Introduction to Operations Research
Prof. Fernando Augusto Silva Marins
www.feg.unesp.br/~fmarins
fmarins@feg.unesp.br
1
What Is Management Science
(Operations Research, Operational
Research ou ainda Pesquisa Operacional)?

Management Science is the discipline that adapts
the scientific approach for problem solving to help
managers make informed decisions.
The
goal of management science is to recommend
the course of action that is expected to yield the best
outcome with what is available.
2
What Is Management Science?

3
The basic steps in the management science problem
solving process involves
–
Analyzing business situations (problem identification)
–
Building mathematical models to describe them
–
Solving the mathematical models
–
Communicating/implementing recommendations based
on the models and their solutions (reports)
The Management Science
Process

The four-step management science process
Problem definition
Mathematical modeling
Solution of the model
Communication/implementation
of results
4
The Management Science
Process
5

Management Science is a discipline that adopts the
scientific method to provide management with key
information needed in making informed decisions.

The team concept calls for the formation of
(consulting) teams consisting of members who come
from various areas of expertise.
The Management Science
Approach
6

Logic and common sense are basic components
in supporting the decision making process.

The use of techniques such as:
– Statistical inference
– Mathematical programming
– Probabilistic models
– Network and computer science
– Simulation
Using Spreadsheets in
Management Science Models
Spreadsheets have become a powerful tool in
management science modeling.
 Several reasons for the popularity of spreadsheets:
– Data are submitted to the modeler in spreadsheets
– Data can be analyzed easily using statistical (Data
Analysis Statistical Package) and mathematical
tools (Solver Optimization Package) readily
available in the spreadsheet.
– Data and information can easily be displayed using
graphical tools.

7
Classification of Mathematical
Models

Classification by the model purpose
– Optimization models
– Prediction models
 Classification by
the degree of certainty of the data in
the model
– Deterministic models (Mathematical Programming)
– Probabilistic (stochastic) models (Simulation)
8
Examples of Management
Science Applications
9

Linear Programming was used by Burger King to
find how to best blend cuts of meat to minimize
costs.

Integer Linear Programming model was used by
American Air Lines to determine an optimal flight
schedule.

The Shortest Route Algorithm was implemented
by the Sony Corporation to developed an onboard
car navigation system.
Examples of Management Science
Applications
 Project Scheduling Techniques were used by a
contractor to rebuild Interstate 10 damaged in the 1994
earthquake
in
the
Los
Angeles
area.

Decision Analysis approach was the basis for the
development of a comprehensive framework for
planning environmental policy in Finland.

