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OPERATIONS RESEARCH An Intro By Farizal, PhD

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OPERATIONS RESEARCH

An Intro

By

Farizal, PhD

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2

Operations Research, OR

What is It?

Operations research (also known as

management science, MS) is a

collection of techniques based onmathematics and other scientific

approaches a problem within a

system to yield the optimal

solution.

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History of O.R.

• World War II-research on military operation.

• 1947-simplex method by George Dantzig.

1950-LP , DP , Queueing Theory , andInventory Theory.

• Computer revolution.

1980s-software package.

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Optimisation = Efficiency + Savings

• Kellogg’s  – The largest cereal producer in the world. – LP-based operational planning (production, inventory, distribution) system

saved $4.5 million in 1995.

• Procter and Gamble – A large worldwide consumer goods company. – Utilised integer programming and network optimization worked in concert

with Geographical Information System (GIS) to re-engineering productsourcing and distribution system for North America.

 – Saved over $200 million in cost per year.

• Hewlett-Packard – Robust supply chain design based on advanced inventory optimization

techniques. – Realized savings of over $130 million in 2004

Source: Interfaces

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Properties of O.R.

• O.R. is concerned with OPTIMAL decision

making in, and modeling of, deterministic &

probabilistic systems that originate from Real

life.

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Properties of O.R.

• Creative scientific research into the

fundamental properties of operations.

• Search for optimality.

• Team approach-involving the backgrounds of 

mathematics, statistics & probability theory,

economics, business administration, electronic

computing, engineering & physics, and

behavior sciences etc.

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Distinct Nature of OR

• Simultaneously analyzes all variables

• Seeks global, balanced solutions definedby:

 – Multiple criteria – Multiple, conflicting objectives

• Helps mitigate risk and reduces uncertainty

by modeling different scenarios• Goes beyond single-issue management

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Contribution of O.R.

• The structuring of the real life situation into a

mathematical model, abstracting the essential elements

so that a solution relevant to the decision maker’s

objectives can be sought. This involves looking at the

problem in the context of the ENTER SYSTEM.

• Exploring the STRUCTURE of such solutions & developing

systematic procedures for obtaining them.

•Developing a solution, including the mathematical theory,if necessary that yields an optimal measure of 

DESIRABILITY.

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What in O.R.?

Deterministic

Problem

LP , DP , NLP ,

IP ,Inventory ,Network 

Scheduling,

PERT/CPM

Stochastic

Problem

D.S.

Simulation.

Queueing ,Game Theory

Forecasting ,

Decision

Analysis,Markov Chain

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Deterministic versus Stochastic• Two broad categories of optimization models exist

 – deterministic

• parameters/data known with certainty

 – stochastic

• parameters/data know with uncertainty

• Deterministic models are easier to solve. we

pretend we know the parameter/input with

certainty).

• Stochastic model are difficult to solve. In reality, we

know a distribution about our demand. We get

around this in real life by re-optimizing.

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Deterministic versus Stochastic

• Deterministic optimization ignores risk of beingwrong about parameter/data estimates.

• No commercial software packages are currently

available to do generalized, stochasticoptimization.

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Linear Programs

A Major Tool of OR

• Linear Programs (LPs) are a special type of mathematical model where all relationshipsbetween parts of the system being modeled can be

represented linearly (a straight line).• Not always realistic, but we know how to solve LPs.

• May need to approximate a relationship that is

slightly non-linear with a linear one.• When to use: if a problem has too many

dimensions and alternative solutions to evaluate allmanually, use an LP to evaluate.

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General Optimization Model

Problem(1):

Min f ( x )

s.t. g( x 

)

0 --------(1) x   0

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Linear Programs

• LPs can evaluate thousands, millions, etc. of different alternatives to find the one that best

meets the objective of the business problem.

