An evaluation framework for equipment contribution rate to system of systems based on operation loop and improved Shapley value

doi:10.23919/JSEE.2026.000117

Abstract

Abstract:

The evaluation of the equipment contribution rate to system-of-systems (CRSoS) is crucial for optimizing the armament system-of-systems structure and enhancing combat effectiveness. The traditional relative contribution rate method poses limitations by focusing on individual equipment evaluation without considering the interrelations between equipment. In response to the issue, this study proposes a framework based on operation loop and improved Shapley value (OLISV) for analysis to ananlyze the equipment CRSoS. Specifically, a multi-layer network model is first constructed based on complex heterogeneous network and operation loop theory. Next, information entropy and evidence theory are used for the edges of the functional node layer, while improving the parallel node structure within the network. Subsequently, an improved Shapley value contribution rate method based on non-efficiency influencing factors is proposed. Finally, the rationality and effectiveness of the OLISV are illustrated through a case study.

Key words: operation loop, heterogeneous networks, armament system-of-systems, contribution rate, improvement of Shapley value

Cancan HU, Yaping WANG. An evaluation framework for equipment contribution rate to system of systems based on operation loop and improved Shapley value[J]. Journal of Systems Engineering and Electronics, 2026, 37(3): 974-992.

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Entity	Capability symbol	Indicator	Indicator symbol	Unit	Type
S	$ {C}_\text{S} $	Maneuvering speed	$ c_\text{S}^{1} $	$ {\mathrm{km/h}} $	Benefit
		Maximum detecting distance	$ c_\text{S}^{2} $	$ {\mathrm{km}} $	Benefit
		Detecting accuracy	$ c_\text{S}^{3} $	$ {{\mathrm{m}}}^{2} $	Benefit
		Identifying probability	$ c_\text{S}^{4} $	$ 0-1 $	Benefit
		Tracking probability	$ c_\text{S}^{5} $	$ 0-1 $	Benefit
D	$ {C}_\text{D} $	Decision making time	$ c_\text{D}^{1} $	$ {\mathrm{s}} $	Cost
		Decision- making accuracy	$ c_\text{D}^{2} $	$ 0-1 $	Benefit
		Assisted decision making	$ c_\text{D}^{3} $	$ 0-1 $	Benefit
I	$ {C}_\text{I} $	Maneuvering speed	$ c_\text{I}^{1} $	$ {\mathrm{km/h}} $	Benefit
		Damaging radius	$ c_\text{I}^{2} $	$ {\mathrm{m}} $	Benefit
		Striking accuracy	$ c_\text{I}^{3} $	$ 0-1 $	Benefit
		Effective range	$ c_\text{I}^{4} $	$ {\mathrm{km}} $	Benefit
T	$ {C}_\text{T} $	Maneuvering speed	$ c_\text{T}^{1} $	$ {\mathrm{km/h}} $	Benefit
		Warning time	$ c_\text{T}^{2} $	$ {\mathrm{s}} $	Benefit
		Destructive capability	$ c_\text{T}^{3} $	$ 0-1 $	Benefit
		Stealth capability	$ c_\text{T}^{4} $	$ 0-1 $	Benefit
Communication	$ {C}_\text{C} $	Communication coverage	$ c_\text{C}^{1} $	$ {\mathrm{km}} $	Benefit
		Communication rate	$ c_\text{C}^{2} $	$ {\mathrm{Mb/s}} $	Benefit
		Communication capacity	$ c_\text{C}^{3} $	${\mathrm{ bit/s/Hz}} $	Benefit
		Communication delay	$ c_\text{C}^{4} $	$ {\mathrm{ms}} $	Cost
		Communication quality	$ c_\text{C}^{5} $	$ 0-1 $	Cost

Edge type	Meaning
$ {\mathrm{T}}\rightarrow {\mathrm{S}} $	Intelligence acquisition
$ {\mathrm{S}}\rightarrow {\mathrm{S}} $	Intelligence sharing
$ {\mathrm{S}}\rightarrow {\mathrm{D}} $	Information uploading
$ {\mathrm{D}}\rightarrow {\mathrm{D}} $	Collaboration between decision
$ {\mathrm{D}}\rightarrow {\mathrm{I}} $	Fire control
$ {\mathrm{I}}\rightarrow {\mathrm{T}} $	Destroy enemy targets

