Resources:

### Giapetto’s problem

# Giapetto's problem
# This finds the optimal solution for maximizing Giapetto's profit

/* Set of toys */
set TOY;

/* Parameters */
param Finishing_hours {i in TOY};
param Carpentry_hours {i in TOY};
param Demand_toys     {i in TOY};
param Profit_toys     {i in TOY};

/* Decision variables */
var x {i in TOY} >=0;

/* Objective function */
maximize z: sum{i in TOY} Profit_toys[i]*x[i];

/* Constraints */
s.t. Fin_hours : sum{i in TOY} Finishing_hours[i]*x[i] <= 100;
s.t. Carp_hours : sum{i in TOY} Carpentry_hours[i]*x[i] <= 80;
s.t. Dem {i in TOY} : x[i] <= Demand_toys[i];

data;
/* data  section */

set TOY := soldier train;

param Finishing_hours:=
soldier         2
train           1;

param Carpentry_hours:=
soldier         1
train           1;

param Demand_toys:=
soldier        40
train    6.02E+23;

param Profit_toys:=
soldier         3
train           2;

end;
• Statements are semicolon terminated
• set and param are to specify static data
• var is to introduce variables in optimization
• Objective is preceded with keyword maximize or minimize
• Constraints are specified with subject to, subj to, s.t. or even omitted at all
• Array subscripts are declared with braces, so as iteration variables in summation
• Objective and constraints must be named, optionally carrying an expression for subscripts
• Data section starts with the statement data, in the above, parameters are 1D numeric with subscripts over the set TOY

### Diet problem

# Diet problem
#
# This finds the optimal solution for minimizing the cost of my diet
#

/* sets */
set FOOD;
set NEED;

/* parameters */
param NutrTable {i in FOOD, j in NEED};
param Cost {i in FOOD};
param Need {j in NEED};

/* decision variables: x1: Brownie, x2: Ice cream, x3: soda, x4: cake*/
var x {i in FOOD} >= 0;

/* objective function */
minimize z: sum{i in FOOD} Cost[i]*x[i];

/* Constraints */
s.t. const{j in NEED} : sum{i in FOOD} NutrTable[i,j]*x[i] >= Need[j];

/* data section */
data;

set FOOD := Brownie "Ice cream" soda cake;
set NEED := Calorie Chocolate Sugar Fat;

param NutrTable: Calorie        Chocolate       Sugar   Fat:=
Brownie          400            3               2       2
"Ice cream"      200            2               2       4
soda             150            0               4       1
cake             500            0               4       5;

param Cost:=
Brownie         0.5
"Ice cream"     0.2
soda            0.3
cake            0.8;

param Need:=
Calorie         500
Chocolate       6
Sugar           10
Fat             8;

end;
• Parameter NutrTable is a 2D numeric parameter
• Recalling a value of a 2D array: NutrTable[i,j]
• Strings with space need to be double quoted

### Knapsack problem

# Size of knapsack
param c;

# Items: index, size, profit
set I, dimen 3;

# Indices
set J := setof{(i,s,p) in I} i;

# Assignment
var a{J}, binary;

maximize obj :
sum{(i,s,p) in I} p*a[i];

s.t. size :
sum{(i,s,p) in I} s*a[i] <= c;

solve;

printf "The knapsack contains:\n";
printf {(i,s,p) in I: a[i] == 1} " %i", i;
printf "\n";

data;

# Size of the knapsack
param c := 100;

# Items: index, size, profit
set I :=
1 10 10
2 10 10
3 15 15
4 20 20
5 20 20
6 24 24
7 24 24
8 50 50;

end;
• Set J is build by extracting the index part of set I, through the setof operation
• The summation is to scan over set I, but only use the index part
• Output is manipulated by the printf statements

### Assignment problem

/* ASSIGN, Assignment Problem */
/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
/* The assignment problem is one of the fundamental combinatorial
optimization problems.

