8.0 LINGO TUTORIAL
Created by:
Kris Thornburg
Anne Hummel
Table of Contents
Introduction to LINGO 8.0..2
Creating a LINGO Model3
Solving a LINGO Model.4
Using Sets in LINGO..6
The LINGO Data Section8
Variable Types in LINGO.10
Navigating the LINGO Interface...11
LINGO Operators and Functions...14
Common LINGO Error Messages.16
LINGO Programming Examples...17
Introduction to LINGO 8.0
LINGO is a software tool designed to efficiently build and solve linear, nonlinear, and integer
optimization models.
LINGO 8.0 includes several new features, including:
A new global solver to confirm that the solution found is the global optimum,
Multistart capability to solve problems more quickly,
Quadratic recognition and solver to identify quadratic programming (QP) problems,
A faster and more robust Dual Simplex solver,
An improved integer solver to enhance performance in solving many types of problems,
Linearization capability to transform common nonsmooth functions to a series of linear
functions,
Infeasible and unbounded analytical tools to help identify model definition problems,
A decomposition feature to identify if a model contains independent submodels,
A threadsafe DLL for various classes of models, and
More fun than ever before!
1
Creating a LINGO Model
An optimization model consists of three parts:
Objective function This is single formula that describes exactly what the model should
optimize. A general manufacturing example of an objective function would be to
minimize the cycle time for a given product.
Variables These are the quantities that can be changed to produce the optimal value of
the objective function. For example, when driving a car, the duration of the trip (t) and
the speed at which it is taken (v) determine the distance (d) that can be traveled.
Constraints These are formulas that define the limits on the values of the variables. If
an ice cream store is determining how many flavors it should offer, only a positive
number of flavors is feasible. This constraint could be expressed as
Flavors >= 0;
A sample model for cookie production by two bakers at a bakery is given by:
! A cookie store can produce drop cookies and decorated cookies, which sell
for $1 and $1.50 apiece, respectively. The two bakers each work 8 hours per
day and can produce up to 400 drop cookies and 200 decorated cookies. It
takes 1 minute to produce each drop cookie and 3 minutes to produce each
decorated cookie. What combination of cookies produced will maximize the
baker's profit? ;
MAX = 1*Drop + 1.5*Deco;
Drop <= 400;
Deco <= 200;
1/60*Drop + 3/60*Deco <=16;
Several other items must be noted about this model:
Comments in the model are initiated with an exclamation point (!) and appear in green
text.
LINGO specified operators and functions appear in blue text.
All other text is shown in black.
Each LINGO statement must end in a semi-colon (;).
Variable names are not case-sensitive and must begin with a letter (A-Z). Other
characters in the variable name may be letters, numbers (0-9), or the underscore character
(_). Variable names can be up to 32 characters in length.
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Solving a LINGO Model
Once the LINGO model has been entered into the LINGO Model window, the model can be
solved by clicking the Solve button on the toolbar, by selecting LINGO | Solve from the menus,
or by using the ctrl + s keyboard shortcut.
LINGO will notify you of any errors it has encountered. The best way to get information about
these errors is to consult the Error Messages section in the softwares proprietary tutorial.
If no errors are found, then the LINGO Solver Status window appears.
This window provides information on the number of nonlinear, integer, and total variables in the
model; the nonlinear and total number of constraints used in the model; and the number of
nonlinear and total nonzero variable coefficients used. The Solver Status box in this window
details the model classification (LP, QP, ILP, IQP, NLP, etc.), state of the current solution (local
or global optimum, feasible or infeasible, etc.), the value of the objective function, the
infeasibility of the model (amount constraints are violated by), and the number of iterations
required to solve the model. The Extended Solver Status box details similar information for the
more advanced branch-and-bound, global, and multistart solvers.
By closing this window, you can then view the Solution Report window.
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This window shows the values of each variable that will produce the optimal value of the
objective function.
The reduced cost for any variable that is included in the optimal solution is always zero. For
variables not included in the optimal solution, the reduced cost shows how much the value of the
objective function would decrease (for a MAX problem) or increase (for a MIN problem) if one
unit of that variable were to be included in the solution. For example, if the reduced cost of a
certain variable was 5, then the optimal value of the MAX problem would decrease by 5 units if
1 unit of the variable were to be added.
The Slack or Surplus column in the Solution Report shows how tight the constraint is. If a
constraint is completely satisfied as an equality, then slack/surplus is zero. If slack/surplus is
positive, then this tells you how many more units of the variable could be added to the optimal
solution before the constraint becomes an equality. If slack/surplus is negative, then the
constraint has been violated.
The Dual Price column describes the amount to which the value of the objective function would
improve if the constraining value is increased by one unit.
