... wäre hier vielleicht eine Möglichkeit in Verbindung mit dem verb "to decrease/reduce", oder so etwas wie "to (greatly) simplify logistics processes/workflows".

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.. is the usual term

Example from link below:
"This initial four-month scientific effort will be mostly composed of Hawaiian investigators in order to minimize logistic effort.

I agree that Aufwand is ambiguous in German, but it doesn't have to be a problem. I usually translate it as costs, because if you expend less effort on something, then by definition you'll reduce your costs, even if only by freeing you up to do other things. I feel that "logistics effort" sounds a bit woolly.

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Note added at 4 hrs (2013-01-09 17:47:06 GMT)
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And I disagree with your customer: if you have fewer parts to transport, your logistics costs will tend to be lower. They're not giving a contractual guarantee, just making a general statement.

therefore minimising logistics usage (or utilisation)

eases/minimises/significantly reduces the logistics burden on

Aufwand can be a difficult word to deal with. This is therefore just one suggestion that may work in your context. Like Aufwand, burden can be ambiguous, i.e. a burden in terms of cost or time or effort, etc.

Some examples:

The decision to adopt a single or multiple sourcing strategy for each material input is a strategic choice for the MNE plant and may be driven by cost, quality or logistic considerations. For example one Southern MNE plant indicated their intention to move from multiple sourcing to single sourcing to reduce the logistics burden of dealing with many suppliers of less important inputs.
http://eservices.afbini.gov.uk/erini/pdf/NIERCBooks36.pdf

For years PENCOM has been providing kitting and assembly services saving customers the time and expense of ordering, receiving, handling and tracking the same multiple parts over and over again. By ordering pre–counted and labeled packages, customers achieve lean manufacturing and cost reduction objectives while consolidating vendors and relieving procurement personnel from the logistical burden of managing multiple individual components.
https://docs.google.com/viewer?a=v&q=cache:SS7zR-t9LiAJ:www....

There are many other options for Aufwand, e.g.
minimises/reduces (the) logistics/logistical difficulties/problems/complexity for, etc.