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1%* glpk08.tex *%
2
3\chapter{MPS Format}
4\label{champs}
5
6\section{Fixed MPS Format}
7\label{secmps}
8
9The MPS format\footnote{The MPS format was developed in 1960's by IBM
10as input format for their mathematical programming system MPS/360.
11Today the MPS format is a most widely used format understood by most
12mathematical programming packages. This appendix describes only the
13features of the MPS format, which are implemented in the GLPK package.}
14is intended for coding LP/MIP problem data. This format assumes the
15formulation of LP/MIP problem (1.1)---(1.3) (see Section \ref{seclp},
16page \pageref{seclp}).
17
18{\it MPS file} is a text file, which contains two types of
19cards\footnote{In 1960's MPS file was a deck of 80-column punched cards,
20so the author decided to keep the word card'', which may be understood
21as line of text file''.}: indicator cards and data cards.
22
23Indicator cards determine a kind of succeeding data. Each indicator card
24has one word in uppercase letters beginning in column 1.
25
26Data cards contain problem data. Each data card is divided into six
27fixed fields:
28
29\begin{center}
30\begin{tabular}{lcccccc}
31& Field 1 & Field 2 & Field 3 & Field 4 & Field 5 & Feld 6 \\
32\hline
33Columns & 2---3 & 5---12 & 15---22 & 25---36 & 40---47 & 50---61 \\
34Contents & Code & Name & Name & Number & Name & Number \\
35\end{tabular}
36\end{center}
37
38On a particular data card some fields may be optional.
39
40Names are used to identify rows, columns, and some vectors (see below).
41
42Aligning the indicator code in the field 1 to the left margin is
43optional.
44
45All names specified in the fields 2, 3, and 5 should contain from 1 up
46to 8 arbitrary characters (except control characters). If a name is
47placed in the field 3 or 5, its first character should not be the dollar
48sign \verb|$|'. If a name contains spaces, the spaces are ignored. 49 50All numerical values in the fields 4 and 6 should be coded in the form 51$sxx$\verb|E|$syy$, where$s$is the plus \verb|+|' or the minus 52\verb|-|' sign,$xx$is a real number with optional decimal point, 53$yy$is an integer decimal exponent. Any number should contain up to 12 54characters. If the sign$s$is omitted, the plus sign is assumed. The 55exponent part is optional. If a number contains spaces, the spaces are 56ignored. 57 58If a card has the asterisk \verb|*|' in the column 1, this card is 59considered as a comment and ignored. Besides, if the first character in 60the field 3 or 5 is the dollar sign \verb|$|', all characters from the
61dollar sign to the end of card are considered as a comment and ignored.
62
63MPS file should contain cards in the following order:
64
65$\bullet$ NAME indicator card;
66
67$\bullet$ ROWS indicator card;
68
69$\bullet$ data cards specifying rows (constraints);
70
71$\bullet$ COLUMNS indicator card;
72
73$\bullet$ data cards specifying columns (structural variables) and
74constraint coefficients;
75
76$\bullet$ RHS indicator card;
77
78$\bullet$ data cards specifying right-hand sides of constraints;
79
80$\bullet$ RANGES indicator card;
81
82$\bullet$ data cards specifying ranges for double-bounded constraints;
83
84$\bullet$ BOUNDS indicator card;
85
86$\bullet$ data cards specifying types and bounds of structural
87variables;
88
89$\bullet$ ENDATA indicator card.
90
91{\it Section} is a group of cards consisting of an indicator card and
92data cards succeeding this indicator card. For example, the ROWS section
93consists of the ROWS indicator card and data cards specifying rows.
94
95The sections RHS, RANGES, and BOUNDS are optional and may be omitted.
96
97\section{Free MPS Format}
98
99{\it Free MPS format} is an improved version of the standard (fixed)
100MPS format described above.\footnote{This format was developed in the
101beginning of 1990's by IBM as an alternative to the standard fixed MPS
102format for Optimization Subroutine Library (OSL).} Note that all
103changes in free MPS format concern only the coding of data while the
104structure of data is the same for both fixed and free versions of the
105MPS format.
