Discussion on aligning Logistic Invoice Verification Transaction document numbers in MM and FI

How many times have you wondered why the same invoice document from SAP MM, that got posted to SAP FI-AP (through a logistics invoice verification-LIV transaction), will have a document number in FI different from the one in MM?, and how much of your precious time is spent on drill downs from FI in order to retrieve the corresponding MM document?
The logistic invoice verification process results in the creation of two separate documents, one MM invoice document and one FI invoice document for a single procurement transaction. Ordinarily these documents should have the same document numbers, because they are outcomes of a single event. This, quite often is not the case, to the inconvenience of users.
Inconvenient because any attempt to process this document for any reason like reversing it or retrieving it (say by the accounts payable guys) brings about the issue of identifying the corresponding logistic/MM document which has a different number.
This post discusses how we can align these two types of documents (the MM and the FI document of a single transaction) to avoid this inconvenience and confusion.
Now, how is it possible that the same transaction in an integrated system like SAP can result in two different document numbers?
In order to understand why this happens, we have to grasp the following:

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full instructions

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