Chapter 1 - Introduction to Or

8/13/2019 Chapter 1 - Introduction to Or

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BAHIRDAR UNIVERSITYINSTITUTE OF TECHNOLOGY

SCHOOL OF MECHANICAL AND INDUSTRIAL ENGINEERING

Pr oduct i on Engineer i ng and Management M.Sc Pr ogr am

ADVANCED OPERATIONS RESEARCH

ns ruc or : mare a e u r

PhD in Industrial Engineering

.

B.Sc in Textile Engineering


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OBJECTIVES OF THE COURSE

1. Introduce different quantitative techniques fordecision making process.

2. Provide rational basis for decision making by

seeking to understand and structure complexsituations.

3. Introduce advanced concepts to optimize the

scarce resources.

Amare Matebu (Dr.) - BDU IOT


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What is expected from you?

You are encouraged to study with friends, but you are

expected to compose your own reports.

ou are expecte to respon to quest ons as e y t e

Instructors) during lectures/tutorials.

ugges ons, commen s:

Directly to Instructors via Student Representative

I dont expect any specific knowledge, but I do expect an

open a u e o ngs.

I expect you to read moreand know by your own effort.



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History of Operations Research

s a re a ve y new sc p ne.

It is generally agreed that OR came into existence as a

critical need to manage scarce resources.

and engineers to analyze several military problems

Management of convoy, bombing, antisubmarine,

. The result was called Military Operations Research,



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ORoriginated in Great Britain during World War IIto bring mathematical or quantitative approaches

.

( Started in the UK and developed in the USA)

sta s ment o teams o sc ent sts to stu y t e

strategic and tactical problems involved in military

operations.

The ob ective was to find the most effective

utilization of limited military resources by the use

.



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There arethree important factors behind the rapid

1. The economic and industrial boom after World war II

resulted in continuous mechanization and

automation.2. Many Operations Researchers continued their

research after World war II.

3. Analytic power was made available by high-speedcomputers.

During 1950s, there was substantial progress in the

app ication o OR tec niques or Civi ian activitiesalong with great interest in the professional

.



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In 1948, OR club was formed in England which later

of UK.

Durin OR foundation, its rimar a lications

were: To support military operations: such as to support

radar systems, against submarine, etc.

Following the war, numerous peacetime applicationsemer e , ea n o e use o an mana emen

science in many industries and occupations.

,(ORSA) was founded.

B 1960s OR rou s were formed in several

organizations.Amare Matebu (Dr.) - BDU IOT


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This analytical approach is known by several different

names:

Operations Research (OR)

Operational Research (UK)

Systems Science

Mathematical Modeling

Industrial Engineering

Critical Systems strategic thinking

Success Science S and S stems Anal sis and Desi n



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Because ofOR s multi-disciplinary characterandapplication in varied fields,it has a bright future,

prov e peop e evo e o s u y can e p

meet the needs of society.

However, in order to make the future of OR

brighter, its specialists have to make good use of

the opportunities available to them.



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Definition of Operations ResearchDefinition of Operations Research

1. OR is the application of scientific methods, techniques

and tools to problems involving the operations of

systems so as to provide those in control of the

.

2. OR is the application of the scientific method to the

study of the operations of large, complex organizations

or activities.

3. OR is the application of the scientific method to the



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Definition - summary

ApplicationApplication of SCIENTIFICof SCIENTIFIC METHODMETHOD

StudyStudy ofof LARGE and COMPLEXLARGE and COMPLEX SYSTEMSSYSTEMS

na ys sna ys s oo



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Application of Operations Research

- Aggregate production planning, assembly line, blending, inventory control

- Employment, training, layoffs and quality control

- Transportation, planning and scheduling

Facilities planning

- Location and size of warehouse or new lant

- Logistics, layout and engineering design

- Transportation, planning and scheduling

- Capital budgeting, cost allocation and control, and financial planning

Marketing

- Sales effort allocation and assignment- Predicting customer loyalty

Purchasing, procurement and Exploration

- Optimal buying and reordering with or without price quantity discount



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Successful OR Applications

Com an Year Problem Techni ues Used

Annual

av ngs

Hewlett Packard 1998Designing buffers into

production li neQueuing models $280 mil l ion

Taco Bell 1998 Employee schedulingIP, Forecasting ,

$13 mil lion

Proctor & Gamble 1997

Redesign production &

distributon system Transportation models $200 mi ll ion

Delta Airl ines 1994 Assigning planes to routes Integer Programming $100 mil l ion

Queuin models,

Simulation

Yellow Freight

Systems, Inc.1992 Design trucking network

Network m odels,

Forecasting, Simulation$17.3 mil li on

San Francisco Police

Dept.

Bethlehem Steel 1989 Design an Ingot Mold Stripper Integer Programming $8 mil l ion

North Ameri can Van

Lines1988 Assigning loads to drivers Network modeling $2.5 mill ion

Citgo Petroleum 1987

distribution

,

Forecasting $70 mil lion

United Airl ines 1986Schedul ing reservation

personnelLP, Queuing, Forecasting $6 mi ll ion

Dair man's Creamer 1985 O timal roduction levels Linear Pro rammin $48 000

Phil l ips Petroleum 1983 Equipment replacement Network modeling $90,000



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Basics of Operations Research

Operations Research - Characteristics

Managerial decision making

System approach

Computers

Mathematical models



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OR provides rational basis for decision making:Solves the type of complex problems that turn

Builds mathematical and computer models of

organizational systems composed of people,

,

Uses analytical and numerical techniques to

make predictions and decisions based on these

models



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Why must we learn the decision-making process?"

