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Short-Term Borrowings and Long-Term Debt - Schedule of Line of Credit Facilities (Details) - Syndicated line of credit - USD ($)	Sep. 02, 2023	Jun. 03, 2023	Jul. 31, 2021	Jun. 30, 2021
Line of Credit Facility [Line Items]
Syndicated revolving line of credit borrowing capacity			$ 725,000,000	$ 500,000,000
Syndicated revolving line of credit, due July 2026
Line of Credit Facility [Line Items]
Syndicated revolving line of credit borrowing capacity	$ 725,000,000.0	$ 725,000,000.0
Less: Borrowings under the syndicated revolving line of credit	367,300,000	426,700,000
Less: Outstanding letters of credit	13,900,000	14,100,000
Available borrowings under the syndicated revolving line of credit	$ 343,800,000	$ 284,200,000

X

- Definition

The total amount of the contingent obligation under letters of credit outstanding as of the reporting date.

+ References

No definition available.

+ Details

Name:	us-gaap_LettersOfCreditOutstandingAmount
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

X

- Definition

The carrying value as of the balance sheet date of the current and noncurrent portions of long-term obligations drawn from a line of credit, which is a bank's commitment to make loans up to a specific amount. Examples of items that might be included in the application of this element may consist of letters of credit, standby letters of credit, and revolving credit arrangements, under which borrowings can be made up to a maximum amount as of any point in time conditional on satisfaction of specified terms before, as of and after the date of drawdowns on the line. Includes short-term obligations that would normally be classified as current liabilities but for which (a) postbalance sheet date issuance of a long term obligation to refinance the short term obligation on a long term basis, or (b) the enterprise has entered into a financing agreement that clearly permits the enterprise to refinance the short-term obligation on a long term basis and the following conditions are met (1) the agreement does not expire within 1 year and is not cancelable by the lender except for violation of an objectively determinable provision, (2) no violation exists at the BS date, and (3) the lender has entered into the financing agreement is expected to be financially capable of honoring the agreement.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22))
-SubTopic 10
-Topic 210
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147480566/210-10-S99-1

Reference 2: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 944
-SubTopic 210
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.7-03(a)(16)(a)(2))
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479440/944-210-S99-1

Reference 3: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 942
-SubTopic 210
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.9-03(16))
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479853/942-210-S99-1

+ Details

Name:	us-gaap_LineOfCredit
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

X

- Definition

Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 235
-SubTopic 10
-Name Accounting Standards Codification
-Section S99
-Paragraph 1
-Subparagraph (SX 210.4-08(f))
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147480678/235-10-S99-1

+ Details

Name:	us-gaap_LineOfCreditFacilityLineItems
Namespace Prefix:	us-gaap_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

X

- Definition

Maximum borrowing capacity under the credit facility without consideration of any current restrictions on the amount that could be borrowed or the amounts currently outstanding under the facility.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Topic 210
-SubTopic 10
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02.19(b),22(b))
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147480566/210-10-S99-1

+ Details

Name:	us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

X

- Definition

Amount of borrowing capacity currently available under the credit facility (current borrowing capacity less the amount of borrowings outstanding).

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Topic 210
-SubTopic 10
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02.19(b),22(b))
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147480566/210-10-S99-1

+ Details

Name:	us-gaap_LineOfCreditFacilityRemainingBorrowingCapacity
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

X

- Details

Name:	us-gaap_CreditFacilityAxis=us-gaap_DomesticLineOfCreditMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

X

- Details

Name:	us-gaap_LongtermDebtTypeAxis=mlkn_SyndicatedRevolvingLineOfCreditDueJuly2026Member
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type: