30 jui 1971 · A matrix-generator language for use in structuring and inputting linear programming problem matrices is described The generator is based
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LIBRARYOFTHE
UNIVERSITYOFILLINOIS
ATURBANA-CHAMPAIGN
510.84
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no.I-10AUG51976
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sponsibleforitsreturntothelibraryfrom whichitwaswithdrawnonorbeforetheLatestDatestampedbelow.
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arereasonsfordisciplinaryactionandmay resultindismissalfromtheUniversity. f." uliviiLL^ CI!JUL07JECT,
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DigitizedbytheInternetArchive
in2012withfundingfromUniversityofIllinoisUrbana-Champaign
GQHFtRENCEESSEi
,SSAS)CACDocumentNo.k
AMATRIX-GENERATORSYSTEMFOR
LINEARPROGRAMMINGPROBLEMS
byIan¥.Marceau
andThomasW.Mason
CenterforAdvancedComputation
UniversityofIllinoisatUrbana-Champaign
Urbana,Illinois61801
June30,1971
ABSTRACT
given.TABLEOFCONTENTS
PageINTRODUCTION1
COSTEFFECTIVENESSOFAMATRIXGENERATOR2
MATRIXGENERATORLANGUAGE6
ExamplesofDataStructures.8
UseoftheDataStructures11
EXAMPLEMODELlk
TheTableauIk
TheMatrixGeneratorStatements20
Figure121
Figure223
Figure3••2k
Figurek2k
Figure525
CONCLUSION27
REFERENCES28
INTRODUCTION
modelfrom5000cardsto70cards[l]. ofmatrix-generatorparameters -1-COSTEFFECTIVENESSOFAMATRIXGENERATOR
timetoinputthedata. acreageconstraints ^(k)Ex.'111 11....1 -2- usingthematrixgeneratorwithonestatement: DIAGONAL(m):1*n.
structure thematrixgeneratorandcanbeexpressedas Cv(f-N),k2m
k. systemasforthematrixgenerator), requiredfortheconventionalsystem,and C(tI-(T+et)N)c2m
c T e whereC_isthecostofcomputertime, permatrixgeneratorstatement,and isthetimetointernallygenerateonevalue. -3- C[(T-2t)J-TN].c2m
ingettingthedataintothecomputeris o(1-u)Ir2m whereCisCT,thecostofreadingacard,re'to discussion, S=(C.+C)(|-N)xkr
v2m S=C(|-|),or"
J S-of{Sf).I
NNotethatC - equalsthetotalcostofkeypunchingandenteringdataby tionalsystemcost. -h- generatorincreaseswiththesizeofthemodel. -5- MATRIXGENERATORLANGUAGE
-6- Table1
GeometricStructuresandGeneratorNames
StructureGeneratorName
1.point,orsinglevalue
2.row 3.column
h diagonal 5.bandmatrix:valuesaboveand
belowmaindiagonal 6.lowerbandmatrix:valuesonly
belowmaindiagonal 7.upperbandmatrix:valuesonly-
abovemaindiagonal 8.uppertriangularmatrix
9-lowertriangularmatrix
10.parallelogram
11.rectangle
12 square POINT ROW COLUMN
DIAGONAL
(R)BAND (R)LOBAND (R)HIBAND (R)HITRI (R)LOTRI (R)PARM RECTANGLE
SQUARE
-7- oftheform: Table2
CharacterizationofDataStructures
Dimension
Values
SimpleCompound
Single
POINT,SQUARE,
(R)LOTRI,(R)HITRI ROW,COLUMN,
DIAGONAL
Double
RECTANGLE(r)band,(r)loband,
(r)hiband (r)parm .1 ExamplesofDataStructures
1.Singledimension,simplevaluelist:
-8- POINT:1*5
11111
SQUAEE(3):2
222
222
LOTRI(^):1*2
I1 II11 III111
11111111
RLOTRI.(4):1*2
11 1111
111111
11111111
HITRl(if):1*2
11111111
111111
1111
11 RHITRI(^):1*2
11111111
111111
1111
11 2.Singledimension,compoundvaluelist:
R0W(5):1,2,3*2
1233312333
COIAMN(5):1,2,3*2
11 22
33
33
33
DIAGONAL^):1,2,3,^5*3111
222
333
kkk 555
3.Doubledimension,simplevaluelist:
RECTANGLE(3,5):1
11111
11111
11111
k.Doubledimension,compoundvaluelist: -9- EAND(6,5):l,2,3,k,5
321
U321 5^321 5^321 5^32 5^3 BAND(6,5):1,2,3
