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Aplikasi Riset Operasional

Dalam MATLABArif Rahman, ST MT

OptimtoolOptimization Toolbox™ Optimtool

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Linear ProgrammingVariables

Decision variablesStructural variablesAuxiliary variablesSlack variablesArtificial variables

CoefficientsCost coefficientsTechnological coefficientsConstraint parameter or Right Hand Side value

FunctionObjective or criterion functionRestriction or functional constraintsNonnegativity constraints

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Linear ProgrammingMinimize f(x) = c1x1 + c2x2 + … + cnxn

Subject toa11x1 + a12x2 + … + a1nxn b1

a21x1 + a22x2 + … + a2nxn b2

am1x1 + am2x2 + … + amnxn bm

andx1 0; x2 0; … ; xn 0

Maximize or Minimize

or or =

0 or 0 or unrestricted

Linear Programming[x,fval,exitflag] = linprog(f,A,b,Aeq,beq,lb)Di mana

f Vector koefisien fungsi tujuan (cost coefficient).A Matrix koefisien fungsi kendala pertidaksamaan (technological coefficient).b Vector parameter batas fungsi kendala pertidaksamaan(right-hand side or constraint parameter).Aeq Matrix koefisien fungsi kendala persamaan (technological coefficient).beq Vector parameter batas fungsi kendala persamaan(right-hand side or constraint parameter).lb Vector batas bawahvariabel keputusanub Vector batas bawahvariabel keputusan

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Linear ProgrammingMinimize f(x) = –5x1 – 4x2 – 6x3

Subject to : x1 – x2 + x3 20

3.x1 + 2.x2 + 4.x3 42

3.x1 + 2.x2 30

where x1 0; x2 0; x3 0

Ketikkan perintah berikut di dalam command window: f = [-5; -4; -6]; x = 0.000 x1

A = [1 -1 1; 3 2 4; 3 2 0]; 15.000 x2

b = [20; 42; 30]; 3.000 x3

lb=zeros(3,1); fval = -78.000[x,fval,exitflag]= linprog(f,A,b,[],[],lb)

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7Binary Integer ProgrammingMinimize f(x) = c1x1 + c2x2 + … + cnxn

Subject toa11x1 + a12x2 + … + a1nxn b1

a21x1 + a22x2 + … + a2nxn b2

am1x1 + am2x2 + … + amnxn bm

andx1; … ; xn {0;1}

Maximize or Minimize

or or =

0 or 1

Binary Integer Programming

x = bintprog(f,A,b)Di mana

f Vector koefisien fungsi tujuan (cost coefficient).A Matrix koefisien fungsi kendala (technological coefficient).b Vector parameter batas fungsi kendala (right-hand side or constraint parameter).

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Binary Integer ProgrammingMinimize f(x) = –9x1 – 5x2 – 6x3 – 4x4,

Subject to :6.x1 + 3.x2 + 5.x3 + 2.x4 9

x3 + x4 1

-x1 + x3 0

-x2 + x4 0

where x1, x2, x3, and x4 are binary integers

Ketikkan perintah berikut di dalam command window: f = [-9; -5; -6; -4]; x = 1 x1

A = [6 3 5 2; 0 0 1 1; -1 0 1 0; 0 -1 0 1]; 1 x2

b = [9; 1; 0; 0]; 0 x3

x = bintprog(f,A,b) 0 x4

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Quadratic ProgrammingMinimize f(x) = c1x1

2 + c2x22 + c3x1x2 + c4x1 + c5x2 …

Subject toa11x1 + a12x2 + … + a1nxn b1

a21x1 + a22x2 + … + a2nxn b2

am1x1 + am2x2 + … + amnxn bm

andx1 0; x2 0; … ; xn 0

Maximize or Minimize

or or =

0 or 0 or unrestricted

Quadratic Programmingx = quadprog(H,f,A,b)Di mana

H Matrix bujursangkar simetris koefisien interaksi fungsi tujuan (cost coefficient).f Vector koefisien fungsi tujuan (cost coefficient).A Matrix koefisien fungsi kendala (technological coefficient).b Vector parameter batas fungsi kendala (right-hand side or constraint parameter).

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Quadratic ProgrammingMinimize f(x) = 1/2.x1

2 + x22 – x1x2 – 2.x1 – 6.x2,

Subject to : x1 + x2 2

-x1 + 2.x2 2

2.x1 + x2 3

where x1 0; x2 0

Ketikkan perintah berikut di dalam command window: H = [1 -1; -1 2];f = [-2; -6]; x = 0.6667 x1

A = [1 1; -1 2; 2 1]; 1.3333 x2

b = [2; 2; 3];x = quadprog(H, f, A, b)

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Goal ProgrammingMultiobjective goal attainmentA weighting vector to control the relative underattainment or overattainment of the objectives in fgoalattain.When the values of goal are all nonzero, to ensure the same percentage of under- or overattainment of the active objectives, set the weighting function to abs(goal).The active objectives are the set of objectives that are barriers to further improvement of the goals at the solution.

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Goal Programmingx = fgoalattain(fun,x0,goal,weight,A,b)

Di manafun function dengan vector x menghasilkan vector F, dengan objective functions dievaluasi pada vector x x0 Vector initial x.

goal Vector koefisien fungsi tujuan (cost coefficient)weight Vector pembobotan dari underattainment atau overattainment dari tujuanA Matrix koefisien fungsi kendala (technological coefficient).b Vector parameter batas fungsi kendala (right-hand side or constraint parameter).

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Minimax Programmingminimizes the worst-case (largest) value of a set of multivariable functions, starting at an initial estimate.

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Minimax Programmingx = fminimax(fun,x0,A,b)

Di manafun function dengan vector x menghasilkan vector F, dengan objective functions dievaluasi pada vector x x0 Vector initial x.

A Matrix koefisien fungsi kendala (technological coefficient).b Vector parameter batas fungsi kendala (right-hand side or constraint parameter).

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Minimax ProgrammingMinimax f(x) = {f1(x); f2(x); f3(x); f4(x); f5(x)}

Where :f1 (x) =2.x1

2 + x22 – 48.x1 – 40.x2 + 304

f2 (x) =-x12 – 3.x2

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f3 (x) =x1 + 3.x2 – 18

f4 (x) =-x1 – x2

f5 (x) =x1 + x2 – 8

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Minimax ProgrammingKetikkan perintah berikut di dalam command window:

function f = myfun(x)f(1)= 2*x(1)^2+x(2)^2-48*x(1)-40*x(2)+304;f(2)= -x(1)^2 - 3*x(2)^2;f(3)= x(1) + 3*x(2) -18;f(4)= -x(1)- x(2);f(5)= x(1) + x(2) – 8

x0 = [0.1; 0.1];[x,fval] = fminimax(@myfun,x0)

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