Workflow Processes [1] as they represent basic routings encountered in Workflow Processes. Moreover, Petri nets can be used for specifying the real time characteristics of Workflow Management Systems (in the time Petri net case) resource allocation mechanisms. As a matter of fact, late deliveries in an organization are generally due to the problem of resource overload. The model used at a Workflow Management System level should consider resource allocation mechanisms. In particular, time management of Workflow Processes is crucial for improving the efficiency of Business Processes within an organization. The dynamic behavior of a system imposes a scheduling of control flow. The scheduling problem [5] consists of organizing in time the sequence of activities considering time constraints (time intervals) constraints of shared resources utilization necessary for activity execution. From the traditional point of view of Software Engineering, the scheduling problem is similar to the activity of scenario execution. A scenario execution becomes a kind of simulation which shows the system s behavior in real time. In the real time system case, several scenarios (several cases in a Workflow Management System) can be executed simultaneously conflict situations which have to be solved in real time (without a backtrack mechanism) can occur if a same non-preemptive resource is called at the same time for the execution of activities which belong to different scenarios. One of the fundamental differences between a traditional production System [10] a Workflow Management System is the nature of the resources used to treat activities. In the Production System case, resources represent physical equipment are represented by simple tokens in places. They are discrete type resources. In the Workflow Management System case, resources can represent physical equipment as well as human employees. For example, it is possible to allocate a nurse in a hospital health care service to take care of several patients at the same time during her working day. In this case, the nurse could not be seen as a simple discrete token a model based on an ordinary Petri net would not be able to represent the real features which exist at a Workflow Management System level. In [6], an approach based on a p-time hybrid Petri net model was proposed to solve the scheduling problem of Workflow Management Systems. In particular, the proposed approach used a Hybrid Petri net [3] with discrete transitions to model hybrid resource allocation mechanisms. Discrete resources represented equipment continuous resources employees availability. The proposal of this work is to express in a more realistic way the resource allocation mechanisms when human behavior is considered. For that, Fuzzy sets delimited by possibility distributions [4] will be associated with the Petri net models that represent human type resource allocation mechanisms. II. WORKFLOW MODELING An activity of a Workflow Process can be represented by a specific place of an ordinary Petri net [3] with an input transition which shows the beginning of the activity an output transition which shows the end of the activity. For example, The Hle Complaint Process presented in [6] can be used to illustrate the basic routings generally encountered in Workflow Processes. In this process, first, an incoming complaint is recorded. Then, the client who has complained the department affected by the complaint are contacted. The client is approached for more information.

