Multi-Layer Product-IF Formula

I've got a matrix of "Items" (1-10) that have various dollar values across multiple "Dimensions" (1-4). The dollar values for an Item within each Dimension are made up of certain "Group" Costs (1-4). The Dimensions each have a percentage breakdown of Group Costs that they are made up of. As an example, Dim 1 cost is made up 100% by Group 1, but Dim 3 cost is 50% Group 1 and 50% Group 2.

What I'd like to do is summarize the Group Costing for each line Item. Thus, I need a sort-of multi-layered Product-IF or Match/Multiply/Sum, etc. I've attached a sample sheet for reference, where the green area would contain the formula I'm looking for. I've filled in the first cell long-hand (yellow), but I know there's got to be a Product(If( or Match-type Array out there to do it more elegantly. TIA for your help.