106
107In free MPS format indicator and data records\footnote{{\it Record} in
108free MPS format has the same meaning as {\it card} in fixed MPS format.}
109may have arbitrary length not limited to 80 characters. Fields of data
110records have no predefined positions, i.e. the fields may begin in any
111position, except position 1, which must be blank, and must be separated
112from each other by one or more blanks. However, the fields must appear
113in the same order as in fixed MPS format.
114
115Symbolic names in fields 2, 3, and 5 may be longer than 8
116characters\footnote{GLPK allows symbolic names having up to 255
117characters.}
118and must not contain embedded blanks.
119
120Numeric values in fields 4 and 6 are limited to 12 characters and must
121not contain embedded blanks.
122
123Only six fields on each data record are used. Any other fields are
124ignored.
125
126If the first character of any field (not necessarily fields 3 and 5)
127is the dollar sign (\$), all characters from the dollar sign to the end 128of record are considered as a comment and ignored. 129 130\section{NAME indicator card} 131 132The NAME indicator card should be the first card in the MPS file (except 133optional comment cards, which may precede the NAME card). This card 134should contain the word \verb|NAME| in the columns 1---4 and the problem 135name in the field 3. The problem name is optional and may be omitted. 136 137\section{ROWS section} 138\label{secrows} 139 140The ROWS section should start with the indicator card, which contains 141the word \verb|ROWS| in the columns 1---4. 142 143Each data card in the ROWS section specifies one row (constraint) of the 144problem. All these data cards have the following format. 145 146\verb|N|' in the field 1 means that the row is free (unbounded): 147$$-\infty < x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} 148< +\infty;$$ 149 150\verb|L|' in the field 1 means that the row is of less than or equal 151to'' type: 152$$-\infty < x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} 153\leq b_i;$$ 154 155\verb|G|' in the field 1 means that the row is of greater than or 156equal to'' type: 157$$b_i \leq x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} 158< +\infty;$$ 159 160\verb|E|' in the field 1 means that the row is of equal to'' type: 161$$x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} \leq 162b_i,$$ 163where$b_i$is a right-hand side. Note that each constraint has a 164corresponding implictly defined auxiliary variable ($x_i$above), whose 165value is a value of the corresponding linear form, therefore row bounds 166can be considered as bounds of such auxiliary variable. 167 168The filed 2 specifies a row name (which is considered as the name of 169the corresponding auxiliary variable). 170 171The fields 3, 4, 5, and 6 are not used and should be empty. 172 173Numerical values of all non-zero right-hand sides$b_i$should be 174specified in the RHS section (see below). All double-bounded (ranged) 175constraints should be specified in the RANGES section (see below). 176 177\section{COLUMNS section} 178 179The COLUMNS section should start with the indicator card, which contains 180the word \verb|COLUMNS| in the columns 1---7. 181 182Each data card in the COLUMNS section specifies one or two constraint 183coefficients$a_{ij}$and also introduces names of columns, i.e. names 184of structural variables. All these data cards have the following format. 185 186The field 1 is not used and should be empty. 187 188The field 2 specifies a column name. If this field is empty, the column 189name from the immediately preceeding data card is assumed. 