ScienceDirect. Using Search Algorithms for Modeling Economic Processes

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1 vailable online at ScienceDirect Procedia Economics and Finance 6 ( 03 ) International Economic onerence o Sibiu 03 Post risis Economy: hallenges and Opportunities, IES 03 Using Search lgorithms or Modeling Economic Processes Marian Pompiliu ristescu a,*, Eduard lexandru Stoica a, b a Lucian Blaga University o Sibiu, Faculty o Econom, Sibiu 55034, Romania b The Bucharest University o Economic Studies, Faculty o ybernetics, Statistics and Economic Inormatics, 5- bstract Economic issues are placed in ormal practice, when is desired a modelling o the economic process, a manuacturing process, a device, etc.. Each share o that economic process is denoted by a, b, c, d, these actions with deined time periods and action pairs are ormed strings o the orm, ab * cab * bc..., ab, bb, bc. so or them there are no other restrictions. I the graph is viewed as a system image, nodes representing components, then an immediate interpretation o an arc (xi, xj) are the component xi that is said to directly inluence component xj. I nodes have the signiicance o possible states o a system when a spring (xi.xj) means that, the system can jump rom state xi in xj state. In this paper, we present an algorithm that aims to ind all roads in a directed graph with a inite number o nodes. 03 The uthors. Published by Elsevier B.V. 03 The uthors. Published by Elsevier B.V. Open access under BY-N-ND license. Selection Selection and and peer-review peer-review under under responsibility responsibility o o Faculty Faculty o o Economic Economic Sciences, Sciences, Lucian Lucian Blaga Blaga University University o Sibiu. o Sibiu. Keywords: lgorithm, Graph, Model, Economic Process.. Introduction production process can be simulated or modeled eectively using linear bounded automated. But i every time stock elements is bounded, then a inite state automaton can simulate the complexity o machine depends essentially on the size o graphs that describe technology products recipes. The unit o time is greater, the more the grammars associated with a system are simple, and the system is easier modeling. * orresponding author addresses: marian.cristescu@ulbsibiu.ro (M.P. ristescu), eduard.stoica@ulbsibiu.ro (E.. Stoica), laurentiu.ciovica@gmail.com (L. ) The uthors. Published by Elsevier B.V. Open access under BY-N-ND license. Selection and peer-review under responsibility o Faculty o Economic Sciences, Lucian Blaga University o Sibiu. doi:0.06/s-567(3)0096-

2 Marian Pompiliu ristescu et al. / Procedia Economics and Finance 6 ( 03 ) Operation o many devices due to continuous processes and signal deviations rom normality can be simulated using sequential transducers. Through the practical consequences o these results can be listed: the ordering, production planning and programming, ormal demonstration o the need or top-down design and implementation o inormation systems or manage the economic systems, necessity hierarchical management o socio-economic systems. mathematical model known as the graph can be successully used in investment, organization o production, activities o economic analysis, transport etc. Graph is a igure composed o points connected by arrows. Points symbolize dierent elements depending on the modeled phenomenon and arrows represent the established links between elements.. Methods or speciying economic systems using ormal elements onsidering vocabulary V as a set o labels. To view the original state machine on this multi-graph, an arrow is drawn rom the outside toward the node associated with the initial state and the inal states are encircled with two lines. Even i the automaton is deterministic, the multi-graph () can actually be a multi-graph and not a marked graph. This thing can be explained as: Let L = {a, b}{c*}{a, b} be a regular language recognized by the given deterministic inite automaton: ( K, V,, s0, F), where K s, s, s }, V { a, b}, F { }, () { 0 s s 0, a) ( s0, b), ( s, c) s, ( 0, a) ( s, b) s ( s the associated multi-graph is: ( 0 0 s s, () ) ( K,{( s, a, s ),( s, b, s ),( s, c, s ),( s, a, s ),( s, b, )}). (3) The () multi-graph associated with a inite automaton ( K, V,, s0, F) is a graph marked only i or any: s, s K, a, a V, i s ( s, a) and s ( s, a), then a a. Thus, we conclude that: the language o () is ininite i and only i the () multi-graph contains circuits. ny path to the multi-graph with the initial node in s 0 point corresponds to a range rom the T ( (, s0)) language and reverse. In order to ind all paths that unite a node s i with a node s j, will take s i as initial state and s j, as inal state (F = s j), then the language T( (, s, s)) will indicate exactly the set o all paths in the orm o rows o labels. I ( X, U) is a graph, then it can be considered U and ( s i, s j ) which implies marking each arc with itsel, in which case the previous automaton will recognize the paths written in the orm o arcs. nother way to address these problems and namely, to obtain the paths rom a graph using the grammar is by using derivation trees. Let G = (V N, V T, S,P) be a context-ree grammar, so that to every D derivation in grammar G, a derivation tree is associated, thereby:. is marked with the S the tree root;. i a tree node is marked with a nonterminal and within the D derivation this nonterminal is rewritten using

