## Complexity Assessment of Assembly Supply Chains from the Sustainability Viewpoint

The main point of the paper is to address supply chain networks in terms of sustainability. How can customization of physical networks help to better manage demand?

### Definition of Testing Rules for ASC Complexity Indicators

To prove the selected complexity indicators, all of them were verified through defined criteria for complexity measures that might also be taken into account to assess their validity. Such rules were specified and proposed for static complexity metrics by Deshmuk et al. and Olbrich et al. Deshmuk et al. analyzed different factors influencing the static complexity of manufacturing systems, and defined four conditions that the static complexity metric must satisfy. They are as follows:**Rule#1:** Static complexity should increase with the number of parts and the number of machines and operations required to process the part mix.

**Rule#2:** Static complexity should increase with increases in sequence flexibility for the parts in the production batch.

**Rule#3:** Static complexity should increase as the sharing of resources by parts increases.

**Rule#4:** If the original part mix is split into two or more groups, then the complexity of processing should remain constant.

Due to the fact that Rule#3 and Rule#4 are not relevant for assembly supply chains, only two of them were adopted in terms of ASC systems.

Olbrich et al. studied how static complexity measures behave if the system size is increased and explored three special cases of adding an independent element, two independent subsystems, and two identical copies. In this context, the authors proposed three specific requirements or rules for a reliable complexity measure that must be met:**Rule#1:** Additional independent element: The element has no structure itself, so it has no complexity of its own. Because it is independent of the rest of the system, the complexity should not change.

**Rule#2:** Union of two independent systems: Because there are no dependencies between the two systems, the complexity of the union should be simply the sum of the complexities of the subsystems.

**Rule#3:** Union of two identical copies: Because there is no need for additional information to describe the second system, one could argue that the complexity should be equal to the complexity of one system. One has, however, to include the fact in the description that the second system is a copy of the first one. At least this part should not be extensive with respect to the system size.

However, any of the three rules appear to be impactful for ASC complexity metrics, and therefore were not directly used for the purpose of testing and comparing the complexity indicators.

After deeper consideration, the two rules by Deshmuk et al., namely Rules 1 and 2, were adopted by us into the four testing criteria (C). Moreover, we added one additional criterion (C#5) based on the increasing number of echelons. The criteria are summarily shown as follows:**C#1: **Static complexity should increase with the number of parts required to process the part mix.**C#2: **Static complexity should increase with the number of machines required to process the part mix.**C#3: **Static complexity should increase with the number of operations required to process the part mix.**C#4: **Static complexity should increase with increases in sequence flexibility for the parts in the production batch.**C#5: **Static complexity should increase with the number of echelons while the number of parts, machines, and operations is constant.