15.057 Spring 03 Vande Vate1 Practice Final: Linear? param T; param Demand{1..T}; var InitialLevel;...
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Transcript of 15.057 Spring 03 Vande Vate1 Practice Final: Linear? param T; param Demand{1..T}; var InitialLevel;...
![Page 1: 15.057 Spring 03 Vande Vate1 Practice Final: Linear? param T; param Demand{1..T}; var InitialLevel; var GrowthRate; var Estimate{1..T}; minimize TotalError:](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ed55503460f94be5a88/html5/thumbnails/1.jpg)
15057 Spring 03 Vande Vate 1
Practice Final Linear
param Tparam Demand1Tvar InitialLevel var GrowthRate var Estimate1T minimize TotalError sumt = 1T (Estimate[t] -Demand[t]) (Estimate[t] -Demand[t]) st DefineEstimate t in 1T Estimate[t] = InitialLevel + GrowthRatet
15057 Spring 03 Vande Vate 2
Linear
Set QUESTIONS The set of questions Param AvgPctCorrectQUESTIONS Param MinAverage minimum average score Param MaxAverage maximum average score Var PointsQUESTIONS gt= 0 Var TooHigh gt= 0Var TooLow gt= 0Minimize Error
TooHigh + TooLow st Nearly100TotalPoints sumq in QUESTIONS Points[q] = 100+TooHigh -TooLow st MeetMin (sumq in QUESTIONS AvgPctCorrect[q]Points[q]) (100+TooHigh -TooLow) gt= MinAverage st MeetMax (sumq in QUESTIONS AvgPctCorrect[q]Points[q]) (100+TooHigh -TooLow) lt= MaxAverage
15057 Spring 03 Vande Vate 3
Linear
Set POOLS Set STORES Set DCS Set PRODUCTS Param DemandPRODUCTS STORES Param CubicCapacity Param WeightLimit Param MaxFillTimeDCS POOLS Param WeightPRODUCTS Param CubePRODUCTS var WeighOutDCS POOLS binary var UseEdgeDCS POOLS binary var Flow[PRODUCTS DCS POOLS gt= 0 var AssignPOOLS STORES binary
15057 Spring 03 Vande Vate 4
Linear continued
st ImposeTrailerFillbyWeightdc in DCS pool in POOLS MaxFillTime[dc pool]sumprd in PRODUCTS Weight[prd]Flow[prd dc pool] gt= WeightLimitWeighOut[dc pool]UseEdge[dc pool] st ImposeTrailerFillbyCubedc in DCS pool in POOLS MaxFillTime[dc pool]sumprd in PRODUCTS Cube[prd]Flow[prd dc pool] gt= CubicCapacityCubeOut[dc pool]UseEdge[dc pool] st DefineUseEdgeprd in PRODUCTS dc in DCS pool in POOLS Flow[prd dc pool] lt= (sumstore in STORES Demand[prd store])UseEdge[dc pool] st WeightOrCubedc in DCS pool in POOLS WeighOut[dc pool] + CubeOut[dc pool] = 1
15057 Spring 03 Vande Vate 5
LinearldquoAndrdquo CubeOut[dc pool]UseEdge[dc pool] equivalent to
CubeOut[dc pool] AND
UseEdge[dc pool]
Linear Version
(CubeOut[dc pool] + UseEdge[dc pool] -1)
Or more precisely
Var Both[dc pool] binary
Both[dc pool] gt= CubeOut[dc pool] +UseEdge[dc pool] -1
Both[dc pool] lt= CubeOut[dc pool]
Both[dc pool] lt= UseEdge[dc pool]
15057 Spring 03 Vande Vate 6
Sensitivity Analysis
15057 Spring 03 Vande Vate 7
Questions
What is the minimum amount by which the selling price of C would have to change before it would be attractive to produce Blend C
15057 Spring 03 Vande Vate 8
What are the Shadow Prices of the 4 vintagesWhat are the units of these prices
15057 Spring 03 Vande Vate 9
What would be the impact of losing 100 gals of Vintage 3
15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
- Slide 1
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![Page 2: 15.057 Spring 03 Vande Vate1 Practice Final: Linear? param T; param Demand{1..T}; var InitialLevel; var GrowthRate; var Estimate{1..T}; minimize TotalError:](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ed55503460f94be5a88/html5/thumbnails/2.jpg)
15057 Spring 03 Vande Vate 2
Linear
Set QUESTIONS The set of questions Param AvgPctCorrectQUESTIONS Param MinAverage minimum average score Param MaxAverage maximum average score Var PointsQUESTIONS gt= 0 Var TooHigh gt= 0Var TooLow gt= 0Minimize Error