3 734 Marian Pompiliu ristescu et al. / Procedia Economics and Finance 6 ( 03 ) x x... x x rules like: r, i being terminal or non-terminal symbols, then node has r descendants marked rom let to right with the symbols x x... xr. In order to generate all possible itineraries that meet the conditions o a problem, it is necessary to construct a linear grammar G( ) or each itinerary, in order to have all itineraries that start with a given point o departure, corresponding to that itinerary. For inding a minimum total path duration, going once through each point o the graph, there is at least one solution ound through Hamiltonian paths o a graph. 3. Types o algorithms used in the modeling business processes The ollowing presented algorithm is used to ind all roads in a graph with a inite number o nodes (ormen, 00; 8). Step : We build the Boolean matrix o direct adjacent corresponding to the graph, noted with. Here, are ollowed all the roads o length. Step : alculate the powers o, successively, up to power n- Step 3: Proceed to the matrix calculation D = n- In case we aim only the existence o the paths between nodes, and not their numbers, we use the Boolean multiplication and addition in accordance with the above observation: Then, we notice that: (4) ( I) n 0 n n 3... n n 3 D... n n n (5) It is deducted that it is necessary only the operation which calculates only the power n- or the matrix + I and then we multiply it with. The main advantage, proposed by this method, it reers to time economy, stressed also by the ollowing observation: i D contains those pairs o arcs or which highlighted the existence o a path then: D n n n n k (... )... D whatever it would be k 0 n k n n n n k ( I) (... )... D n n k n ( I) ( I) ( I) whatever it would be k 0 (6) which means that starting with the power k = n-, absolutely all k matrixes are equal. Thereore, is directly calculated, any o the powers o + I that are greater or equal with n-. above procedure is providing only the inding out whatever or not there is a path between two nodes, possibly what is the length and how many o this length. However, in practical problems the most important thing is to know which actually these paths are - decomposition o a graph in elementary paths, in the practical problems they are generally the problems o interest, the ollowing steps o the algorithm are dedicated in their inding and decomposition. In order to ind them it is used the graph representation through the Latin matrix. Here, it is worth mentioning that the two mathematicians who introduced in the operational calculus the concept o the Latina matrix were. Kaumann and J. Malgrange. The Latin matrix participate on the relationship o deining a graph. Sequences o nodes o an oriented vertically graph can be characterized by certain properties. The graphoriented nodes which have the same property and which succeed in a certain compatible order with the order o the graph is called a sequence. The operation that can be realized with the sequences, having the same property, is called

