Parameters in AMPL

1. Description
2. Declaring a Parameter
3. Parameter Types
4. Parameter Bounds
5. Default Values
6. Defining a Parameter
7. Accessing a Parameter

Description

Parameters hold "hard" values in AMPL. The values of parameters can be defined and changed in AMPL, but a solver will not change them while looking for an optimal solution.

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Declaring a Parameter

AMPL parameters are created in a similar way to AMPL variables, using the param keyword followed by a label.

param <paramname>;

Example

param MinProtein;

Like variables parameters are often defined over a set and may have several attributes:

param <paramname> [{<indexname>}] [<attributes>];

Example

param ProteinPercent {INGREDIENTS} >= 0 <= 100;

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Parameter Types

Parameters have the same possible types as variables. However, since parameter values are defined (not searched for), declaring a parameter type means that AMPL will ensure that the parameter is only ever assigned that type of data. For example, you cannot assign an integer parameter the value 1.5. This behaviour is very useful for automatically checking the validity of data files.

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Parameter Bounds

As well as using parameter types to check the validity of data, real and integer parameters can also have bounds set during their declaration. These bounds will be checked by AMPL any time the value of the parameter changes and, if they are violated, and error will be generated.

Example

param counter integer >= 0;

let counter := -1; # This generates an error as counter is < 0

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Default Values

Default parameter values can be used to quickly set a large number of parameter values automatically. If a parameter is used without being explicitly assigned a value the default value is used for that parameter. AMPL uses a default value of 0 if no default value is given.

Example

set DIGITS := 1..5;
param isok {DIGITS} binary default 1;

let isok[3] := 0;

display {i in DIGITS} isok[i];

# Result
# ======
# isok[i] [*] :=
# 1  1
# 2  1
# 3  0
# 4  1
# 5  1
# ;

The AMPL macros Infinity and -Infinity are useful as defaults for parameters that act as bounds (Infinity as a default upper bound, 0 or -Infinity as a default lower bound).

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Defining a Parameter

Once a parameter has been declared it is usually defined in a data file. This is done simply for a single value using the assignment operator :=:

param MinProtein := 8.0 ;

For parameters declared over a 1-dimensional set this can be done using default values and a list for those parameters that don't take default values:

model;

param Min {REQUIREMENTS} default -Infinity;

data;

param Min :=
PROTEIN 8.0
FAT     6.0
 ;

Defining 2-Dimensional Parameters

In a similar way to 2-dimensional sets, there are three different ways to define 2-dimensional sets.

1. Using a List For any parameter values that don't take the default value, you list the set element and value for that parameter.

   model;
   
   param Min {ARCS} integer, default 0;
   
   data;
   
   param                      Min :=
   Youngstown  'Kansas City'  1000
   Pittsburgh  'Kansas City'  2000
   Cincinnati   Albany        1000
   

2. Using a Table To define parameter data in a table format you use the param keyword and the parameter's name followed by the : operator, a list of the second index set elements followed by the := operator, then rows of the table with an element of the first index set followed by the values corresponding to the second index set's element in that column.

   param <paramname> :
         <j1>            <j2>            ... <jn> :=
   <i1>  <value(i1, j1)> <value(i1, j2)> ... <value(i1, jn)>
   ...
   <im>  <value(im, j1)> <value(im, j2)> ... <value(im, jn)>
    ;
   

   If the element does not exist or the default value is correct then place a . in the table. Otherwise, put the parameter value.

   param Cost:   Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary :=
   Youngstown        350        450       375    500        .      .    .
   Pittsburgh        350        450       400     .         .      .   450
   Cincinnati         .          .         .     350       550     .    .
   'Kansas City'      .          .         .      .        375    650   .
   Chicago            .          .         .      .         .     600  120 ;
   

   You can also define parameter data in a transposed table using almost the same syntax, but with the (tr) keyword and reversing the indexing sets

   param <paramname> (tr) :
         <i1>            <i2>            ... <im> :=
   <j1>  <value(i1, j1)> <value(i2, j1)> ... <value(im, j1)>
   ...
   <jn>  <value(i1, jn)> <value(i2, jn)> ... <value(im, jn)>
    ;
   

3. Using an Array You define a list of column indices and values for each row index.

   param           Cost :=
   [Youngstown, *] Cincinnati    350 'Kansas City' 450 ...
   ...
   [Chicago, *]    ...                Gary         120 ;
   

   Note The row indices have [=} and =] around them (as opposed to ( and ) for sets).

Defining Multi-Dimensional Parameters

Since we have multi-dimensional sets, we might need multi-dimensional parameters, e.g., Cost {TIME_ARCS} has four dimensions. We can define these parameters in a similar way to multi-dimensional sets:

1. Using a List

   param Cost :=
   Youngstown April Albany     April 0.5    # = 500 / 1000
   Youngstown April Youngstown May   0.015  # =  15 / 1000
   ... ;
   

2. Using a Table

   param Cost : =
   [*, May, *, May]  Cincinnati 'Kansas City' Albany ... :=
   Youngstown         0.35      0.45          0.5    ...
   Pittsburgh         0.35      0.45          .      ...
   ... ;
   

   Notice the [ ] around *, May, *, May as opposed to the ( ) for sets!
3. Using an Array

   set TIME_ARCS :=
   (*, May, *, May) (Youngstown, Cincinnati) 0.35
   ... ;
   

   or

   set TIME_ARCS :=
   (Youngstown, May, *, May) Cincinnati 0.35 'Kansas City' 0.45 ...
   ... ;
   

??? Up to here ???

Defining Multiple Parameters

Using the {\tt :} operator, multiple parameters may be defined at once. Simply state the names of the parameters and the {\tt :=} operator. Then list the set elements and values on the following rows.

\begin{verbatim} param: ... : <value1,1> <value1,2> ... <value2,1> <value2,2> ... ; \end{verbatim}

If a parameter is not defined or the default value is sufficient, use the {\tt .} operator.

\begin{verbatim} model;

# The lower and upper bounds on the requirements param Min {REQUIREMENTS} default -Infinity; param Max {REQUIREMENTS} default Infinity;

data; param: Min Max:= PROTEIN 8.0 . FAT 6.0 . FIBRE . 2.0 SALT . 0.4 ; \end{verbatim}

This approach also works for 2-dimensional parameters and lists, for the American Steel problem this allows us to "cut-and-paste" the list of arc properties

\begin{verbatim} From node To node Cost Minimum Maximum Youngstown Albany 500 - 1000 Youngstown Cincinnati 350 - 3000 Youngstown Kansas City 450 1000 5000 Youngstown Chicago 375 - 5000 etc \end{verbatim}

becomes

\begin{verbatim} param: Cost Min Max:= Youngstown Cincinnati 350 0 3000 Youngstown 'Kansas City' 450 1000 5000 ... Chicago Gary 120 0 4000 ; \end{verbatim}

Accessing a Parameter

Parameter values are accessed by specifying the indices of the parameter you want to access within {\tt [} and {\tt ]}.

Examples

See {\tt Cost} and {\tt Contributes} below.

\begin{verbatim} # Objective: minimise the cost per (100g) can minimize TotalCost: sum {i in INGREDIENTS} Cost[i] * Percentage[i];

# Constraints: Meet the nutritional requirements

subject to MeetRequirement {r in REQUIREMENTS}: Min[r] <= sum {i in INGREDIENTS} Contributes[i, r] * Percentage[i] <= Max[r]; \end{verbatim}