TooHigh + TooLow st Nearly100TotalPoints sumq in QUESTIONS Points[q] = 100+TooHigh -TooLow st MeetMin (sumq in QUESTIONS AvgPctCorrect[q]Points[q]) (100+TooHigh -TooLow) gt= MinAverage st MeetMax (sumq in QUESTIONS AvgPctCorrect[q]Points[q]) (100+TooHigh -TooLow) lt= MaxAverage
15057 Spring 03 Vande Vate 3
Linear
Set POOLS Set STORES Set DCS Set PRODUCTS Param DemandPRODUCTS STORES Param CubicCapacity Param WeightLimit Param MaxFillTimeDCS POOLS Param WeightPRODUCTS Param CubePRODUCTS var WeighOutDCS POOLS binary var UseEdgeDCS POOLS binary var Flow[PRODUCTS DCS POOLS gt= 0 var AssignPOOLS STORES binary
15057 Spring 03 Vande Vate 4
Linear continued
st ImposeTrailerFillbyWeightdc in DCS pool in POOLS MaxFillTime[dc pool]sumprd in PRODUCTS Weight[prd]Flow[prd dc pool] gt= WeightLimitWeighOut[dc pool]UseEdge[dc pool] st ImposeTrailerFillbyCubedc in DCS pool in POOLS MaxFillTime[dc pool]sumprd in PRODUCTS Cube[prd]Flow[prd dc pool] gt= CubicCapacityCubeOut[dc pool]UseEdge[dc pool] st DefineUseEdgeprd in PRODUCTS dc in DCS pool in POOLS Flow[prd dc pool] lt= (sumstore in STORES Demand[prd store])UseEdge[dc pool] st WeightOrCubedc in DCS pool in POOLS WeighOut[dc pool] + CubeOut[dc pool] = 1
15057 Spring 03 Vande Vate 5
LinearldquoAndrdquo CubeOut[dc pool]UseEdge[dc pool] equivalent to
CubeOut[dc pool] AND
UseEdge[dc pool]
Linear Version
(CubeOut[dc pool] + UseEdge[dc pool] -1)
Or more precisely
Var Both[dc pool] binary
Both[dc pool] gt= CubeOut[dc pool] +UseEdge[dc pool] -1
Both[dc pool] lt= CubeOut[dc pool]
Both[dc pool] lt= UseEdge[dc pool]
15057 Spring 03 Vande Vate 6
Sensitivity Analysis
15057 Spring 03 Vande Vate 7
Questions
What is the minimum amount by which the selling price of C would have to change before it would be attractive to produce Blend C
15057 Spring 03 Vande Vate 8
What are the Shadow Prices of the 4 vintagesWhat are the units of these prices
15057 Spring 03 Vande Vate 9
What would be the impact of losing 100 gals of Vintage 3
15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
- Slide 1
- Slide 2
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- Slide 10
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![Page 3: 15.057 Spring 03 Vande Vate1 Practice Final: Linear? param T; param Demand{1..T}; var InitialLevel; var GrowthRate; var Estimate{1..T}; minimize TotalError:](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ed55503460f94be5a88/html5/thumbnails/3.jpg)
15057 Spring 03 Vande Vate 3
Linear
Set POOLS Set STORES Set DCS Set PRODUCTS Param DemandPRODUCTS STORES Param CubicCapacity Param WeightLimit Param MaxFillTimeDCS POOLS Param WeightPRODUCTS Param CubePRODUCTS var WeighOutDCS POOLS binary var UseEdgeDCS POOLS binary var Flow[PRODUCTS DCS POOLS gt= 0 var AssignPOOLS STORES binary
15057 Spring 03 Vande Vate 4
Linear continued
st ImposeTrailerFillbyWeightdc in DCS pool in POOLS MaxFillTime[dc pool]sumprd in PRODUCTS Weight[prd]Flow[prd dc pool] gt= WeightLimitWeighOut[dc pool]UseEdge[dc pool] st ImposeTrailerFillbyCubedc in DCS pool in POOLS MaxFillTime[dc pool]sumprd in PRODUCTS Cube[prd]Flow[prd dc pool] gt= CubicCapacityCubeOut[dc pool]UseEdge[dc pool] st DefineUseEdgeprd in PRODUCTS dc in DCS pool in POOLS Flow[prd dc pool] lt= (sumstore in STORES Demand[prd store])UseEdge[dc pool] st WeightOrCubedc in DCS pool in POOLS WeighOut[dc pool] + CubeOut[dc pool] = 1
15057 Spring 03 Vande Vate 5
LinearldquoAndrdquo CubeOut[dc pool]UseEdge[dc pool] equivalent to
CubeOut[dc pool] AND
UseEdge[dc pool]
Linear Version
(CubeOut[dc pool] + UseEdge[dc pool] -1)
Or more precisely
Var Both[dc pool] binary
Both[dc pool] gt= CubeOut[dc pool] +UseEdge[dc pool] -1
Both[dc pool] lt= CubeOut[dc pool]
Both[dc pool] lt= UseEdge[dc pool]
15057 Spring 03 Vande Vate 6
Sensitivity Analysis
15057 Spring 03 Vande Vate 7
Questions
What is the minimum amount by which the selling price of C would have to change before it would be attractive to produce Blend C