4 Marian Pompiliu ristescu et al. / Procedia Economics and Finance 6 ( 03 ) Step 4: It builds that Latin matrix which can be associated the graph and noted with L, where: (7), and the matrix ~ L, deined by: (8), called Latin reduced matrix. Step 5: alculate successively the matrices: Using the multiplication and addition Latin operations, the alphabet represents the set o graph nodes, where the multiplication operation is modiied in a nonessential way, the product o two o the elements o the matrices is 0, in case one o the items is 0, or the elements have in common a node that represents the Latin product, otherwise. 3.. lgorithms to optimize economic lows based on graph theory Given the act that economic lows can be associated with lows rom the classical theory o graphs, in this paper, we propose the appealing to a set o evolved algorithms develop in operational research. From these algorithms, were chosen the Ford-Fulkerson algorithm. Ford-Fulkerson method solves the problem o maximum low. This method is based on three important ideas that go beyond the algorithm and are used in other low problems: residual networks, improving paths and cuts (ormen, Ford-Fulkerson method is iterative. It starts with a low or like a irst low o value 0. For each iteration step will increase the low value to ind a "way o improvement," which is a path along which the low can be increased, and hence the value. These steps are repeated until no longer a way to improve can be ound Residual network transport network and a low, is said to be a residual network i she is composed o arcs that admit the great low. I there is a transport network o the orm with the source s and destination t, with the low in and consider a pair o vertices the quantity o additional low that can be transported rom the la u to v, without exceeding the capacity, is the residual capacity o the arc deined by (ormen, 00): Given a transmission network G ( V, E) G V, E ), where ( E c ( u, c( u, ( u, (0) and low the residual network o G induced by is {( u, V V : c ( u, 0} () Each arc o the residual network, or residual arc, admits a strictly positive low growth. It can be seen that the residual network G is a transmission network with the unction o capacity c. (9)

5 736 Marian Pompiliu ristescu et al. / Procedia Economics and Finance 6 ( 03 ) Roads o improvement Having a transmission network G ( V, E) and a low a way o improvement p is a simple way on s to t in residual network G. ter the deinition o the residual network, each arc ( u, on a road to improvement admits a positive low additional, without breaching the restriction o capacity (ormen, 00). The residual capacity o p is the maximum capacity o the low that can be transported along the road o improvement p given by the ormula: uts in the low networks c ( p) min{ c ( u, : ( u, is on the road p} () Ford Fulkerson method repeatedly increases the low along the augmenting path up to the point where a maximum low will be reached. The maximum low minimum cut theory proves that a low is maximum i and only i its residual network does not include any augmenting path. cut (S,T) o a low network G = (V, E) is a part o V set in the sets S and T = V-S so that and. I is a low, than the cut low (S,T) is deined as being equal to (S,T). The capacity o the cut(s,t) is c(s,t). minimum cut is the cut which has the lowest capacity o all the cuts o the network Ford Fulkerson lgorithm In any iteration o Ford Fulkerson method, we seek an augmenting path p and we increase the low along the path p with the residual capacity c (p). The implementation o the method calculates the maximum low in graph G = (V,E), updating the low [u,v] between any two peaks which are connected through arch. I u and v are not connected through an arch in any direction, we assume that [u,v] = 0. The value o capacity o u and v peaks is given by the unction c ( u, calculable in constant time, c ( u, 0 i ( u, E. The execution time o Ford Fulkerson algorithm depends on the determination meaner o the augmenting path p. I the path is aultily selected, the algorithm may not stop: the low value will successively increase, but it does * not converge to the maximum value. The execution time o Ford Fulkerson algorithm is given by O ( E ), where * is the maximum low obtained by the algorithm. 4. onclusions This paper presented the stages through which a ormalism is associated to the model o an economic process. The mathematical model is described through a graph, which is going to be used to shape various economic processes, as investments, organization o production, economic analysis activity, etc. Thus, the paper analyzed the economic problems, which can be set in ormal practice when there is no possibility to use one o the known techniques: when we shape an economic process, a manuacturing process, etc. The problem is simpliied to such an extent that the simple notations as: a, b, c, d, represent actions with well deined time rames, Using the interpretation according to which the graph is seen as a system where the nodes are the components o the system, two algorithms were presented: the algorithm through which we ind all the paths in an oriented graph with a inished number o nodes and the elaboration algorithm o Latin matrix where the alphabet represents the set o graph nodes. ll the paths within a graph can be decomposed in elementary roads, the being the primordial element sought in practice. The execution o the decomposition, in elementary roads, is perormed with the Latin matrix. Reerences BMPI (003), Business Process Modeling Language, Business Process Management Institute, January 4;

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