In its most general form, the problem is as follows:

There are a number of agents and a number of tasks. Any agent can be
assigned to perform any task, incurring some cost that may vary
depending on the agent-task assignment. It is required to perform all
tasks by assigning exactly one agent to each task in such a way that
the total cost of the assignment is minimized.

param m, integer, > 0;  /* number of agents */
param n, integer, > 0;  /* number of tasks */
set I := 1..m;          /* set of agents */
set J := 1..n;          /* set of tasks */

param c{i in I, j in J}, >= 0;
/* cost of allocating task j to agent i */

var x{i in I, j in J}, >= 0;
/* x[i,j] = 1 means task j is assigned to agent i
note that variables x[i,j] are binary, however, there is no need to
declare them so due to the totally unimodular constraint matrix */

s.t. phi{i in I}: sum{j in J} x[i,j] <= 1;
/* each agent can perform at most one task */

s.t. psi{j in J}: sum{i in I} x[i,j] = 1;
/* each task must be assigned exactly to one agent */

minimize obj: sum{i in I, j in J} c[i,j] * x[i,j];
/* the objective is to find a cheapest assignment */

solve;

printf "\n";
printf{i in I} "%5d %5d %10g\n", i, sum{j in J} j * x[i,j],
sum{j in J} c[i,j] * x[i,j];
printf "----------------------\n";
printf "     Total: %10g\n", sum{i in I, j in J} c[i,j] * x[i,j];
printf "\n";

data;

/* These data correspond to an example from [Christofides]. */

/* Optimal solution is 76 */

param m := 8;

param n := 8;

param c : 1  2  3  4  5  6  7  8 :=
1  13 21 20 12  8 26 22 11
2  12 36 25 41 40 11  4  8
3  35 32 13 36 26 21 13 37
4  34 54  7  8 12 22 11 40
5  21  6 45 18 24 34 12 48
6  42 19 39 15 14 16 28 46
7  16 34 38  3 34 40 22 24
8  26 20  5 17 45 31 37 43 ;

end;
• Integer constraint on variables or parameters are specified with keyword integer

### Bin packing problem

/* BPP, Bin Packing Problem */
/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
/* Given a set of items I = {1,...,m} with weight w[i] > 0, the Bin
Packing Problem (BPP) is to pack the items into bins of capacity c
in such a way that the number of bins used is minimal. */

param m, integer, > 0;   /* number of items */
set I := 1..m;           /* set of items */
param w{i in 1..m}, > 0; /* w[i] is weight of item i */
param c, > 0;            /* bin capacity */

/* We need to estimate an upper bound of the number of bins sufficient
to contain all items. The number of items m can be used, however, it
is not a good idea. To obtain a more suitable estimation an easy
heuristic is used: we put items into a bin while it is possible, and
if the bin is full, we use another bin. The number of bins used in
this way gives us a more appropriate estimation. */

param z{i in I, j in 1..m} :=
/* z[i,j] = 1 if item i is in bin j, otherwise z[i,j] = 0 */

if i = 1 and j = 1 then 1
/* put item 1 into bin 1 */

else if exists{jj in 1..j-1} z[i,jj] then 0
/* if item i is already in some bin, do not put it into bin j */

else if sum{ii in 1..i-1} w[ii] * z[ii,j] + w[i] > c then 0
/* if item i does not fit into bin j, do not put it into bin j */

else 1;
/* otherwise put item i into bin j */

check{i in I}: sum{j in 1..m} z[i,j] = 1;
/* each item must be exactly in one bin */

check{j in 1..m}: sum{i in I} w[i] * z[i,j] <= c;
/* no bin must be overflowed */

param n := sum{j in 1..m} if exists{i in I} z[i,j] then 1;
/* determine the number of bins used by the heuristic; obviously it is
an upper bound of the optimal solution */

display n;

set J := 1..n;  /* set of bins */

var x{i in I, j in J}, binary;
/* x[i,j] = 1 means item i is in bin j */

var used{j in J}, binary;
/* used[j] = 1 means bin j contains at least one item */

s.t. one{i in I}: sum{j in J} x[i,j] = 1;
/* each item must be exactly in one bin */

s.t. lim{j in J}: sum{i in I} w[i] * x[i,j] <= c * used[j];
/* if bin j is used, it must not be overflowed */

minimize obj: sum{j in J} used[j];
/* objective is to minimize the number of bins used */

data;

/* The optimal solution is 3 bins */

param m := 6;

param w := 1 50, 2 60, 3 30, 4 70, 5 50, 6 40;

param c := 100;

end;
• Check statement is to verify the on-the-fly parameter construction is correct
• Display statement is a simple way to print data
• The sum statement in defining parameter n: No else part because the if acts as a filter

