$*************************************************************
$ Problem of Determining the optimal positioning of a new
$   product in a multiattribute space
$
$ M.A. Duran and I.E. Grossmann, "An outer-approximation
$   algorithm for a class of mixed-integer nonlinear programs"
$   Math. Prog. 1986, 36, 307--339.
$
$ Optimal solution:  -8.0641
$*************************************************************

OPTION {{
  MINLP = "OAER";
  MINOS = "Iterations Limit 10000";
  MINOS = "Major Iterations Limit 250";
}}

DECLARATION {{
  INDEX {i,j,t};
  SET I = |1:25|;
  SET J = |1:10|;
  SET T = |1:5|;
  XVAR {x(T)};
  YVAR {y(I)};
  BINA {y(I)};
  XLBD {  2,0,3,0, 4};
  XUBD {4.5,8,9,5,10};

# z: Ideal points
  PARAM z(I,T) = {2.26, 5.15, 4.03, 1.74, 4.74,
                  5.51, 9.01, 3.84, 1.47, 9.92,
                  4.06, 1.80, 0.71, 9.09, 8.13,
                  6.30, 0.11, 4.08, 7.29, 4.24,
                  2.81, 1.65, 8.08, 3.99, 3.51,
                  4.29, 9.49, 2.24, 9.78, 1.52,
                  9.76, 3.64, 6.62, 3.66, 9.08,
                  1.37, 6.99, 7.19, 3.03, 3.39,
                  8.89, 8.29, 6.05, 7.48, 4.09,
                  7.42, 4.60, 0.30, 0.97, 8.77,
                  1.54, 7.06, 0.01, 1.23, 3.11,
                  7.74, 4.40, 7.93, 5.95, 4.88,
                  9.94, 5.21, 8.58, 0.13, 4.57,
                  9.54, 1.57, 9.66, 5.24, 7.90,
                  7.46, 8.81, 1.67, 6.47, 1.81,
                  0.56, 8.10, 0.19, 6.11, 6.40,
                  3.86, 6.68, 6.42, 7.29, 4.66,
                  2.98, 2.98, 3.03, 0.02, 0.67,
                  3.61, 7.62, 1.79, 7.80, 9.81,
                  5.68, 4.24, 4.17, 6.75, 1.08,
                  5.48, 3.74, 3.34, 6.22, 7.94,
                  8.13, 8.72, 3.93, 8.80, 8.56,
                  1.37, 0.54, 1.55, 5.56, 5.85,
                  8.79, 5.04, 4.83, 6.94, 0.38,
                  2.66, 4.19, 6.49, 8.04, 1.66};
# w: attribute weight
PARAM w(I,T) = {9.57, 2.74, 9.75, 3.96, 8.67,
                8.38, 3.93, 5.18, 5.20, 7.82,
                9.81, 0.04, 4.21, 7.38, 4.11,
                7.41, 6.08, 5.46, 4.86, 1.48,
                9.96, 9.13, 2.95, 8.25, 3.58,
                9.39, 4.27, 5.09, 1.81, 7.58,
                1.88, 7.20, 6.65, 1.74, 2.86,
                4.01, 2.67, 4.86, 2.55, 6.91,
                4.18, 1.92, 2.60, 7.15, 2.86,
                7.81, 2.14, 9.63, 7.61, 9.17,
                8.96, 3.47, 5.49, 4.73, 9.43,
                9.94, 1.63, 1.23, 4.33, 7.08,
                0.31, 5.00, 0.16, 2.52, 3.08,
                6.02, 0.92, 7.47, 9.74, 1.76,
                5.06, 4.52, 1.89, 1.22, 9.05,
                5.92, 2.56, 7.74, 6.96, 5.18,
                6.45, 1.52, 0.06, 5.34, 8.47,
                1.04, 1.36, 5.99, 8.10, 5.22,
                1.40, 1.35, 0.59, 8.58, 1.21,
                6.68, 9.48, 1.60, 6.74, 8.92,
                1.95, 0.46, 2.90, 1.79, 0.99,
                5.18, 5.10, 8.81, 3.27, 9.63,
                1.47, 5.71, 6.95, 1.42, 3.49,
                5.40, 3.12, 5.37, 6.10, 3.71,
                6.32, 0.81, 6.12, 6.73, 7.93};
# del: co-ordinate of existing products
  PARAM del(J,T) = {0.62, 5.06, 7.82, 0.22, 4.42,
                    5.21, 2.66, 9.54, 5.03, 8.01,
                    5.27, 7.72, 7.97, 3.31, 6.56,
                    1.02, 8.89, 8.77, 3.10, 6.66,
                    1.26, 6.80, 2.30, 1.75, 6.65,
                    3.74, 9.06, 9.80, 3.01, 9.52,
                    4.64, 7.99, 6.69, 5.88, 8.23,
                    8.35, 3.79, 1.19, 1.96, 5.88,
                    6.44, 0.17, 9.93, 6.80, 9.75,
                    6.49, 1.92, 0.05, 4.89, 6.43};
  PARAM c(I) = {-1.0, -0.2, -1.0, -0.2, -0.9,
                -0.9, -0.1, -0.8, -1.0, -0.4,
                -1.0, -0.3, -0.1, -0.3, -0.5,
                -0.9, -0.8, -0.1, -0.9, -1.0,
                -1.0, -1.0, -0.2, -0.7, -0.7};
  PARAM r(I) = <i E I| min[j E J| <<t E T| w(i,t)*((del(j,t)-z(i,t))^2)>>]>;
  SHOW {r(I)}; 
}}

MODEL {{
  MIN: <<i E I| c(i)*y(i)>> + 0.6*x(1)^2 - 0.9*x(2) - 0.5*x(3) + 0.1*x(4)^2 + x(5);
  nsum1(i E I): <<t E T| w(i,t)*(x(t)-z(i,t))^2>> - r(i) + 1000*y(i) =l= 1000;
  lcon1: x(1) - x(2) + x(3) + x(4) + x(5) =L= 10;
  lcon2: 0.6*x(1) - 0.9*x(2) - 0.5*x(3) + 0.1*x(4) + x(5) =L= -0.64;
  lcon3: x(1) - x(2) + x(3) - x(4) + x(5) =G= 0.69;
  lcon4: 0.157*x(1) + 0.05*x(2) =L= 1.5;
  lcon5: 0.25*x(2) + 1.05*x(4) - 0.3*x(5) =G= 4.5;
  ldsmy: <<i E I| y(i)>> =l= 7;
}}