Functional nodes	Equipment	Functional nodes	Equipment	Functional nodes	Equipment
$ \text{S}_{1} $	$ \text{UE}_{1} $	$ \text{S}_{9} $	$ \text{E}_{6} $	$ \text{S}_{8} $	$ \text{E}_{5} $
$ \text{I}_{3} $	$ \text{UE}_{1} $	$ \text{I}_{14} $	$ \text{E}_{6} $	$ \text{I}_{11} $	$ \text{E}_{5} $
$ \text{S}_{2} $	$ \text{UE}_{2} $	$ \text{S}_{10} $	$ \text{E}_{7} $	$ \text{S}_{3} $	$ \text{UE}_{3} $
$ \text{I}_{12} $	$ \text{UE}_{2} $	$ \text{I}_{6} $	$ \text{E}_{7} $	$ \text{D}_{5} $	$ \text{C}^{3}\text{S}_{5} $
$ \text{S}_{4} $	$ \text{E}_{1} $	$ \text{S}_{11} $	$ \text{E}_{8} $	$ \text{I}_{1} $	$ \text{UE}_{4} $
$ \text{I}_{8} $	$ \text{E}_{1} $	$ \text{I}_{5} $	$ \text{E}_{8} $	$ \text{I}_{2} $	$ \text{UE}_{5} $
$ \text{S}_{5} $	$ \text{E}_{2} $	$ \text{S}_{12} $	$ \text{E}_{9} $	$ \text{I}_{4} $	$ \text{UE}_{6} $
$ \text{I}_{9} $	$ \text{E}_{2} $	$ \text{I}_{13} $	$ \text{E}_{9} $	$ \text{I}_{15} $	$ \text{UE}_{7} $
$ \text{S}_{6} $	$ \text{E}_{3} $	$ \text{D}_{1} $	$ \text{C}^{3}\text{S}_{1} $	$ \text{I}_{16} $	$ \text{UE}_{8} $
$ \text{I}_{10} $	$ \text{E}_{3} $	$ \text{D}_{2} $	$ \text{C}^{3}\text{S}_{2} $	$ \text{I}_{17} $	$ \text{UE}_{9} $
$ \text{S}_{7} $	$ \text{E}_{4} $	$ \text{D}_{3} $	$ \text{C}^{3}\text{S}_{3} $	$ \text{I}_{18} $	$ \text{UE}_{10} $
$ \text{I}_{7} $	$ \text{E}_{4} $	$ \text{D}_{4} $	$ \text{C}^{3}\text{S}_{4} $	$ \text{S}_{13} $	$ \text{UE}_{11} $

Number	Operation loop	Combat effectiveness
1	$ \text{T}_{3}\rightarrow \text{S}_{2}\rightarrow \text{D}_{5}\rightarrow \text{D}_{2}\rightarrow \text{D}_{5}\rightarrow \text{I}_{12}\rightarrow \text{T}_{3} $	0.265
2	$ \text{T}_{3}\rightarrow \text{S}_{2}\rightarrow \text{D}_{5}\rightarrow \text{D}_{2}\rightarrow \text{D}_{3}\rightarrow \text{I}_{4}\rightarrow \text{T}_{3} $	0.271
3	$ \text{T}_{3}\rightarrow \text{S}_{2}\rightarrow \text{D}_{5}\rightarrow \text{I}_{5}\rightarrow \text{T}_{3} $	0.318
4	$ \text{T}_{3}\rightarrow \text{S}_{3}\rightarrow \text{D}_{4}\rightarrow \text{D}_{2}\rightarrow \text{D}_{3}\rightarrow \text{I}_{4}\rightarrow \text{T}_{3} $	0.288
5	$ \text{T}_{3}\rightarrow \text{S}_{3}\rightarrow \text{D}_{4}\rightarrow \text{D}_{2}\rightarrow \text{I}_{5}\rightarrow \text{T}_{3} $	0.336
6	$ \text{T}_{3}\rightarrow \text{S}_{3}\rightarrow \text{D}_{4}\rightarrow \text{D}_{2}\rightarrow \text{D}_{5}\rightarrow \text{I}_{12}\rightarrow \text{T}_{3} $	0.292
		
96	$ \text{T}_{9}\rightarrow \text{S}_{13}\rightarrow \text{D}_{5}\rightarrow \text{D}_{1}\rightarrow \text{I}_{16}\rightarrow \text{T}_{9} $	0.294

Node number	RCR	SCR	ISCR	Change rate/%
$ \text{S}_{3} $	0.7737	0.2181	0.2223	1.9
$ \text{S}_{2} $	0.6644	0.1996	0.1954	−2.1
$ \text{D}_{4} $	0.7737	0.2181	0.2184	0.1
$ \text{D}_{2} $	0.7737	0.2181	0.2176	−0.2
$ \text{D}_{5} $	−	1	1	0.0
$ \text{I}_{12} $	−	1	1	0.0