15057 Spring 03 Vande Vate 8
What are the Shadow Prices of the 4 vintagesWhat are the units of these prices
15057 Spring 03 Vande Vate 9
What would be the impact of losing 100 gals of Vintage 3
15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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-
![Page 4: 15.057 Spring 03 Vande Vate1 Practice Final: Linear? param T; param Demand{1..T}; var InitialLevel; var GrowthRate; var Estimate{1..T}; minimize TotalError:](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ed55503460f94be5a88/html5/thumbnails/4.jpg)
15057 Spring 03 Vande Vate 4
Linear continued
st ImposeTrailerFillbyWeightdc in DCS pool in POOLS MaxFillTime[dc pool]sumprd in PRODUCTS Weight[prd]Flow[prd dc pool] gt= WeightLimitWeighOut[dc pool]UseEdge[dc pool] st ImposeTrailerFillbyCubedc in DCS pool in POOLS MaxFillTime[dc pool]sumprd in PRODUCTS Cube[prd]Flow[prd dc pool] gt= CubicCapacityCubeOut[dc pool]UseEdge[dc pool] st DefineUseEdgeprd in PRODUCTS dc in DCS pool in POOLS Flow[prd dc pool] lt= (sumstore in STORES Demand[prd store])UseEdge[dc pool] st WeightOrCubedc in DCS pool in POOLS WeighOut[dc pool] + CubeOut[dc pool] = 1
15057 Spring 03 Vande Vate 5
LinearldquoAndrdquo CubeOut[dc pool]UseEdge[dc pool] equivalent to
CubeOut[dc pool] AND
UseEdge[dc pool]
Linear Version
(CubeOut[dc pool] + UseEdge[dc pool] -1)
Or more precisely
Var Both[dc pool] binary
Both[dc pool] gt= CubeOut[dc pool] +UseEdge[dc pool] -1
Both[dc pool] lt= CubeOut[dc pool]
Both[dc pool] lt= UseEdge[dc pool]
15057 Spring 03 Vande Vate 6
Sensitivity Analysis
15057 Spring 03 Vande Vate 7
Questions
What is the minimum amount by which the selling price of C would have to change before it would be attractive to produce Blend C
15057 Spring 03 Vande Vate 8
What are the Shadow Prices of the 4 vintagesWhat are the units of these prices
15057 Spring 03 Vande Vate 9
What would be the impact of losing 100 gals of Vintage 3
15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
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![Page 5: 15.057 Spring 03 Vande Vate1 Practice Final: Linear? param T; param Demand{1..T}; var InitialLevel; var GrowthRate; var Estimate{1..T}; minimize TotalError:](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ed55503460f94be5a88/html5/thumbnails/5.jpg)
15057 Spring 03 Vande Vate 5
LinearldquoAndrdquo CubeOut[dc pool]UseEdge[dc pool] equivalent to
CubeOut[dc pool] AND
UseEdge[dc pool]
Linear Version
(CubeOut[dc pool] + UseEdge[dc pool] -1)
Or more precisely
Var Both[dc pool] binary
Both[dc pool] gt= CubeOut[dc pool] +UseEdge[dc pool] -1
Both[dc pool] lt= CubeOut[dc pool]
Both[dc pool] lt= UseEdge[dc pool]
15057 Spring 03 Vande Vate 6
Sensitivity Analysis
15057 Spring 03 Vande Vate 7
Questions
What is the minimum amount by which the selling price of C would have to change before it would be attractive to produce Blend C
15057 Spring 03 Vande Vate 8
What are the Shadow Prices of the 4 vintagesWhat are the units of these prices
15057 Spring 03 Vande Vate 9
What would be the impact of losing 100 gals of Vintage 3
15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
-
![Page 6: 15.057 Spring 03 Vande Vate1 Practice Final: Linear? param T; param Demand{1..T}; var InitialLevel; var GrowthRate; var Estimate{1..T}; minimize TotalError:](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ed55503460f94be5a88/html5/thumbnails/6.jpg)
15057 Spring 03 Vande Vate 6
Sensitivity Analysis
15057 Spring 03 Vande Vate 7
Questions
What is the minimum amount by which the selling price of C would have to change before it would be attractive to produce Blend C
15057 Spring 03 Vande Vate 8
What are the Shadow Prices of the 4 vintagesWhat are the units of these prices
15057 Spring 03 Vande Vate 9