2 The department is informed of the complaint may be asked for its initial reaction. These two tasks may be performed in parallel. After this, the data is registered a decision is made. Depending upon the decision, either a compensation payment is made or a letter is sent. Finally, the complaint is filed. Fig. 1. Hle Complaint Process The corresponding ordinary Petri net model for the Hle Complaint Process is the one shown in figure 1. Activities are represented by places Ai (for i=1 to 8) waiting times between the sequential activities are represented by places Ej (for j=0 to 9). As a matter of fact, at the end of an activity, the next one can only be initiated if the necessary resource to execute the corresponding activity is immediately available, which is not necessarily the case. As the actual time taken by an activity in a Workflow Management System is non-deterministic not easily predictable, a time interval can be assigned to every workflow activity. As was shown in [7], explicit time constraints which exist in a Real Time System can be formally defined using a p-time Petri net model. In particular, the static definition of a p-time Petri net [8] is based on static intervals which represent the permanency duration (sojourn time) of tokens in places the dynamic evolution of a p-time Petri net depends on the time situation of the tokens (time intervals associated with the tokens). III. RESOURCE ALLOCATION MECHANISM Resources in Workflow Management Systems are nonpreemptive [1] ones : once a resource has been allocated to a specific activity, it cannot be released before ending the corresponding activity. As already mentioned, there exists different kinds of resources in Workflow Processes. Some of them are of discrete type can be represented by a simple token. For example, a printer used to treat a specific class of documents will be represented as a non-preemptive resource could be allocated to a single document at the same time. On the contrary, some other resources cannot be represented by a simple token. This is the case of most human type resources. As a matter of fact, it is not unusual for an employee who works in an administration to treat simultaneously several cases. For example, in an insurance company, one employee could treat normally several documents during a working day not necessarily in a pure sequential way. In this case, a simple token could not model human behavior in a proper manner. The different kinds of allocation mechanisms will be formalized in the following sections. A. Discrete resource allocation mechanism A discrete resource allocation mechanism can be defined by the marked ordinary Petri net model [3] C DR =< A DR, T DR, P re DR, P os DR, M DR > with: A DR = N DR α=1 A α {R D } where R D represents the discrete resource place, A α an activity place N DR the number of activities which are connected to the discrete resource place R D. T DR = N DR α=1 T in α N DR α=1 T out α where T inα represents the input transition of the activity A α T outα represents the output transition of the activity A α. P re DR : A DR T DR {0, 1} the input incidence application such as P re DR (R D, T inα ) = 1 P re DR (A α, T outα ) = 1 (other combinations of P os DR : A DR T DR {0, 1} the output incidence application such as P os DR (R D, T outα ) = 1 P os DR (A α, T inα ) = 1 (other combinations of M DR : R D N + the initial marking application such as M DR (R D ) = m D the number of discrete resources of the same type. Fig. 2. Discrete Allocation Resource If one assumes that in the Hle Complaint Process an employee of the Complaint Department is used to treat

3 the activities Contact-Client, Contact-Department Send-Letter, the example of discrete resource allocation mechanism given in figure 2 is then obtained. In this figure, it is clear that if the token in R D is used to realize the activity A 2, then the activities A 3 A 7 could only be initiated after the end of activity A 2. This means that resource R D could only be used on a pure non-preemptive way. In particular, once the activity A 2 initiated, if the employee cannot enter in contact immediately with the client, he could not use his available time (waiting for an answer from the client) to initiate another activity, like sending a letter for example (activity