321
3321
33321
33321
3332
333
RBAND(6,5):
123
123k
123^5
123^5
23^5
3^5 LOBA.ND(6,5)
1 21
321
*J-321 5^321 5^321 1,2,3,^,5
1,2,3,^,5
L0BA.ND(6,2):1*2
I1 II11 1111
1111
1111
1111
RLOBAWD(6,3):1,2,3
1 12 123
123
123
123
HIBAND(6,5):1,2,3,^,5
5^321 5^321 5^32 5^3 5 RHIBA.ro(6,5):1,2,3
12333
12333
2333
333
33
3 parm(6,i+): I1 II11 III111
11111111
11111111
11111111
111111
1111
11 1*2PARM(k,6)
2 22
222
2222
2222
222
22
2 RPARM(^,6)
2 22
:2 2222
-10- 22
2 UseoftheDataStructures
Activities
Constraints
Obtain1unit
inperiod 1p Use1unit
inperiod 1 - p Save1unit
inperiod 1p Totalunits
accumulated Initial
inventoryMonthly salary Disposeor-1
Accumulate
Disposal
Requirements
Additionto
savings'1-quotesdbs_dbs14.pdfusesText_20
11....1 -2- usingthematrixgeneratorwithonestatement: DIAGONAL(m):1*n.
structure thematrixgeneratorandcanbeexpressedas Cv(f-N),k2m
k. systemasforthematrixgenerator), requiredfortheconventionalsystem,and C(tI-(T+et)N)c2m
c T e whereC_isthecostofcomputertime, permatrixgeneratorstatement,and isthetimetointernallygenerateonevalue. -3- C[(T-2t)J-TN].c2m
ingettingthedataintothecomputeris o(1-u)Ir2m whereCisCT,thecostofreadingacard,re'to discussion, S=(C.+C)(|-N)xkr
v2m S=C(|-|),or"
J S-of{Sf).I
NNotethatC - equalsthetotalcostofkeypunchingandenteringdataby tionalsystemcost. -h- generatorincreaseswiththesizeofthemodel. -5- MATRIXGENERATORLANGUAGE
-6- Table1
GeometricStructuresandGeneratorNames
StructureGeneratorName
1.point,orsinglevalue
2.row 3.column
h diagonal 5.bandmatrix:valuesaboveand
belowmaindiagonal 6.lowerbandmatrix:valuesonly
belowmaindiagonal 7.upperbandmatrix:valuesonly-
abovemaindiagonal 8.uppertriangularmatrix
9-lowertriangularmatrix
10.parallelogram
11.rectangle
12 square POINT ROW COLUMN
DIAGONAL
(R)BAND (R)LOBAND (R)HIBAND (R)HITRI (R)LOTRI (R)PARM RECTANGLE
SQUARE
-7- oftheform: Table2
CharacterizationofDataStructures
Dimension
Values
SimpleCompound
Single
POINT,SQUARE,
(R)LOTRI,(R)HITRI ROW,COLUMN,
DIAGONAL
Double
RECTANGLE(r)band,(r)loband,
(r)hiband (r)parm .1 ExamplesofDataStructures
1.Singledimension,simplevaluelist:
-8- POINT:1*5
11111
SQUAEE(3):2
222
222
LOTRI(^):1*2
I1 II11 III111
11111111
RLOTRI.(4):1*2
11 1111
111111
11111111
HITRl(if):1*2
11111111
111111
1111
11 RHITRI(^):1*2
11111111
111111
1111
11 2.Singledimension,compoundvaluelist:
R0W(5):1,2,3*2
1233312333
COIAMN(5):1,2,3*2
11 22
33
33
33
DIAGONAL^):1,2,3,^5*3111
222
333
kkk 555
3.Doubledimension,simplevaluelist:
RECTANGLE(3,5):1
11111
11111
11111
k.Doubledimension,compoundvaluelist: -9- EAND(6,5):l,2,3,k,5
321
U321 5^321 5^321 5^32 5^3 BAND(6,5):1,2,3
321
3321
33321
33321
3332
333
RBAND(6,5):
123
123k
123^5
123^5
23^5
3^5 LOBA.ND(6,5)
1 21
321
*J-321 5^321 5^321 1,2,3,^,5
1,2,3,^,5
L0BA.ND(6,2):1*2
I1 II11 1111
1111
1111
1111
RLOBAWD(6,3):1,2,3
1 12 123
123
123
123
HIBAND(6,5):1,2,3,^,5
5^321 5^321 5^32 5^3 5 RHIBA.ro(6,5):1,2,3
12333
12333
2333
333
33
3 parm(6,i+): I1 II11 III111
11111111
11111111
11111111
111111
1111
11 1*2PARM(k,6)
2 22
222
2222
2222
222
22
2 RPARM(^,6)
2 22
:2 2222
-10- 22
2 UseoftheDataStructures
Activities
Constraints
Obtain1unit
inperiod 1p Use1unit
inperiod 1 - p Save1unit
inperiod 1p Totalunits
accumulated Initial
inventoryMonthly salary Disposeor-1
Accumulate
Disposal
Requirements
Additionto
savings'1-quotesdbs_dbs14.pdfusesText_20
DIAGONAL(m):1*n.