4
Using Sets in LINGO
LINGO allows you to group many instances of the same variable into sets. For example, if a
model involved 27 delivery trucks, then these 27 trucks could be described more simply as a
single set. Sets may also include attributes for each member, such as the hauling capacity for
each delivery truck.
Sets may be either primitive or derived. A primitive set is one that contains distinct members. A
derived set, however, contains other sets as its members.
To use sets in a model, the special section called the SETS section must be defined before any of
the set members are used in the models constraints. This section begins with the tag SETS:
and ends with the tag ENDSETS.
A primitive set could be defined as follows:
SETS:
Trucks/TR1..TR27/:Capacity;
ENDSETS
This set is given the setname Trucks and contains 27 members, identified by TR1 TR27.
The attributes for each member are called Capacity.
The derived set is defined similarly, but must also include the parent set list. An example of a
derived set could be:
SETS:
Product/X Y/;
Machine/L M/;
Make(Product Machine)/X L, X M,
Y M/;
ENDSETS
This set declaration defines two primitive sets, Product and Machine, and one derived set called
Make. The Make set is derived from the parent sets Product and Machine. Members are
specified as shown. Notice that a fourth Product-Machine combination, Y L, could be
theoretically possible. This example does not allow for such a combination. If all combinations
of the parent sets are possible, then no member set need be defined. An attribute list for the
derived set can also be included in the same way as for a primitive set.
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Several set looping functions are also available for use in LINGO. These functions are as
follows:
@FOR generates constraints over members of a set.
@SUM sums an expression over all members of the set.
@MIN computes the minimum of an expression over all members of the set.
@MAX computes the maximum of an expression over all members of the set.
Each of the above looping functions has a similar form of syntax and the looping functions can
even be nested. Examples of expressions using each type of looping function are as follows:
This @FOR statement sets the hauling capacity for all 27 delivery trucks in the Trucks
set to at most 3000 pounds:
@FOR(Trucks(T): Capacity(T)<=3000);
This @SUM statement calculates the total hauling capacity from the individual trucks:
TOTAL_HAUL=@SUM(Trucks(J): Capacity(J));
These @MIN and @MAX statements find the extreme hauling capacity levels from the
individual delivery trucks:
MIN_HAUL = @MIN(Trucks(J): Capacity(J));
MAX_HAUL = @MAX(Trucks(J): Capacity(J));
6
The LINGO Data Section
LINGO provides a separate section called the DATA section in which values can be defined for
different variables. Set members can be initialized in this section, attributes of the sets can be
defined, or scalar variable parameters can be assigned values as well.
The DATA section is defined after the SETS section is defined in the model. The section begins
with the tag DATA: and ends with the tag ENDDATA. Statements within the DATA section
follow the syntax: object_list = value_list;
The object list contains the names of the attributes or of the set whose values are being
initialized. The value list assigns the values to the specified members of the object list.
The following examples show two ways to use the DATA section in LINGO. In each example,
the X and Y attributes of SET1 are being initialized to [1, 2, 3] and [4, 5, 6], respectively. The
first example defines values for each attribute separately:
SETS:
SET1 /A, B, C/: X, Y;
ENDSETS
DATA:
X = 1, 2, 3;
Y = 4, 5, 6;
ENDDATA
The next example shows how one statement can be used to assign values to the two attributes
simultaneously. Each row assigns different values to the X, Y pair:
SETS:
SET1 /A, B, C/: X, Y;
ENDSETS
DATA:
X, Y = 1, 4,
2, 5,
3, 6;
ENDDATA
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When parameters or attributes are defined in the DATA section of a model, a feature called Whatif Analysis can be used to examine the effects of varying the value of the parameter or attribute.
For example, if the inflation rate is most likely going to fall between 2% and 6%, the parameter
can be defined as follows in the DATA section:
DATA:
INFLATION_RATE = ?;
ENDDATA
When LINGO encounters the ? in the DATA section, it will prompt the user to enter a value for
the parameter. The user can then enter values between 2% and 6%, and LINGO will solve the
model using that what-if value.