A 7 ). It is evident that in practice, such a situation will not happen. If the client is not available at a given instant, the employee will use his available time to execute another task. B. Continuous resource allocation mechanism A continuous allocation mechanism can be defined by the marked hybrid Petri net model [3] C CR =< A CR, T CR, P re CR, P os CR, M CR > with: A CR = N CR α=1 A α {R C } where R C represents the continuous resource place, A α an activity place N CR the number of activities which are connected to the continuous resource place R C. T CR = N CR α=1 T in α N CR α=1 T out α where T inα represents the discrete input transition of the activity A α T outα represents the discrete output transition of the activity A α. P re CR : A CR T CR R + the input incidence application such as P re CR (R C, T inα ) = X α with X α R + P re CR (A α, T outα ) = 1 (other combinations of P os CR : A CR T CR R + the output incidence application such as P os CR (R C, T outα ) = X α P os CR (A α, T inα ) = 1 (other combinations of M CR : R C R + the initial marking application such as M CR (R C ) = m C the availability (in percentage) of the continuous resource. available for answering the questions of the employee, this employee could use his available time (waiting for an answer from the client) to initiate another activity, like sending a letter for example (activity A 7 ). As a matter of fact, 50% of the employee availability is necessary for the activity Send- Letter, after the beginning of the activity Contact- Client, the employee is still 70% available. The limitation of such a model is related with the fact that the representation of human behavior in term of availability in a practical situation will be known only as an uncertain value (a fuzzy percentage). C. Fuzzy continuous resource allocation mechanism 1) Fuzzy sets possibility measures: The notion of fuzzy set has been introduced by Zadeh [12] in order to represent the gradual nature of human knowledge. For example, the size of a man could be considered by the majority of a population as small, normal, tall, etc. A certain degree of belief can be attached to each possible interpretation of a symbolic information can simply be formalized by a fuzzy set F of a reference set X that can be defined by a membership function µ F (x) [0, 1]. In particular, for a given element x X, µ F (x) = 0 denotes that x is not a member of the set F, µ F (x) = 1 denotes that x is definitely a member of the set F, intermediate values denotes the fact that x is more or less an element of F. Normally, a fuzzy set is represented by a trapezoid A = [a1, a2, a3, a4] as that represented in figure 4 where the smallest subset corresponding to the membership value equal to 1 is called the core, the largest subset corresponding to the membership value greater than 0 is called the support. Fig. 4. Representation of a fuzzy set Fig. 3. Continuous Allocation Resource An example of continuous resource is given in figure 3. This figure shows that only 30% of the employee availability is necessary to realize the activity Contact-Client. It will be then possible for the employee to treat simultaneously more than one activity. For example, even after the beginning of the activity Contact-Client, if the client is not immediately There exist three particular cases of fuzzy sets that are generally considered: the triangular form where a2=a3, the imprecise case where a1=a2 a3=a4, the precise case where a1=a2=a3=a4. When considering two distinct fuzzy sets A B, the basic operations are the following ones [9]: the fuzzy sum A B defined as : [a1, a2, a3, a4] [b1, b2, b3, b4] = [a1 + b1, a2 + b2, a3 + b3, a4 + b4], the fuzzy substraction A B defined as : [a1, a2, a3, a4] [b1, b2, b3, b4] = [a1 b4, a2 b3, a3 b2, a4 b1], the fuzzy product A B defined as :

