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Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions...
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![Page 1: Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm.](https://reader036.fdocuments.in/reader036/viewer/2022062718/56649e7c5503460f94b7e3cf/html5/thumbnails/1.jpg)
Spring 2015Mathematics in
Management Science
Transportation Problems
Tableaus
Shipping Solutions
NorthWest Corner Rule
Stepping Stone Algorithm
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Transportation Problems
Problems where group of suppliers must meet the needs/demands of a group of users of these supplies. (Think corporate sized users, not individuals.)
Have cost associated with shipping from a particular supplier to particular “consumer”.
Objective is to minimize total shipping cost.
![Page 3: Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm.](https://reader036.fdocuments.in/reader036/viewer/2022062718/56649e7c5503460f94b7e3cf/html5/thumbnails/3.jpg)
Example
The military buys fuel from many refineries, and needs to it delivered to multiple bases.
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Example
During World War II, supplies needed to be shipped from multiple US ports to many different European ports.
(This is the origin of these types of problems.)
![Page 5: Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm.](https://reader036.fdocuments.in/reader036/viewer/2022062718/56649e7c5503460f94b7e3cf/html5/thumbnails/5.jpg)
Example
A supermarket chain buys bread from a supplier that makes its bread in multiple bakeries.
The bread must be shipped from individual bakeries to individual stores.
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Bakery ExampleChain gets bread deliveries from a bakery chain that does its baking in different places.
Each store needs certain number of loaves/day.Supplier bakes enough to exactly meet demands.
How many loaves ship (b to s) to stay within the demands and to minimize the cost?
![Page 7: Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm.](https://reader036.fdocuments.in/reader036/viewer/2022062718/56649e7c5503460f94b7e3cf/html5/thumbnails/7.jpg)
Transportation Problems
Have suppliers, users, shipping costs.
Each supplier has a fixed amount that they can supply.
Each user has a fixed amount they need/demand.
The total capacity of the suppliers exactly matches the needs of the users.
![Page 8: Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm.](https://reader036.fdocuments.in/reader036/viewer/2022062718/56649e7c5503460f94b7e3cf/html5/thumbnails/8.jpg)
Transportation Problems
Inputs:• individual capacities of the suppliers,• individual needs of the users, and• shipping costs from any one supplier
to any one user.
This information is collected in a table called a tableau.
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Transportation Problems
Have suppliers, users, shipping costs.
Tableau: displays info with rim conditions.
Each cell has two entries: costs and amount to ship.
Shipping solution: obeys all constraints.
NorthWest Corner Rule: an algorthim for obtaining a shipping solution.
Indicator Values.
Stepping Stone Algorithm.
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Mines & Plants
Two mines (extract/supply ore) and three plants (process/demand ore).
Mine A supplies 7 m tons, B 3 m tons.
Plant 1 needs 2 tons, 2&3 each 4 tons.
Shipping costs are
from A to P1,P2,P3 --- 7, 1, 3
from B to P1,P2,P3 --- 9, 5, 12
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Tableaus
The tableau is a table with one row per supplier (mines—labeled with A,B) and one column per consumer/demander (plants—labeled with 1,2,3).
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Plant1
Plant2
Plant3
Mine A
MineB
Suppliers on left; demanders on top.
Next add rim conditions:
Supply amounts in last column.
Demand amounts in last row.
Check that amounts agree!
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Plant1
Plant2
Plant3
MineA
7
MineB
3
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7
MineB
3
2 4 4 10
Next add shipping costs.
Cell (m,p) is cost to ship
from mine m to plant p.
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Plant1
Plant2
Plant3
MineA
7 1 3
7
MineB
9 5 12
3
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7 1 3
7
MineB
9 5 12
3
2 4 4 10
Need shipping solution to start with.
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NorthWest Corner Rule
Locate top far-left-hand cell. Ship via this cell with smaller of the two rim cells (call the value s) ; circle entry in tableau.
Cross out row or column with rim value s, & reduce other rim value for this cell by s.
When a single cell remains, shud be tie for rim conditions of row and column involved; put this amount into cell and circle.
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Plant1
Plant2
Plant3
MineA
7 1 3
7
MineB
9 5 12
3
2 4 4 10
Here!
