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mps - file format for linear programming problems
The MPS file format was introduced for an IBM program, but has also
been accepted by most subsequent linear programming codes.
One standard form for a linear programming problem is as follows:
under the constraints
Ax = b
and x >= 0
where x is a vector of unknowns, c is the cost (or objective) vector,
c' is the transpose of c, and A is a constraint matrix with m rows and
Alternately, the constraints may be defined as
Ax < b
and the goal may be to maximize c'x. Unfortunately, nothing in the MPS
file format specifies whether the objective is to be minimized or
maximized, and different programs have different defaults for this. On
the other hand, it is trivial to restate a maximization problem as a
minimization problem: just reverse the sign of each element of c.
The feasible region described by the constraints is a polytope, or
simplex, and at least one member of the solution set lies at a vertex
of this polytope.
The MPS file format is column-oriented, designed for use with punched
cards. All numerical values should include a decimal point. MPS files
are typically all upper-case, though many MPS readers accept mixed case
anywhere except the headers, and some accept mixed case anywhere. The
file layout is suggested in the following table:
Field: 1 2 3 4 5 6
Columns: 2-3 5-12 15-22 25-36 40-47 50-61
col_name row_name value row_name value
rhs_name row_name value row_name value
range_name row_name value row_name value
type bound_name col_name value
Here are the details on each of the seven sections:
This section consists of a single card, with "NAME" in columns 1-4 and
the title of the problem in columns 15-22.
This section describes the rows of the constraint matrix, and the
objective function. It starts with a card with "ROWS" in columns 1-4.
There is an additional card for each row in the constraint matrix, plus
one for the objective function. Each of these cards has a type in
column 2 or 3, as follows:
L Less than or equal
G Greater than or equal
N no restriction. The first N-type row encountered is regarded as
the objective, unless it is explicitly identified in the control
Linear combinations of rows may also be specified. In this case the
above row types are denoted respectively by the codes DE, DL, DG, and
DN, in columns 2-3. Field 2 contains the linear combination rowname.
Fields 3-6 contain the rowname(s) (fields 3 and 5) and their
multiplier(s) (fields 4 and 6) which form the combination. A linear
combination of three or more rows requires additional cards, following
the first card contiguously. In the additional cards field 1 is empty.
(The right-hand sides of a linear combination row must be specified in
the RHS section, described below.)
The order of the cards in the ROWS section is not significant.
This section defines the names of the variable, the coefficients of the
objective, and all the nonzero matrix elements Aij. The section starts
with a card with COLUMNS in columns 1-7, followed by data cards which
may have one or two matrix elements per card. The data are entered
column by column, and all the data cards for the nonzero entries in
each column must be grouped together contiguously. Within a column,
the order of the entries is irrelevant. Rows not mentioned are assumed
to have coefficients of zero.
The data card has the column label in field 2 (columns 5-12), the row
label in field 3 (columns 15-22), and the value of the coefficient Aij
(or cj) in field 4 (columns 25-36). Remember that the coefficient
should include a decimal point. If more than one nonzero row entry for
the same column is to be made on the card, then field 5 (columns 40-47)
has the next row label and field 6 (columns 50-61) has its
corresponding coefficient value. It should be emphasized that the use
of fields 5 and 6 is optional.
There is no need to specify columns for slack variables; this is taken
care of automatically having defined the row types.
A mixed integer program requires the specification of which variables
are required to be integer. Markers are placed in the COLUMNS section
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.
This section supplies the elements of the right-hand side. The section
starts with a card with "RHS" in columns 1-3. Since the right-hand side
can be regarded as another column of the matrix, the data cards
specifying the nonzero entries are in exactly the same format as the
COLUMNS data cards, except that field 2 (columns 5-12) has a label for
the right-hand side. More than one right-hand side may thus be
specified in this section; the one to be used for the current run is
specified separately. Rows not mentioned in the RHS section are
assumed to have a right-hand-side of zero.
The RANGES section is for constraints of the form
hi <= Ai1 x1 + Ai2 x2 + ... Ain xn <= ui
i.e. both an upper and lower bound exist for the row. The range of the
ri = ui - hi
The value of ui or hi is specified in the RHS section data, and the
value of ri is specified in the RANGES section data. This information,
plus the row type specified in the ROWS section, defines the bounds ui
If bi is the number entered in the RHS section and ri is the number
specified in the RANGES section, the ui and hi are defined as follows:
Row type Sign of ri Lower limit, hi Upper limit, ui
G (>=) + or - bi bi + |ri|
L (<=) + or - bi - |ri| bi
E (=) + bi bi + |ri|
E (=) - bi - |ri| bi
The section starts with a card with "RANGES" in columns 1-6. The data
cards specifying the values of ri are in the same format as the COLUMNS
data cards, except field 2 (columns 5-12) has a label for the column of
ranges (which can also be regarded as another column of the matrix).
More than one column of ranges may be specified, but all the data cards
for each column must be grouped together contiguously.
The BOUNDS section specifies bounds on the variables. This is an
alternative to defining extra rows in the matrix. The section starts
with a card with "BOUNDS" in columns 1-6. Each card has a type code in
field 1 (columns 2-3). The type codes, and the resulting bounds, are
LO Lower bound: value <= x (< infinity)
UP Upper bound: (0 <=) x <= value
FX Fixed variable: x = value
FR Free variable
MI Lower bound is minus infinity: -infinity <= x (<= 0)
PL upper bound is plus infinity (default): (0 <=) x < infinity
BV Binary variable: x = 0 or 1
Field 2 (columns 5-12) specifies, a bounds row name. Field 3
(columns 15-22) specifies a column label j, corresponding to the
variable xj. Field 4 (columns 25-36) specifies a bound value
bj. Fields 5 and 6 are blank.
When bounds are not specified for a column, or the entire BOUNDS
section is omitted, the usual bounds, 0 <= xi <= infinity, are assumed.
More than one bound for a given variable may be entered, i.e. both a
lower and an upper bound. When only one is specified the other is
assumed to be one of the default values of 0 or infinity, as shown in
This section consists of a single card with "ENDATA" in columns 1-6.
Note the odd spelling.
Suppose we want to minimize
XONE + 4 YTWO + 9 ZTHREE (COST)
XONE + YTWO <= 5 (LIM1)
XONE + ZTHREE >= 10 (LIM2)
- YTWO + ZTHREE = 7 (MYEQN)
0 <= XONE <= 4
-1 <= YTWO <= 1
This problem is represented by the following MPS file:
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