190 191The field 3 specifies a row name defined in the ROWS section. 192 193The field 4 specifies a numerical value of the constraint coefficient 194$a_{ij}$, which is placed in the corresponding row and column. 195 196The fields 5 and 6 are optional. If they are used, they should contain 197a second pair row name---constraint coefficient'' for the same column. 198 199Elements of the constraint matrix (i.e. constraint coefficients) should 200be enumerated in the column wise manner: all elements for the current 201column should be specified before elements for the next column. However, 202the order of rows in the COLUMNS section may differ from the order of 203rows in the ROWS section. 204 205Constraint coefficients not specified in the COLUMNS section are 206considered as zeros. Therefore zero coefficients may be omitted, 207although it is allowed to explicitly specify them. 208 209\section{RHS section} 210 211The RHS section should start with the indicator card, which contains the 212word \verb|RHS| in the columns 1---3. 213 214Each data card in the RHS section specifies one or two right-hand sides 215$b_i$(see Section \ref{secrows}, page \pageref{secrows}). All these 216data cards have the following format. 217 218The field 1 is not used and should be empty. 219 220The field 2 specifies a name of the right-hand side (RHS) 221vector\footnote{This feature allows the user to specify several RHS 222vectors in the same MPS file. However, before solving the problem a 223particular RHS vector should be chosen.}. If this field is empty, the 224RHS vector name from the immediately preceeding data card is assumed. 225 226The field 3 specifies a row name defined in the ROWS section. 227 228The field 4 specifies a right-hand side$b_i$for the row, whose name is 229specified in the field 3. Depending on the row type$b_i$is a lower 230bound (for the row of \verb|G| type), an upper bound (for the row of 231\verb|L| type), or a fixed value (for the row of \verb|E| 232type).\footnote{If the row is of {\tt N} type,$b_i$is considered as 233a constant term of the corresponding linear form. Should note, however, 234this convention is non-standard.} 235 236The fields 5 and 6 are optional. If they are used, they should contain 237a second pair row name---right-hand side'' for the same RHS vector. 238 239All right-hand sides for the current RHS vector should be specified 240before right-hand sides for the next RHS vector. However, the order of 241rows in the RHS section may differ from the order of rows in the ROWS 242section. 243 244Right-hand sides not specified in the RHS section are considered as 245zeros. Therefore zero right-hand sides may be omitted, although it is 246allowed to explicitly specify them. 247 248\section{RANGES section} 249 250The RANGES section should start with the indicator card, which contains 251the word \verb|RANGES| in the columns 1---6. 252 253Each data card in the RANGES section specifies one or two ranges for 254double-side constraints, i.e. for constraints that are of the types 255\verb|L| and \verb|G| at the same time: 256$$l_i \leq x_i = a_{i1}x_{m+1} + a_{i2}x_{m+2} + \dots + a_{in}x_{m+n} 257\leq u_i,$$ 258where$l_i$is a lower bound,$u_i$is an upper bound. All these data 259cards have the following format. 260 261The field 1 is not used and should be empty. 262 263The field 2 specifies a name of the range vector\footnote{This feature 264allows the user to specify several range vectors in the same MPS file. 265However, before solving the problem a particular range vector should be 266chosen.}. If this field is empty, the range vector name from the 267immediately preceeding data card is assumed. 