3 lp simplex maximization
DESCRIPTION
metode kuantitatif simplex maximizatinTRANSCRIPT
Budi Harsanto blogs.unpad.ac.id/budiharsanto
2012
LP Simplex Maximization
Course : Quantitative Method / Operations Research
Introduction
Graphical analysis is only for 2 variables.
In reality, LP problem is too complex to solved by graphical analysis.
Simplex method can accomodate 2 variables or more.
For Example We Use 2 Variables: We’ll Solve Use Graphical & Simplex
Sebuah perusahaan furnitur hendak memproduksi 2 buah produk, yaitu Meja dan Kursi yang masing-masing memberikan laba bersih per unit sebesar €7 & €5. Kedua produk tersebut diproses oleh dua divisi, yaitu divisi Perkayuan dan divisi Perakitan. Jam kerja yang tersedia di divisi Perkayuan tersedia maksimal 100 jam sedangkan jam kerja divisi Perakitan tersedia tidak lebih dari 240 jam. Setiap unit Meja harus diproses selama 2 jam kerja di divisi Perkayuan dan 4 jam kerja di divisi Perakitan. Sedangkan setiap unit kursi memerlukan proses di divisi Perkayuan selama 1 jam dan divisi Perakitan selama 3 jam.
Determine the best combination!
Matrix & Mathematical Formulation
Division Meja (M) Kursi (K) Quantity
C 2 1 < 100
D 4 3 < 240
Profit (€) 7 5 - -
Objective: Max Profit Z = €7M + €5K
Constraits:
2M + 1K < 100
4M + 3K < 240
Status: M; K > 0
Preparation: Simplex Formulation
Change inequality in constraints function become equality.
The consequence, (≤) change with adding
slack variable for every slack. Slack variable is unused resources.
Simplex Formulation
Objective: Maks Laba: Z = €7M + €5K + €0S1 + €0S2
Constraint:
2M + 1K + 1S1 + 0S2 = 100
4M + 3K + 0S1 +1 S2 = 240
Status: M; K; S1 ;S2 > 0
Basic Variable
• 1st Iteration Like at (0,0) point.
• Assume all product are zero.
1st Iteration
Cj
Solution Mix M K S1 S2 Quantity
€7 €5 €0 €0
2 1 1 0
4 3 0 1
€0 €0 €0 €0
€7 €5 €0 €0
€0
€0
S1
S2
Zj
Cj - Zj
100
240
€0
€0
5 Steps
5 Langkah:
1. Entering variable: choose the biggest positive Cj - Zj
2. Leaving variable: choose the smallest non negative ratio.
3. New Pivot Row. Old pivot row divided by pivot number.
4. Other row.
5. Cj - Zj non positive mean optimal. If not optimal, back to the first step for the next iteration.
Cj
Solution Mix M K S1 S2 Quantity
€7 €5 €0 €0
2 1 1 0
4 3 0 1
€0 €0 €0 €0
€7 €5 €0 €0
€0
€0
S1
S2
Zj
Cj - Zj
100
240
€0
€0
Pivot Number
Pivot column
Biggest positive Cj - Zj
Pivot Row
Pivot!
Other Row Equation
= - x 0 1 -2 1
40
4 3 0 1
240
(4) (4) (4) (4) (4)
(1) (1/2) (1/2) (0)
(50)
= - x
= - x
= - x
-
new row
the in
number
Corresponding
number pivot
above Number
row
old in
Numbers
Numbers Row
New
or below
= - x
=
2nd Iteration
Cj
Solution Mix M K S1 S2 Quantity
€7 €5 €0 €0
1 1/2 1/2 0
0 1 -2 1
€7 €7/2 €7/2 €0
€0 €3/2 -€7/2 €0
€7
€0
M
S2
Zj
Cj - Zj
50
40
€350
Cj
Solution Mix M K S1 S2 Quantity
€7 €5 €0 €0
1 1/2 1/2 0
0 1 -2 1
€7 €7/2 €7/2 €0
€0 €3/2 -€7/2 €0
€7
€0
T
S2
Zj
Cj - Zj
50
40
€350 (Total Profit)
Pivot row
Pivot number
Pivot column
= - x 1 0
3/2 -1/2 30
1 1/2 1/2 0
50
(1/2) (1/2) (1/2) (1/2) (1/2)
(0) (1) (-2) (1)
(40)
= - x
= - x
= - x
-
=
new row
the in
number
Corresponding
number pivot
above Number
row
old in
Numbers
Numbers
Row
New
or below
3rd Iteration
Cj
Solution Mix M K S1 S2 Quantity
€7 €5 €0 €0
1 0 3/2 -1/2
0 1 -2 1
€7 5 €1/2 €3/2
€0 €0 -€1/2 -€3/2
€7
€5
M
K
Zj
Cj - Zj
30
40
€410
Optimal
T = 30 units C = 40 units Pofit = €410
Graphical Vs Simplex K
urs
i
100
80
60
40
20 0 20 40 60 80 100 X
X2
Meja
B = (0,80)
C = (30,40)
D = (50,0)
Daerah Layak
240 4T + 3C <
2T + 1C 100 <
A = (0,0)
References
1. Render, Barry; Stair, Jr Ralph M & Hanna, Michael E, Quantitative Analysis for Management, Latest Edition.
2. Taylor III, Bernard W, Introduction to Management Science, Latest Edition.
3. Taha, Hamdy A., Operation Research An Introduction, Latest Edition.
4. Internet