All the elements of an attribute can be initialized to a single value using the DATA section as
well. following The example shows how to assign the value of 20 to all seven members of the
NEEDS attribute and 100 to all seven members of the COST attribute:
SETS:
DAYS / MO, TU, WE, TH, FR, SA,
SU/: NEEDS, COST;
ENDSETS
DATA:
NEEDS, COST = 20, 100;
ENDDATA
Data values can also be omitted from the DATA section of a LINGO model to indicate that
LINGO is free to determine the values of those attributes itself. The following example shows
that the first two values of the attribute CAPACITY have been initialized to 34, but the last three
variables have not been defined:
SETS:
YEARS /1..5/: CAPACITY;
ENDSETS
DATA:
CAPACITY = 34, 34, , , ;
ENDDATA
8
Variable Types in LINGO
All variables in a LINGO model are considered to be non-negative and continuous unless
otherwise specified. LINGOs four variable domain functions can be used to override the default
domain for given variables. These variable domain functions are:
@GIN any positive integer value
@BIN a binary value (ie, 0 or 1)
@FREE any positive or negative real value
@BND any value within the specified bounds
Similar syntax is used for the @GIN, @BIN, and @FREE variable domain functions. The
general form for the declaration of a variable x using any of these functions is
@FUNCTION(X);
The @BND function has a slightly modified syntax, which includes the upper and lower bounds
for the acceptable variable values. The general form for the declaration of a variable x between a
lower bound and an upper bound is given by
@BND(lowerBound, X, upperBound);
9
Navigating the LINGO Interface
Operations in LINGO can be carried out using commands from the menus, toolbars, or shortcut
keys.
There are five menus in the main LINGO window. They are the File, Edit, LINGO, Window,
and Help menus.
The following list details the commands in the File menu. Shortcut keys are included in
parentheses when available:
New (F2)
Open (F3)
Save (F4)
Save As (F5)
Close (F6)
Print (F7)
Print Setup (F8)
Print Preview (Shift+F8)
Log Output (F9)
Take Commands (F11)
Import LINDO File (F12)
Export File
License
Database User Info
Exit (F10)
Open a new model window
Open a saved file
Save a current model
Save a current model to a new filename
Close the current model
Prints the current windows content
Configures printer preferences
Displays the window content as it would look if printed
Opens a log file to log output to the command window
Runs a command script contained in a file
Converts a LINDO file into a LINGO model
Exports a model in MPS or MPI file format
Prompts user for new license password to upgrade system
Prompts for a user id and password for database access via the
@ODBC() function
Closes LINGO
The Edit menu contains the following commands:
Undo (ctrl+z)
Redo (ctrl+y)
Cut (ctrl+x)
Copy (ctrl+c)
Paste (ctrl+v)
Paste Special
Undoes the last action
Redoes the last undo command
Copies and deletes highlighted text
Copies highlighted text to the clipboard
Pastes the clipboards contents into the document
Pastes the clipboards content into the document, in a userspecified manner
Select All (ctrl+a)
Selects all of the contents of the current window
Find (ctrl+f)
Searches the document for a specific text string
Find Next (ctrl+n)
Searches the document for the next occurrence of a specific text
string
Replace (ctrl+h)
Replaces a specific text string with a new string
Go To Line (ctrl+t)
Moves the cursor to a certain line number
Match Parenthesis (ctrl+p) Finds the selected parenthesiss mate
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Paste Function
Select Font (ctrl+j)
Insert New Object
Links
Object Properties
Pastes a syntax template for a specific LINGO @function
Configures the font for a selected portion of text
Puts an OLE object in the document
Creates links to external objects
Defines the properties of an embedded object
The LINGO menu contains the following commands:
Solve (ctrl+s)
Solution (ctrl+o)
Range (ctrl+r)
Options (ctrl+i)
Generate
Picture (ctrl+k)
Debug (ctrl+d)
Model Statistics (ctrl+e)
Look (ctrl+l)
Solves the model
Makes a solution report window for the current model
Creates a range analysis report for the current window
Sets system options
Generates the algebraic form of the model
Displays a graphic of the models matrix
Identifies errors in infeasible and unbounded models
Reports the technical details the model
Creates a formulation report for the current window
The Window menu contains the following commands:
Command Window (ctrl+1)
Status Window (ctrl+2)
Send to Back (ctrl+b)
Close All (ctrl+3)
Tile (ctrl+4)
Cascade (ctrl+5)
Arrange Icons (ctrl+6)
Opens a window for command-line operation of LINGO
Opens the solver's status window
Places the current window behind all other open windows
Closes all open windows
Places open windows in a tiled arrangement
Places all open windows in a cascaded arrangement
Aligns icon windows at the bottom of the main LINGO window
The Help window contains the following commands:
Help Topics
Register
AutoUpdate
About LINGO
Opens LINGOs manual
Registers your version of LINGO online
Provides prompts to download software updates
Displays software information
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A single toolbar located at the top of the main LINGO window contains many of the same
commands as listed above. These commands can be accessed simply by using the mouse to click
on the icon representing them. The following pictures detail which icons correspond to which
commands.
New
Save
Open
Find
Cut
Print
Paste
Copy
Redo
Undo
Match
Close
Parenthesis Solution Options All
Go To Line Solve Matrix
Picture
Help
Topics
Send to Tile
Back
Help
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LINGO Operators and Functions
LINGO provides a vast array of operators and functions, making it a useful problem-solving tool.
A selection of the primary operators and functions is given below.
There are three types of operators that LINGO uses: arithmetic, logical, and relational operators.
