There are $ N $ bins with equal capacity $ C $. In addition, there are $ N $ objects $x_1, x_2, \dots, x_N $ that need to be accommodated using the least amount of bins. Each object $x_i$ has a volume $ v_i < C $. However, there is a penalization $ p_{ij} $ for accommodating any two objects $x_i, x_j$ together. This penalization is related to the shape incompatibility among objects, which causes capacity waste.
To summarize, the objective is to minimize the number of bins used while taking into consideration the capacity constraints and penalization due to grouping.
I need help to formulate this problem. My ultimate objective is to use dynamic programming to solve this problem but I cannot come up with a formulation, mainly due to the penalization policy. Also, if you have seen a similar problem before, please point me to the source.
Further information: For instance, if three objects $ x_1, x_2, x_3$ are put together in the same bin, the overall occupied volume is $ v_1 + v_2 + v_3 + p_{12} + p_{13} + p_{23}$.
Update: I just want to thank you all for your help. Thank you @MarcoLübbecke for the paper you shared about bin packing with conflicts. Thank you @RenaudM for the two formulations you proposed and for pointing me to this article in your blog https://orandtricks.wordpress.com/2013/08/12/planning-your-wedding-is-not-easy-and-i-am-not-even-speaking-of-handling-the-brides-moods/ about the issue with symmetric formulations. Thank you @RobPratt for your simplified formulation. I have implemented @RobPratt's formulation and @RenaudM's first approach, and for small-scale problems they perform the same. I aim at developing a dynamic programming solution for either of these two formulations. If I succeed I will post the solution. Thank you again for your time and help in this matter.