 –

Fleet Assignment Model - assign aircraft to flightlegs to minimize cost and maximize revenue

 – Revenue Management - set bid prices to maximize

revenue and/or minimize spill

 – Crew Scheduling - schedule crew members to

minimize number of crew needed and maximize

utilization

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Other Types of Linear Optimizations

• MIP (Mixed Integer Programming)

 – is similar to LP but at least one decision variable is

required to be a integer value

 – violates the LP rule that decision variables be

continuous – is solved by “branch and bound” - solving a series of 

LPs that fix the integer decision variables to various

integer values and comparing the resulting objective

function values

 – is done in a smart way to avoid enumerating all

possibilities

 –is useful, since you can not have .3 of an aircraft

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Other Types of Linear Optimization

• Network problem

 – is a special form of LP which turns out to be

“naturally integer” 

 – can be solved faster than an LP, using a special

network optimization algorithm – is very restrictive on types of constraints that can be

present in the problem

Shortest Path – finds the shortest path from the source (start) to

sink (end) nodes, along connecting arcs, each having

a cost associated with them

 – is used in many applications

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Other Optimization Models

• Quadratic Program

 – has a quadratic objective function with linear

constraints

 – can be applied to revenue management, because itallows fare to rise with demand within a problem

• price(OD) = 50 + [5*numpax(OD)]

• max revenue = price * numpax

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Other Optimization Models

• Non-linear Program (NLP)

 – can have either non-linear objective function or

non-linear constraints or both

 – feasible region is generally not convex

 – much more difficult to solve

 – but it is worth our time to learn to solve them since

world is actually non-linear most of the time – some non-linear programs can be solved with LPs or

MIPs using piecewise linear functions

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Applications of O.R.

• Inventory & Production Problem

• Maximization Problem

Minimization Problem• Work-Force Planning Problem

• Waiting Line Problem

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Application Details

• Accounting:

 – Cash flow planning

 – Credit policy

 –Strategy planning

• Facilities Planning

 – Location & size

 – Logistics systems

 – Transportation Planning

 – Hospital planning

• Manufacturing

 – Production scheduling

 – Production-marketing

balance• Organization Behavior

 – Employee recruiting

 – Skills balancing

 – Training programs

scheduling

 – Manpower justification \

planning

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版權所有:巫沛倉 

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OR/MS Successes

Best cases from the annual INFORMS

Edelman Competition

2002: Continental Airlines Survives 9/112001: Merrill Lynch Integrated Choice

2001: NBC’s Optimization of Ad Sales 

2000: Ford Motor Prototype Vehicle Testing1996: Procter & Gamble Supply Chain

1991: American Airlines Revolutionizes Pricing

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Case 4: Ford Motor Prototype Vehicle

Testing

• Business Problem: Developing prototypes

for new cars and modified products is

enormously expensive. Ford sought toreduce costs on these unique, first-of-a-

kind creations.

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Ford Motor (con’t) 

• Model Structure: Ford and a team fromWayne State University developed a

Prototype Optimization Model (POM) to

reduce the number of prototype vehicles.The model determines an optimal set of 

vehicles that can be shared and used to

satisfy all testing needs.• Project Value: Ford reduced annual

prototype costs by $250 million.

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Case 5: Procter & Gamble Supply

Chain

• Business Problem: To ensure smart growth,

P&G needed to improve its supply chain,

streamline work processes, drive out non-value-added costs, and eliminate

duplication.

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P&G Supply Chain (con’t) 

Model Structure: The P&G operations researchdepartment and the University of Cincinnati created

decision-making models and software. They

followed a modeling strategy of solving two easier-

to-handle subproblems:

 – Distribution/location

 – Product sourcing

• Project Value: The overall Strengthening GlobalEffectiveness (SGE) effort saved $200 million a year

before tax and allowed P&G to write off $1 billion of 

assets and transition costs.

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P&G Supply Chain (con’t) 

• Project Value: The overall Strengthening

Global Effectiveness (SGE) effort saved

$200 million a year before tax and allowedP&G to write off $1 billion of assets and

transition costs.

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Case 6: American Airlines

Revolutionizes Pricing

• Business Problem: To compete effectively in

a fierce market, the company needed to“sell the right seats to the right customers

at the right prices.” 

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American Airlines (con’t) 

• Model Structure: The team developed yield

management, also known as revenue management

and dynamic pricing. The model broke down the

problem into three subproblems: – Overbooking

 – Discount allocation

 – Traffic management

The model was adapted to American Airlinescomputers.

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American Airlines (con’t) 

• Project Value: In 1991, American Airlines

estimated a benefit of $1.4 billion over the

previous three years. Since then, yieldmanagement was adopted by other

airlines, and spread to hotels, car rentals,

and cruises, resulting in added profits going

into billions of dollars.

