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160
DualityinLinearProgramming
FinalTableau
Basic
Current
variables
values
x
1
x
2
x
3
x
4
x
5
x
6
x
1
4
3
1
7
3
5
3
4
15
1
15
x
4
1
15
1
30
1
30
1
1
150
2
75
(−
z
)
56
3
20
3
10
3
44
15
4
15
c) Verifythecomplementary-slacknessconditions.
4. InthesecondexerciseofChapter1,wegraphicallydeterminedtheshadowpricestothefollowinglinearprogram:
Maximizez
=
2x
1
+
x
2
,
subjectto:
12x
1
+
x
2
6
,
3x
1
+
x
2
7
,
x
2
10
,
x
1
0
,
x
2
0
.
a) Formulatethedualtothislinearprogram.
b) Showthattheshadowpricessolvethedualproblem.
5. Solvethelinearprogrambelowasfollows:First,solvethedualproblemgraphically. . Thenusethesolutiontothe
dualproblemtodeterminewhichvariablesintheprimalproblemarezerointheoptimalprimalsolution. [Hint:
Invokecomplementaryslackness.] Finally,solvefortheoptimalbasicvariablesintheprimal, , usingtheprimal
equations.
Primal
Maximize
4x
2
+
3x
3
+
2x
4
8x
5
,
subjectto:
3x
1
+
x
2
+
2x
3
+
x
4
=
3
,
x
1
x
2
+
x
4
x
5
2
,
x
j
0
(
j
=
1
,
2
,
3
,
4
,
5
).
6. Adieticianwishestodesignaminimum-costdiettomeetminimumdailyrequirementsforcalories,protein,car-
bohydrate,fat,vitaminAandvitaminBdietaryneeds.Severaldifferentfoodscanbeusedinthediet,withdataas
specifiedinthefollowingtable.
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GameTheory
161
Contentandcostsperpoundconsumed
Food
Food
....
Food
....
Food
Daily
1
2
j
n
requirements
Calories
a
11
a
12
a
1j
a
1n
b
1
Protein
a
21
a
22
a
2j
a
2n
b
2
(grams)
Carbohydrate
a
31
a
32
a
3j
a
3n
b
3
(grams)
Fat
a
41
a
42
a
4j
a
4n
b
4
(grams)
VitaminA
a
51
a
52
a
5j
a
5n
b
5
(milligrams)
VitaminB
a
61
a
62
a
6j
a
6n
b
6
(milligrams)
Costs
c
1
c
2
c
j
c
n
(dollars)
a) Formulatealinearprogramtodeterminewhichfoodstoincludeintheminimumcostdiet. . (Morethanthe
minimumdailyrequirementsofanydietaryneedcanbeconsumed.)
b) Statethedualtothedietproblem,specifyingtheunitsofmeasurementforeachofthedualvariables. . Interpret
thedualproblemintermsofadruggistwhosetspricesonthedietaryneedsinamannertoselladietarypillwith
b
1
,b
2
,b
3
,b
4
,b
5
,andb
6
unitsofthegivendietaryneedsatmaximumprofit.
7. Inordertosmoothitsproductionscheduling,afootwearcompanyhasdecidedtouseasimpleversionofalinear
costmodelforaggregateplanning.Themodelis:
Minimizez
=
N
i
=
1
T
t
=
1
(v
i
X
it
+
c
i
I
it
)+
T
t
=
1
(
rW
t
+
oO
t
),
subjectto:
X
it
+
I
i
,
t
1
I
it
=
d
it
t
=
1
,
2
,...,
T
i
=
1
,
2
,...,
N
N
i
=
1
k
i
X
it
W
t
O
t
=
0
t
=
1
,
2
,...,
T
0
W
t
≤(
rm
)
t
=
1
,
2
,...,
T
pW
t
+
O
t
0
t
=
1
,
2
,...,
T
X
it
,
I
it
0
i
=
1
,
2
,...,
N
t
=
1
,
2
,...,
T
withparameters:
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162
DualityinLinearProgramming
v
i
=Unitproductioncostforproductiineachperiod,
c
i
=Inventory-carryingcostperunitofproductiineachperiod,
r=Costperman-hourofregularlabor,
o=Costperman-hourofovertimelabor,
d
it
=Demandforproductiinperiodt,
k
i
=Man-hoursrequiredtoproduceoneunitofproducti,
(
rm
)
=Totalman-hoursofregularlaboravailableineachperiod,
p=Fractionoflaborman-hoursavailableasovertime,
T=Timehorizoninperiods,
N=Totalnumberofproducts.
