Soundness Verification of Data-Aware Process Models with Variable-to-Variable Conditions

Traditionally Business Process Modeling has only focused on the control-flow perspective, thus allowing process designers to specify the constraints on the activities of the process: the order and potential concurrency of their execution, their mutual exclusivity, the possibility of being repeated, etc. However, activities are executed by different resources, manipulate data objects and are constrained by the state of such objects. This requires that the traditional notion of soundness, typically introduced for control-flow-only models, is extended so as to consider data. Intuitively, a (data-aware) process model is sound if (1) it does not contain deadlocks, (2) no more activities are enabled when the process instance is marked as completed and finally (3) there are no parts of the model that cannot be executed. Although several data-aware notations have been introduced in the literature, not all of these are given a formal semantics. In this paper, we propose a technique for checking the data-aware soundness for a specific class of such integrated models, with a simple syntax and semantics, building on Data Petri Nets (DPNs). These are Petri nets enriched with case variables, where transitions are guarded by formulas that inspect and update such variables, and are of the form variable-operator-variable or variable-operator-constant. Even though DPNs are less expressive than Petri nets where data are carried by tokens, they elegantly capture business processes operating over simple case data, allowing to model complex data-aware decisions. We show that, if a DPN is data-aware sound, the Constraint Graph is a finite-state automaton; however, a finite-state Constraint Graph does not guarantee data-aware soundness, but provides a finite structure through which this property can be checked. Finally, we investigate further properties beyond data-aware soundness, such as the problem of verifying that an actor participating in the business process can unilaterally enforce data-aware soundness by restricting the possible executions of a bounded DPN, assuming this actor to be able to control the firing of some transitions and decide the value of some of the case variables whenever these are updated.

VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

Bounded solutions of a difference equation with finite number of jumps of operator coefficient

We study the problem of existence of a unique bounded solution of a difference equation with variable operator coefficient in a Banach space. There is well known theory of such equations with constant coefficient. In that case the problem is solved in terms of spectrum of the operator coefficient. For the case of variable operator coefficient correspondent conditions are known too. But it is too hard to check the conditions for particular equations. So, it is very important to give an answer for the problem for those particular cases of variable coefficient, when correspondent conditions are easy to check. One of such cases is the case of piecewise constant operator coefficient. There are well known sufficient conditions of existence and uniqueness of bounded solution for the case of one jump. In this work, we generalize these results for the case of finite number of jumps of operator coefficient. Moreover, under additional assumption we obtained necessary and sufficient conditions of existence and uniqueness of bounded solution.

A decision model for kitting and line stocking with variable operator walking distances

Purpose – The purpose of this paper is to propose a decision model to choose between kitting and line stocking at the level of single parts, while taking into account the variable operator walking distances. Different ways of feeding assembly lines, such as kitting and line stocking not only have an impact on in-plant logistics flows but also determine the amount of stock that is available at the line. This, in turn, has an impact on operator walking distances during assembly. Design/methodology/approach – A mixed integer linear programming model is developed for the assignment of parts to one of both methods, and to be able to extensively test the model, an algorithm is created for the construction of representative datasets. Findings – Parts are often kitted because of a space constraint at the line, but even without a space constraint, the shorter walking distances might give preference to kitting. An analysis is presented that demonstrates how specific part characteristics influence the chances of a part being kitted. Research limitations/implications – Our research model can be extended to include, e.g., the study of alternative in-plant logistic designs and the outsourcing of kitting to a third-party logistics provider (3PL) or to the suppliers. Practical implications – The objective assignment model and the insights obtained from it are valuable for logistics and production engineers that otherwise have to rely solely on intuition. In situations with thousands of components, intuition mostly falls far short. Originality/value – First, existing models do not consider variable walking distances, which are shown to have a crucial impact on the decision. Second, the data instances created allow for a systematic comparison of future research in the field.