Introduction to AMPL

Syntax

• declare parameters, sets, and variables using keywords param, set and var. Notice that each statement ends with semicolon.

param parameter1; var variable1 >=0; set S:= 1..s;

• express objective function using keyword minimize or maximize; constraints after using the keyword subject to. Both objective function and constraints must have names such as objectivef, cons1. Notice colons after names.

minimize objective f: 100*x1 + 75*x2;

subject to
cons1: x1 <= 10;

• AMPL statements always end with a semicolon. Any line or part of a line after “#” is only a comment. Blank lines have no impact¹.

• sum{i in 1..5}does summation over index i. \(\sum_{i=1}^5\)

• In all cases where objects are defined over index ranges, the range is shown within “{ }” brackets¹. Example:

var x{1..5};

• Subscripts are called out within square “[]” brackets¹. Example:

x[1], x[5]

• The symbol “:=” is used to indicate assignment of a constant’s value¹. Example:

param l:= 5;

• No commas or other delimiters separate data values¹.

param a:= 1 10 2 5; here a is a vector of length 2 and it needs to be declared before assigning values.

• Single-subscript parameters are assigned in a list of subscript-value pairs¹.

• Multiple-subscript parameters are assigned in a table with columns corresponding to the last of the subscripts and rows for each combination of the other indices¹. Example:

param p:= 1 2 3:=

1 10 20 30

2 12 14 16;

this is 2x3 matrix data.

• You can use console of AMPL software to code line by line or better use a txt (other extensions exist too) file to write and modify your code and read this txt file into AMPL. model keyword sources the txt (or other extension) file.

• Once a model is coded, data can be sparely entered after the keyword data.

• The keyword solve is used to call a solver to solve the model.

• display displays the solution in terms of variables or objective function (whichever is written after).
• warning!: use reset keyword to delete the existing model in AMPL.

Two Crude Example

• define variables x1, x2

var x1 >=0; # non-negativity is expressed here var x2 >=0

• express the objective function explicitly
  

minimize z: 100*x1 + 75*x2; #notice the name z and colon after

• express main constraints

subject to

c1: 0.3*x1 + 0.4*x2 >= 2;

c2: 0.4*x1 + 0.2*x2 >= 1.5;

c3: 0.2*x1 + 0.3*x2 >= 0.5;

c4: x1 <= 9; c5: x2 <= 6;

• solve and display results

solve; display x1,x2;

Generalized Model

• define parameters to be used in sets

param m; param n;

• define sets to be used for indexing

set R = 1..m; set A = 1..n;

• define cost coefficients to be used in objective function

param cost{A};

• define right hand side of requirement constraints

param requireRHS{R};

• define coefficients for unit contribution to requirement satisfaction

param require{R,A};

• define availability right hand sides

param availRHS{A};

• define variables

var x{A} >= 0;

• express objective function
  

minimize z: sum{j in A}cost[j]*x[j]; #look like

\(\sum_{j=1}^ncost_jx_j\)

• express requirement constraints

subject to

c1{i in R}: sum{j in A}require[i,j]*x[j] >= requireRHS[i]; #look like

\(\sum_{j=1}^nrequire_{i,j}x_i\ge requireRHS_i \ \forall i\)

• express availability constraints

c2{j in A}: x[j] <= availRHS[j]; #look like

\(x_j\le availRHS_j \ \forall j\)

• enter data

data;

param m:= 3; param n:= 2;

param cost:= 1 100 2 75;

param requireRHS:= 1 2 2 1.5 3 0.5;

param require: 1 2:=

1 0.3 0.4

2 0.4 0.2

3 0.2 0.3;

param availRHS:= 1 9 2 6;

• link to cplex solver

option solver cplex

• solve and display results

solve; display x;

Practice Example

\(maximize\ 3x_1 + 4x_2 + 2x_3\)

\(subject\ to\)

\(x_1 + 0.5x_2 + 5x_3 \le 2\)

\(2x_1 -x_2 +3x_3 \le 3\)

\(x1,x2,x3 \ge 0\)

• define parameter n which will indicate number of variables–in this example the value of n is three.

  param n;

• define a parameter m which will indicate number of main constraints–in this example the value of m is two.

  param m;

• define a set which will have elements from 1 to n;

  set V:= 1..n;

• define a set which will have elements from 1 to m;

  set R:= 1..m;

• define a profit vector to account for the objective function coefficients

  param profit{V};

• define constraint coefficient matrix;

  param coeff{R,V};

• define RHS of constraints

  param RHS{R};

• define variable vector

  var x{V} >=0;

• express objective function

  maximize z: sum{i in V}profit[i]*x[i];

• express contraints

  subject to

  const{j in R}: sum{i in V}coeff[j,i]*x[i] <= RHS[j];

• enter data

  data;

  param n:= 3; param m:=2;

  param profit:= 1 3 2 4 3 -2;

  param coeff: 1 2 3:=

  1 1 0.5 5

  2 2 -1 3;
  param RHS:= 1 2 2 3;

• set solver and solve, display results

  option solver cplex; solve; display x;

Display Options

show lists the components of the model

Also try: show vars, obj, constr;

xref displays dependencies on a specific model component: xref variable_1;

expand displays the model component explicitly: expand z;

option omit_zero_rows 1 suppresses 0 values.

option show_stats 1 shows model summary.

print, printf allows flexibility in displaying modified results: printf “Total revenue is $%6.2f.\n”, sum {p in PROD, t in 1..T} revenue[p,t]*Sell[p,t];

Total revenue is $787810.00 [this example is from the AMPL book page 239].

• ‘>’ or ‘>>’ writes results into a file.

display supply > multi.out [page 251]

• read{range} x[.] < foo.txt reads from an external file.

read{i in I}RHS[i] < RHS.txt

Please see the AMPL book and colab notebooks in Black Board to practice more on AMPL. Here is the link to the AMPL book: AMPL book

Footnotes

1. R. Rardin, Optimization in Operations Research