Thedecisionvariablesare:
X
it
=Unitsofproductitobeproducedinperiodt,
I
it
=Unitsofproductitobeleftoverasinventoryattheendofperiodt,
W
t
=Man-hoursofregularlaborusedduringperiod(fixedworkforce),
O
t
=Man-hoursofovertimelaborusedduringpeirodt.
Thecompanyhastwomajorproducts,bootsandshoes,whoseproductionitwantstoscheduleforthenextthree
periods.Itcosts$10tomakeapairofbootsand$5tomakeapairofshoes.Thecompanyestimatesthatitcosts$2
tomaintainapairofbootsasinventorythroughtheendofaperiodandhalfthisamountforshoes. Averagewage
rates,includingbenefits,arethreedollarsanhourwithovertimepayingdouble. Thecompanyprefersaconstant
laborforceandestimatesthatregulartimewillmake2000man-hoursavailableperperiod. Workersarewilling
toincreasetheirworktimeupto25%forovertimecompensation. Thedemandforbootsandshoesforthethree
periodsisestimatedas:
Period
Boots
Shoes
1
300pr.
3000pr.
2
600pr.
5000pr.
3
900pr.
4000pr.
a) Setupthemodelusing1man-hourand
1
2
man-hourastheeffortrequiredtoproduceapairofbootsandshoes,
respectively.
b) Writethedualproblem.
c) Definethephysicalmeaningofthedualobjectivefunctionandthedualconstraints.
8. Incapital-budgetingproblemswithinthefirm,thereisadebateastowhetherornottheappropriateobjectivefunction
shouldbediscounted.Theformulationofthecapital-budgetingproblemwithoutdiscountingisasfollows:
Maximize
v
N
,
subjectto
Shadow
prices
J
j
=
1
(−
c
ij
x
j
)≤
f
i
(
i
=
0
,
1
,
2
,...,
N
1
)
J
j
=
1
(−
c
Nj
x
j
)+v
N
f
N
,
y
N
0
x
j
u
j
(
j
=
1
,
2
,...,
J
),
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GameTheory
163
wherec
ij
isthecashoutflow
(
c
ij
<
0
)
orinflow
(
c
ij
>
0
)
inperiodiforproject j;therighthand-sideconstant f
i
isthenetexogenousfundsmadeavailableto
(
f
i
>
0
)
orwithdrawnfrom
(
f
i
<
0
)
adivisionofthefirminperiod
i;thedecisionvariablex
j
isthelevelofinvestmentinprojectj;u
j
isanupperboundonthelevelofinvestmentin
projectj;and
v
N
isavariablemeasuringthevalueoftheholdingsofthedivisionattheendoftheplanninghorizon.
Iftheundiscountedproblemissolvedbytheboundedvariablesimplexmethod,theoptimalsolutionisx
j
for
j
=
1
,
2
,...,
Jwithassociatedshadowprices(dualvariables)y
i
fori
=
0
,
1
,
2
,...,
N.
a) Showthaty
N
=
1.
b) Thediscountfactorfortimeiisdefinedtobethepresentvalue(time i
=
0)ofonedollarreceivedattimei.
Showthatthediscountfactorfortimei,
ρ
i
isgivenby:
ρ
i
=
y
i
y
∗0
(
i
=
0
,
1
,
2
,...,
N
).
(Assumethattheinitialbudgetconstraintisbindingsothaty
0
>
0.)
c) Whyshould
ρ
0
=
1?
