I was personally involved in a parts cost reduction program about ten years ago with one of Toyota's suppliers. It was pretty interesting, since I had never seen it done this way before. Compared to the way the Big 3 had been doing it, this was a breath of fresh air.
First Toyota went to the part supplier and asked for a reduction in part cost while also maintaining part quality. So they wanted to pay less per part but maintain, and if possible, improve quality. Obviously the first thing that was said was "impossible!". And here was the interesting part. Toyota said they not only would help, but they would "give" the part supplier a couple of their engineers to help them during the process. This was unheard of...at least from my experience in doing similar projects with the Big 3. Not only did Toyota supply the engineers, but they did so at no cost to the supplier. At the time, Toyota really seemed to understand how important their suppliers were to their overall operation. They helped the company introduce a Lean Manufacturing model based on Toyota's own system and implemented their "kaizen" mindset. Kaizen basically means "continuous improvement". It was a very interesting project and I had never seen one like it since then.
I'm not sure how they'll do it this time around. But they would be wise to avoid taking the route the Big 3 have done in the past which was to simply bark at your suppliers to reduce cost at any cost.