PENELITIAN OPERASIONAL I LINEAR · PDF filePENELITIAN OPERASIONAL I ... Contoh: Shadow Price...
Transcript of PENELITIAN OPERASIONAL I LINEAR · PDF filePENELITIAN OPERASIONAL I ... Contoh: Shadow Price...
13/11/2013
1
PENELITIAN OPERASIONAL I
(TIN 4109)
Lecture 8
LINEAR PROGRAMMING
Lecture 8
• Outline: – Duality – Analisa Sensitivitas
• References: – Frederick Hillier and Gerald J. Lieberman. Introduction
to Operations Research. 7th ed. The McGraw-Hill Companies, Inc, 2001.
– Hamdy A. Taha. Operations Research: An Introduction. 8th Edition. Prentice-Hall, Inc, 2007.
DUALITAS
Hubungan PRIMAL – DUAL
Bila x adalah feasible terhadap PRIMAL dan y feasible terhadap DUAL, maka cx yb
Nilai objektif problem Max Nilai objektif problem Min
DUAL Constraint y A c
x 0 y Ax cx
Ax b y b cx
Teorema Dualitas
● Bila x* adalah penyelesaian dari PRIMAL dan y* adalah penyelesaian dari DUAL, maka cx* = y*b
● Bila x0 feasible terhadap PRIMAL dan y0 feasible terhadap DUAL sedemikian hingga cx0 = y0b, maka x0 dan y0 adalah penyelesaian optimal
Menyelesaikan
PRIMAL
Menyelesaikan
DUAL
z DUAL FR
PRIMAL FR
Optimal
(PRIMAL – DUAL FEASIBLE)
13/11/2013
2
Teorema Dualitas
1. P optimal D optimal
2. P tak terbatas
D tak terbatas
D tidak feasible
P tidak feasible
3. P tidak feasible
D tidak feasible
D tak terbatas/tidak feasible
P tak terbatas/tidak feasible
Both Primal and Dual Infeasible.
, ,
Dual Certificate
Someone tells us x* is an optimal solution. Do we trust them? • Check if x* is feasible? • How to check if it is optimal?
Dual Certificates
• Given x (Claimed to be primal optimal solution) and
• Given y (Claimed to be dual optimal solution).
We can use both to convince ourselves.
1. Check feasibility of x using primal problem
2. Form dual problem and check feasibility of y
3. Check that primal objective value is equal to dual objective value.
Complementary Variable Pairs
Primal Decision Variables
Dual Slack Variables
Primal Slack Variables
Dual Decision Variables
Contoh
0 ,
1
832
Subject to
3 Max
21
21
21
21
xx
xx
xx
xx
0 , , ,
1
8 32
Subject to
3 Max
4321
421
321
21
xxxx
xxx
xxx
xx
z x 1 x 2 x 3 x 4 RHS
z 1 -1 -3 0 0 0
x 3 0 2 3 1 0 8
x 4 0 -1 1 0 1 1
z 1 -4 0 0 3 3
x 3 0 5 0 1 -3 5
x 2 0 -1 1 0 1 1
z 1 0 0 0.8 0.6 7
x 1 0 1 0 0.2 -0.6 1
x 2 0 0 1 0.2 0.4 2
13/11/2013
3
Pada Problem Dual
Hubungan Primal - Dual
Primal
Dual
z x 1 x 2 x 3 x 4 RHS
z 1 -1 -3 0 0 0
x 3 0 2 3 1 0 8
x 4 0 -1 1 0 1 1
z 1 -4 0 0 3 3
x 3 0 5 0 1 -3 5
x 2 0 -1 1 0 1 1
z 1 0 0 0.8 0.6 7
x 1 0 1 0 0.2 -0.6 1
x 2 0 0 1 0.2 0.4 2
ANALISA SENSITIVITAS
Shadow Price
• It is often important for managers to determine how a change in a constraint’s right-hand side changes the LP’s optimal z-value.
• With this in mind, we define the shadow price for the ith constraint of an LP to be the amount by which the optimal z-value is improved—increased in a max problem and decreased in a min problem—if the right-hand side of the ith constraint is increased by 1.
• This definition applies only if the change in the right-hand side of Constraint i leaves the current basis optimal.