9. Thediscountedformulationofthecapital-budgetingproblemdescribedinthepreviousexercisecanbestatedas:
Maximize
J
j
=
1
 N
i
=
0
ρ
i
c
ij
x
j
,
subjectto
Shadow
prices
J
j
=
1
(−
c
ij
x
j
)≤
f
i
(
i
=
0
,
1
,
2
,...,
N
),
λ
i
0
x
j
u
j
(
j
=
1
,
2
,...,
J
),
wheretheobjective-functioncoefficient
N
i
=
0
ρ
i
c
ij
representsthediscountedpresentvalueofthecashflowsfrominvestingatunitlevelinproject j, and
λ
i
for
i
=
1
,
2
,...,
Naretheshadowpricesassociatedwiththefunds-flowconstraints.
Supposethatwewishtosolvetheabovediscountedformulationofthecapital-budgetingproblem,usingthe
discountfactorsdeterminedbytheoptimalsolutionofthepreviousexercise,thatis,setting:
ρ
i
i
=
y
i
y
0
(
i
=
0
,
1
,
2
,...,
N
).
Showthattheoptimalsolutionx
j
for j
=
1
,
2
,...,
J,determinedfromtheundiscountedcaseintheprevious
exercise,isalsooptimaltotheabovediscountedformulation,assuming:
ρ
i
i
(
i
=
0
,
1
,
2
,...,
N
).
[Hint: Writetheoptimalityconditionsforthediscountedproblem,usingshadow-pricevalues
λ
i
=
0fori
=
1
,
2
,...,
N. Doesx
j
(
j
=
1
,
2
,...,
J
)
satisfytheseconditions? Dotheshadowpricesontheupperbounding
constraintsforthediscountedmodeldifferfromthoseoftheundiscountedmodel?]
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164
DualityinLinearProgramming
10. AnalternativeformulationoftheundiscountedproblemdevelopedinExercise8istomaximizethetotalearnings
ontheprojectsratherthanthehorizonvalue.Earningsofaprojectaredefinedtobethenetcashflowfromaproject
overitslifetime.Thealternativeformulationisthen:
Maximize
J
j
=
1
 N
i
=
0
c
ij
x
j
,
subjectto
Shadow
prices
J
j
=
1
(−
c
ij
x
j
)≤
f
i
(
i
=
0
,
1
,
2
,...,
N
)
,
y
i
0
x
j
u
j
(
j
=
1
,
2
,...,
J
)
.
Letx
j
forj
=
1
,
2
,...,
Jsolvethisearningsformulation,andsupposethatthefundsconstraintsarebindingatall
timesi
=
0
,
1
,...,
Nforthissolution.
a) Showthatx
j
forj
=
1
,
2
,...,
JalsosolvesthehorizonformulationgiveninExercise8.
b) Denotetheoptimalshadowpricesofthefundsconstraintsfortheearningsformulationasy
i
fori
=
0
,
1
,
2
,...,
N.
Showthaty
i
=
1
+
y
i
fori
=
0
,
1
,
2
,...,
Nareoptimalshadowpricesforthefundsconstraintsofthehorizon
formulation.
11. SupposethatwenowconsideravariationofthehorizonmodelgiveninExercise8,thatexplicitlyincludesone-period
borrowingb
i
atrater
b
andone-periodlending
i
atrater
.Theformulationisasfollows:
Maximize
v
N
,
subjectto
Shadow
prices
J
j
=
1
(−
c
0j
x
j
)+
0
b
0
f
0
,
y
0
J
j
=
1
(−
c
ij
x
j
)−(
1
+
r
)
i
1
+
i
+(
1
+
r
b
)
b
i
1
b
i
f
i
(
i
=
1
,
2
,...,
N
1
)
y
i
J
j
=
1
(−
c
Nj
x
j
)−(
1
+
r
)
N
1
+(
1
r
b
)
b
N
1
+v
N
f
N
,
y
N
b
i
B
i
(
i
=
0
,
1
,...,
N
1
)
,
w
i
0
x
j
u
j
(
j
=
1
,
2
,...,
J
)
,
b
i
0
, 
i
0
(
i
=
0
,
1
,...,
N
1
)
.
a) Supposethattheoptimalsolutionincludeslendingineverytimeperiod;thatis,
i
>
0fori
=
0
,
1
,...,
N
1.
