Quark, the design pressure is equal to or less than the MAWP of the vessel. By 1.5, do you mean the hydrotest pressure (or 1.3 as allowed under the latest code version)?
Maybe we are talking about something different. To me as a process engineer, the design pressure is what I put on the data sheet initally, say 500 psig. When the vessel is actually designed and available actual head, shell and nozzle thicknesses selected, the vessel will likely be able to withstand somewhat more than my design pressure (unless the available plate for a component is exactly what I calculate I need, somewhat unlikely OR my design conditions are set by flange limits).
BAck to the OP. For pressure vessels in the US, you can not operate the vessel greater than its MAWP on a continuous basis. This is also the highest pressure you can set a single PSV protecting the vessel (if you have more than 1 PSV, the others can be set higher than MAWP but not more than 105% of MAWP). There is nothing stopping you from setting a PSV lower than MAWP however.
When looking at PSVs and allowable relieving pressures, you need to look at the ASME code (or other relevant codes for your area) for what you are trying to achieve. For the ASME code, for a single PSV other than fire, the code requires the PSV to prevent the vessel's pressure from rising more than 110% of MAWP. That means if the PSV is set at MAWP then it must have sufficient area to relieve the gas without the pressure exceeding 110%. Will the pressure actually reach 110%?
Consider that when you size a PSV you calculate the required area and then pick the next larger size PSV. If your relieving requirements were precisely correct, you could put the actual area back into the PSV equation and solve for the relieving pressure which would be lower than 110% of MAWP. However, picking the next largest orifice area (rather than having an orifice custom made for the precise area you calculated) provides a margin of safety in case your estimates of relieving flow rate, temperature, molecular weight, compressibility or other factors will have some errors in them.
Now, if you set a PSV lower than MAWP, you still only need to prevent the vessel from exceeding 110% of MAWP (for a single relief valve other than fire). The %accumulation for the valve itself could be higher than 110%, it's the vessel accumulation that is critical to ensure it won't be exceeded. Similar logic applies to multiple relief valves or fire cases.
If the vessel is properly designed for the conditions, 110%, 116% or 120% is unlikely to cause a vessel failure. First, the ASME code already has a substantial margin of safety built into it when you calculate the required thicknesses. Secondly, the vessel has been hydrotested to (typically in the US) 150% (or 130% today) of design pressure so you know it's seen and withstood higher pressures although not necessarily at the relieving temperatures.