Transportation Problems
Tableaus
Shipping Solutions
NorthWest Corner Rule
Stepping Stone Algorithm
Spring 2015
Mathematics in
Management Science
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.
Example
The military buys fuel from many
refineries, and needs to it delivered to
multiple bases.
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.)
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.
Bakery Example
Chain 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?
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.
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.
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.
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
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).
Plant
1
Plant
2
Plant
3
Suppliers on left; demanders on top.
Mine
A
Next add rim conditions:
Supply amounts in last column.
Demand
amounts
in
last
row.
Mine
B
Check that amounts agree!
Plant
1
Plant
2
Plant
3
7
Mine
A
3
Mine
B
2
4
4
10
Plant
1
Mine
A
Plant
2
Plant
3
Next add shipping costs.
Cell (m,p) is cost to ship
from mine m to plant p.
Mine
B
2
4
4
7
3
10
Plant
1
Plant
2
7
Plant
3
1
3
7
Mine
A
9
12
5
3
Mine
B
2
4
4
10
Plant
1
Plant
2
7
Mine
A
Plant
3
1
3
Need shipping solution to start with.
9
7
12
5
3
Mine
B
2
4
4
10
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.
Plant
1
Plant
2
7
Mine
A
Plant
3
1
3
7
Here!
9
12
5
3
Mine
B
2
4
4
10
Plant
1
Plant
2
7
Mine
A
Plant
3
1
3
2
5
9
12
5
3
Mine
B
2
4
4
10
Plant
1
Plant
2
7
Mine
A
Plant
3
1
3
4
2
9
1
12
5
3
Mine
B
2
4
4
10
Plant
1
Plant
2
7
Mine
A
1
4
2
9
3
1
4
1
12
5
Mine
B
2
Plant
3
3
3
4
10
Plant
1
Plant
2
7
Mine
A
1
4
2
9
Mine
B
Plant
3
1
4
1
12
5
Cost=14+4+3+36=57
2
3
3
3
4
10
Plant
1
Plant
2
7
Mine
A
1
4
2
9
Mine
B
Plant
3
1
4
1
12
5
Check Indicator Values
2
3
3
3
4
10
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.
Tableau
The tableau for this problem is the following:
1
Bakeries
I
II
III
Stores
2
8
3
9
3
15
1
12
1
3
5
3
7
8
1
2
1
Each square is called a cell and the top-right corner of each cell
shows the shipping cost for the given supplier → user.
1
Bakeries
I
II
III
3
8
Stores
2
9
3
3
15
1
12
1
3
5
7
8
1
2
1
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.
Plant
1
Plant
2
7
Mine
A
1
4
2
9
Mine
B
Plant
3
1
4
1
12
5
Check Indicator Values
2
3
3
3
4
10
Plant
1
Plant
2
7
Mine
A
1
4
2
9
3
1
4
1
12
5
Mine
B
2
Plant
3
3
3
4
10
Plant
1
Plant
2
7
Mine
A
1
4
2
9
3
1
4
1
12
5
Mine
B
2
Plant
3
3
3
4
10
Plant
1
Plant
2
7
Mine
A
Plant
3
1
1
2
9
3
4
12
5
3
Mine
B
2
4
1
3
4
10
Plant
1
Plant
2
7
1
3
Mine
A
9
Mine
B
Plant
3
3
4
12
5
2
1
2
4
1
3
4
10
Plant
1
Plant
2
7
Plant
3
1
3
7
Mine
A
9
12
5
3
Mine
B
2
4
4
10
Plant
1
Plant
2
7
Plant
3
1
3
7
Mine
A
9
12
5
3
Mine
B
2
4
4
10
Plant
1
Plant
2
7
Plant
3
1
3
7
Mine
A
9
12
5
3
Mine
B
2
4
4
10
Plant
1
Plant
2
7
Plant
3
1
3
7
Mine A
9
12
5
3
Mine B
2
4
4
10
Plant
1
Plant
2
7
Plant
3
1
3
7
Mine A
9
12
5
3
Mine B
2
4
4
10
Plant
1
Plant
2
7
Plant
3
1
3
7
Mine A
9
12
5
3
Mine B
2
4
4
10
Plant
1
Plant
2
7
Plant
3
1
3
7
Mine A
9
12
5
3
Mine B
2
4
4
10
Plant
1
Plant
2
3
Plant
3
1
6
Mine
A
5
5
2
3
Mine
B
4
1
7
8
Mine
C
1
1
2
7
10
Plant
1
Plant
2
3
Plant
3
1
6
Mine
A
5
5
2
3
Mine
B
4
1
7
8
Mine
C
1
1
2
7
10