Leaving side the -Mn term, which is a constant, the minimum possible objective value appears to be zero, and that is achieved with mu_n = 0 and any feasible Pi and Pj. Therefore, because mu_n does not appear in any constraints other than being nonnegative, It looks like mu_n = 0 is optimal, together with any value of Pi and Pj which satisfy the constraints
Am I missing something?