### Curve fitting by least square

/*Curve fitting problem by Least Squares
Nigel_Galloway@operamail.com
October 1st., 2007
*/
set Sample;
param Sx {z in Sample};
param Sy {z in Sample};

var X;
var Y;
var Ex{z in Sample};
var Ey{z in Sample};

/* sum of variances is zero for Sx*/
variencesX{z in Sample}: X + Ex[z] = Sx[z];
zumVariancesX: sum{z in Sample} Ex[z] = 0;
/* sum of variances is zero for Sy*/
variencesY{z in Sample}: Y + Ey[z] = Sy[z];
zumVariancesY: sum{z in Sample} Ey[z] = 0;

solve;

param b1 := (sum{z in Sample} Ex[z]*Ey[z])/(sum{z in Sample} Ex[z]*Ex[z]);
printf "\nbest linear fit is:\n\ty = %f %s %fx\n\n", Y-b1*X, if b1 < 0 then "-" else "+", abs(b1);

data;

param:
Sample:   Sx    Sy :=
1         0    1
2       0.5  0.9
3         1  0.7
4       1.5  1.5
5       1.9    2
6       2.5  2.4
7         3  3.2
8       3.5    2
9         4  2.7
10       4.5  3.5
11         5    1
12       5.5    4
13         6  3.6
14       6.6  2.7
15         7  5.7
16       7.6  4.6
17       8.5    6
18         9  6.8
19        10  7.3
;

end;
• In the data section, the parameters Sx, Sy and the set Sample are defined in one statement using the tabbing construct. Which Sample is a set of numbers 1..19 and Sx, Sy are corresponding floating point values

### Graph coloring problem

/* COLOR, Graph Coloring Problem */
/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
/* Given an undirected loopless graph G = (V, E), where V is a set of
nodes, E <= V x V is a set of arcs, the Graph Coloring Problem is to
find a mapping (coloring) F: V -> C, where C = {1, 2, ... } is a set
of colors whose cardinality is as small as possible, such that
F(i) != F(j) for every arc (i,j) in E, that is adjacent nodes must
be assigned different colors. */

param n, integer, >= 2;   /* number of nodes */
set V := {1..n};          /* set of nodes */
set E, within V cross V;  /* set of arcs */

check{(i,j) in E}: i != j;  /* there must be no loops */

/* We need to estimate an upper bound of the number of colors |C|.
The number of nodes |V| can be used, however, for sparse graphs such
bound is not very good. To obtain a more suitable estimation we use
an easy "greedy" heuristic. Let nodes 1, ..., i-1 are already
assigned some colors. To assign a color to node i we see if there is
an existing color not used for coloring nodes adjacent to node i. If
so, we use this color, otherwise we introduce a new color. */

set EE := setof{(i,j) in E} (i,j) union setof{(i,j) in E} (j,i);
/* symmetrisized set of arcs */

param z{i in V, case in 0..1} :=
/* z[i,0] = color index assigned to node i
z[i,1] = maximal color index used for nodes 1, 2, ..., i-1 which are
(  if case = 0 then
(  /* compute z[i,0] */
min{c in 1..z[i,1]}
(  if not exists{j in V: j < i and (i,j) in EE} z[j,0] = c then
c
else
z[i,1] + 1
)
)
else
(  /* compute z[i,1] */
if not exists{j in V: j < i} (i,j) in EE then
1
else
max{j in V: j < i and (i,j) in EE} z[j,0]
)
);

check{(i,j) in E}: z[i,0] != z[j,0];
/* check that all adjacent nodes are assigned distinct colors */

param nc := max{i in V} z[i,0];
/* number of colors used by the heuristic; obviously, it is an upper
bound of the optimal solution */

display nc;

var x{i in V, c in 1..nc}, binary;
/* x[i,c] = 1 means that node i is assigned color c */

var u{c in 1..nc}, binary;
/* u[c] = 1 means that color c is used, i.e. assigned to some node */

s.t. map{i in V}: sum{c in 1..nc} x[i,c] = 1;
/* each node must be assigned exactly one color */

s.t. arc{(i,j) in E, c in 1..nc}: x[i,c] + x[j,c] <= u[c];
/* adjacent nodes cannot be assigned the same color */

minimize obj: sum{c in 1..nc} u[c];
/* objective is to minimize the number of colors used */

solve;

printf "Total color: %d\n", (sum{c in 1..nc} u[c]);
printf "Node  Color\n";
printf{i in V, c in 1..nc: x[i,c]=1} "%4d  %5d\n", i, c;

data;

