APL (Dyalog Classic), 16 13 12 bytes

-3 thanks to @Adám

⍳∘≢+.×3⌽⊢+⌊/

Try it online!

⊢+⌊/ arg + min(arg)

3⌽ rotate 3 to the left, i.e. 1 to the right

+.× inner product, i.e. multiply element by element and sum

⍳∘≢ the indices of the array: 1 2 3 4 (uses 1-based indexing, ⎕io←1)

APL (Dyalog Classic), 16 13 12 bytes

-3 thanks to @Adám

(+/+\)1⌽⌽+⌊/

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⌽+⌊/ reverse(arg) + min(arg)

1⌽ rotate 1 to the left

+\ partial sums

+/ sum

APL (Dyalog Classic), 16 13 bytes

-3 thanks to @Adám

(1⌽⍳4)+.×⊢+⌊/

Try it online!

⊢+⌊/ arg + min(arg)

⍳4 1 2 3 4

1⌽⍳4 2 3 4 1

+.× inner product, i.e. multiply element by element and sum

APL (Dyalog Classic), 16 13 12 bytes

-3 thanks to @Adám

⍳∘≢+.×3⌽⊢+⌊/

Try it online!

⊢+⌊/ arg + min(arg)

3⌽ rotate 3 to the left, i.e. 1 to the right

+.× inner product, i.e. multiply element by element and sum

⍳∘≢ the indices of the array: 1 2 3 4 (uses 1-based indexing, ⎕io←1)

APL (Dyalog Classic), 16 13 bytes

(1⌽⍳4)+.×⊢+⌊/

Try it online!

APL (Dyalog Classic), 16 13 bytes

-3 thanks to @Adám

(1⌽⍳4)+.×⊢+⌊/

Try it online!

⊢+⌊/ arg + min(arg)

⍳4 1 2 3 4

1⌽⍳4 2 3 4 1

+.× inner product, i.e. multiply element by element and sum