4 [a1, a2, a3, a4] [b1, b2, b3, b4] = [a1.b1, a2.b2, a3.b3, a4.b4]. A fuzzy set F can be used to define a possibility distribution [4], [2] Π f, such as : x X, Π f (x) = µ F (x). Given a possibility distribution Π a (x), the measures of possibility Π(S) necessity N(S) that data a belongs to a crisp set S of X are defined by Π(S) = sup x S Π a (x) N(S) = inf x S (1 Π a (x)) = 1 Π(S). If Π(S) = 0, it is impossible that a belongs to S. If Π(S) = 1, it is possible that a belongs to S, but it also depends on the value of N(S). If N(S) = 1, it is certain (the larger the value of N(S), the greater the credibility given to it). In particular, there exists a duality relationship between the modalities of the possible the necessary which postulates that an event is necessary when its contrary is impossible. Some practical examples of possibility necessity measures are presented in [4]. Fig. 5. Possibility Measure Given two pieces of data a b caracterized by two fuzzy sets A B as shown in figure 5, the measure of possibility necessity of having a b are defined as: Π(a b) = sup x y (min(π a (x), min(π b (y))) = max([a, + [ ], B]) N(a b) = 1 sup x y (min(π a (x), min(π b (y))). 2) Static dynamic definition: A fuzzy continuous allocation mechanism can be defined by the marked fuzzy hybrid Petri net model C F CR =< A F CR, T F CR, P re F CR, P os F CR, M F CR > with: A F CR = N F CR α=1 A α {R F C } where R F C represents the fuzzy continuous resource place, A α an activity place N F CR the number of activities which are connected to the fuzzy continuous resource place R F C. T F CR = N F CR α=1 T inα N F CR α=1 T outα where T inα represents the discrete input transition of the activity A α T outα represents the discrete output transition of the activity A α. P re F CR : A F CR T F CR F the input incidence application such as P re F CR (R F C, T inα ) = [w1, w2, w3, w4] with w2=w3 P re F CR (A α, T outα ) = [1, 1, 1, 1] (other combinations of place/transition are equal to zero) with F the set of fuzzy numbers of the triangular form. P os F CR : A F CR T F CR F the output incidence application such as P os F CR (R F C, T outα ) = [w1, w2, w3, w4] with w2=w3 P os F CR (A α, T inα ) = [1, 1, 1, 1] (other combinations of M F CR : R F C F the initial marking application such as M F CR (R F C ) = [m1, m2, m3, m4] the fuzzy availability (in percentage) of the fuzzy continuous resource. Fig. 6. Fuzzy Continuous Resource An example of fuzzy continuous resource is given in figure 6. For example, this figure shows that 30% ± 10% of the resource availability R F C is necessary to realize the activity A2 (Contact-Client). The behavior of a fuzzy continuous resource allocation model can be defined through the concepts of enabled transition fundamental equation. In an ordinary Petri net, a transition t is enabled if only if for all the input places p of the transition, M(p) P re(p, t), which means that the number of tokens in each input place is greater or equal to the weight associated with the arcs which connect the input places to the transition t. With a fuzzy continuous resource allocation mechanism, considering a transition t, the marking of an input place p the weight associated with the arc which connects this place to the transition t are defined through different fuzzy sets. In this case, a transition t is enabled if only if (for all the input places of the transition t): Π t = Π(P re F CR (p, t) M F CR (p)) > 0 For example, the transition t3 in figure 6 is enabled because Π t3 = Π(P re F CR (R F C, t3) M F CR (R F C )) = 1 > 0 as shown in figure 7 (a = P re F CR (R F C, t3) b = M F CR (R F C )). Fig. 7. Possibility Measure of t3 For an ordinary Petri net, once a transition is enabled by a marking M, it can be fired a new marking M is obtained according to the fundamental equation : M (p) = M(p) P re(p, t) + P os(p, t) With a fuzzy continuous resource allocation model, the marking evolution is defined through the following fundamental equation:

5 M F CR (p) = M F CR(p) P re F CR (p, t) P os F CR (p, t) The operation corresponds to the fuzzy substraction. The operation, when considering the sum of two fuzzy sets, is different from the one given in fuzzy logic is defined as: [a1, a2, a3, a4] [b1, b2, b3, b4] = [a1 + b4, a2 + b3, a3 + b2, a4 + b1] This difference is due to the fact that the fuzzy operation does not maintain the marking of the fuzzy continuous resource allocation model invariant (the p-invariant property of the Petri net theory [11]). As a matter of fact, after realizing different activities, the resource s availability must go back to 100 %, even in the fuzzy case. To a certain extent, from the point of view of the fuzzy continuous resource allocation mechanism, the operation can be seen as a kind of defuzzyfication operation. In particular, using this operation, it will be possible to find a linear expression of the fuzzy marking which will always be constant which will correspond to the following expression: M F CR (R F C )) (w 1 M F CR (A1)) (w 2 M F CR (A2)) (w NF CR M F CR (A NF CR )) = CONST with w α = P re F CR (R F C, t inα ) = P os F CR (R F C, t outα ) for α = 1 to N F CR. 3) Example: To illustrate the fuzzy concepts of enabled transition, transition firing invariant marking, the firing sequence t 3 t 4 t 14 t 5 t 6 t 16 will be considered when considering the fuzzy resource allocation mechanism in figure 6. Firing of t 3 : the possibility measure of t 3 is : Π t3 = Π(P re F CR (R F C, t3) M F CR (R F C )) = 1 > 0 as shown in figure 7 (with a = P re F CR (R F C, t3) b = M F CR (R F C )). After the firing of t 3, the new markings of R F C A2 are: M F CR (R F C) = M F CR (R F C ) P re F CR (R F C, t3) = [20, 30, 30, 40] = [60, 70, 70, 80] M F CR (A2) = M F CR(A2) P os F CR (A2, t3) = [0, 0, 0, 0] [1, 1, 1, 1] = [1, 1, 1, 1] [60, 70, 70, 80] ([20, 30, 30, 40] [1, 1, 1, 1]) ([30, 40, 40, 50] [0, 0, 0, 0]) ([40, 50, 50, 60] [0, 0, 0, 0]) = [60, 70, 70, 80] [20, 30, 30, 40] = Firing of t 4 : the possibility measure of t 4 is : Π t4 = Π(P re F CR (R F C, t4) M F CR (R F C )) = 1 > 0 as shown in figure 8 (with a = P re F CR (R F C, t4) b = M F CR (R F C )). Fig. 8. Possibility Measure of t4 After the firing of t 4, the new markings of R F C A3 are: M F CR (R F C) = M F CR (R F C ) P re F CR (R F C, t4) = [60, 70, 70, 80] [30, 40, 40, 50] = [10, 30, 30, 50] M F CR (A3) = M F CR(A3) P os F CR (A3, t4) = [0, 0, 0, 0] [1, 1, 1, 1] = [1, 1, 1, 1] [10, 30, 30, 50] ([20, 30, 30, 40] [1, 1, 1, 1]) ([30, 40, 40, 50] [1, 1, 1, 1]) ([40, 50, 50, 60] [0, 0, 0, 0]) = [10, 30, 30, 50] [20, 30, 30, 40] [30, 40, 40, 50] = [50, 60, 60, 70] [30, 40, 40, 50] = Firing of t 14 : the possibility measure of t 14 is : Π t4 = Π(P re F CR (R F C, t14) M F CR (R F C )) = 0, 33 > 0 as shown in figure 9 (with a = P re F CR (R F C, t14) b = M F CR (R F C )). Fig. 9. Possibility Measure of t14 After the firing of t 14, the new markings of R F C A3 are: M F CR (R F C) = M F CR (R F C ) P re F CR (R F C, t14) = [10, 30, 30, 50] [40, 50, 50, 60] = [ 50, 20, 20, 10] M F CR (A7) = M F CR(A7) P os F CR (A7, t14) = [0, 0, 0, 0] [1, 1, 1, 1] = [1, 1, 1, 1] [ 50, 20, 20, 10] ([20, 30, 30, 40] [1, 1, 1, 1]) ([30, 40, 40, 50] [1, 1, 1, 1]) ([40, 50, 50, 60] [1, 1, 1, 1]) = [ 50, 20, 20, 10]

6 [20, 30, 30, 40] [30, 40, 40, 50] ([40, 50, 50, 60] = [ 10, 10, 10, 30] [30, 40, 40, 50] ([40, 50, 50, 60] = [40, 50, 50, 60] ([40, 50, 50, 60] = Firing of t 5 : the possibility measure of t 5 is : Π t5 = Π(P re F CR (A2, t5) M F CR (A2)) = 1 > 0 After the firing of t 5, the new markings of R F C A2 are: M F CR (R F C) = M F CR (R F C ) P os F CR (R F C, t5) = [ 50, 20, 20, 10] [20, 30, 30, 40] = [ 10, 10, 10, 30] M F CR (A2) = M F CR(A2) P re F CR (A2, t5) = [1, 1, 1, 1] [1, 1, 1, 1] = [0, 0, 0, 0] [ 10, 10, 10, 30] ([20, 30, 30, 40] [0, 0, 0, 0]) ([30, 40, 40, 50] [1, 1, 1, 1]) ([40, 50, 50, 60] [1, 1, 1, 1]) = [ 10, 10, 10, 