Criteria for the determination of a basic Clark s flow time distribution function in network planning

Transcription

1 International Journal of Engineering & Technology IJET-IJENS Vol:4 No:0 60 Criteria for the determination of a basic Clark s flo time distribution function in netork planning Dusko Letić, Branko Davidović, Nikola Gogić Abstract In the ork are presented the results of the theoretical - experimental researches of the quantification for the superposed flo time of the to local - autonomous flos in the net on the basic Clark s equations. The computer-based solving of this basic variant of the general flo model through the net is performed through methods of numerical simulation Monte Carlo. The numerical experiment is realized using the program tool Mathcad Professional. Index Term Mathematical model, Clark s equations, Simulation Monte Carlo I. INTRODUCTION The basic model for hich Clark s equation of the resulting activity (time flo is defined, consists of an oriented graph, here the to activities occur in parallel, have a common beginning ( i and flo to one terminal occurrence ( j of netork planning in project management. ith to parallel flos of activity (Fig. has a key role in netork planning of manufacturing (and business flos. Both methods, analytical (Clark s and numerical (Monte Carlo, are poerful enough in order to study and solve the other processes established on graphic models, as are flos of activities, resources, traffic, energy, fluids or phenomena like queuing, reliability of technical systems, biological processes and the like. These problems are, as e kno from experience 3, 4, of stochastic nature and their solution through analytical methods, ithout approximation, is often unfeasible. In this ay, motivated by the researches of Fishman [3], Slyke [5], Vosea [6], Heizer, J. H. and Render, B. [7], Hillier, F. S. and Hillier M. S [8], Krajeski, L. J. and Ritzman L. P [9], Dodin [0], Bezak, S. and Nahod, M. [], Buehler, R. Griffin, D. and Ross, M. [], Haga, W and O keefe, T. [3], Malvin H. K. and Whitlock, A. P. [4], Plaskota, L. and Woźniakoski, H. [5] and others, e ere able to define and solve a netork model of the activity flos, and e hope that e ill contribute to the development of the algorithm for solving general methods for critical periodical flos of material, established on ordinal-parallel structures of an oriented graph [6]. Here a normal distribution of the ends of some activities is supposed, having a characteristic of mean value and an adequate deviation of their realization time. Fig.. The flo netork ith to local - autonomous critical flos of activity The activities in the netork are generally of stochastic character. Parallel activities can be independent from each other (locally-autonomous or dependent (measured e.g. by correlation. In this case, Clark s equations have been developed for the resulting flo. The results of the equations in this paper are being applied and compared to the results of the Monte Carlo numerical simulation 3. The basic model Dusko Letic, University of Novi Sad, Technical Faculty M. Pupin, Zrenjanin, Serbia ( dletic@open.telekom.rs. Branko Davidovic is ith Technical High School, Kragujevac, Serbia ( itbg@beotel.net. Corresponding author. Gogić Nikola,is ith University of Beograd, Faculty of Transport and Traffic Engineering, Srbija ( gogicnikola@yahoo.com. II. FLOWS WITH CRITICAL ACTIVITIES The uniform solution of a critical flo of activities, and at the same time of the resulting flo rate (time, using the expected times of single activities during production, is one of the most problematic effects of netork planning application based on stochastic netork techniques. Netorks of stochastic (but also deterministic activities, formed e.g. on the basis of an ADM (Arro Diagram Method structure (Fig., can, in some planning cases, be very complex. There are examples of netork diagrams in mechanical engineering, here the number of activities and occurrences amounts to almost several thousands 7, ith several hundred identified critical and subcritical resulting activities (flos. Common for these parallel flos are: the initial and final occurrence, as ell as the same or substantially different time intervals of their realization. In that case, the competent occurrence k is the moment hen both activities that "flo into it" are realized. Generally, e can, ith good reason, raise the folloing question: ho high is the certainty or risk (as ell as the distribution of probability that the resulting flo rate time of all T activities ill end in a certain planned time