Diet Problem Data
FOODS
Diet Problem Data
Food Calories
Total_Fat Protein Vit_A Vit_C
Calcium Price
Peppers 20 0.1 0.7 467.7 66.1 6.7 0.8
Potatoes, Baked 171.5 0.2 3.7 0 15.6 22.7 0.5
Tofu 88.2 5.5 9.4 98.6 0.1 121.8 1.1
Couscous 100.8 0.1 3.4 0 0 7.2 1
White Rice 102.7 0.2 2.1 0 0 7.9 0.4
Macaroni,Ckd 98.7 0.5 3.3 0 0 4.9 0.2
Peanut Butter 188.5 16 7.7 0 0 13.1 0.6
Nutrient Min Max
Calories 2000 2250
Total_Fat 0 65
Protein 50 100
Vit A 5000 50000
Vit C 50 20000
Calcium 800 1600
13/11/2013
4
Diet Problem Setup Diet Problem Dual
What does the dual mean?
Food Calories
Total_Fat Protein Vit_A Vit_C
Calcium Price
Peppers 20 0.1 0.7 467.7 66.1 6.7 0.8
Potatoes, Baked 171.5 0.2 3.7 0 15.6 22.7 0.5
Tofu 88.2 5.5 9.4 98.6 0.1 121.8 1.1
Couscous 100.8 0.1 3.4 0 0 7.2 1
White Rice 102.7 0.2 2.1 0 0 7.9 0.4
Macaroni,Ckd 98.7 0.5 3.3 0 0 4.9 0.2
Peanut Butter 188.5 16 7.7 0 0 13.1 0.6
Nutrient Min Max
Calories 2000 2250
Total_Fat 0 65
Protein 50 100
Vit A 5000 50000
Vit C 50 20000
Calcium 800 1600
Optimal Solutions
Food Opt. Amt. Nutrient Dual (yU) Dual (yL)
Peppers 9.55 Calories 0.000 0.002
Potatoes, Baked 0.95 Total_Fat 0.000 0.000
Tofu 5.39 Protein 0.021 0.000
Couscous 0.00 Vit A 0.000 0.002
White Rice 0.00 Vit C 0.000 0.000
Macaroni,Ckd 11.86 Calcium 0.000 0.008
Peanut Butter 0.00
Shadow Cost of Constraints
How does a “small” change in bi affect the total optimal value?
Linear Programming Problem
(6,11)
13/11/2013
5
Dual Optimum Geometric View
Sensitivity Analysis
1. x* and y* be optimal primal/dual from final dictionary 2. dictionary is assumed non-degenerate.
For a infinitesimally small change d in bj (I.e, bj changes to bj +d) the objective changes by yj * d
What does the dual mean?
Food Calories
Total_Fat Protein Vit_A Vit_C
Calcium Price
Peppers 20 0.1 0.7 467.7 66.1 6.7 0.8
Potatoes, Baked 171.5 0.2 3.7 0 15.6 22.7 0.5
Tofu 88.2 5.5 9.4 98.6 0.1 121.8 1.1
Couscous 100.8 0.1 3.4 0 0 7.2 1
White Rice 102.7 0.2 2.1 0 0 7.9 0.4
Macaroni,Ckd 98.7 0.5 3.3 0 0 4.9 0.2
Peanut Butter 188.5 16 7.7 0 0 13.1 0.6
Nutrient Min Max
Calories 2000 2250
Total_Fat 0 65
Protein 50 100
Vit A 5000 50000
Vit C 50 20000
Calcium 800 1600
Optimal Solutions
Food Opt. Amt. Nutrient Dual (yU) Dual (yL)
Peppers 9.55 Calories 0.000 0.002
Potatoes, Baked 0.95 Total_Fat 0.000 0.000
Tofu 5.39 Protein 0.021 0.000
Couscous 0.00 Vit A 0.000 0.002
White Rice 0.00 Vit C 0.000 0.000
Macaroni,Ckd 11.86 Calcium 0.000 0.008
Peanut Butter 0.00
Primal Solution Dual Solution
13/11/2013
6
31
Contoh: Shadow Price Contoh: Shadow Price
Shadow price jika resource
b1 bertambah 1 unit
b1 ditambah 1 unit menjadi 9
Nilai z bertambah 4/5 (shadow
price) 7 + 4/5 = 39/5
Max x1 3x2
ST 2x1 3x2 x3 8
x1 x2 x4 1
x1 , x2 , x3, x4 0
Soal:
The Dakota Furniture Company manufactures desk, tables, and chairs. The manufacture of each type of furniture lumber and two types of skilled labor: finishing and carpentry. The amount of each resource needed to make each type of furniture is given in Table
At present, 48 bard feet of lumber, 20 finishing hours, and 8 carpentry hours are available. A desk sells for $60, and a table for $30, and a chair for $20. Dakota believes that demand for desks, chairs and tables is unlimited. Since available resource have already been purchased. Dakota wants to maximize total revenue. Do the sensitivity analysis for Dakota problem.
