m

number of decision alternatives

p

number of objectives

S

set of decision alternatives

z_i

objective i

$z_i^k$

performance measure of alternative k on objective i

u_i(.)

marginal utility function for objective i

g(.)

decision maker’s utility or value function

lt_i

lower threshold of objective i

ut_i

upper threshold of objective i

u_i1(z_i)

marginal utility function for increasing objectives with thresholds and for increasing side of target objective i

u_i2(z_i)

marginal utility function for decreasing objectives with threshold and for decreasing side of target objective i

I_i

ideal value for objective i

N_i

nadir value for objective i

A_i

aspiration value for objective i

${z^{\prime }}_i^k$

modified value of alternative k on objective i

m_i

number of distinct values for objective i between N_i and I_i, inclusive

M

index set of alternatives

w_i

weight of objective i in the simulated DM preference function

B_j

set of out-ranking alternatives to alternative j

c

maximum allowable normalized difference for objectives where alternative k has better scores than alternative j

${\alpha }_i^{kj}$

binary function equal 1 if modified value of alternative k is greater than or equal to j with respect to objective i

${\beta }_i^{kj}$

difference between normalized values of alternative j and k with respect to objective i