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33 Pages
iastate-part4
Course: IE 361, Spring 2006School: Iowa State
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4 IEalid? Iftheparameterschange... ...howmuchdoestheobjectivefunctionchange? ...howmuchdotheoptimalvaluesofthedecision variableschange? Notes WhySensitivityAnalysis Sofar:findanoptimiumsolutiongivencertain constantparameters(costs,demand,etc) Howwelldoweknowtheseparameters? Usuallynotveryaccuratelyroughestimates Doourresultsremainv 312 1 GeneralOptimizationProblem...
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4
IEalid?
Iftheparameterschange...
...howmuchdoestheobjectivefunctionchange?
...howmuchdotheoptimalvaluesofthedecision
variableschange?
Notes WhySensitivityAnalysis
Sofar:findanoptimiumsolutiongivencertain
constantparameters(costs,demand,etc)
Howwelldoweknowtheseparameters?
Usuallynotveryaccuratelyroughestimates
Doourresultsremainv 312
1
GeneralOptimizationProblem
Minimizesomecostormaximizebenefit
Constraints:
=Bothsupplyrestrictionanddemandrequirement
Restrictiononsatisfyingdemandforsomeresource
Restrictionsonsupplyofsomeresource
Variabletypeconstraints
Decisionvariabledeterminethelevelsofsome
activity
Coefficients=perunitimpactofactivities
Notes 4
IE 312
2
ChangingConstraints
Relaxingconstraints:
Optimalvaluesameorbetter
Tighteningconstraints:
Optimalvaluesameorworse
Original
Notes 4
Relaxed
IE 312
Tightened
3
CrudeOilModel
min 20 x1 + 15 x2
s.t. 0.3 x1 + 0.4 x2 2.0 (gasoline)
Satisfy
Demand
0.4 x1 + 0.2 x2 1.5 (jet fuel)
0.2 x1 + 0.3 x2 0.5 (lubricants)
x1 9
Supply
Restriction
x2 6
x1 , x2 2
Notes 4
IE 312
4
Solution(LINDO) 1
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
X1
X2
Notes 4
ROW
2)
3)
4)
5)
6)
7)
8)
112.5000
VALUE
2.000000
3.500000
SLACK OR SURPLUS
0.000000
0.000000
0.950000
7.000000
2.500000
0.000000
1.500000
IE 312
REDUCED COST
0.000000
0.000000
DUAL PRICES
-37.500000
0.000000
0.000000
0.000000
0.000000
-18.750000
0.000000
5
160
Sensitivity
140
120
80
60
Plenty of this crude
40
20
5.
Crude Supply
Notes 4
IE 312
6
6
75
5
5.
25
5.
5
75
4.
5
4.
25
4.
4
75
3.
5
3.
25
3.
3
75
2.
5
2.
2.
25
0
2
Cost
100
RHSCoefficients
Constraint
RHS
RHS
Type
Increase Decrease
Supply (<)
Relax
Tighten
Demand (>) Tighten
Relax
Notes 4
IE 312
7
LHSCoefficients
Constraint
LHS
LHS
Type
Increase Decrease
Supply (<)
Tighten
Relax
Demand (>)
Relax
Tighten
Notes 4
IE 312
8
NewConstraints
Addingconstraintstightensthefeasibleset
Removingconstraintsrelaxesthefeasibleset
Whataboutunmodeledconstraints?
Notes 4
IE 312
9
RateofChange
Supply
Demand
Optimal
Value
Optimal
Value
Maximize
RHS
RHS
Optimal
Value
Optimal
Value
Minimize
RHS
RHS
Notes 4
IE 312
10
ObjectiveFunctionChanges
Model
Coefficient
Coefficient
Form
Increase
Decrease
Maximize Same or better Same or worse
Minimze Same or worse Same or better
Notes 4
IE 312
11
CrudeOil:Changingx1Coefficient
140
120
100
Cost
80
60
40
20
0
0
5
10
15
20
25
30
35
Coefficient
Notes 4
IE 312
12
RateofChange
Maximize
Minimize
Optimal
Value
Optimal
Value
Coefficient.
Coefficient.
Notes 4
IE 312
13
NewActivities
Addingactivities
Optimalvaluesameorbetter
Removingactivities
Notes 4
Optimalvaluesameorworse
IE 312
14
QuantifyingEffects
Nowknowthequalitativeeffectsof
ChangingRHScoefficients
ChangingLHScoefficients
Changingobjectivefunctioncoefficients
Adding/deletingconstraints
Adding/deletingactivities
Howmuchdoestheobjectivechange?
Quantitativechange
Notes 4
IE 312
15
BacktoCrudeOilExample
min 20 x1 + 15 x2
s.t. 0.3 x1 + 0.4 x2 2.0 (gasoline)
0.4 x1 + 0.2 x2 1.5 (jet fuel)
0.2 x1 + 0.3 x2 0.5 (lubricants)
x1 9
x2 6
x1 , x2 2
Decreasing RHS
will make objective
better or no worse,
but by how much?
How much are we willing to pay to have one more barrel available?
Notes 4
IE 312
16
AnswerusingtheDual
max 2v1 + 1.5v2 + 0.5v3 + 9v4 + 6v5
s.t. + 0.3v1 0.4v2 + 0.2v3 + 1v4 20
0.4v1 + 0.2v2 + 0.3v3 + 1v5 15
v1 , v2 , v3 0
v4 , v5 0
Notes 4
IE 312
17
LINDOSolution
LP OPTIMUM FOUND AT STEP
4
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
V1
V2
V3
V4
V5
Notes 4
92.50000
VALUE
20.000000
35.000000
0.000000
0.000000
0.000000
IE 312
REDUCED COST
0.000000
0.000000
0.950000
0.000000
0.000000
18
Interpretation
Ourcostwillbereducedby$20or$35,
respectively,ifthedemandforgasoline
orjetfuelisoneunitless.
