MPS file format
The main things to know about MPS format are that it is column oriented
(as opposed to entering the model as equations), and everything
(variables, rows, etc.) gets a name.
MPS is an old format, so it is set up as though you were using punch
cards, and is not free format. Fields start in column 2, 5, 15, 25, 40
and 50. Sections of an MPS file are marked by so-called header cards,
which are distinguished by their starting in column 1. Although it is
typical to use upper-case throughout the file (like I said, MPS has
long historical roots), many MPS-readers will accept mixed-case for
anything except the header cards, and some allow mixed-case anywhere.
The names that you choose for the individual entities (constraints or
variables) are not important to the solver; you should pick names that
are meaningful to you, or will be easy for a post-processing code to
Here is a little sample model written in MPS format (explained in more
XONE COST 1 LIM1 1
XONE LIM2 1
YTWO COST 4 LIM1 1
YTWO MYEQN -1
ZTHREE COST 9 LIM2 1
ZTHREE MYEQN 1
RHS1 LIM1 5 LIM2 10
RHS1 MYEQN 7
UP BND1 XONE 4
LO BND1 YTWO -1
UP BND1 YTWO 1
For comparison, here is the same model written out in lp-format:
min: +XONE +4 YTWO +9 ZTHREE;
LIM1: +XONE +YTWO <= 5;
LIM2: +XONE +ZTHREE >= 10;
MYEQN: -YTWO +ZTHREE = 7;
XONE <= 4;
YTWO >= -1;
YTWO <= 1;
Strangely, there is nothing in MPS format that specifies the direction
of optimisation. And there really is no standard "default" direction;
some LP codes will maximize if you don't specify otherwise, others will
minimize, and still others put safety first and have no default and
require you to specify it somewhere in a control program or by a
calling parameter. If you have a model formulated for minimization
and the code you are using insists on maximization (or vice versa), it
may be easy to convert: just multiply all the coefficients in your
objective function by (-1). The optimal value of the objective
function will then be the negative of the true value, but the values of
the variables themselves will be correct.
Any line with an asterisk (*) in Column 1 is treated as a comment.
The eight character names used to specify variables, constraints and
other entities are fixed format. Names are not automatically justified,
so blanks are treated just like other characters. For example "ROW1 "
is not the same as " ROW1 ". (Note that some optimisers do not permit
blanks in names.) No case conversion is performed, so "row1 " is
different from "ROW1 ".
Floating point numbers may be specified in free format within the 12
character field (including embedded blanks). The following list describes
the possible ways of writing a number.
+ or - optional sign character (no sign indicates a positive number)
digits optional integer part of the mantissa
. optional decimal point (if not present, a decimal point will
be assumed after the mantissa digit)
digits optional fraction part of the mantissa -the mantissa must
contain at least one digit
D or E exponent leader
+ or - optional exponent sign
digits exponent digits
Numbers with an absolute value greater than 1010 or less than 10-10 are rejected.
The NAME card can have anything you want, starting in column 15. The
ROWS section defines the names of all the constraints; entries in
column 2 or 3 are E for equality rows, L for less-than ( <= ) rows, G
for greater-than ( >= ) rows, and N for non-constraining rows (the
first of which would be interpreted as the objective function). The
order of the rows named in this section is unimportant.
The largest part of the file is in the COLUMNS section, which is the
place where the entries of the A-matrix are put. All entries for a given
column must be placed consecutively, although within a column the
order of the entries (rows) is irrelevant. Rows not mentioned for a
column are implied to have a coefficient of zero.
The RHS section allows one or more right-hand-side vectors to be
defined; most people don't bother having more than one. In the above
example, the name of the RHS vector is RHS1, and has non-zero values
in all 3 of the constraint rows of the problem. Rows not mentioned in
an RHS vector would be assumed to have a right-hand-side of zero.
The optional BOUNDS section lets you put lower and upper bounds on
individual variables (no * wild cards, unfortunately), instead of
having to define extra rows in the matrix. All the bounds that have
a given name in column 5 are taken together as a set. Variables not
mentioned in a given BOUNDS set are taken to be non-negative (lower
bound zero, no upper bound). A bound of type UP means an upper bound
is applied to the variable. A bound of type LO means a lower bound is
applied. A bound type of FX ("fixed") means that the variable has
upper and lower bounds equal to a single value. A bound type of FR
("free") means the variable has neither lower nor upper bounds.
There is another optional section called RANGES that I won't go into
here. The final card must be ENDATA, and yes, it is spelled funny.
MPS input format was originally introduced by IBM to express linear
and integer programs in a standard way. The format is a fixed column
format, so care must be taken that all information is placed in the
correct columns as described below.
