multipleDropdownField selection issue

Try This. 

a!localVariables(
local!designatedBusiness: {
'type!{urn:com:appian:types:LCM}LCM_DesignatedBusinesses'('dbkey': 1, 'dbname': "Test-1"),
'type!{urn:com:appian:types:LCM}LCM_DesignatedBusinesses'('dbkey': 2, 'dbname': "Test-2"),
'type!{urn:com:appian:types:LCM}LCM_DesignatedBusinesses'('dbkey': 3, 'dbname': "Test-3"),
'type!{urn:com:appian:types:LCM}LCM_DesignatedBusinesses'('dbkey': 4, 'dbname': "Test-4")
},
local!selectedDesignatedBusinesses,
a!formLayout(
label: "Form",
contents: {
a!sectionLayout(
contents: {
a!multipleDropdownField(
label: "Designated Business/Function",
labelPosition: "ABOVE",
helpTooltip: "Designated Business/Function",
placeholder: "Select a Designated Business/Function",
choiceLabels: index(local!designatedBusiness, "dbname", ""),
choiceValues: index(local!designatedBusiness, "dbkey", ""),
value: local!selectedDesignatedBusinesses,
saveInto: {
local!selectedDesignatedBusinesses,

},
required: true
)
}
),

},
buttons: a!buttonLayout(
primaryButtons: {
a!buttonWidget(
label: "Submit",
submit: true,
style: "PRIMARY"
)
},
secondaryButtons: {
a!buttonWidget(
label: "Cancel",
value: true,
saveInto: ri!cancel,
submit: true,
style: "NORMAL",
validate: false
)
}
)
)
)

When the page loads, I will have few values in the below variable 

local!selectedDesignatedBusinesses

I have to be in a position to retain those values in addition to new selections.

Try this:

a!localVariables(
  local!dbkeysFromRi: ri!lcmIntakeDesignatedBusinesses.dbkey,
  local!designatedBusiness: {
    'type!{urn:com:appian:types:LCM}LCM_DesignatedBusinesses'(dbkey: 1, dbname: "Test-1"),
    'type!{urn:com:appian:types:LCM}LCM_DesignatedBusinesses'(dbkey: 2, dbname: "Test-2"),
    'type!{urn:com:appian:types:LCM}LCM_DesignatedBusinesses'(dbkey: 3, dbname: "Test-3"),
    'type!{urn:com:appian:types:LCM}LCM_DesignatedBusinesses'(dbkey: 4, dbname: "Test-4")
  },
  local!selectedDesignatedBusinesses: if(
    a!isNullOrEmpty(ri!lcmIntakeDesignatedBusinesses),
    null(),
    ri!lcmIntakeDesignatedBusinesses.dbkey
  ),
  local!addItems: difference(
    local!selectedDesignatedBusinesses,
    local!dbkeysFromRi
  ),
  local!removeItems: wherecontains(
    difference(
      local!dbkeysFromRi,
      local!selectedDesignatedBusinesses
    ),
    ri!lcmIntakeDesignatedBusinesses.dbkey
  ),
  a!formLayout(
    label: "Form",
    contents: {
      a!sectionLayout(
        contents: {
          a!multipleDropdownField(
            label: "Designated Business/Function",
            labelPosition: "ABOVE",
            helpTooltip: "Designated Business/Function",
            placeholder: "Select a Designated Business/Function",
            choiceLabels: index(local!designatedBusiness, "dbname", ""),
            choiceValues: index(local!designatedBusiness, "dbkey", ""),
            value: local!selectedDesignatedBusinesses,
            saveInto: local!selectedDesignatedBusinesses,
            required: true
          )
        }
      ),
      
    },
    buttons: a!buttonLayout(
      primaryButtons: {
        a!buttonWidget(
          label: "Submit",
          submit: true,
          style: "PRIMARY",
          saveInto: {
            a!forEach(
              local!addItems,
              a!save(
                ri!lcmIntakeDesignatedBusinesses,
                append(
                  ri!lcmIntakeDesignatedBusinesses,
                  'type!{urn:com:appian:types:LCM}LCM_IntakeDesignatedBusiness'(
                    intakedesignatedbusinesskey: null,
                    intakeheaderkey: null,
                    dbkey: fv!item
                  )
                )
              )
            ),
            a!save(
              ri!lcmIntakeDesignatedBusinesses,
              remove(
                ri!lcmIntakeDesignatedBusinesses,
                local!removeItems
              )
            )
          }
        )
      },
      secondaryButtons: {
        a!buttonWidget(
          label: "Cancel",
          value: true,
          saveInto: ri!cancel,
          submit: true,
          style: "NORMAL",
          validate: false
        )
      }
    )
  )
)

I have saved the desired values (addition/removal) in the rule input on Submit button saveInto 

Thank you so much Sanchit and salute to the login which you have provided to the code.

It is a great help. 

Glad to hear that

Glad to hear that