/* These data correspond to the instance myciel3.col from:
http://mat.gsia.cmu.edu/COLOR/instances.html */

/* The optimal solution is 4 */

param n := 11;

set E :=
1 2
1 4
1 7
1 9
2 3
2 6
2 8
3 5
3 7
3 10
4 5
4 6
4 10
5 8
5 9
6 11
7 11
8 11
9 11
10 11
;

end;
• Edges are pair of nodes, thus defined as within V cross V
• EE is defined as the directed edge, which counts both way of E, defined by a union operation
• Max color is found by a heuristic on z, which is by a nested if statement

### Sudoku

/* SUDOKU, Number Placement Puzzle */

/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */

/* Sudoku, also known as Number Place, is a logic-based placement
puzzle. The aim of the canonical puzzle is to enter a numerical
digit from 1 through 9 in each cell of a 9x9 grid made up of 3x3
subgrids (called "regions"), starting with various digits given in
some cells (the "givens"). Each row, column, and region must contain
only one instance of each numeral.

Example:

+-------+-------+-------+
| 5 3 . | . 7 . | . . . |
| 6 . . | 1 9 5 | . . . |
| . 9 8 | . . . | . 6 . |
+-------+-------+-------+
| 8 . . | . 6 . | . . 3 |
| 4 . . | 8 . 3 | . . 1 |
| 7 . . | . 2 . | . . 6 |
+-------+-------+-------+
| . 6 . | . . . | 2 8 . |
| . . . | 4 1 9 | . . 5 |
| . . . | . 8 . | . 7 9 |
+-------+-------+-------+

param givens{1..9, 1..9}, integer, >= 0, <= 9, default 0;
/* the "givens" */

var x{i in 1..9, j in 1..9, k in 1..9}, binary;
/* x[i,j,k] = 1 means cell [i,j] is assigned number k */

s.t. fa{i in 1..9, j in 1..9, k in 1..9: givens[i,j] != 0}:
x[i,j,k] = (if givens[i,j] = k then 1 else 0);
/* assign pre-defined numbers using the "givens" */

s.t. fb{i in 1..9, j in 1..9}: sum{k in 1..9} x[i,j,k] = 1;
/* each cell must be assigned exactly one number */

s.t. fc{i in 1..9, k in 1..9}: sum{j in 1..9} x[i,j,k] = 1;
/* cells in the same row must be assigned distinct numbers */

s.t. fd{j in 1..9, k in 1..9}: sum{i in 1..9} x[i,j,k] = 1;
/* cells in the same column must be assigned distinct numbers */

s.t. fe{I in 1..9 by 3, J in 1..9 by 3, k in 1..9}:
sum{i in I..I+2, j in J..J+2} x[i,j,k] = 1;
/* cells in the same region must be assigned distinct numbers */

/* there is no need for an objective function here */

solve;

for {i in 1..9}
{  for {0..0: i = 1 or i = 4 or i = 7}
printf " +-------+-------+-------+\n";
for {j in 1..9}
{  for {0..0: j = 1 or j = 4 or j = 7} printf(" |");
printf " %d", sum{k in 1..9} x[i,j,k] * k;
for {0..0: j = 9} printf(" |\n");
}
for {0..0: i = 9}
printf " +-------+-------+-------+\n";
}

data;

/* These data correspond to the example above. */

param givens : 1 2 3 4 5 6 7 8 9 :=
1   5 3 . . 7 . . . .
2   6 . . 1 9 5 . . .
3   . 9 8 . . . . 6 .
4   8 . . . 6 . . . 3
5   4 . . 8 . 3 . . 1
6   7 . . . 2 . . . 6
7   . 6 . . . . 2 8 .
8   . . . 4 1 9 . . 5
9   . . . . 8 . . 7 9 ;

end;
• Solving for any feasible solution, without optimizing
• First constraint: The cell at (i,j) with “given”: x[i,j,k] is fixed for all k