268 269The field 3 specifies a row name defined in the ROWS section. 270 271The field 4 specifies a range value$r_i$(see the table below) for the 272row, whose name is specified in the field 3. 273 274The fields 5 and 6 are optional. If they are used, they should contain 275a second pair row name---range value'' for the same range vector. 276 277All range values for the current range vector should be specified before 278range values for the next range vector. However, the order of rows in 279the RANGES section may differ from the order of rows in the ROWS 280section. 281 282For each double-side constraint specified in the RANGES section its 283lower and upper bounds are determined as follows: 284 285\begin{center} 286\begin{tabular}{cccc} 287Row type & Sign of$r_i$& Lower bound & Upper bound \\ 288\hline 289{\tt G} &$+$or$-$&$b_i$&$b_i + |r_i|$\\ 290{\tt L} &$+$or$-$&$b_i - |r_i|$&$b_i$\\ 291{\tt E} &$+$&$b_i$&$b_i + |r_i|$\\ 292{\tt E} &$-$&$b_i - |r_i|$&$b_i$\\ 293\end{tabular} 294\end{center} 295 296\noindent 297where$b_i$is a right-hand side specified in the RHS section (if$b_i$298is not specified, it is considered as zero),$r_i$is a range value 299specified in the RANGES section. 300 301\section{BOUNDS section} 302\label{secbounds} 303 304The BOUNDS section should start with the indicator card, which contains 305the word \verb|BOUNDS| in the columns 1---6. 306 307Each data card in the BOUNDS section specifies one (lower or upper) 308bound for one structural variable (column). All these data cards have 309the following format. 310 311The indicator in the field 1 specifies the bound type: 312 313\begin{tabular}{@{}ll} 314\verb|LO| & lower bound; \\ 315\verb|UP| & upper bound; \\ 316\verb|FX| & fixed variable (lower and upper bounds are equal); \\ 317\verb|FR| & free variable (no bounds); \\ 318\verb|MI| & no lower bound (lower bound is minus infinity''); \\ 319\verb|PL| & no upper bound (upper bound is plus infinity''); \\ 320\end{tabular} 321 322The field 2 specifies a name of the bound vector\footnote{This feature 323allows the user to specify several bound vectors in the same MPS file. 324However, before solving the problem a particular bound vector should be 325chosen.}. If this field is empty, the bound vector name from the 326immediately preceeding data card is assumed. 327 328The field 3 specifies a column name defined in the COLUMNS section. 329 330The field 4 specifies a bound value. If the bound type in the field 1 331differs from \verb|LO|, \verb|UP|, and \verb|FX|, the value in the field 3324 is ignored and may be omitted. 333 334The fields 5 and 6 are not used and should be empty. 335 336All bound values for the current bound vector should be specified before 337bound values for the next bound vector. However, the order of columns in 338the BOUNDS section may differ from the order of columns in the COLUMNS 339section. Specification of a lower bound should precede specification of 340an upper bound for the same column (if both the lower and upper bounds 341are explicitly specified). 342 343By default, all columns (structural variables) are non-negative, i.e. 344have zero lower bound and no upper bound. Lower ($l_j$) and upper 345($u_j$) bounds of some column (structural variable$x_j$) are set in the 346following way, where$s_j$is a corresponding bound value explicitly 347specified in the BOUNDS section: 348 349\begin{tabular}{@{}ll} 350\verb|LO| & sets$l_j$to$s_j$; \\ 351\verb|UP| & sets$u_j$to$s_j$; \\ 352\verb|FX| & sets both$l_j$and$u_j$to$s_j$; \\ 353\verb|FR| & sets$l_j$to$-\infty$and$u_j$to$+\infty$; \\ 354\verb|MI| & sets$l_j$to$-\infty$; \\ 355\verb|PL| & sets$u_j$to$+\infty$. \\ 356\end{tabular} 357 358\section{ENDATA indicator card} 359 360The ENDATA indicator card should be the last card of MPS file (except 361optional comment cards, which may follow the ENDATA card). This card 362should contain the word \verb|ENDATA| in the columns 1---6. 