optimization.ppt

7/28/2019 OPTIMIZATION.ppt

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IE 312 Optimization

Siggi Olafsson

3018 [email protected]


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Optimization In this class you will learn to solve industrial

engineering problems by modeling them as

optimization problems You will understand common optimization

algorithms for solving such problems

You will learn the use of software for solvingcomplex problems, and you will work as partof a team to address complex ill-structuredproblems with multiple solutions


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Optimization Formulation

integer

05.1580.520.455.335.375.215.2s.t.

max

appetizerofNumber

654321

6

1

i

i

i

i

x

xxxxxx

x

ix

Decision variables:

Integer programming problem:

This is a variant of what is calledthe knapsack problem


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How about the traveling

salesman?

What is the shortest route that visits each city exactly once?


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Problem Formulation What is the objective function?

Maximize profit,

Minimize inventory, ...

What are the decision variables?

Capacity, routing, production and stock levels

What are the constraints? Capacity is limited by capital

Production is limited by capacity


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Mathematical ProgramsObjective/constraints

Linear Non-linear

Continuous

LinearProgramming (LP)

Non-linearProgramming(NLP)

Discrete

IntegerProgramming(IP), Mixed IP(MIP)

NLIP, NLMIP


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Analysis Optimization Algorithm

Computer Implementation

Excel (or other spreadsheet)

Optimization software (e.g., LINDO)

Modeling software (e.g., LINGO)

Increasing

Complexity


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Optimization Algorithms Find an initial solution

Loop: Look at neighbors of current solution

Select one of those neighbors

Decide if to move to selected solution

Check stopping criterion


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In This Class You Will

Learn problem formulation (modeling)

Learn selecting appropriate algorithms

Learn using those algorithms


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Academic HonestyYou are expected to be honest in all of your actions and

communications in this class.

Students suspected of committing academic dishonesty will

be referred to the Dean of Students Office as per

University policy.

For more information regarding Academic Misconduct see

http://www.dso.iastate.edu/ja/academic/misconduct.html
http://www.dso.iastate.edu/ja/academic/misconduct.htmlhttp://www.dso.iastate.edu/ja/academic/misconduct.html


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Professionalism

You are expected to behave in a

professional manner during this class.


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The OR ProcessProblem (System)

Model

Conclusions

Problem Formulation

Analysis


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Problem Formulation Capture the essence of the system

Variables

Relationships

Ask ourselves:

What is the objective?

What are the decision variables?

What are the constraints?


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Mortimer Middleman Wholesale diamond business

sale price $900/carat

average order 55 carats/week

International market

purchase price $700/carat

minimum order 100 carats/trip

trip takes one week and costs $2000


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Inventory Problem Cost of keeping inventory

insurance

tied up capital

0.5% of wholesale value/week

Cost of not keeping inventory

lost sales (no backordering)


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The Current Situation Holding cost of $38,409 in past year

Unrealized profits of $31,600

Resupply travel cost $24,000

Total of $94,009

Can we do better? How do we start answering that question?


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Problem Formulation What are the decision variables?

When should we order?

Reorder point r(quantity that trigger order)

How much should we order?

Order quantity q

Note that this grossly simplifies the reality ofMortimers life!


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Problem Formulation What are the constraints?

What is the objective? Minimize cost

Holding cost Replenishment cost

Lost-sales cost

0

100

r

q


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Relationships

(System Dynamics)Assumptions

Constant-rate demand

Is this a strong or weak assumption?

Is this assumption realistic?


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System Dynamics

r

1 week

r

1 week

r

1 week

With safety stock No safety stock or lost sales With lost sales

Why might we get lost sales despite our planning?

Can we ignore this?


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No lost sales implies r55

Cycle length

Average inventory

Assuming No Lost Sales

55ratedemand

quantityorder q

2

)55(2

)55()55( qr

qrr


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Optimization Model

55

100

55/2000

2)55(50.3

cost/weekentReplenishmcost/weekHolding

r

q

qqr

Subject to

Minimize


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Solving the Problem (Analysis) Feasible solution

Any set of values that satisfies the constraints

Optimal solution A feasible solution that has the best possible

objective function value

Algorithms:

Find a feasible solution

Try to improve on it


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A Solution for Mortimer What are some feasible solutions?

The smallest feasible value ofris 55

What happens if we change it to 56?

