TIN206 9 Stepping Stone
description
Transcript of TIN206 9 Stepping Stone
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.
4
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?
5
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
6
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
7
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
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