**6.4. Develop a workflow logic for a given business**

134 Petri Nets – Manufacturing and Computer Science

**Figure 23.** Reduction Process

Let T = {t1, t2,…tn} and B = {b1, b2, …bm} be respectively the set of business tasks with < as its immediate successor relation, and the set of blanks on the B-form for a well-designed business. There should be a fixed correspondence between T and B to make clear responsibilities for each of the tasks in T. This correspondence will not be mentioned any more.

Let Σ= (S,T;F,K,W,M0) be the net system such that T is the task set and S = {p1, p2, …pk}∪{s0} ∪E where for each i, i = 1, 2, …, k, k is the number of non-end tasks t1, t2, …tk in T, . pi = {ti}, pi. is the set of immediate successors of ti. Let ai be the number of tasks in this immediate successor set, ai' is the intended number of tasks to be executed after ti, then K(pi) = ai', and W(ti, pi) = ai' for every immediate successor t of ti, W(pi, t) = 1. It is easy to see that pi is a synchronizer: pi = ({ti}, pi . ' ( 1, ai')).

The place named s0 is the unique start place: it has an empty pre-set and its post-set contains the unique start task with arc weight 1.

For every end-task in T, there is a unique end place e in E with 1 as the arc weight between them. E contains nothing else.

M0(s0) = 1 and all other elements in S have no token.

As a workflow logic of a business process, transition rules for RP/T-systems are assumed for Σ.

Σ must be well-ordered since < is exactly its next relation among transitions.

Σ may be not well-structured as shown by Figure 12 (a). In this case, measures must be taken as suggested from Figure 12 (a) to (b), then to Figure 13.

Redundant synchronizers may be removed by Reduction Rule 3.

As an example, Let T = {t1, t2, t3, t4, t5} and <= {(t1, t2), (t1, t3), (t2, t4), (t2, t5), (t3, t4), (t3, t5)}. There are 3 non-end tasks in T, namely t1, t2 and t3. So there 3 synchronizers: p1, p2 and p3. Figure 24 shows the developed workflow logic.

**Figure 24.** A developed Workflow Logic

There is redundancy in the developed system at p2 and p3. By Reduction Rule 3, these two synchronizers are reduced to a single synchronizer p = ({t2, t3}, {t4, t5}, (2, 1)). This is the workflow logic for insurance claim, as shown in Figure 22, there it was reduced to harmony.
