  We investigate a wireless power transfer (WPT)-based backscatter-mobile edge
computing (MEC) network with a {reconfigurable intelligent surface (RIS)}.In
this network, wireless devices (WDs) offload task bits and harvest energy, and
they can switch between backscatter communication (BC) and active transmission
(AT) modes. We exploit the RIS to maximize energy efficiency (EE). To this end,
we optimize the time/power allocations, local computing frequencies, execution
times, backscattering coefficients, and RIS phase shifts.} This goal results in
a multi-objective optimization problem (MOOP) with conflicting objectives.
Thus, we simultaneously maximize system throughput and minimize energy
consumption via the Tchebycheff method, transforming into two single-objective
optimization problems (SOOPs). For throughput maximization, we exploit
alternating optimization (AO) to yield two sub-problems. For the first one, we
derive closed-form resource allocations. For the second one, we design the RIS
phase shifts via semi-definite relaxation, a difference of convex functions
programming, majorization minimization techniques, and a penalty function for
enforcing a rank-one solution. For energy minimization, we derive closed-form
resource allocations. We demonstrate the gains over several benchmarks. For
instance, with a $20$-element RIS, EE can be as high as 3 (Mbits/Joule), a
150\% improvement over the no-RIS case (achieving only 2 (Mbits/Joule)).