Showthatthepresentvalueofadollarinperiodiis
ρ
i
=
1
1
+
r
i
.
b) Supposethatthereisnoupperboundonborrowing;thatis, B
i
=+∞
fori
=
0
,
1
,...,
N
1,andthatthe
borrowingrateequalsthelendingrate,r
=
r
b
=
r
.Showthat
y
i
1
y
i
=
1
+
r
,
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GameTheory
165
andthatthepresentvalueofadollarinperiodiis
ρ
i
=
1
1
+
r
i
.
c) Let
w
i
betheshadowpricesontheupper-boundconstraintsb
i
B
i
,andletr
=
r
b
=
r
.Showthattheshadow
pricessatisfy:
1
+
r
y
i
1
y
i
1
+
r
+w
i
1
.
d) Assumethattheborrowingrateisgreaterthanthelendingrate,r
b
>
r
;showthatthefirmwillnotborrowand
lendinthesameperiodifitusesthislinear-programmingmodelforitscapitalbudgetingdecisions.
12. Asanexampleofthepresent-valueanalysisgiveninExercise8,considerfourprojectswithcashflowsandupper
boundsasfollows:
Endof
Endof
Endof
Upper
Project
year0
year1
year2
bound
A
1
.
00
0.60
0.60
B
1
.
00
1.10
0
500
C
0
1
.
00
1.25
D
1
.
00
0
1.30
Anegativeentryinthistablecorrespondstoacashoutflowandapositiveentrytoacashinflow.
Thehorizon-valuemodelisformulatedbelow,andtheoptimalsolutionandassociatedshadowpricesaregiven:
a) Explainexactlyhowoneadditionaldollarattheendofyear2wouldbeinvestedtoreturntheshadowprice1.25.
Similarly,explainhowtousetheprojectsintheportfoliotoachieveareturnof1.35foranadditionaldollarat
theendofyear0.
b) Determinethediscountfactorsforeachyear,fromtheshadowpricesontheconstraints.
c) Findthediscountedpresentvalue
N
i
=
0
ρ
i
c
ij
ofeachofthefourprojects. Howdoesthesignofthediscounted
presentvalueofaprojectcomparewiththesignofthereducedcostforthatproject? Whatistherelationship
betweenthetwo?Candiscountinginterpretedinthiswaybeusedfordecision-making?
d) Considerthetwoadditionalprojectswithcashflowsandupperboundsasfollows:
Endof
Endof
Endof
Upper
Project
year0
year1
year2
bound
E
1
.
00
0.75
0.55
F
1
.
00
0.30
1.10
Whatisthediscountedpresentvalueofeachproject?Bothappearpromising.Supposethatfundsaretransferred
fromthecurrentportfoliointoprojectE,willprojectFstillappearpromising? Why? Dothediscountfactors
change?
166
DualityinLinearProgramming
13. IntheexercisesfollowingChapter1,weformulatedtheabsolute-valueregressionproblem:
Minimize
m
i
=
1
|
y
i
x
i1
β
1
x
i2
β
2
−···−
x
in
β
n
|
asthelinearprogram:
Minimize
m
i
=
1
(
P
i
+
N
i
),
subjectto:
x
i1
β
1
+
x
i2
β
2
+···+
x
in
β
n
+
P
i
N
i
=
y
i
fori
=
1
,
2
,...,
m
,
P
i
0
,
N
i
0
fori
=
1
,
2
,...,
m
.
Inthisformulation,they
i
aremeasurements ofthedependentvariable(e.g., income), whichisassumedtobe
explainedbyindependentvariables(e.g.,levelofeducation,parents’income,andsoforth),whicharemeasuredas
x
i1
,
x
i2
,...,
x
in
.Alinearmodelassumesthatydependslinearlyuponthe
β
’s,as:
ˆ
y
i
=
x
i1
β
1
+
x
i2
β
2
+···+
x
in
β
n
.
(29)
Givenanychoiceoftheparameters
β
1
2
,...,β
n
y
i
isanestimateofy
i
.Theaboveformulationaimstominimize
thedeviationsoftheestimatesof
ˆ
y
i
from y
i
asmeasuredbythesumofabsolutevaluesofthedeviations. The
variablesinthelinear-programmingmodelaretheparameters
β
1
2
,...,β
n
aswellastheP
i
andN
i
.Thequantities
y
i
,
x
i1
,
x
i2
,...,
x
in
areknowndatafoundbymeasuringseveralvaluesforthedependentandindependentvariables
forthelinearmodel(29).
