Spring 2015Mathematics in

Management Science

Transportation Problems

Tableaus

Shipping Solutions

NorthWest Corner Rule

Stepping Stone Algorithm

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 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?

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).

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!

Plant1

Plant2

Plant3

MineA

7

MineB

3

2 4 4 10

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.

Plant1

Plant2

Plant3

MineA

7 1 3

7

MineB

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

MineA

7 1 3

7

MineB

9 5 12

3

2 4 4 10

Need shipping solution to start with.

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.

Plant1

Plant2

Plant3

MineA

7 1 3

7

MineB

9 5 12

3

2 4 4 10

Here!

Plant1

Plant2

Plant3

MineA

7 1 3

52

MineB

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

MineA

7 1 3

12 4

MineB

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

MineA

7 1 3

12 4 1

MineB

9 5 12

33

2 4 4 10

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

Plant1

Plant2

Plant3

MineA

7 1 3

12 4 1

MineB

9 5 12

33

2 4 4 10

Check Indicator Values

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.

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.

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.

Plant1

Plant2

Plant3

MineA

7 1 3

12 4 1

MineB

9 5 12

33

2 4 4 10

Check Indicator Values

Plant1

Plant2

Plant3

MineA

7 1 3

12 4 1

MineB

9 5 12

33

2 4 4 10

Plant1

Plant2

Plant3

MineA

7 1 3

12 4 1

MineB

9 5 12

33

2 4 4 10

Plant1

Plant2

Plant3

MineA

7 1 3

12 1 4

MineB

9 5 12

33

2 4 4 10

Plant1

Plant2

Plant3

MineA

7 1 3

13 4

MineB

9 5 12

32 1

2 4 4 10

Plant1

Plant2

Plant3

MineA

7 1 3

7

MineB

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

MineA

7 1 3

7

MineB

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

MineA

7 1 3

7

MineB

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

Mine A

7 1 3

7

Mine B

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

Mine A

7 1 3

7

Mine B

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

Mine A

7 1 3

7

Mine B

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

Mine A

7 1 3

7

Mine B

9 5 12

3

2 4 4 10

Plant1

Plant2

Plant3

MineA

3 1 6

5

MineB

5 2 3

4

MineC

1 7 8

1

1 2 7 10

Plant1

Plant2

Plant3

MineA

3 1 6

5

MineB

5 2 3

4

MineC

1 7 8

1

1 2 7 10