363 364\section{Specifying objective function} 365 366It is impossible to explicitly specify the objective function and 367optimization direction in the MPS file. However, the following implicit 368rule is used by default: the first row of \verb|N| type is considered 369as a row of the objective function (i.e. the objective function is the 370corresponding auxiliary variable), which should be {\it minimized}. 371 372GLPK also allows specifying a constant term of the objective function 373as a right-hand side of the corresponding row in the RHS section. 374 375\section{Example of MPS file} 376\label{secmpsex} 377 378In order to illustrate what the MPS format is, consider the following 379example of LP problem: 380 381\medskip 382\noindent minimize 383$$384value = .03\ bin_1 + .08\ bin_2 + .17\ bin_3 + .12\ bin_4 + .15\ bin_5 385+ .21\ al + .38\ si 386$$ 387 388\noindent subject to linear constraints 389$$390\begin{array}{@{}l@{\:}l@{}} 391yield &= \ \ \ \ \;bin_1 + \ \ \ \ \;bin_2 + \ \ \ \ \;bin_3 + 392 \ \ \ \ \;bin_4 + \ \ \ \ \;bin_5 + \ \ \ \ \;al + 393 \ \ \ \ \;si \\ 394FE &= .15\ bin_1 + .04\ bin_2 + .02\ bin_3 + .04\ bin_4 + .02\ bin_5 395 + .01\ al + .03\ si \\ 396CU &= .03\ bin_1 + .05\ bin_2 + .08\ bin_3 + .02\ bin_4 + .06\ bin_5 397 + .01\ al \\ 398MN &= .02\ bin_1 + .04\ bin_2 + .01\ bin_3 + .02\ bin_4 + .02\ bin_5 399 \\ 400MG &= .02\ bin_1 + .03\ bin_2 401\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + .01\ bin_5 \\ 402AL &= .70\ bin_1 + .75\ bin_2 + .80\ bin_3 + .75\ bin_4 + .80\ bin_5 403 + .97\ al \\ 404SI &= .02\ bin_1 + .06\ bin_2 + .08\ bin_3 + .12\ bin_4 + .02\ bin_5 405 + .01\ al + .97\ si \\ 406\end{array} 407$$ 408and bounds of (auxiliary and structural) variables 409$$410\begin{array}{r@{\ }l@{\ }l@{\ }l@{\ }rcr@{\ }l@{\ }l@{\ }l@{\ }r} 411&&yield&=&2000&&0&\leq&bin_1&\leq&200\\ 412-\infty&<&FE&\leq&60&&0&\leq&bin_2&\leq&2500\\ 413-\infty&<&CU&\leq&100&&400&\leq&bin_3&\leq&800\\ 414-\infty&<&MN&\leq&40&&100&\leq&bin_4&\leq&700\\ 415-\infty&<&MG&\leq&30&&0&\leq&bin_5&\leq&1500\\ 4161500&\leq&AL&<&+\infty&&0&\leq&al&<&+\infty\\ 417250&\leq&SI&\leq&300&&0&\leq&si&<&+\infty\\ 418\end{array} 419$$ 420 421A complete MPS file which specifies data for this example is shown 422below (the first two comment lines show card positions). 423 424\begin{verbatim} 425*000000001111111111222222222233333333334444444444555555555566 426*234567890123456789012345678901234567890123456789012345678901 427NAME PLAN 428ROWS 429 N VALUE 430 E YIELD 431 L FE 432 L CU 433 L MN 434 L MG 435 G AL 436 L SI 437COLUMNS 438 BIN1 VALUE .03000 YIELD 1.00000 439 FE .15000 CU .03000 440 MN .02000 MG .02000 441 AL .70000 SI .02000 442 BIN2 VALUE .08000 YIELD 1.00000 443 FE .04000 CU .05000 444 MN .04000 MG .03000 445 AL .75000 SI .06000 446 BIN3 VALUE .17000 YIELD 1.00000 447 FE .02000 CU .08000 448 MN .01000 AL .80000 449 SI .08000 450 BIN4 VALUE .12000 YIELD 1.00000 451 FE .04000 CU .02000 452 MN .02000 AL .75000 453 SI .12000 454 BIN5 VALUE .15000 YIELD 1.00000 455 FE .02000 CU .06000 456 MN .02000 MG .01000 457 AL .80000 SI .02000 458 ALUM VALUE .21000 YIELD 1.00000 459 FE .01000 CU .01000 460 AL .97000 SI .01000 461 SILICON VALUE .38000 YIELD 1.00000 462 FE .03000 SI .97000 463RHS 464 RHS1 YIELD 2000.00000 FE 60.00000 465 CU 100.00000 MN 40.00000 466 SI 300.00000 467 MG 30.00000 AL 1500.00000 468RANGES 469 RNG1 SI 50.00000 470BOUNDS 471 UP BND1 BIN1 200.00000 472 UP BIN2 2500.00000 473 LO BIN3 400.00000 474 UP BIN3 800.00000 475 LO BIN4 100.00000 476 UP BIN4 700.00000 477 UP BIN5 1500.00000 478ENDATA 479\end{verbatim} 480 481\section{MIP features} 482 483The MPS format provides two ways for introducing integer variables into 484the problem. 