Inpractice,thenumberofobservationsmfrequentlyismuchlargerthanthenumberofparametersn. Show
howwecantakeadvantageofthispropertybyformulatingthedualtotheabovelinearprograminthedualvariables
u
1
,
u
2
,...,
u
m
.Howcanthespecialstructureofthedualproblembeexploitedcomputationally?
14. Thefollowingtableauisincanonicalformformaximizingz,exceptthatonerighthand-sidevalueisnegative.
Basic
Current
variables
values
x
1
x
2
x
3
x
4
x
5
x
6
x
1
14
1
2
3
2
x
2
6
1
1
2
2
x
3
10
1
1
2
0
(−
z
)
40
1
4
2
However,thereducedcostsofthenonbasicvariablesallsatisfytheprimaloptimalityconditions.Findtheoptimal
solutiontothisproblem,usingthedualsimplexalgorithmtofindafeasiblecanonicalformwhilemaintainingthe
primaloptimalityconditions.
15. InChapter2wesolvedatwo-constraintlinear-programmingversionofatrailer-productionproblem:
Maximize6x
1
+
14x
2
+
13x
3
,
subjectto:
1
2
x
1
+
2x
2
+
4x
3
24 (Metalworkingcapacity)
,
x
1
+
2x
2
+
4x
3
60 (Woodworkingcapacity)
,
x
1
0
,
x
2
0
,
x
3
0
,
obtaininganoptimaltableau:
GameTheory
167
Basic
Current
variables
values
x
1
x
2
x
3
x
4
x
5
x
1
36
1
6
4
1
x
3
6
1
1
1
1
2
(−
z
)
294
9
11
1
2
Supposethat, informulatingthisproblem, weignoredaconstraintlimitingthetimeavailable inthe shopfor
inspectingthetrailers.
a) Ifthesolutionx
1
=
36
,
x
2
=
0,andx
3
=
6totheoriginalproblemsatisfiestheinspectionconstraint,isit
necessarilyoptimalfortheproblemwhenweimposetheinspectionconstraint?
b) Supposethattheinspectionconstraintis
x
1
+
x
2
+
x
3
+
x
6
=
30
,
wherex
6
isanonnegativeslackvariable.Addthisconstrainttotheoptimaltableauwithx
6
asitsbasicvariable
andpivottoeliminatethebasicvariablesx
1
andx
3
fromthisconstraint. Isthetableaunowindualcanonical
form?
c) Usethedualsimplexmethodtofindtheoptimalsolutiontothetrailer-productionproblemwiththeinspection
constraintgiveninpart(b).
d) Cantheideasusedinthisexamplebeappliedtosolvealinearprogramwheneveranewconstraintisaddedafter
theproblemhasbeensolved?
16. Applythedualsimplexmethodtothefollowingtableauformaximizingwithnonnegativedecisionvariables
x
1
,
x
2
,...,
x
5
.
Basic
Current
variables
values
x
1
x
2
x
3
x
4
x
5
x
1
3
1
1
2
2
x
2
7
1
3
4
8
(−
z
)
5
1
5
6
Istheproblemfeasible?Howcanyoutell?
17. Considerthelinearprogram:
Minimizez
=
2x
1
+
x
2
,
subjectto:
4x
1
+
3x
2
x
3
16
,
x
1
+
6x
2
+
3x
3
12
,
x
i
0 fori
=
1
,
2
,
3
.
a) Writetheassociateddualproblem.
b) Solvetheprimalproblem,usingthedualsimplexalgorithm.
c) Utilizingthefinaltableaufrompart(b), , findoptimalvaluesforthedualvariables
y
1
andy
2
. Whatisthe
correspondingvalueofthedualobjectivefunction?
18. Forwhatvaluesoftheparameter
θ
isthefollowingtableauincanonicalformformaximizingtheobjectivevaluez?