Increased holding cost!

Clearly the optimal replenishment

point is 55* r


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New Optimization Problem

100

552000

2

50.3


qq

q

Subject to

Minimize

Differentiate the objective function and set equal to zero:

0552000

2

50.32

q


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Optimal Solution

7.2507.250

50.35520002

50.3

5520002

552000

2

50.3

*

2

2

q

q

q

Optimum


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Economic Order Quantity (EOQ)

ldr

h

fd

q

*

* 2

Classical result in inventory theory:

Lead time

(replenishment)

Weekly

demand

Holding cost

Cost of

replenishment


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Discussion Sensitivity Analysis

Exploring how the results change if parameters

change Why is this important?

Closed-Form Solutions

Final result a simple formula in terms of the input

variables Very fast computationally

Makes sensitivity analysis very easy


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Evaluating the Model Tractability of Model

Ease by which we can analyze the model

Validity

The degree by which inferences drawnfrom model also hold for actual system

Trade-Off!AGood Model is Tractable and Valid


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Model ValidityA model is valid for a specific purpose

It only has to answer the questions we ask

correctly!

Recipe for a Good Model

Start with a simple model

Evaluate assumptions Does adding complexity change the outcome?

Relax assumption/add constraint


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Validity of Our Model Customer demand (average 55)

0

20

40

60

80

100

120

1 3 5 7 911

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

51

Week

Demand


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Simulation Analysis Using the same conceptual model, we

can simulate the performance using the

historical data Check if Mortimer is due with a shipment

Check if a new trip is warranted

Reduce inventory by actual demand

Simulation of our policy q=251, r=55implies a cost of $108,621


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Evaluation of Cost Our predicted cost is

Mortimers current cost is $94,009

The simulated cost is $108,621

630,45$251552000

225150.3



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Simulation Validity Which model should we trust:

The simulation model that predicts a performance of

$108,621, or the EOQ model that predicts a performance of

$45,630?

Examine the assumptions made:

EOQ model: constant demand Simulation: future identical to past

In general, simulation models have a highdegree of validity


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Descriptive vs Prescriptive How about tractability?

What does the simulation tell us? How

easy is it to do sensitivity analysis?

Descriptive models

Only evaluate an alternative or solution

Prescriptive models

Suggest a good (or optimal) alternative


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Numerical Search We can use our descriptive simulation

model to look for a better solution

Algorithm: Start with an initial (good) solution

Checksimilar solution (neighbors)

Select one of the neighbors

Repeat until a stopping criterion is satisfied


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Search for Reorder Point

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250 300

q

r

$64,242

$63,054

$63,254

$108,421

$108,621


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Search for Order Quantity

0

10

20

30

40

50

6070

80

90

100

240 245 250 255 260 265

q

r

$63,054 $95,193$72,781

Best Point Found


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Evaluation We have found a solution q=251, r=85

with cost $63,054

Better than current ($94,009) andpreviously obtained solution ($108,621)

Is this the best solution?

We don

t know! What if we started with a different

initial solution?


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A New Search

55

75

95

115

135

155

250 255 260 265 270 275

q

r

56,904

59,539

56,900

59,732

54,193 58,467

Initial

Solution

Best Point Found


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Heuristic vs Optimal Optimal Solution

Solution that is gives the best objective

function value Heuristic Solution

Agood feasible solution

Should we demand optimality? Inaccurate search vs approximation in

model


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Deterministic vs Stochastic

Models Our deterministic simulation model

assumed future identical to past

Not true! Demand is random Stochastic simulation fits a random

distribution to the historical data

The world is stochastic Why not always use stochastic models?

Tractability versus validity


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Mathematical Programming Deterministic models

Assume all data known with certainty

Validity Often produce valid results

Tractability

Easier than stochastic models Known as mathematical

programming


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Problem Formulation Decision variables

Constraints

Variable-type constraints

Main constraints

Objective function


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Two Crude Petroleum

Saudi Arabia

Venezuela

Refinery

Gasoline

Jet Fuel

Lubricants

$20

9000 barrels/day

$156000 barrels/day

2000

barrels

1500

barrels

500

barrels


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Oil Processing Data Barrel of Saudi Crude

0.3 barrels of gasoline

0.4 barrels of jet fuel 0.2 barrels of lubricant

Barrel of Venezuela Crude

0.4 barrels of gasoline 0.2 barrels of jet fuel

0.3 barrels of lubricant


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Decision Variables What can we control or decide upon?