485 486The first way is most general and based on using special marker cards 487INTORG and INTEND. These marker cards are placed in the COLUMNS section. 488The INTORG card indicates the start of a group of integer variables 489(columns), and the card INTEND indicates the end of the group. The MPS 490file may contain arbitrary number of the marker cards. 491 492The marker cards have the same format as the data cards (see Section 493\ref{secmps}, page \pageref{secmps}). 494 495The fields 1, 2, and 6 are not used and should be empty. 496 497The field 2 should contain a marker name. This name may be arbitrary. 498 499The field 3 should contain the word \verb|'MARKER'| (including 500apostrophes). 501 502The field 5 should contain either the word \verb|'INTORG'| (including 503apostrophes) for the marker card, which begins a group of integer 504columns, or the word \verb|'INTEND'| (including apostrophes) for the 505marker card, which ends the group. 506 507The second way is less general but more convenient in some cases. It 508allows the user declaring integer columns using three additional types 509of bounds, which are specified in the field 1 of data cards in the 510BOUNDS section (see Section \ref{secbounds}, page \pageref{secbounds}): 511 512\begin{tabular}{@{}lp{112.3mm}@{}} 513\verb|LI| & lower integer. This bound type specifies that the 514corresponding column (structural variable), whose name is specified in 515field 3, is of integer kind. In this case an lower bound of the 516column should be specified in field 4 (like in the case of \verb|LO| 517bound type). \\ 518\verb|UI| & upper integer. This bound type specifies that the 519corresponding column (structural variable), whose name is specified in 520field 3, is of integer kind. In this case an upper bound of the 521column should be specified in field 4 (like in the case of \verb|UP| 522bound type). \\ 523\end{tabular} 524 525\pagebreak 526 527\begin{tabular}{@{}lp{112.3mm}@{}} 528\verb|BV| & binary variable. This bound type specifies that the 529corresponding column (structural variable), whose name is specified in 530the field 3, is of integer kind, its lower bound is zero, and its upper 531bound is one (thus, such variable being of integer kind can have only 532two values zero and one). In this case a numeric value specified in the 533field 4 is ignored and may be omitted.\\ 534\end{tabular} 535 536Consider the following example of MIP problem: 537 538\medskip 539 540\noindent 541\hspace{1in} minimize 542$$Z = 3 x_1 + 7 x_2 - x_3 + x4$$ 543\hspace{1in} subject to linear constraints 544$$545\begin{array}{c} 546\nonumber r_1 = 2 x_1 - \ \ x_2 + \ \ x_3 - \ \;x_4 \\ 547\nonumber r_2 = \ \;x_1 - \ \;x_2 - 6 x_3 + 4 x_4 \\ 548\nonumber r_3 = 5 x_1 + 3 x_2 \ \ \ \ \ \ \ \ \ + \ \ x_4 \\ 549\end{array} 550$$ 551\hspace{1in} and bound of variables 552$$553\begin{array}{cccl} 554\nonumber 1 \leq r_1 < +\infty && 0 \leq x_1 \leq 4 &{\rm(continuous)}\\ 555\nonumber 8 \leq r_2 < +\infty && 2 \leq x_2 \leq 5 &{\rm(integer)} \\ 556\nonumber 5 \leq r_3 < +\infty && 0 \leq x_3 \leq 1 &{\rm(integer)} \\ 557\nonumber && 3 \leq x_4 \leq 8 &{\rm(continuous)}\\ 558\end{array} 559$$ 560 561The corresponding MPS file may look like the following: 562 563\begin{verbatim} 564NAME SAMP1 