The following is not intended as a complete description of MPS format,
but only as a brief introduction. For more information, the reader is
"Advanced Linear Programming," by Bruce A. Murtagh
"Computer Solutions of Linear Programs," by J.L. Nazareth
It may be useful to look at an example MPS file while reading this
The following template is a guide for the use of MPS format:
Field: 1 2 3 4 5 6
Columns: 2-3 5-12 15-22 25-36 40-47 50-61
NAME problem name
column row value row value
name name name
rhs row value row value
name name name
range row value row value
name name name
type bound column value
type CaseName SOSName SOSpriority
CaseName VarName1 VarWeight1
CaseName VarName2 VarWeight2
CaseName VarNameN VarWeightN
A. In the ROWS section, each row of the constraint matrix must have a
row type and a row name specified. The code for indicating row type
is as follows:
L less than or equal
G greater than or equal
N no restriction
B. In the COLUMNS section, the names of the variables are defined along
with the coefficients of the objective and all the nonzero constraint
matrix elements. It is not necessary to specify columns for slack or
surplus variables as this is taken care of automatically.
C. The RHS section contains information for the right-hand side of the problem.
D. The RANGES section is for constraints of the form: h <= constraint <= u .
The range of the constraint is r = u - h . The value of r is specified
in the RANGES section, and the value of u or h is specified in the RHS
section. If b is the value entered in the RHS section, and r is the
value entered in the RANGES section, then u and h are thus defined:
row type sign of r h u
G + or - b b + |r|
L + or - b - |r| b
E + b b + |r|
E - b - |r| b
E. In the BOUNDS section, bounds on the variables are specified. When
bounds are not indicated, the default bounds ( 0 <= x < infinity )
are assumed. The code for indicating bound type is as follows:
LO lower bound b <= x (< +inf)
UP upper bound (0 <=) x <= b
FX fixed variable x = b
FR free variable -inf < x < +inf
MI lower bound -inf -inf < x (<= 0)
PL upper bound +inf (0 <=) x < +inf
BV binary variable x = 0 or 1
LI integer variable b <= x (< +inf)
UI integer variable (0 <=) x <= b
SC semi-cont variable x = 0 or l <= x <= b
l is the lower bound on the variable
If none set then defaults to 1
F. Sections RANGES and BOUNDS are optional as are the fields 5 and 6.
Everything else is required. In regards to fields 5 and 6, consider
the following 2 constraints:
const1: 2x + 3y <= 6
const2: 5x + 8y <= 20
Two ways to enter the variable x in the COLUMNS section are:
(Field: 2 3 4 5 6 )
1. x const1 2.0 const2 5.0
2. x const1 2.0
x const2 5.0
G. A mixed integer program requires the specification of which variables
are required to be integer. Markers are used to indicate the start
and end of a group of integer variables. The start marker has its
name in field 2, 'MARKER' in field 3, and 'INTORG' in field 5. The
end marker has its name in field 2, 'MARKER' in field 3, and 'INTEND'
in field 5. These markers are placed in the COLUMNS section.
H. A specially ordered set of degree N is a collection of variables where
at most N variables may be non-zero. The non-zero variables must be
contiguous (neighbours) sorted by the ascending value of their respective
unique weights. In lp_solve, specially ordered sets may be of any
cardinal type 1, 2, and higher, and may be overlapping. The number of
variables in the set must be equal to, or exceed the cardinal SOS order.
Below is a representation of a SOS in an MPS file, where each SOS is
defined in its own SOS section, which should follow the BOUNDS section.
0 1 2 3 4
Sx CaseName SOSName. SOSpriority.
CaseName VarName1 VarWeight1..
CaseName VarName2 VarWeight2..
CaseName VarNameN VarWeightN..
x at the second line, position 3, defines is the order of the SOS.
Due to limitations in the MPS format, N is restricted to the 1..9 range.
Each SOS should be given a unique name, SOSName. lp_solve does not
currently use case names for SOS'es and the CaseName could be any non-empty
value. The SOSpriority value determines the order in which multiple SOS'es
are analysed in lp_solve.
See also Interpolation with GAMS.
x1 obj -1 c1 -1
x1 c2 1
x2 obj -2 c1 1
x2 c2 -3 c3 1
x3 obj -3 c1 1
x3 c2 1
x4 obj -1 c1 10
x4 c3 -3.5
x5 obj 0
rhs c1 30 c2 30
UP BOUND x1 40
LI BOUND x4 2
UI BOUND x4 3
S2 SET SOS2 10
SET x1 10000
SET x2 20000
SET x4 40000
SET x5 50000