Resource Desk Table Chair
Lumber 8 board ft 6 board ft 1 board ft
Finishing hours 4 hours 2 hours 1.5 hours
Carpentry hours 2 hours 1.5 hours 0.5 hours
0,,
85.05.12
205.1 24
48 6 8 ..
203060
321
321
321
321
321
xxx
xxx
xxx
xxxts
xxxzMax
0,,
205.05.1
305.1 26
60 2 4 8 ..
82048
321
321
321
321
321
yyy
yyy
yyy
yyyts
yyywMin
085.005.1228
085.1022420
2481062848
8 ,0 ,2 ,280
3
2
1
321
s
s
s
xxxz
020105.0105.101
530105.110206
06010210408
10 ,10 ,0 ,280
3
2
1
321
e
e
e
yyyw
Graphical Sensitivity Analysis
• Sensitivity Analysis:
– the investigation of the effect of making changes in the model parameters on a given optimum LP solution.
• Changes in objective coefficients
• Changes in right-hand side of the constraints
Graphical Sensitivity Analysis
Example: Stereo Warehouse
Let x = number of receivers to stock
y = number of speakers to stock
Maximize 50x + 20y gross profit
Subject to 2x + 4y 400 floor space
100x + 50y 8000 budget
x 60 sales limit
x, y 0
13/11/2013
7
Graphical Sensitivity Analysis • Example: Stereo Warehouse
0
50
100
150
200
0 50 100 150 200
Z=2000
Z=3000
Z=3600
Z=3800
A B
C
D
E
Optimal solution ( x = 60, y = 40)
Graphical Sensitivity Analysis Objective-Function Coefficients
0
50
100
150
200
0 50 100 150 200
z = 50x + 20y
x 60 (constraint 3 )
2 4 400x y (constraint 1)
100 50 8000x y (constraint 2)
A B
C
D
D(40, 80)
Graphical Sensitivity Analysis Right-Hand-Side Ranging
0
50
100
150
200
0 50 100 150 200
x 60 (constraint 3 )
100 50 8000x y (constraint 2)
A B I
C
D H
H(60, 280)
DUAL SIMPLEX
Dual Simplex -basic concept-
Variation of simplex method
Dual feasible but not primal feasible
Mirror image of simplex method terkait dengan penentuan leaving dan entering
variable
Mengeliminasi penggunaan artificial variable
Digunakan dalam sensitivity analysis
Hanya digunakan sebagai pelengkap solusi pada dual problem
Dual Simplex -contoh-
Primal Problem Dual Problem
13/11/2013
8
Dual Simplex -langkah pengerjaan-
1. Initialization. – Convert constraints in ≥ to ≤ (by multiplying both sides by -1)
– Add slack variables as needed
– Find a basic solution (Optimal solution is feasible if the values are zero for basic variables and nonnegative for non basic variables)
– Go to feasiblity test.
2. Feasibility test. – If all basic variables are nonnegative, then it is feasible, therefore optimal
– Otherwise, go to iteration.
3. Iterasi a. Determine the leaving variable. Select basic variable with most negative
value.
b. Determine the entering variable. Select non basic variable with most negative coefficient in the leaving variable row.
c. Determine the new basic solution. Solve by Gaussian elimination.
d. Return to feasibility test.
Dual Simplex -latihan soal-
● Contoh:
Min 2x1 + 3x2 + 4x3
s.t.
x1 + 2x2 + x3 3
2x1 – x2 + 3x3 4
x1 , x2 , x3 0
Max -2x1 – 3x2 – 4x3
s.t.
-x1 – 2x2 – x3 + x4 -3
-2x1 + x2 – 3x3 + x5 -4
xi 0
Dual Simplex -latihan soal-
z x1 x2 x3 x4 x5 RHS
z 1 2 3 4 0 0 0
x4 0 -1 -2 -1 1 0 -3
x5 0 -2 1 -3 0 1 -4
z x1 x2 x3 x4 x5 RHS
z
x…
x…
z x1 x2 x3 x4 x5 RHS
z
x…
x…
Lecture 9 – Preparation
• Materi:
– Integer Linear Programming
13/11/2013
9