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Phases of an OR Project

 – Define the problem

 – Develop math model to represent the

system

 – Solve and derive solution from model

 – Test/validate model and solution

 –

Establish controls over the solution – Put the solution to work

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Mathematics in Operation

Mathematical Solution Method (Algorithm) 

Real Practical Problem 

Mathematical (Optimization) Problem x2 

Computer Algorithm 

Human Decision-Maker 

Decision Support Software System 

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Problem Solving Stages

Mathematical Solution Method (Algorithm) 

Real Practical Problem 

Mathematical (Optimization) Problem 

Computer Algorithm 

Human Decision-Maker 

Decision Support Software System 

Staff Rostering at

Childcare Centre

Mathematical

Programming

CPLEX

XpressMP

LINGO

Excel with VBA

Childcare Centre

Manager

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General Optimization Model

Problem(1):

Min f ( x )

s.t. g( x )0 --------(1)

 x   0

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• The Objective Function is an expression that

defines the optimal solution, out of the many

feasible solutions. We can either – MAXimize - usually used with revenue or profit or

 – MINimize - usually used with costs

• Feasible solutions must satisfy the constraints of the problem. LPs are used to allocate scarce

resources in the best possible manner.

Constraints define the scarcity.• The scarcity in this problem involves a fixed

number of seats and scarce high paying

customers.

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• Rules for Constraints

 – must be a linear expression

 – decision variables can be summed together but notmultiplied or divided by each other

 – have relational operators of =, <=, or >=

 –

must be continuous• Constraints define the “feasible region” - all

points within the feasible region satisfy the

constraints.• The feasible region is convex.

• The optimal solution lies at an extreme point of 

the feasible region

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ExampleA company with three plants produces two products.

First plant that operates for 4 hours produces only

product 1. Second plant operates for 12 hours but

produces only product 2. The last plant operates

for 18 hours and produces both products. It takesone hour to produce product 1 at plant 1 and 3

hours at plant 3 while product 2 needs 2 hour to be

produced at available facilities. If the selling price

for product 1 and 2 is $3,000 and $5,000,respectively. Find how many product 1 and product

2 should be made to maximize the profit? How

much the profit?

S l ti

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Solution:

Problem Articulation

Plant Product Product time

available1 2

1

2

3

1

0

3

0

2

2

4 hours

12 hours

18 hours

Profit $3,000 $5,000

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Linear Programming Formulation

Define:

: number of batches of product i produce per week

: total profit per week form purchasing this two

products

Objective function: to maximize

Max 3 X1 + 5 X2

Constraints: production time available

X1 ≤ 4 

i x

 z

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Complete LP Model

Max z =

s.t21 53 x x

0,0

1823122

4

21

21

2

1

 x x

 x x

 x

 x

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Example-ExtendedA company with three plants produces two products.

First plant that operates for 4 hours produces onlyproduct 1. Second plant operates for 12 hours but

produces only product 2. The last plant operates for

18 hours and produces both products. It takes one

hour to produce product 1 at plant 1 and 3 hours atplant 3 while product 2 needs 2 hour to be produced

at available facilities. If the selling price for product 1

and 2 is $3,000 and $5,000, respectively, and if 

product 1 in order to be economical should be

produced at least 3, find how many product 1 and

product 2 should be made to maximize the profit?

How much the profit?

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Complete LP Model

Max z =

s.t21 53 x x

0,0

1823

122

3

4

21

21

2

1

1

 x x

 x x

 x

 x

 x

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Example-ExtendedA company with three plants produces two products .

First plant that operates for 4 hours produces only

product 1. Second plant operates for 12 hours but

produces only product 2. The last plant operates for

18 hours and produces both products. It takes one

hour to produce product 1 at plant 1 and 3 hours at

plant 3 while product 2 needs 2 hour to be produced

at available facilities. If the selling price for product 1

and 2 is $3,000 and $5,000, respectively. Find howmany product 1 and product 2 should be made to

maximize the profit. How much the profit?

Determine which plants produce the products?

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References

• F.S. Hillier, Introduction to Operations

Research, McGraw Hill 2009

• H.A. Taha, Operations Research an

Introduction, Prentice Hall 2008

• F.S. Hillier and Lieberman

• Winston