168
DualityinLinearProgramming
Basic
Current
variables
values
x
1
x
2
x
3
x
4
x
5
x
1
2
1
1
2
3
x
2
1
1
1
0
1
(−
z
)
20
3
−θ
4
−θ
6
Startingwiththistableau,usetheparametricprimal-dualalgorithmtosolvethelinearprogramat
θ=
0.
19. Aftersolvingthelinearprogram:
Maximizez
=
5x
1
+
7x
2
+
2x
3
,
subjectto:
2x
1
+
3x
2
+
x
3
+
x
4
=
5
,
1
2
x
1
+
x
2
+
x
5
=
1
,
x
j
0
(
j
=
1
,
2
,...,
5
),
andobtainingtheoptimalcanonicalform
Basic
Current
variables
values
x
1
x
2
x
3
x
4
x
5
x
3
1
1
1
1
4
x
1
2
1
2
0
2
(−
z
)
12
1
2
2
wediscoverthattheproblemwasformulatedimproperly.Theobjectivecoefficientfor x
2
shouldhavebeen11(not
7)andtherighthandsideofthefirstconstraintshouldhavebeen2(not5).
a) Howdothesemodificationsintheproblemformulationalterthedatainthetableauabove?
b) Howcanweusetheparametricprimal-dualalgorithmtosolvethelinearprogramafterthesedatachanges,
startingwithx
1
andx
3
asbasicvariables?
c) Findanoptimalsolutiontotheproblemafterthedatachangesaremade, , usingtheparametricprimal-dual
algorithm.
20. Rock,Paper,andScissorsisagameinwhichtwoplayerssimultaneouslyrevealnofingers(rock),onefinger(paper),
ortwofingers(scissors).Thepayofftoplayer1forthegameisgovernedbythefollowingtable:
Notethatthegameisvoidifbothplayersselectthesamealternative:otherwise,rockbreaksscissorsandwins,
scissorscutpaperandwins,andpapercoversrockandwins.
Usethelinear-programmingformulationofthisgameandlinear-programmingdualitytheory,toshowthatboth
players’optimalstrategyistochooseeachalternativewithprobability 1
3
.
21. Solveforanoptimalstrategyforbothplayer1andplayer2inazero-sumtwo-persongamewiththefollowing
payofftable:
Istheoptimalstrategyofeachplayerunique?
GameTheory
169
22. Inagameoftic-tac-toe,thefirstplayerhasthreedifferentchoicesatthefirstmove: : thecentersquare,acorner
square,orasidesquare. Thesecondplayerthenhasdifferentalternativesdependinguponthemoveofthefirst
player. Forinstance,ifthefirstplayerchoosesacornersquare,thesecondplayercanselectthecentersquare,the
oppositecorner,anadjacentcorner,anoppositesidesquare,oranadjacentsidesquare.Wecanpicturethepossible
outcomesforthegameinadecisiontreeasfollows:
FigureE4.1
Nodes
1correspondtodecisionpointsforthefirstplayer;nodes
2aredecisionpointsforthesecondplayer.
Thetreeisconstructedinthismannerbyconsideringtheoptionsavailabletoeachplayeratagiventurn,untileither
oneplayerhaswonthegame(i.e.,hasselectedthreeadjacenthorizontalorverticalsquaresorthesquaresalongone
ofthediagonals),orallninesquareshavebeenselected.Inthefirstcase,thewinningplayerreceives1point;inthe
secondcasethegameisadrawandneitherplayerreceivesanypoints.
Thedecision-treeformulationofagamelikethatillustratedhereiscalledanextensiveformformulation.Show
howthegamecanberecastinthelinear-programmingform(calledanormalform)discussedinthetext. Donot
attempttoenumerateeverystrategyforbothplayers.Justindicatehowtoconstructthepayofftables.
[Hint: Canyouextractastrategyforplayer1byconsideringhisoptionsateachdecisionpoint? ? Doeshis
strategydependuponhowplayer2reactstoplayer1’sselections?]
Theremainingexercisesextendthetheorydevelopedinthischapter,providingnewresultsandrelatingsomeofthe
dualityconceptstootherideasthatariseinmathematicalprogramming.
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