379 Class 24 Transportation

8/2/2019 379 Class 24 Transportation

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MIE 379 Deterministic Operations

Research

Todays Goals

Formulate and initialize TransportationProblems

Hillier and Lieberman Sec. 8.1

Signed draft of case study due Today

Case Study due Th. Dec. 11!

Homework #11 Due Thursday. Dec. 4th: 8.1-2ab, 8.1-3a, 8.2-1c


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Review

Transportation problems are a special type of LP.

We use a streamlined version of the simplex method tosolve.

Classic problem involves transporting items from oneset of nodes to another.

But other problems can be modeled as transportation

problems and solved more easily.


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Review

Basically a transportation problem is one inwhich you are asking:

How many of what should go where? How many goods should go from which factory to which

warehouse?

How many goods should go from which warehouse to whichstore?

Inventory: How many goods should be made in month i to beavailable in month j? (i.e. how many goods should go from month ito month j?)

Assigning students to schools.


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Review

Transportation Problem Components:

A source set (Supply Centers / Factories)

A destination set (Distribution Ctrs / Customers)

A cost matrix (Transportation or storage costs)

These are the only parameters in a

transportation problem.

cij : the cost to move from source i to

destination j

si: the supply available at source i

dj: the demand at destination j.


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Review

Everything you need is in this parameter table

Dest1

Dest2

Dest3

Dest4

Supply

Source 1 c11 c12 c13 c14 s1



Demand d1 d2 d3 d4


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Review

A basic transportation problem has fixed supplyequal to fixed demand and the goal is to find theleast cost way to deliver all the goods from

supply nodes to demand nodes. But, we can also handle:

Excess supply (or demand) by adding a dummydemand (or supply)

Infeasible routes (make the cost $M)

Flexible demand (split into two parts)


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Review - Example 1 from last class

The Childfair company has 3 plants producing strollers that areshipped to 4 distribution centers. Plants 1, 2, and 3 can produce14, 19, and 12 shipments per month. Each distribution centerneeds 10 shipments per month. The freight cost for each

shipment is $100 plus $0.50 per mile. The distance is given intable below.

Dist 1 Dist 2 Dist 3 Dist 4

plant 1 800miles

1300 400 700

plant 2 1100 1400 600 1000

plant 3 600 1200 800 900


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Review - Example 1 from last class

Formulate as a transportation problem:

Dist 1 Dist 2 Dist 3 Dist 4 Dummy

Demand

Supply

plant 1 $500 $750 $300 $450 $0 14

plant 2 $650 $800 $400 $600 $0 19

plant 3 $400 $700 $500 $550 $0 12

Demand 10 10 10 10 5

Why Dummy Demand? We have 5 units of excess supply.


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Transportation Problem

Example 2 The plants produce 60, 80 and 40 units. Firm has committed to

sell 40 units to customer 1, 60 units to customer 2, and at least30to customer 3. Customers 3 and 4 want as much as they can get.

How do we represent these parameters?

Cust 1 Cust 2 Cust 3 Cust 4 output

plant 1 $800 $700 $500 $200 60

plant 2 $500 $200 $100 $300 80

plant 3 $600 $400 $300 $500 40

Demand 40 60 30 + ? ?


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Solution

Cust 1 Cust 2 Cust 3 Cust 3excess Cust 4 Output

plant 1 $800 $700 $500 $500 $200 60

plant 2 $500 $200 $100 $100 $300 80plant 3 $600 $400 $300 $300 $500 40

Dummy

plant

-$M -$M -$M $0 $0 50

Demand 40 60 30 50 50


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Inventory Problems

In inventory problems you are asking how muchshould I make when to sell when?

The sources are the periods in which you produce

the goods. The destinations are the periods in which you sell the

goods.

Do the 2nd in-class example problem.

If a particular allocation is impossible, use a big M. (forexample, you cant make a boat in Quarter 3 and sell itin Quarter 2).


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Transportation Problems

Inventory Problems

How many sailboats to produce each of four quarters?

Demand is projected to be 20, 30, 55 and 35

Demand must be met on time

They start with 10 sailboats

Can produce 40 each quarter

Holding costs are $20/boat/quarter

Total capacity (supply) =

Total demand =

In-Class Example Problem 2


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1st Q 2nd Q 3rd Q 4th Q Unusedcapacity

Supply

Initialinventory

Capacity 1

Capacity 2

Capacity 3

Capacity 4

Demand


Inventory Problems


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Supply

Initialinventory

$0 $20 $40 $60 $M 10

Capacity 1 $0 $20 $40 $60 $0 40

Capacity 2 $M $0 $20 $40 $0 40

Capacity 3 $M $M $0 $20 $0 40

Capacity 4 $M $M $M $0 $0 40

Demand 20 30 55 35 30


Inventory Problems


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Transportation Problems: Algorithm

Transportation problems use a slimmed downversion of the simplex method.

Because they have equality constraints, we needto find an initial feasible solution.