Smallerdemandforlubricantshasno
effectontheobjective
Wearenotwillingtopayanythingfor
availabilityofmorecrude!
Notes 4
IE 312
19
WhatistheDual?
Theprimalistheoriginaloptimizationproblem
ThedualisanLPdefinedonthesameinput
parametersbutcharacterizingthesensitivitiesof
theprimal
Thereisonedualvariableforeachmain
constraint
Primal
< constraint > constraint = constraint
vi 0
vi 0
Minimize objective
Unrestricted
vi 0
Maximize objective vi 0
Unrestricted
Notes 4
IE 312
20
Interpretation
Thedualvariablesprovideimplicit
pricesformarginalunitsoftheresource
modeledbytheconstraint
Notes 4
Zerounlessactive
Howmuchwearewillingtopayformoreof
aresource(supplyconstraint)
Howmuchwebenefitfromnothavingto
satisfyarequirement(demandconstraint)
IE 312
21
WhattoOptimize?
Implicitmarginalvalue(minimization
primal)orprice(maximizationprimal)is
a
i, j
vj
All
activities i
Maximizevalueorminimizeprice!
Notes 4
IE 312
22
DualConstraints
Foreachactivityxjinaminimization
primalthereisamaindualconstraint
a
v cj
i, j i
i
Foramaximizationprimal,eachxj 0
hasamaindualconstraint
ai , j vi c j
i
Notes 4
IE 312
23
OptimalSolution
Ifprimalhasoptimalsolution
c x = b v
j
j
*
j
*
ii
i
Eithertheprimaloptimalmakesamaininequality
activeorthecorrespondingdualiszero
Eitheranonnegativeprimalvariablehasoptimal
valuexj=0orthecorrespondingdualpricevjmust
makethejthdualconstraintactive
Notes 4
IE 312
24
DualofaMinPrimal
b v
max
ii
i
s.t.
a
v cj
i, j i
i
vi 0
vi 0
vi URS
Notes 4
IE 312
for all activities j
for all primal ' s i
for all primal ' s i
for all primal =' s i
25
DualofaMaxPrimal
b v
min
ii
i
s.t.
a
v cj
i, j i
i
vi 0
vi 0
vi URS
Notes 4
IE 312
for all activities j
for all primal ' s i
for all primal ' s i
for all primal =' s i
26
TopBrass
max 12 x1 + 9 x2
x1 1000 (brass footballs)
s.t.
x2 1500 (brass soccer balls)
x1 + x2 1750 (brass plaques)
4 x1 + 2 x2 4800 (feet of wood)
x1 , x2 0
Notes 4
IE 312
27
GraphicalSolution
4 x1 + 2 x2 4800
2000
x1 1000
x2 1500
1500
Optimal Solution
1000
x1 0
x1 + x2 1750
500
x2 0
Notes 4
500
1000
IE 312
1500
2000
28
Dual
min 1000v1 + 1500v2 + 1750v3 + 4800v4
v1 + v3 + 4v4 12
s.t.
v2 + v3 + 2v4 9
v1 , v2 , v3 , v4 0
Notes 4
IE 312
29
LindoSolution
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
V1
V2
V3
V4
Notes 4
17700.00
VALUE
0.000000
0.000000
6.000000
1.500000
IE 312
REDUCED COST
350.000000
400.000000
0.000000
0.000000
30
Interpretation
Wearewillingtopayupto
$6/eachforadditionalbrass Our objective is
sensitive to these
plaques
estimates!
Wearewillingtopayupto
$1.5/footformorewood
Wedontneedanymore
brassfootballsorsoccerballs
Notes 4
IE 312
31
FormulatingDuals
max
6 x1 x2 + 13 x3
s.t. 3 x1 + x2 + 2 x3 = 7
5 x1 x2 6
x2 + x3 2
x1 0
x2 0
Notes 4
IE 312
32
FormulatingDuals
min
7 x1 + 44 x3
s.t. 2 x1 4 x2 + x3 15
x1 + 4 x2 5
5 x1 x2 + 3 x3 = 11
x1 0
x2 0
Notes 4
IE 312
33
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TUTORIAL 4 SOLUTIONS27.2Potential ordinary shares are dilutive if the EPS is recalculated on the basis that thesecurities were converted to ordinary shares, and the recalculated diluted EPS figureis less than the basic EPS (which is based on the ordin
UNSW - ACCT - 2542
1SOLUTIONS TUTORIAL 7Chapter 22 Controlled entities: the consolidation methodDISCUSSION QUESTIONS21.RequiredDiscuss the potential for Gilder Ltd being classified as a subsidiary of Croc Ltd.Part ACroc LtdGlider Ltd40%- NCI = 60%- no other part
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1TUTORIAL 8 SOLUTIONSChapter 24 Consolidation: intragroup transactionsDISCUSSION QUESTIONS1. Why is it necessary to make adjustments for intragroup transactions?The consolidated financial statements are the statements of the group, an economic entity
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1TUTORIAL 9 SOLUTIONSChapter 25: Consolidation: Non-controlling interestDISCUSSION QUESTIONS1.If a step approach is used in the calculation of the NCI share of equity, what are the stepsinvolved?Step 1: share of subsidiary equity at acquisition dat
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1TUTORIAL 10 SOLUTIONSChapter 26: Consolidation: indirect ownership interestsDISCUSSION QUESTIONS1.What is the difference between direct and indirect NCI?Direct NCI (DNCI) own shares directly in an entity. Indirect NCI are DNCIshareholders in an en