2 International Journal of Engineering & Technology IJET-IJENS Vol:4 No:0 6 period Tp, considering that such an activity graph can be composed of one, to or a greater number of parallel and ordinal flos. For the correct anser to the previous question, e need to exactly define the criteria and algorithm for the quantification of influence, primarily of critical and subcritical flos on the forming of the resulting, i.e. superposing flo rate time of activities in material flos of production. III. OBJECTIVE OF THIS PAPER The essential purpose of the paper is the quantification of influence of to or more critical flos on the forming of the resulting flo time. The complementary purpose refers to the development of criteria for determining the equivalence ( of such parallel flos. By solving them, e can create a basis for the determination of the probability distribution function, and for noticing the relativities of these flos through the application of methods, based on Clark s equations and numerical simulation. IV. THE BASIC TIME PARAMETERS FOR AUTONOMOUS CRITICAL FLOWS As the researches sho 7, 7 a superposition of the intervals of critical and subcritical flo rate times and their deviations and reduction to an equivalent flo, can be performed: Through analytical methods: ith Clark s equations for solving parallel flos, based on the central limit theorem, for solving ordinal flos and Through numerical methods: ith Monte-Carlo simulation (for parallel and ordinal flos. Based on fuzzy modeling. In order to illustrate the application of the basic specified algorithms, e shall use the basic ADM netork and (Fig.. A. The superposed time and the flo variance In algorithm structuring for analytical solving of this critical flo variant, e start from Clark s authentic equations. With these equations the flo ( parameters are being solved as follos: the superposed flo time T, and its variants ( T, (or. For a basic oriented graph ith to parallel flos, from the initial i to the terminal j event (Fig., the flo time values T,, are []: B. The mean superposed flo time in the form of the folloing Clark s equation: T T T ( ( (4. (, Where: ( ( / exp( t dt - Laplace integral, / ( ( exp( t - function of the centered normal ( T distribution and T (, that is ( T T - the parameter, and variables, of Clark s functions. In addition to it, there are usually taken the predicted or mean values of time intervals: T, T and standard deviations ( T, ( T and according to it follos that [, 3]: The mean superposed flo time is: ( ( (4. (, The superposed dispersion is illustrated by Clark s second equation: ( (, ( ( (4.3 With these equations e can describe the characteristics of one equivalent flo instead of the previous to flos (Fig.. Fig.. The parallel flos and superposed or the equivalent flo of activity C. The groth of the superposed flo time in relation to the critical flo On the basic of the ne superposed function of the time distribution T, ith the characteristics N ~ [,,, ] one can quantify the time groth T, in relation to the single time T ( T or T ( T, dependent on hich of them has a critical feature. For the elementary netork ith autonomous flos and, that groth or the superposed extract, after a more calculation is equal to: ( ( (4.4 (, Meanhile, in the case of the reversed choice, it follos: ( ( (4.5,, (,, In addition, these values are by their nature alays negative, i.e.: 0 and 0 (4.6,