structure thematrixgeneratorandcanbeexpressedasCv(f-N),k2m
k. systemasforthematrixgenerator), requiredfortheconventionalsystem,andC(tI-(T+et)N)c2m
c T e whereC_isthecostofcomputertime, permatrixgeneratorstatement,and isthetimetointernallygenerateonevalue. -3-C[(T-2t)J-TN].c2m
ingettingthedataintothecomputeris o(1-u)Ir2m whereCisCT,thecostofreadingacard,re'to discussion,S=(C.+C)(|-N)xkr
v2mS=C(|-|),or"
JS-of{Sf).I
NNotethatC - equalsthetotalcostofkeypunchingandenteringdataby tionalsystemcost. -h- generatorincreaseswiththesizeofthemodel. -5-MATRIXGENERATORLANGUAGE
-6-Table1
GeometricStructuresandGeneratorNames
StructureGeneratorName
1.point,orsinglevalue
2.row3.column
h diagonal5.bandmatrix:valuesaboveand
belowmaindiagonal6.lowerbandmatrix:valuesonly
belowmaindiagonal7.upperbandmatrix:valuesonly-
abovemaindiagonal8.uppertriangularmatrix
9-lowertriangularmatrix
10.parallelogram
11.rectangle
12 square POINT ROWCOLUMN
DIAGONAL
(R)BAND (R)LOBAND (R)HIBAND (R)HITRI (R)LOTRI (R)PARMRECTANGLE
SQUARE
-7- oftheform:Table2
CharacterizationofDataStructures
Dimension
Values
SimpleCompound
Single
POINT,SQUARE,
(R)LOTRI,(R)HITRIROW,COLUMN,
DIAGONAL
Double
RECTANGLE(r)band,(r)loband,
(r)hiband (r)parm .1ExamplesofDataStructures
1.Singledimension,simplevaluelist:
-8-POINT:1*5
11111SQUAEE(3):2
222222
LOTRI(^):1*2
I1 II11III111
11111111
RLOTRI.(4):1*2
11 1111111111
11111111
HITRl(if):1*2
11111111
111111
111111
RHITRI(^):1*2
11111111
111111
111111
2.Singledimension,compoundvaluelist:
R0W(5):1,2,3*2
1233312333
COIAMN(5):1,2,3*2
11 2233
33
33
DIAGONAL^):1,2,3,^5*3111
222333
kkk 555
3.Doubledimension,simplevaluelist:
RECTANGLE(3,5):1
1111111111
11111
k.Doubledimension,compoundvaluelist: -9-
EAND(6,5):l,2,3,k,5
321U321 5^321 5^321 5^32 5^3
BAND(6,5):1,2,3
3213321
33321
33321
3332
333
RBAND(6,5):
123123k
123^5
123^5
23^5
3^5
LOBA.ND(6,5)
1 21321
*J-321 5^321 5^321
1,2,3,^,5
1,2,3,^,5
L0BA.ND(6,2):1*2
I1 II11 11111111
1111
1111
RLOBAWD(6,3):1,2,3
1 12 123123
123
123
HIBAND(6,5):1,2,3,^,5
5^321 5^321 5^32 5^3 5RHIBA.ro(6,5):1,2,3
1233312333
2333
333
33
3 parm(6,i+): I1 II11
III111
11111111
11111111
11111111
111111
111111
1*2PARM(k,6)
2 22222
2222
2222
222
22
2
RPARM(^,6)
2 22:2 2222
-10- 22
2