How much of each crude

Thus, define

Clearly define what you mean!

nds)(in thousacrude/dayVenezuelanofbarrels

nds)(in thousacrude/daySaudiofbarrels

2

1

x

x


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ConstraintsVariable-type constraints

Domain of decision variables

(most often a range)

Very simple here:

0,21

xx


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Main Constraints (Dynamics) Must meet our volume for each type

Availability

s)(lubricant5.03.02.0

fuel)(jet5.12.04.0

(gasoline)0.24.03.0

21

21

requiredbarrelselyield/barr

21

xx

xx

xx

6

9

2

1

x

x


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Objective Function What are we going to use to evaluate a pair

of values for the decision variables?

Minimize total cost

Note that since sales and production are fixed

this maximizes profit

21 1520min xx


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Standard Model

2,

69

s)(lubricant5.03.02.0

fuel)(jet5.12.04.0

(gasoline)0.24.03.0s.t.

21

2

1

21

21

21

xx

xx

xx

xx

xx

21 1520min xx


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Next Constraint

1x 67.13.0

5.0,0

50.22.0

5.0,0

5.03.02.0

50.72.05.1,0

75.34.0

5.1,0

5.12.04.0

21

12

21

21

12

21

xx

xx

xx

xx

xx

xx2x


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Optimal Solution

1x

2x

5.925.315220

1520 21

xx

21 x

5.31 x


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Variant

1x

2x

21 x

5.31 x

Many Optimal Solutions!


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Exercise

0,1..

3max

21

21

21

wwwwts

ww

0,1..

3max

21

21

21

wwwwts

ww

Q. Which of the above has and optimal solution

and which is unbounded?


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Last Time

Formulated Two Crude PetroleumProblem

Solved (Analyzed) the Model Graphed the constraints

Found the most desirable extreme point

Can this approach be generalized torealistic problem?


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Pi Hybrids

Actual application

Manufacturer of Corn Seed

l=20production facilities

m=25 hybrid varieties

n=30 sales regions

Want to look at the production and

distribution operations


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Constants

Cost per bag of each hybrid at each facility

Processing capacity of each facility (bushels)

Bushels of corns needed for a bag of hybrid

Bags of hybrid demanded by each region

Cost per bag of shipping each hybrid from

each facility to each customer region


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Defining Cost

Define

Then the total production cost is

fhp hf facilityathybridproducingofbagpercost,

l

f

m

h

hfhf xp1 1

,,

Summing over

all facilities

Summing over

all hybrids


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Objective Function

HaveTotal Cost = Production Cost + Shipping Cost

which implies an objective function

l

f

m

h

n

r

rhfrhf

l

f

m

h

hfhf ysxp1 1 1

,,,,

1 1

,,min


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Constraints

Capacity constraints

Demands must be met Balance between production and distribution

m

h

fhfh uxa1

bags

,

gbushels/ba

.,...,2,1 lf


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Model Size

Variables

500 x-variables

1500 y-variables Constraints

l=20 capacity constraints

mn=750 demand constraints

lm= 500 balance constraints

lm + lmn= 15500 non-negativity constraints


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Tractability and Validity

Is this a tractable model?Yes, it is actually not very big at all!

Is this a valid model? Constant production cost

Reasonable for certain range of production

Deterministic demand Reasonable for known products/markets

Constant shipping cost Reasonable for certain range of shipped items


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Linear versus Nonlinear

A function is linear if it is a weightedsum of the decision variables, otherwise

nonlinearAlinear program (LP) has a linear

objective function fand constraintfunctions g1,,gm

Anonlinear program has at least onenonlinear function


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Exercise

jy

yd

y

j

j

jj

,1

s.t.

)11(3ymin5

1

41

jy

yd

y

j

j

jj

,1

s.t.

)11(3ymin5

1

2

41

jy

yy

y

j

j

j

,1

100s.t.

min

21

5

1

(a)

(c)

(b)

Linear or nonlinear?


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Classification Summary

Linear Program (LP) ILP

Nonlinear Program (NLP) INLP

Integer Program (IP)

Increased difficulty


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optimization.ppt

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