3 International Journal of Engineering & Technology IJET-IJENS Vol:4 No:0 6 D. Testing the invariance of parallel flos Testing the invariances of both cases (4.6, e ought to prove hether or not these values have still remained unchanged and hether they have been unambiguously quantified hen their order of flos has been changed. It is knon that ith only the to flos, and, ith the to pairs of parameters (, and (,, e already can form nine combinations of resulting flos. Namely, analyzing the folloing relations beteen the expected times and deviations of some flos, as: N [ 0, ], respectively N [ 0, ]. The final simulation results for the to flo rates, in the form of the simulated mean resulting activity T m, and the suitable standard deviation s,, so the superposed distribution of this time ith the characteristics. N m, s ] [,, has been generated. These parameters values are presented in days. and, here the folloing relations have been taken into consideration as (4.7 Since in the theory of probability the folloing variants are knon, and they can be as ell easily proved,,, ( (,, ( and (, (4.8 e get important equivalent values of the basic tested parameters, hich are related to the superposed flo ith the to parallel activities:, ;, (, and T ( T (4.9 There have been formed altogether 3 9 different combinations. Recognizing that: Table I We can conclude that it is irrelevant hich flo, of the to observed ones, ill be found as critical, and hich one be considered as subcritical. This quality of model invariance, 9 is very substantial in developing an equivalence criterion, in case of a greater number of flo rates (. V. THE APPLICATION OF SIMULATIONS METHODS E. The Monte Carlo methods for setting models ith parallel flos On assumption that the elementary activities of the flo rate time have a normal distribution ith the parameters N [, ], (, the simulation algorithm of the resulting time T is simple. For the to flos, in the Fig. 3 presented are the results of numerical simulation of n 0 replica for the chosen characteristics of normal distribution: 5 Fig. 3. The distributions of probability for critical, subcritical and superposed flo time Also obtained are: theoretical values, by Clark s equations: N 0,8906;,305459] [, simulation values (Monte-Carlo: N m 0,8989; s,30344] and [, differences beteen theoretical and simulation values are comparatively small and equal to: 4 8,730 and 3 4,50 Meanhile, as this algorithm is easily defined using a computer, the main point of the problem as no ithin the domain of simulation. Namely, here, in one session of 5 simulation of n 0 replications the testing of only one chosen variant as done, hen it is and of the possible nine ones. F. The application of the Monte Carlo - frame methods in solving the Clark s flo model Extending the domain of the Monte Carlo method and visualizing its results (e.g. through frames can be carried out by changing the values (, and (,, hich creates a more suitable situation for understanding the more extensive class of appearances, than it as in the classical static vie of the simulation process. In that respect, the supposition (5. can be resolved e.g., ith the three variants in one simulation session: and (5.

4 International Journal of Engineering & Technology IJET-IJENS Vol:4 No:0 63 The number of frames depends on the complexity of a problem, i.e. the process that is being studied, and therefore this integrated method Monte Carlo - animation (through frames plays a significant role, and it can be partially presented by the series of selected frames (Fig. 4-8 for illustrating the distribution of the superposed extract function.. Fig. 8. The frame for [ 6, ] and 0, ] [ VI. CRITERIA FOR ESTABILISHING THE EQUIVALENCE OF PARALLEL FLOW RATES Fig. 4. The frame for [ 9, ] and 0, ] [ When using netork (submodels ith parallel and independent flos for, e have to develop criteria that ill be applied hen the superposition problem has been solved analytically. Their definition is valid for the folloing cases: Case : The equivalence condition for the to parallel flos can be expressed on the basis of the to parameters and three set up relations (Fig.. Fig. 5. The frame for [ 0, ] and 0, ] [ ( ( ( (,, (6. This criterion is established on the relations (4.7 and (4.8 on invariance of the to flos, and. Case 3: None of the three parallel flos, and 3 is equivalent to another in the folloing cases (Fig. 9: Fig. 6. The frame for [, ] and 0, ] [ [( 3 ( 3] [( 3 ( 3] ( (,3,3,3,3 (6. Fig. 7. The frame for [ 4, ] and 0, ] [ Fig. 9. The sub-netork ith three parallel flos of similar characteristics