The arithmetic operators are as follows:
Exponentiation: ^
Multiplication: *
Division: /
Addition: +
Subtraction: -
The logical operators are used in set looping functions to define true/false conditions:
#LT#: TRUE if the left argument is strictly less than the right argument, else FALSE
#LE#: TRUE if the left argument is less-than-or-equal-to the right argument, else FALSE
#GT#: TRUE if the left argument is strictly greater than the right argument, else FALSE
#GE#: TRUE if the left argument is greater-than-or-equal-to the right argument, else
FALSE
#EQ#: TRUE if both arguments are equal, else FALSE
#NE#: TRUE if both arguments are not equal, else FALSE
#AND#: TRUE only if both arguments are TRUE, else FALSE
#OR#: FALSE only if both arguments are FALSE, else TRUE
#NOT#: TRUE if the argument immediately to the right is FALSE, else FALSE
The relational operators are used when defining the constraints for a model. They are as follows:
The expression is equal: =
The left side of the expression is less than or equal to the right side: <=
The left side of the expression is greater than or equal to the right side: >=
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The following list contains a sampling of mathematical functions that can be used in LINGO:
@ABS(X) returns the absolute value of X
@SIGN(X) returns -1 if X is negative and +1 if X is positive
@EXP(X) calculates eX
@LOG(X) calculates the natural log of X
@SIN(X) returns the sine of X, where X is the angle in radians
@COS(X) returns the cosine of X
@TAN(X) returns the tangent of X
LINGO also contains a plethora of financial, probability, and import/export functions. These are
commonly used in more advanced models, which are beyond the intended scope of this tutorial.
14
Common LINGO Error Messages
LINGO provides a variety of error messages useful for debugging a developing model. The
most common errors include the following:
7: Unable to open file: filename
o Retype filename correctly
11: Invalid input: A syntax error has occurred
o Check the line LINGO suggests for missing semi-colons, etc.
12: Unmatched parenthesis
o Close the parenthesis set
15: No relational operator found
o Make sure all constraints contain =, <=, >=
44: Unterminated condition
o Put a colon at the end of each conditional statement in a set operator
50: Improper use of the @FOR() function
o @FOR() functions cannot be nested inside other set operators
68: Multiple objective functions in model
o Only one is allowed, please
71: Improper use of a variable domain function (eg, @GIN, @BIN, @FREE, @BND)
o Check the syntax
81: No feasible solution found
o Check models consistency and constraints
82: Unbounded solution
o Add constraints
102: Unrecognized variable name: variable name
o Check spelling
108: The models dimensions exceed the capacity of this version
o Upgrade to full version or use Excel
164: Invalid LINGO name
o Create a name to conform to LINGOs naming conventions
15
LINGO Programming Examples
A common programming model is the Knapsack problem, which deals with maximizing the
utility of loading capacity. This example shows how to properly set up a knapsack problem.
SETS:
ITEMS / ANT_REPEL, BEER, BLANKET,
BRATWURST, BROWNIES, FRISBEE,
SALAD, WATERMELON/:
INCLUDE, WEIGHT, RATING;
ENDSETS
DATA:
WEIGHT RATING =
1
2
3
9
4
3
3
8
3 10
1
6
5
4
10 10;
KNAPSACK_CAPACITY = 15;
ENDDATA
MAX = @SUM( ITEMS: RATING * INCLUDE);
@SUM( ITEMS: WEIGHT * INCLUDE) <=
KNAPSACK_CAPACITY;
@FOR( ITEMS: @BIN( INCLUDE));
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Another common programming model is the transportation problem. Transportation problems
deal with transporting goods from one location to another at minimal cost. This example shows
how to model a simple transportation problem.
MODEL:
! A 6 Warehouse 8 Vendor Transportation Problem;
SETS:
WAREHOUSES / WH1 WH2 WH3 WH4 WH5 WH6/: CAPACITY;
VENDORS / V1 V2 V3 V4 V5 V6 V7 V8/ : DEMAND;
LINKS( WAREHOUSES, VENDORS): COST, VOLUME;
ENDSETS
! The objective;
MIN = @SUM( LINKS( I, J):
COST( I, J) * VOLUME( I, J));
! The demand constraints;
@FOR( VENDORS( J):
@SUM( WAREHOUSES( I): VOLUME( I, J)) =
DEMAND( J));
! The capacity constraints;
@FOR( WAREHOUSES( I):
@SUM( VENDORS( J): VOLUME( I, J)) <=
CAPACITY( I));
! Here is the data;
DATA:
CAPACITY = 60 55 51 43 41 52;
DEMAND = 35 37 22 32 41 32 43 38;
COST = 6 2 6 7 4 2 5 9
49538582
52197433
76739271
23957265
5 5 2 2 8 1 4 3;
ENDDATA
END
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