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Plant1
Plant2
Plant3
MineA
7 1 3
52
MineB
9 5 12
3
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7 1 3
12 4
MineB
9 5 12
3
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7 1 3
12 4 1
MineB
9 5 12
33
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7 1 3
12 4 1
MineB
9 5 12
33
2 4 4 10
Cost=14+4+3+36=57
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Plant1
Plant2
Plant3
MineA
7 1 3
12 4 1
MineB
9 5 12
33
2 4 4 10
Check Indicator Values
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Example
A supermarket chain has three stores. Their bread supplier makes their bread in three bakeries. The
stores require 3 dozen, 7 dozen, and 1 dozen loaves of bread. The bakeries are able to produce 8 dozen, 1
dozen, and 2 dozen loaves.
Note:
(Needs) 3 + 7 + 1 = 11 = 8 + 1 + 2 (Supplies)
The tableau will be a table with one row per supplier (bakeries) labeled with Roman numerals and
one column per consumer (stores) labeled with Arabic numerals.
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TableauThe tableau for this problem is the
following:Store
s 21 3B
ake
ries
I8 9 3
8
II15
1 12 1
III1 3 5
2
3 71
Each square is called a cell and the top-right corner of each cell shows the shipping cost for the given
supplier → user.
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Stores 21 3
Bake
ries
I8 9 3
8
II15
1 12 1
III1 3 5
2
3 71
Cells are labeled by row and column: e.g., I-2 or III-1.Shipping costs are per unit (here 1 dozen loaves).The numbers along the bottom are the
needs of the corresponding users. (Store 2 needs 7 doz.)
The numbers along the right are the capacities of the corresponding suppliers. (Bakery III can supply 2 doz.)
Collectively the numbers along the bottom and right are called the rim conditions.
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Plant1
Plant2
Plant3
MineA
7 1 3
12 4 1
MineB
9 5 12
33
2 4 4 10
Check Indicator Values
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Plant1
Plant2
Plant3
MineA
7 1 3
12 4 1
MineB
9 5 12
33
2 4 4 10
![Page 30: Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm.](https://reader036.fdocuments.in/reader036/viewer/2022062718/56649e7c5503460f94b7e3cf/html5/thumbnails/30.jpg)
Plant1
Plant2
Plant3
MineA
7 1 3
12 4 1
MineB
9 5 12
33
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7 1 3
12 1 4
MineB
9 5 12
33
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7 1 3
13 4
MineB
9 5 12
32 1
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7 1 3
7
MineB
9 5 12
3
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7 1 3
7
MineB
9 5 12
3
2 4 4 10
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Plant1
Plant2
Plant3
MineA
7 1 3
7
MineB
9 5 12
3
2 4 4 10
![Page 36: Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm.](https://reader036.fdocuments.in/reader036/viewer/2022062718/56649e7c5503460f94b7e3cf/html5/thumbnails/36.jpg)
Plant1
Plant2
Plant3
Mine A
7 1 3
7
Mine B
9 5 12
3
2 4 4 10
![Page 37: Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm.](https://reader036.fdocuments.in/reader036/viewer/2022062718/56649e7c5503460f94b7e3cf/html5/thumbnails/37.jpg)
Plant1
Plant2
Plant3
Mine A
7 1 3
7
Mine B
9 5 12
3
2 4 4 10
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Plant1
Plant2
Plant3
Mine A
7 1 3
7
Mine B
9 5 12
3
2 4 4 10
![Page 39: Spring 2015 Mathematics in Management Science Transportation Problems Tableaus Shipping Solutions NorthWest Corner Rule Stepping Stone Algorithm.](https://reader036.fdocuments.in/reader036/viewer/2022062718/56649e7c5503460f94b7e3cf/html5/thumbnails/39.jpg)
Plant1
Plant2
Plant3
Mine A
7 1 3
7
Mine B
9 5 12
3
2 4 4 10
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Plant1
Plant2
Plant3
MineA
3 1 6
5
MineB
5 2 3
4
MineC
1 7 8
1
1 2 7 10
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Plant1
Plant2
Plant3
MineA
3 1 6
5
MineB
5 2 3
4
MineC
1 7 8
1
1 2 7 10
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