5 International Journal of Engineering & Technology IJET-IJENS Vol:4 No:0 64 The table of relational operators for three parallel flos ( 3 ith all combinations of relations beteen the expected values (,3 and the appropriate standard deviations is in Table. The number of these combinations ith, as it has been stated, amounts 9. With the three flos it rises to 8. Generally, the number of combinations ( u, is of exponential nature, and it is equal to: u 3, N (6.3 Table II Case 3: The equivalency of - parallel flos has been realized only in the case hen the folloing conditions (Fig. 0 have been achieved :,, ( (,,,, (6.4 In that sense (6.4 these flos can be regarded as an equivalent flo ith the characteristic Fig.. Subnet ith an equivalent course According to the previous criteria, the equivalence condition for the three flos (Fig. is fulfilled in only one case, shon ith a bolded frame in Table. This superposition model of parallel flos can be, in further analysis, integrated ith the model of serially connected flos, because the character of material flos and the other activities e.g. in machine-building industry is combined in just that ay. The other more complex variances of the resulting flo times have to be solved by simulation [7]. Directions of further research should include: a Solving a more complex Clarks model, hen to parallel activities are in correlation. b Developing an analytical Clark model ith three or more parallel flos converting into an equivalent superposed flo.. VII. CONCLUSION The most significant advantage of the Monte-Carlo method of simulation for activities and occurrences through a netork of activity, presents the possibility for the modeling of the probability distribution function of the basic netork model superposed flo time shon in Fig.. Meanhile, the advantage of the Clark equation - Monte Carlo simulation method has been significantly increased due to the possibility of dynamic flo rate modeling through the netork. The frames give a ne dimension and provide a more reliable basis for further acquirement and extending of knoledge in this field, especially in respect of the critical flo rate relativity. In addition to it, e can clearly notice, from frame to frame (Fig. 4 do 8, the losses of the critical flo rate domination in favor of the subcritical one, if the mean value of the other one is being increased in relation to the first flo rate. Of course, it is possible to also test the other cases by analytical and /or numerical methods, e.g. hen the vector of the folloing relations beteen expected values and the adequate standard deviations is established in the folloing ay: and for, (7. Fig. 0. The sub-netork ith - parallel flos These influences (Fig. 0 and can be characteristically noticed through simulation ith more complex ADM netorks 6, [8]. When e calculate more complex flo rates (Fig., appreciating the developed criteria, e get interesting values. The consequences of lack of knoledge of the essence of the obtained results can be very negative, especially in the cases of planning and control of the complex stochastic activity flos through the netork. Clark s equations are not developed for three and more parallel flos.

6 International Journal of Engineering & Technology IJET-IJENS Vol:4 No:0 65 If they ere developed, they ould give very complex results. In the meantime, the existing equations for to parallel flos can be generalized for resolving as ell as for more complex flo rate cases ( 3, strictly respecting the developed criteria (6., (6. and (6.4. In that case, they are used as recurrent equations, e.g. ith the form given for the last th iteration th iteration: the superposition of the flo and into the,. The superposed free (flo rate time T T, ( T, (,, (, T,, is equal to: (7. The superposed dispersion is in this case is equal to:, ( T, ( T ( (,, T ( T (,, P (7.3 Obviously that ith the increase of the number of flos the number of combinations: expected time / deviation exponentially rises also. So, for, 3, 4, 5 and 6, this series of numbers yields u 9, 7, 8, 43, 79 and 87 etc. Let us emphasize as ell Slake s results obtained by Monte Carlo simulation (Table 3 and the ne ones [7] obtained on the basis of Clark s equations (Table 4. In the second case, the same models of parallel flos are solved more precisely by the algorithms of the method hich is structured on the iterative basis. After all procedures ere performed, the verification of the bye-results by the program packet Mathcad [8] as done. While at it, ten program models ere being used. Fig.. The results of Slakes s model received by the simulation (Monte Carlo and analytical (Clark method for the same flo times v 0 and variances v ( v, 0 It should be emphasized that this calculation did not solve the complex model of the critical flos, but only the parameters of the netork model ith ten parallel flos of the same criticality index, because 0 and 0 0. In complex models, because of the existence of ordinal flos, the mathematical and simulation models have to be complemented ith the results of the central limit theorem [3]. Application is possible in project management, a discipline hich is on the rise. It is more and more burdened ith stochastically based management solving. Clark's equations can provide significant contribution to the phenomenon of superposition, hich has not been sufficiently explained neither in the theoretical nor in the practical domain. It as noticed that the phenomenon of superposition appears in the great majority of continuous and discrete ordinal - parallel resource flos, especially of the time - dependent activities. REFERENCES [] Lock, D.; Project Management, (9 ed. Goer Publishing, Ltd. ISBN , (007. [] Clark, E. C.; The Greatest of Finite Set of Random Variables, // Operation Research, Vol.,, No. 9, (96, p: 45-6 [3] Fishman, S. 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