| Summary: | In this paper, we propose a new hybrid model, multi <i>Q</i>-hesitant fuzzy soft multi-granulation rough set model, by combining a multi <i>Q</i>-hesitant fuzzy soft set and multi-granulation rough set. We demonstrate some useful properties of these multi <i>Q</i>-hesitant fuzzy soft multi-granulation rough sets. Furthermore, we define multi <i>Q</i>-hesitant fuzzy soft (<inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>M</mi> <mi>k</mi> </msup> <mi>Q</mi> <mi>H</mi> <mi>F</mi> <mi>S</mi> </mrow> </semantics> </math> </inline-formula>) rough approximation operators in terms of <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>M</mi> <mi>k</mi> </msup> <mi>Q</mi> <mi>H</mi> <mi>F</mi> <mi>S</mi> </mrow> </semantics> </math> </inline-formula> relations and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>M</mi> <mi>k</mi> </msup> <mi>Q</mi> <mi>H</mi> <mi>F</mi> <mi>S</mi> </mrow> </semantics> </math> </inline-formula> multi-granulation rough approximation operators in terms of <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>M</mi> <mi>k</mi> </msup> <mi>Q</mi> <mi>H</mi> <mi>F</mi> <mi>S</mi> </mrow> </semantics> </math> </inline-formula> relations. We study the main properties of lower and upper <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>M</mi> <mi>k</mi> </msup> <mi>Q</mi> <mi>H</mi> <mi>F</mi> <mi>S</mi> </mrow> </semantics> </math> </inline-formula> rough approximation operators and lower and upper <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>M</mi> <mi>k</mi> </msup> <mi>Q</mi> <mi>H</mi> <mi>F</mi> <mi>S</mi> </mrow> </semantics> </math> </inline-formula> multi-granulation rough approximation operators. Moreover, we develop a general framework for dealing with uncertainty in decision-making by using the multi <i>Q</i>-hesitant fuzzy soft multi-granulation rough sets. We analyze the photovoltaic systems fault detection to show the proposed decision methodology.
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