Or anizations are becomin more com lex.

Environments are changing so rapidly that past

practices are no longer adequate.

The costs of making bad decisions have

increased.



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Optimization is Everywhere

It is embedded in language, and part of the way we

firms want to maximize value to shareholders

people want to make the best choices

When playing games, we want the best strategy

When we have too much to do, we want to

.



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Mathematical Optimization is nearly everywhere.

gr cu ture

Military

Production Management Financial Management

Marketing Management

Personnel Mana ement

Health care

Construction

, .



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Real World

Assumed Real World Model

Levels of abstraction in the model development



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requires the use of one or more mathematical

models.

ma ema ca mo e s a ma ema ca

representation of the actual situation that may

be used to make better decisions or clarify the

.



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Benefits of Modeling

Economy - it is often less costly to analyze decision

roblems usin models.

Timeliness - models often deliver needed information

more quickly than their real-world counterparts.

Feasibility - models can be used to do things that

would be impossible.

decision making.



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analysis

building

analysis tion

Feed back



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Methods of Operations Research



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Different Models

Linear Programming models

A single objective function, representing either a profit

to be maximized or a cost to be minimized, and a set of

.

The objective function and constraints all are linear

functions of the decision variables.

Software has been developed that is capable of solving

problems containing millions of variables and tens of



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Nonlinear Pro rammin models The objective and/or any constraint is nonlinear.

In general, much more difficult to solve than

linear.

Most (if not all) real world applications require a

nonlinear model.

In order to make the problems tractable, we

often approximate using linear functions.



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D namic Pro rammin A DP model describes a process in terms of states,

decisions, transitions and returns.

The rocess be ins in some initial state where a

decision is made.



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Stochastic models

In many practical situations the attributes of a

.

Examples include the number of customers in a

checkout line, congestion on a highway, the

,

a financial security are some of the stochastic.



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A stochastic process that can be observed at

regu ar nterva s suc as every ay or every wee

can be described by a matrix which gives the

probabilities ofmoving to each state from every

o er s a e n one me n erva .



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Simulation

It is often difficult to obtain a closed form

express on or e e av or o a s oc as c sys em.

Simulation is a very general technique for

estimating statistical measures of complex

sys ems.

A system is modeled as if the random variables

were known.



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The OR Problem Solving Schema

Formulation Monitoring

Realization

Modelling Implementation

olut onAnalysis



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St eps f or sol v i ng OR problem

Formulate the problem to be solved

Select appropriate tool necessary to solve the

problem

,

assumptions

Perform the analysis



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Solution:

Define decision variable

Let x1 and x2 be the number of hectors corn and potato grown

respectively, the following decision model represents the problem.

Max {w = 20,000X1 + 10,000X2} objective functions

Subject to:

20X1 + 40X2 200 (labor)

X1 + 4 (Environment) constraints

X2 3 (Crop rotation)

X1 + X2 6 (Land)

We will see more model buildin in the next cha ter.



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Mathematical Modeling and Optimization Building Blocks

data (Actual Situation and Requirements, Control Parameters)

e.g., number of sites, unit capacities, demand forecasts,

available resources

model (variables, constraints, objective function)

e.g., how much to produce, how much to ship, (decision

variables, unknowns)optimization algorithm and solver

e.g., simplex algorithm, B&B algorithm, outer approximation...

optimal solution (Suggested Values of the Variables)e.g., production plan, unit-connectivity, feed concentrations.



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Design Optimization

Optimization is a component of design process

The design of systems can be formulated as

roblems of o timization where a measure of

performance is to be optimized while satisfying

a t e constraints.


D i O ti i ti


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Design Optimization

Design variables a set of parameters that describes

the system (dimensions, material, load, )

Design constraints all systems are designed to

.

constraints must be influenced by the design variables

(max. or min. values of design variables).

Objective function a criterion is needed to judge

whether or not a given design is better than another

, , , , , .


O i i bl l i


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Optimum Design Problem Formulation

The formulation of an optimization problem is

extremely important, care should always be

exercised in defining and developing expressions

for the constraints.

The optimum solution will only be as good as the

.


P bl F l ti (D i f t b t t )


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Problem Formulation (Design of a two -bar structure)

The problem is to design a two-member bracket to support a

force W without structural failure. Since the bracket will be

,

minimize its mass while also satisfying certain fabrication and



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Example Design of a Beer Can

beer and meet other design requirement. The cans will be

,

of manufacturing.Since the cost can be related directly tothe surface area of the sheet metal used, it is reasonable to

minimize the sheet metal required to fabricate the can.


l i f C


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Fabrication, handling, aesthetic, shipping considerations

and customer needs impose the following restrictions on

the size of the can:

1. The diameter of the can should be no more than 8 cm.

Also, it should not be less than 3.5 cm.2. The height of the can should be no more than 18 cm

and no less than 8 cm.

3. The can is required to hold at least 400 ml of fluid.



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Design variables

D= diameter of the can cm

H= height of the can (cm)

Objective function

The design objective is to minimize the surface area

(Non-linear)



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The constraints must be formulated in terms of design variables.

The first constraint is that the can must hold at least 400 ml of fluid.

(Non-linear)

The other constraints on the size of the can are:

The problem has two independent design variable and five

near

exp c cons ra n s. e o ec ve unc on an rs

constraint are nonlinear in design variable whereas the

remaining constraints are linear.


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Chapter 1 - Introduction to Or