Imagine cutting out the time of obtaining quotes, placing orders, tracking delivery and the payment cycle.

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Personally I think the value of procurement cards is overstated. They should have no effect on the time taken to obtain quotes or track delivery. They simplify ordering and payment but the commodity you are purchasing is still the same and the same care should be taken to get the right product at the right price regardless of payment method.
I think a similar process gain can be had by grouping commodities, selecting prefered supplier for each group and have them consolidate invoice at month end.
This reduces transaction cost and reduces number of suppliers, thereby facilitating unit cost reductions. P card reduces transaction cost but tends to drive supplier numbers and unit cost up.