What would be the impact of losing 100 gals of Vintage 3
15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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15057 Spring 03 Vande Vate 7
Questions
What is the minimum amount by which the selling price of C would have to change before it would be attractive to produce Blend C
15057 Spring 03 Vande Vate 8
What are the Shadow Prices of the 4 vintagesWhat are the units of these prices
15057 Spring 03 Vande Vate 9
What would be the impact of losing 100 gals of Vintage 3
15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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15057 Spring 03 Vande Vate 8
What are the Shadow Prices of the 4 vintagesWhat are the units of these prices
15057 Spring 03 Vande Vate 9
What would be the impact of losing 100 gals of Vintage 3
15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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15057 Spring 03 Vande Vate 9
What would be the impact of losing 100 gals of Vintage 3
15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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15057 Spring 03 Vande Vate 10
True or False In solving an integer programming problem Solver and
CPLEX employ genetic algorithms In solving an integer programming problem Solver and
CPLEX employ the simplex method The set of optimal solutions to an integer program
describes a convex set
The function f(x) = x2 (thats xx) is a convex function
15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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15057 Spring 03 Vande Vate 11
True or False
The function f(xy) = x2 + y2 (thats xx + yy) a convex function
All polynomials are convex functions If a real valued function f is a convex function the
n the function g(x) = -f(x) is a convex function
15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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15057 Spring 03 Vande Vate 12
Restrictions amp Relaxations
_____ le _____ le _____ le _____A Optimal objective function value of the linear
programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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15057 Spring 03 Vande Vate 13
Restrictions amp Relaxations
Minimizing __C___ le __A__ le ___B__ le ___D__
A Optimal objective function value of the linear programming relaxation with integrality restrictions on some but not all variables
B Optimal objective function value of the integer program
C Optimal objective function value of the complete linear programming relaxation (no integrality restrictions)
D Optimal objective function value of the integer program if we fix the values of some of the variables
15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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15057 Spring 03 Vande Vate 14
IP Modeling There are N items numbered 1 2 hellip N If we choose
item i we earn r[i] If we select exactly one of the items we must pay b You may introduce other variables if you wish You will certainly need to add an objective (minimize total cost) and constraints
param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item
var x1N binary x[i] is one if we choose item i and 0 otherwise
15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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15057 Spring 03 Vande Vate 15
IP Modeling param N the number of items param r1N the revenue on each item param b the cost incurred if we choose exactly one
item var x1N binary x[i] is one if we choose item i and 0
otherwise Var All binary is 1 if we choose all maximize Revenue sumk = 1N r[k]x[k] -bAll st NotNecessaryButForCompletenessk in 1N
All lt= x[k] st DidWeSelectAll 1048672
NAll gt= sumk in 1N x[k]
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