3 lp simplex maximization

Budi Harsanto blogs.unpad.ac.id/budiharsanto

2012

LP Simplex Maximization

Course : Quantitative Method / Operations Research

budi.harsanto@fe.unpad.ac.id

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.

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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!

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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

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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.

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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

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Basic Variable

• 1st Iteration Like at (0,0) point.

• Assume all product are zero.

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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

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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.

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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!

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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

=

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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

budi.harsanto@fe.unpad.ac.id

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

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= - 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

budi.harsanto@fe.unpad.ac.id

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

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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)

budi.harsanto@fe.unpad.ac.id

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