It turns out this is easy in transportationproblemsyou do not need to use the Big-Mmethod.


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Initial Allocation

Northwest Corner Method

You simply start at the upper left hand cornerand allocate as much as you can;

Then move right as far as you can;

Then down; then right; etc.


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Initial Allocation


plant 1 40 60

plant 2 80

plant 3 40

Dummy

plant

50

Demand 40 60 30 50 50

Note: we are using a table similar to the formulation table. But

the numbers in the boxes now represent the value of the decision

variables, NOT the parameters


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Initial Allocation


plant 1 40 60

plant 2 80

plant 3 40

Dummy

plant

50

Demand 40 60 30 50 50


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Initial Allocation


plant 1 40 20 60

plant 2 80

plant 3 40

Dummy

plant

50

Demand 40 60 30 50 50


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Initial AllocationCust 1 Cust 2 Cust 3 Cust 3

excessCust 4 Output

plant 1 40 20 60

plant 2 40 80

plant 3 40

Dummy

plant

50

Demand 40 60 30 50 50


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excessCust 4 Output

plant 1 40 20 60

plant 2 40 30 80

plant 3 40

Dummy

plant

50

Demand 40 60 30 50 50


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excessCust 4 Output

plant 1 40 20 60

plant 2 40 30 10 80

plant 3 40

Dummy

plant

50

Demand 40 60 30 50 50


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excessCust 4 Output

plant 1 40 20 60

plant 2 40 30 10 80

plant 3 40 40

Dummyplant

50

Demand 40 60 30 50 50


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excessCust 4 Output

plant 1 40 20 60

plant 2 40 30 10 80

plant 3 40 0 40

Dummyplant

50

Demand 40 60 30 50 50

This represents a degenerate variable. It is basic but equal to zero.


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excessCust 4 Output

plant 1 40 20 60

plant 2 40 30 10 80

plant 3 40 0 40

Dummyplant

50 50

Demand 40 60 30 50 50


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Initial Allocation

Cust 1 Cust 2 Cust 3 Cust 3excess

Cust 4 Output

plant 1 40 20 60

plant 2 40 30 10 80

plant 3 40 0 40

Dummy plant 50 50

Demand 40 60 30 50 50

Are we feasible? Yes; The fixed demand is met, cust 3 gets (30 + 50) 80, cust 4gets nothing (why?).Are we optimal? Who knows, probably were not. Costs come in later, this is

only the allocation to find an initial solution.


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Initial Allocation

Cust 1 Cust 2 Cust 3 Cust 3excess

Cust 4 Output

plant 1 40 20 60

plant 2 40 30 10 80

plant 3 40 0 40

Dummy plant 50 50

Demand 40 60 30 50 50

NOTE: This grid looks the same as the grid we used to formulate the

problem. But this is completely different. Here, the numbers in the boxes

in the center represent the value of the decision variable. In the

formulation, these numbers represent costs.


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Formulation vs Allocation

Cus

t 1Cu

st 2Cu

st 3Cust

3exce

ss

Cust

4Outp

ut

plant 1 40 20 60

plant 2 40 30 10 80

plant 3 40 0 40

Dummy

plant

50 50

Deman

d

40 60 30 50 50

Cust

1Cust

2Cust

3Cust

3excess

Cust

4Outp

ut

plant

1

$800 $700 $500 $500 $200 60

plant2

$500 $200 $100 $100 $300 80

plant

3

$600 $400 $300 $300 $500 40

Dum

my

plant

-$M -$M -$M $0 $0 50

Dema

nd

40 60 30 50 50

The yellow numbers represent the cost of

going from plant i to cust j The green numbers represent number ofitems to produce in plant i for cust j


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Inventory Problems

How many sailboats to produce each of four quarters?

Demand is projected to be 20, 30, 55 and 35

Demand must be met on time

They start with 10 sailboats

Can produce 40 each quarter

Holding costs are $20/boat/quarter

Total capacity (supply) =

Total demand =

In-Class Example Problem 2

T i P bl


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Supply

Initialinventory

$0 $20 $40 $60 $M 10

Capacity 1 $0 $20 $40 $60 $0 40

Capacity 2 $M $0 $20 $40 $0 40

Capacity 3 $M $M $0 $20 $0 40

Capacity 4 $M $M $M $0 $0 40

Demand 20 30 55 35 30


Inventory Problems

T i P bl


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Supply

Initialinventory

10 10

Capacity 1 10 30 40

Capacity 2 40 40

Capacity 3 15 25 40

Capacity 4 10 30 40

Demand 20 30 55 35 30


Inventory Problems


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Transportation Problem - Example

Mrs. A has a cookie company along with her 2 friends Mrs. B

and Mr. C. They each can cook up to 40 dozen cookies per day,and then have their teenagers deliver them. They have a

standing order from the 4 local elementary schools of 60,20,15,and 10. The cost of delivery, per dozen, is given by the table.

1 2 3 4

A 1 2 3 4

B 2 1 2 5

C 3 3 1 2

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