30] [30, 40, 40, 50] ([40, 50, 50, 60] = [40, 50, 50, 60] ([40, 50, 50, 60] = Firing of t 6 : the possibility measure of t 6 is : Π t6 = Π(P re F CR (A3, t6) M F CR (A3)) = 1 > 0 After the firing of t 6, the new markings of R F C A3 are: M F CR (R F C) = M F CR (R F C ) P os F CR (R F C, t6) = [ 10, 10, 10, 30] [30, 40, 40, 50] = [40, 50, 50, 60] M F CR (A3) = M F CR(A3) P re F CR (A3, t6) = [1, 1, 1, 1] [1, 1, 1, 1] = [0, 0, 0, 0] [40, 50, 50, 60] ([20, 30, 30, 40] [0, 0, 0, 0]) ([30, 40, 40, 50] [0, 0, 0, 0]) ([40, 50, 50, 60] [1, 1, 1, 1]) = [40, 50, 50, 60] ([40, 50, 50, 60] = Firing of t 16 : the possibility measure of t 16 is : Π t6 = Π(P re F CR (A7, t16) M F CR (A7)) = 1 > 0 After the firing of t 16, the new marking of R F C A7 are: M F CR (R F C) = M F CR (R F C ) P os F CR (R F C, t16) = [40, 50, 50, 60] [40, 50, 50, 60] = M F CR (A7) = M F CR(A7) P re F CR (A7, t16) = [1, 1, 1, 1] [1, 1, 1, 1] = [0, 0, 0, 0] ([20, 30, 30, 40] [0, 0, 0, 0]) ([30, 40, 40, 50] [0, 0, 0, 0]) ([40, 50, 50, 60] [0, 0, 0, 0]) = The negative part of the fuzzy marking of R F C which appears after the firing of t14 simply shows the possibility of overloading the resource (the employee works above his normal capacity). It is important to underline that the negative part of the marking is not inconsistent with the Petri net theory. As a matter of fact, only the positive part of the fuzzy marking can be used to enable a transition of the fuzzy continuous resource model. IV. CONCLUSION In this article, a new fuzzy continuous resource allocation model was presented in order to express human behavior in a more realistic. The corresponding enabled transition definition fundamental equation were defined. A linear expression of the fuzzy marking which remains constant was proposed in order to guarantee that the fuzzy continuous resource allocation model is consistent with the general Petri net theory. Such a model was applied to a Hle Complaint Process when human type resources are considered. As a future work proposal, it will be interesting to show that fuzzy continuous resource allocation mechanisms allow a much more realistic simulation of Workflow Management Systems which use human type resources. In particular, in the extremely urgent cases, it will be possible, using such a model, to overload human type resources as generally happens in real organizations. REFERENCES [1] Aalst, W.v.d., Hee, K.v. (2002). Workflow Management: Models, Methods, Systems. The MIT Press Cambridge, Massachusetts. London, Engl. [2] Cardoso, J., Valette, R., Dubois, D. (1999). Possibilistic Petri Nets. IEEE Trans. on Systems, Man, Cybernetics - Part B. Vol. 29, No. 5. P [3] David, R., Alla, H. (2004). Discrete, Continuous, Hybrid Petri Nets. Springer [4] Dubois, D., Prade, (1988). Possibility theory. Springer. [5] Esquirol, P., Huguet, M.J., Lopez, P. (1995). Modelling managing disjunctions in scheduling problems. Journal of Intelligent Manufacturing 6. pp [6] Julia, S., Francielle, F. (2004). A p-time hybrid Petri net model for the scheduling problem of Workflow Management Systems. IEEE International Conference on Systems, Man Cybernetics. The Hague. [7] Julia, S., Valette, R. (2000). Real Time Scheduling of Batch Systems. Simulation Practice Theory. Elsevier Science. pp [8] Khansa, W., Aygaline, P., Denat, J. P. (1996). Structural analysis of p-time Petri Nets. Symposium on discrete events manufacturing systems. CESA 96 IMACS Multiconference. Lille, France. [9] Klir, G.J., Yuan, B. (1995). Fuzzy Sets Fuzzy Logic - Theory Applications. Imp Upper Saddle River: Prentice Hal. [10] Lee, D.Y., DiCesare, F. (1994). Scheduling flexible manufacturing systems using Petri nets heuristic search. IEEE Trans. on Robotics Automation 10 (3). P [11] Murata, T. (1989). Petri Nets: Properties, analysis applications. Proceedings of the IEEE 77(4). p [12] Zadeh, L.A. (1965). Fuzzy sets. Inform. Contr., vol. 8. pp