Outputting flavor text depending on inputted strings
It seems this could be simplified with a small lookup table and then concatenating strings. Using a table as simple as:

const lookup = {
  blue: 'b',
  green: 'f',
  purple: 'g',
  yellow: 'p'
};


we can then lookup the color values and concatenate them together. Combined with a simple check to make sure the color is valid and the 2 colors aren't the same, and it seems to produce the same result. 

Here's the full function:

mixItUp: function() {
  this.mixArray.sort();
  const lookup = {
    blue: 'b',
    green: 'f',
    purple: 'g',
    yellow: 'p'
  };

  let mix;
  const color1 = lookup[this.mixArray[0]];
  const color2 = lookup[this.mixArray[1]];

  if (color1 &amp;&amp; color2 &amp;&amp; (color1 !== color2)) {
    mix = `${color1}${color2}`;
  } else {
    mix = '??';
  }

  return mix;
}


And here it is in action:



function createMix(color1, color2) {
  return {
    mixArray: [color1, color2],
    mixItUp: function() {
      this.mixArray.sort();
      const lookup = {
        blue: 'b',
        green: 'f',
        purple: 'g',
        yellow: 'p'
      };

      let mix;
      const color1 = lookup[this.mixArray[0]];
      const color2 = lookup[this.mixArray[1]];
      
      if (color1 &amp;&amp; color2 &amp;&amp; (color1 !== color2)) {
        mix = `${color1}${color2}`;
      } else {
        mix = '??';
      }

      return mix;
    }
  }
}


// Handle the form
const form = document.getElementById('mixitup');

form.addEventListener('submit', event =&gt; {
  event.preventDefault();
  const color1 = document.getElementById('color1').value;
  const color2 = document.getElementById('color2').value;

  const mixedArray = createMix(color1, color2)
  console.log(mixedArray.mixItUp())
})
&lt;form id="mixitup"&gt;
  &lt;select id="color1"&gt;
    &lt;option value=""&gt;Select...&lt;/option&gt;
    &lt;option value="blue"&gt;Blue&lt;/option&gt;
    &lt;option value="green"&gt;Green&lt;/option&gt;
    &lt;option value="purple"&gt;Purple&lt;/option&gt;
    &lt;option value="yellow"&gt;Yellow&lt;/option&gt;
  &lt;/select&gt;
  &lt;select id="color2"&gt;
    &lt;option value=""&gt;Select...&lt;/option&gt;
    &lt;option value="blue"&gt;Blue&lt;/option&gt;
    &lt;option value="green"&gt;Green&lt;/option&gt;
    &lt;option value="purple"&gt;Purple&lt;/option&gt;
    &lt;option value="yellow"&gt;Yellow&lt;/option&gt;
  &lt;/select&gt;
  &lt;input type="submit" value="Mix it up!"&gt;
&lt;/form&gt;