A linear programming problem comes to identifing an

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“A Linear Programming problem comes to identifing an extreme (i.e., minimum or maximum) point of a function f(x 1 ,x 2 ,..,x n ), which furthermore satisfies a set of constraints, e.g., g(x 1 ,x 2 ,..,x n )>0. Linear programming is the specialization of mathematical programming to the case where both, function f (objective function) and the problem constraints are linear...” [21]. From an applications perspective, linear programming is an optimization tool. An important factor for the applicability of the linear programming methodology, is the computational tractability of the resulting analytical models. Under the advent of modern computing technology, this tractability requirement translates to the existence of effective and efficient algorithmic procedures able to provide a systematic and fast solution to these models [21]. There are three inputs in LP: variables, an objective function and constraints, where both the objective function and constraints must be linear. The output of LP is the maximum (or minimum) value of the objective function as well as the values of variables at this maximum or minimum point [21].
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18 In order to apply LP optimization technique to selection of an optimal execution plan, the objective function is chosen as equatition (6) [1]: ( 29 ) ( max 1 i n i p Score = (7) Assume that A is the set of all tasks (i.e., basic states) of the statechart. For each task t j , there is a set of Web services S j that can be assigned to this task, but on the end, for each task t j , only one Web service should be selected. Given that y ij : (y ij = 0 or 1) denotes the participation of Web service s ij in the selected plan. This latter fact is captured by the following constraint [1]: 2200 = j S i ij A j y , 1 (8) For example, there are 100 potential Web services that can execute task t j , since only one of them will be selected to execute the task t j , then we have = 100 1 i ij y = 1 . Assume that variable x j represents the earliest start time of task, variable τ j represents the execution duration for task t j , and variable τ ij represents the execution duration for task t j by service s ij . The notation t j t k is used to denote that task t k is task t j 's direct successor task. We have the following constraints [1]: A j y j S i ij ij j 2200 = , τ τ (9) x k – ( τ j + x j ) 0, 2200 t j t k , j,k A (10) Q du – (x j + τ j ) 0, 2200 j A (11) Constraint 9 indicates that the execution duration of a given task t j is equal to the execution duration of one of the Web services in A . Constraint 10 captures the fact that if task t k is a direct successor of task t j , the execution of task t k must start after task t j has been completed. Constraint 11 indicates that the execution of a composite service is completed only when all its tasks are completed [1].
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  • Winter '15
  • MAhmoudali
  • World Wide Web, Web Services, Business process modeling, Web Services Description Language

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