565ROWS 566 N Z 567 G R1 568 G R2 569 G R3 570COLUMNS 571 X1 R1 2.0 R2 1.0 572 X1 R3 5.0 Z 3.0 573 MARK0001 'MARKER' 'INTORG' 574 X2 R1 -1.0 R2 -1.0 575 X2 R3 3.0 Z 7.0 576 X3 R1 1.0 R2 -6.0 577 X3 Z -1.0 578 MARK0002 'MARKER' 'INTEND' 579 X4 R1 -1.0 R2 4.0 580 X4 R3 1.0 Z 1.0 581RHS 582 RHS1 R1 1.0 583 RHS1 R2 8.0 584 RHS1 R3 5.0 585BOUNDS 586 UP BND1 X1 4.0 587 LO BND1 X2 2.0 588 UP BND1 X2 5.0 589 UP BND1 X3 1.0 590 LO BND1 X4 3.0 591 UP BND1 X4 8.0 592ENDATA 593\end{verbatim} 594 595The same example may be coded without INTORG/INTEND markers using the 596bound type UI for the variable$x_2$and the bound type BV for the 597variable$x_3$: 598 599\begin{verbatim} 600NAME SAMP2 601ROWS 602 N Z 603 G R1 604 G R2 605 G R3 606COLUMNS 607 X1 R1 2.0 R2 1.0 608 X1 R3 5.0 Z 3.0 609 X2 R1 -1.0 R2 -1.0 610 X2 R3 3.0 Z 7.0 611 X3 R1 1.0 R2 -6.0 612 X3 Z -1.0 613 X4 R1 -1.0 R2 4.0 614 X4 R3 1.0 Z 1.0 615RHS 616 RHS1 R1 1.0 617 RHS1 R2 8.0 618 RHS1 R3 5.0 619BOUNDS 620 UP BND1 X1 4.0 621 LO BND1 X2 2.0 622 UI BND1 X2 5.0 623 BV BND1 X3 624 LO BND1 X4 3.0 625 UP BND1 X4 8.0 626ENDATA 627\end{verbatim} 628 629%\section{Specifying predefined basis} 630%\label{secbas} 631% 632%The MPS format can also be used to specify some predefined basis for an 633%LP problem, i.e. to specify which rows and columns are basic and which 634%are non-basic. 635% 636%The order of a basis file in the MPS format is: 637% 638%$\bullet$NAME indicator card; 639% 640%$\bullet$data cards (can appear in arbitrary order); 641% 642%$\bullet\$ ENDATA indicator card.
643%
644%Each data card specifies either a pair "basic column---non-basic row"
645%or a non-basic column. All the data cards have the following format.
646%
647%\verb|XL|' in the field 1 means that a column, whose name is given in
648%the field 2, is basic, and a row, whose name is given in the field 3,
649%is non-basic and placed on its lower bound.
650%
651%\verb|XU|' in the field 1 means that a column, whose name is given in
652%the field 2, is basic, and a row, whose name is given in the field 3,
653%is non-basic and placed on its upper bound.
654%
655%\verb|LL|' in the field 1 means that a column, whose name is given in
656%the field 3, is non-basic and placed on its lower bound.
657%
658%\verb|UL|' in the field 1 means that a column, whose name is given in
659%the field 3, is non-basic and placed on its upper bound.
660%
661%The field 2 contains a column name.
662%
663%If the indicator given in the field 1 is \verb|XL|' or \verb|XU|',
664%the field 3 contains a row name. Otherwise, if the indicator is
665%\verb|LL|' or `\verb|UL|', the field 3 is not used and should be
666%empty.
667%
668%The field 4, 5, and 6 are not used and should be empty.
669%
670%A basis file in the MPS format acts like a patch: it doesn't specify
671%a basis completely, instead that it is just shows in what a given basis
672%differs from the "standard" basis, where all rows (auxiliary variables)
673%are assumed to be basic and all columns (structural variables) are
674%assumed to be non-basic.
675%
676%As an example here is a basis file that specifies an optimal basis
677%for the example LP problem given in Section \ref{secmpsex},
678%Page \pageref{secmpsex}:
679%
680%\pagebreak
681%
682%\begin{verbatim}
683%*000000001111111111222222222233333333334444444444555555555566
684%*234567890123456789012345678901234567890123456789012345678901
685%NAME          PLAN
686% XL BIN2      YIELD
687% XL BIN3      FE
688% XL BIN4      MN
689% XL ALUM      AL
690% XL SILICON   SI
691% LL BIN1
692% LL BIN5
693%ENDATA
694%\end{verbatim}
695
696%* eof *%
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