TIN206 9 Stepping Stone

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Analisa Post Optimal Metode Transportasi

description

or stapping stone

Transcript of TIN206 9 Stepping Stone

Analisa Post Optimal Metode Transportasi

Metode ini digunakan untuk menentukan solusi optimal

dengan memeriksa kemungkinan penguranganbiaya untuk kasus minimasi/penambahankeuntungan untuk kasus maksimasi jika sel non basis tertentu berubah menjadi sel basis

Penentuan sel masuk pada metode ini sama dengan metode modifikasi distribusi (MODI), bedanya stepping stone tidak berhubungan sama sekali dengan metode simpleks

Terminologi stepping stone muncul dari analogi berjalan di atas batu yang separuhnya terendam air, dimana kata “water” menunjukkan sel yang belum terisi dan “stone” sebagai sel yang terisi

Introduction

The Stepping-Stone Method

1. Select any unused square to evaluate. 2. Begin at this square. Trace a closed path back to the

original; square via square that are currently being used (only horizontal or vertical moves allowed).

3. Place + in unused square; alternate – and + on each corner square of the closed path.

4. Calculate improvement index: add together the unit figures found in each square containing a +; subtract the unit cost figure in each square containing a -.

5. Repeat steps 1-4 for each unused square.

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Stepping-stone method Let consider the following initial tableau from the Min Cost algorithm

These are basic variables

There are Non-basic variables

Question: How can we introduce a non-basic variable into basic variable?

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Introduce a non-basic variable into basic variables

Here, we can select any non-basic variable as an entry and then using the “+ and –” steps to form a closed loop as follows:

let consider this non basic variable

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Stepping stone

+

- +

-

The above saying that, we add min value of all –ve cells into cell that has “+” sign, and subtracts the same value to the “-ve” cells Thus, max –ve is min (200,25) = 25, and we add 25 to cell A1 and A3, and subtract it from B1 and A3

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Stepping stone

The above tableaus give min cost = 25*6 + 120*10 + 175*11 175*4 + 100* 5 = $4525 We can repeat this process to all possible non-basic cells in that above tableau until one has the min cost! NOT a Good solution method

Stepping-Stone Method: Tracing a Closed Path for the Des Moines to Cleveland route

An Initial Solution by Northwest-Corner (VP- Corner Method)

Total cost = 70($7) + 30($2) + 60($1) + 15($5) + 30($7) + 50($4) = $1,095

Plant Warehouse

Capacity Miami Denver Lincoln Jackson

Chicago 70 7

30 2 4 5

100

Houston 3

60 1

15 5 2

75

Buffalo 6 9

30 7

50 4

80

Requirem

ent 70 90 45 50

255

255

Initial Solution by the “Least Cost Method”

Total Cost = 40*7+15*2+45*4+75*1+30*6+50*4=945<(1095)

Plant Warehouse

Capacity VAM Costs Miami Denver Lincoln Jackson

Chicago 40 7

15 2

45 4 5

100 $2

Houston 3

75 1 5 2

75 $1

Buffalo 30 6 9 7

50 4

80 $2

Require - ments

70 90 45 50 255

255

VAM

Costs $3 $1 $1 $2

Plant Warehouse

Capacity Miami Denver Lincoln Jackson

Chicago +$3 7

55 2 4 5

100

Houston 3

35 1

+$2 5 2

75

Buffalo 6 9

+$1 7

50 4

80

Requirem

ent 70 90 45 50

255

255

$3

40

30 +$5

45

+$1

Total Cost = 40($3) + 30($6) + 55($2) + 35($1) + 45($4) + 50($4) = $825

Optimal Solution

Assignment Problem: A Special Case

W-1 W-2 W-3 W-4 supply

P-1 100

P-2 100

P-3 100

P-4 100

demand 100 100 100 100 (400)

plant

warehouse

3

5

6

9

1 2 4

9 2 1

6 3 4

4 7 2

100=1 track load (can’t be split)