15.057 Spring 03 Vande Vate1 Practice Final: Linear? param T; param Demand{1..T}; var InitialLevel;...

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


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


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


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


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]


Sensitivity Analysis


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


What are the Shadow Prices of the 4 vintagesWhat are the units of these prices


What would be the impact of losing 100 gals of Vintage 3


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


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


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



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





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


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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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


Linear



Linear continued





UseEdge[dc pool]

Linear Version


Or more precisely








Questions













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item








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Linear



Linear continued





UseEdge[dc pool]

Linear Version


Or more precisely








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True or False






















item








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Linear continued





UseEdge[dc pool]

Linear Version


Or more precisely








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item








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UseEdge[dc pool]

Linear Version


Or more precisely








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item








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True or False






















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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;...