reset;

option randseed'';

param N := 10;									            # Number of projects/workers
param w {i in 1..N,j in 1..N} := round(Uniform(5,(i+j)^2));	# willingness of worker j to take project i

###########  1 Linear integer  ###############

var x {i in 1..N,j in 1..N} binary;	                    # whether worker j takes project i

maximize LP1: sum{i in 1..N,j in 1..N} w[i,j]*x[i,j];   # maximize total willingness

subject to NBa1{j in 1..N}: sum{i in 1..N} x[i,j] = 1;  # each project has to be taken
subject to NBb1{i in 1..N}: sum{j in 1..N} x[i,j] = 1;  # each worker has to take one project

objective LP1;
option solver cplex;
solve;

var y {j in 1..N} = sum{i in 1..N} w[i,j]*x[i,j];       # received utility of worker j

display LP1, y, min{j in 1..N} y[j], max{j in 1..N} y[j];

#end;

############  2 Linear continuous (relaxation)  #############

fix x; drop NBa1; drop NBb1;                 # fix variables x, drop constraints of 1st model

var x2 {i in 1..N,j in 1..N} >=0, <=1;	     # whether worker j takes project i (now continuous)

maximize LP2: sum{i in 1..N,j in 1..N} w[i,j]*x2[i,j];   # maximize total willingness

subject to NBa2{j in 1..N}: sum{i in 1..N} x2[i,j] = 1;  # each project has to be taken
subject to NBb2{i in 1..N}: sum{j in 1..N} x2[i,j] = 1;  # each worker has to take one project

objective LP2;
option solver cplex;

var y2 {j in 1..N} = sum{i in 1..N} w[i,j]*x2[i,j];      # received utility of worker j

solve;

display LP2, y2, min{j in 1..N} y2[j], max{j in 1..N} y2[j];    # relaxed solution integer?


#end;

############  3 Nonlinear continuous relaxation (maybe no integer solution!)

fix x2; drop NBa2; drop NBb2;  # fix variables x2, drop constraints of 2nd model

var x3 {i in 1..N,j in 1..N} >=0, <=1;	        # whether worker j takes project i

maximize LP3: sum{i in 1..N} (sum{j in 1..N} w[i,j]*x3[i,j])^0.5;  # utility approach

subject to NBa3{j in 1..N}: sum{i in 1..N} x3[i,j] = 1;  # each project has to be taken
subject to NBb3{i in 1..N}: sum{j in 1..N} x3[i,j] = 1;  # each worker has to take one project

objective LP3;
option solver minos;

var y3 {j in 1..N} = sum{i in 1..N} w[i,j]*x3[i,j];      # received utility of worker j

solve;

display LP3, y3, min{j in 1..N} y3[j], max{j in 1..N} y3[j];


#end;

###########  4 Quadric integer (transformed minimization)  #############

fix x3; drop NBa3; drop NBb3;     # fix variables x3, drop constraints of 3rd model

var x4 {i in 1..N,j in 1..N} binary;	        # whether worker j takes project i

minimize LP4: sum{i in 1..N} (sum{j in 1..N}    # Transformed as quadratic minimization problem
             (max{i2 in 1..N,j2 in 1..N} w[i2,j2] - w[i,j])*x4[i,j])^2;

subject to NBa4{j in 1..N}: sum{i in 1..N} x4[i,j] = 1;  # each project has to be taken
subject to NBb4{i in 1..N}: sum{j in 1..N} x4[i,j] = 1;  # each worker has to take one project

objective LP4;
option solver cplex;                                     # cplex can do quadratic integer

var y4 {j in 1..N} = sum{i in 1..N} w[i,j]*x4[i,j];      # received utility of worker j

solve;

display LP4, y4, min{j in 1..N} y4[j], max{j in 1..N} y4[j];


###########  5 worst case / mixed criteria (linear)   #############

fix x4; drop NBa4; drop NBb4;     # fix variables x4, drop constraints of 4th model

var x5 {i in 1..N,j in 1..N} binary;	        # whether worker j takes project i
var zu;                                         # variable quetscht von unten
var zo;                                         # variable quetscht von oben

maximize LP5: sum{i in 1..N,j in 1..N} w[i,j]*x5[i,j] + 10*zu - 5*zo;   # balance criteria

subject to NBa5{j in 1..N}: sum{i in 1..N} x5[i,j] = 1;  # each project has to be taken
subject to NBb5{i in 1..N}: sum{j in 1..N} x5[i,j] = 1;  # each worker has to take one project
subject to WC  {j in 1..N}: zu<=sum{i in 1..N} w[i,j]*x5[i,j]; # quetschen von unten (for high worst case)
subject to BC  {j in 1..N}: zo>=sum{i in 1..N} w[i,j]*x5[i,j]; # quetschen von oben (for low best case)

objective LP5;
option solver cplex;

var y5 {j in 1..N} = sum{i in 1..N} w[i,j]*x5[i,j];

solve;

display LP5, y5;    # min{j in 1..N} y5[j], max{j in 1..N} y5[j];


#############  Final Comparison of Assigment Solutions  #############

display sum{i in 1..N,j in 1..N} w[i,j]*x[i,j],  min{j in 1..N} y[j],  max{j in 1..N} y[j];
display sum{i in 1..N,j in 1..N} w[i,j]*x2[i,j], min{j in 1..N} y2[j], max{j in 1..N} y2[j];
display sum{i in 1..N,j in 1..N} w[i,j]*x3[i,j], min{j in 1..N} y3[j], max{j in 1..N} y3[j];
display sum{i in 1..N,j in 1..N} w[i,j]*x4[i,j], min{j in 1..N} y4[j], max{j in 1..N} y4[j];
display sum{i in 1..N,j in 1..N} w[i,j]*x5[i,j], min{j in 1..N} y5[j], max{j in 1..N} y5[j];