Queuing models are incorporated into the overall
design plans for Disneyland and Disney World, which
lead to the development of ‘waiting line entertainment’
in order to improve customer satisfaction.
10
Is Operations Research really important?
11
INFORMS 2007
Sucessos da Pesquisa
Operacional em
Logística
61 trabalhos = 42%
12
Todos finalistas
Método
Otimização
Heurísticas
Estatística
Simulação
Mistos
DSS
Contr. estoques
Análise de risco
Revenue mngt
Prog. Dinâmica
Filas
Soft systems
System dynamics
Expert systems
Delphi
Adm. de projetos
DEA
N/D
13
TOTAL
Ocorrências
61
17
12
11
9
5
5
4
4
4
4
2
2
1
1
1
1
1
145
Edelman: métodos
empregados
Somente logística
Método
Transp PCP
Rede Compr. Estoq Armaz. TOTAL
Otimização
11
10
7
4
1
33
Heurísticas
7
5
12
Mistos
2
1
1
4
Contr. estoques
2
2
4
Simulação
1
1
2
Revenue mngt
2
2
Prog. Dinâmica
1
1
2
Expert systems
1
1
Análise de risco
1
1
TOTAL
22
21
8
5
3
2
61
Simulação estocástica
discreta é popular na
indústria...
Ano
1996
1996
1996
1996
1996
1996
1996
1995
1995
1995
1995
1995
1995
1994
1994
1994
1994
1994
1994
1993
1993
1993
1993
1993
1993
14
Empresa
South African National Defense Force*
Título do Trabalho
"Guns or Butter: Decision Support for Determining the Size and Shape of the
South African National Defense Force (SANDF)"
The Finance Ministry of Kuwait
"The Use of Linear Programming in Disentangling the Bankruptcies of al-Manakh
Stock Market Crash
AT&T Capital
"Credit and Collections Decision Automation in AT&T Capital's Small-Ticket
Business"
British National Health Service
"A New Formula for Distributing Hospital Funds in England"
National Car Rental System, Inc.
"Revenue Management Program"
Procter and Gamble
"North American Product Supply Restructuring at Procter & Gamble"
Federal Highway Administration/California Department "PONTIS: A System for Maintenance Optimization and Improvement of U.S.
of Transportation
Bridge Networks "
Harris Corporation/Semiconductor Sector*
"IMPReSS: An Automated Production-Planning and Delivery-Quotation System at
Harris Corporation - Semiconductor Sector"
Israeli Air Force
"Air Power Multiplier Through Management Excellence"
KeyCorp
"The Teller Productivity System and Customer Wait Time Model"
NYNEX
"The Arachne Network Planning System"
Sainsbury's
"An Information Systems Strategy for Sainsbury’s"
SADIA
"Integrated Planning for Poultry Production"
Tata Iron & Steel Company, Ltd.*
"Strategic and Operational Management with Optimization at Tata Steel"
Bellcore
"SONET Toolkit: A Decision Support System for the Design of Robust and CostEffective Fiber-Optic Networks"
Chinese State Planning Commission and the World
"Investment Planning for China’s Coal and Electricity Delivery System"
Digital Equipment Corp.
"Global Supply Chain Management at Digital Equipment Corp."
Hanshin Expressway Public Corporation
"Traffic Control System on the Hanshin Expressway"
U.S. Army
"An Analytical Approach to Reshaping the Army"
AT&T*
"AT&T's Call Processing Simulator (CAPS) Operational Design for Inbound Call
Centers"
Frank Russell Company & The Yasuda Fire and Marine "An Asset/Liability Model for a Japanese Insurance Company Using Multistage
Insurance Co. Ltd.
Stochastic Programming"
North Carolina Department of Public Instruction
"Data Envelopment Analysis of Nonhomogeneous Units: Improving Pupil
Transportation in North Carolina"
National Aeronautic and Space Administration (NASA) "Management of the Heat Shield of the Space Shuttle Orbiter: Priorities and
Recommendations Based on Risk Analysis"
Delta Airlines
"COLDSTART: Daily Fleet Assignment Model"
Bellcore
"An Optimization Approach to Analyzing Price Quotations Under Business Volume
Discounts"
FINALISTAS EDELMAN 1984-2007
FINALISTAS EDELMAN 1984-2007
Ano Empresa
1985 Weyerhaeuser Company*
1985 Canadian National Railways
1985
1985
1985
1985
1984
1984
1984
1984
1984
15
1984
Pacific Gas and Electric Company
New York, NY, Department of Sanitation
Eletrobras and CEPEL, Brazil
United Airlines
Blue Bell, Inc.*
The Netherlands Rijkswaterstaat and the Rand
Austin, Texas, Emergency Medical Services
Pfizer, Inc.
Monsanto Corporation
U.S. Air Force
Título do Trabalho
Weyerhaeuser Decision Simulator Improves Timber Profits
"Cost Effective Strategies for Expanding Rail-Line Capacity Using Simulation and
Parametric Analysis"
"PG&E's State-of-the-Art Scheduling Tool for Hydro Systems"
"Polishing the Big Apple"
Coordinating the Energy Generation of the Brazilian System
United Airlines Station Manpower Planning System
Blue Bell Trims Its Inventory
Planning the Netherlands' Water Resources
Determining Emergency Medical Service Vehicle Deployment
"Inventory Management at Pfizer Pharmaceuticals"
"Chemical Production Optimization"
"Improving Utilization of Air Force Cargo Aircraft"
Optimization Models

Many managerial decision situations lend themselves
to quantitative analyses.

A Mathematical Model consists of
– Objective function with one or more Control
/Decision Variables to be optimised.
–
16
Constraints (Functional constraints “”, “”, “=”
restrictions that involve expressions with one or
more Control /Decision Variables)
The Galaxy Industries Production
Problem
 Galaxy manufactures two toy doll models:
– Space Ray.
– Zapper.
 Resources are
limited to
–1000 pounds of special plastic.
– 40 hours of production time per week.
17
Galaxy Industries Production Problem
Technological input
– Space Rays uses 2 of plastic and 3 min of labor
– Zappers uses 1 of plastic and 4 min of labor
Marketing requirement
– Total production cannot exceed 700 dozens.
–
18
Number of dozens of Space Rays cannot exceed
number of dozens of Zappers by more than 350.
The Galaxy Industries Production
Problem

The current production plan calls for:
– Producing as much as possible of the more profitable
product, Space Ray ($8 profit per dozen).
– Use resources left over to produce Zappers ($5 profit
per dozen), while remaining within the marketing
guidelines.
• The current production plan consists of:
Space Rays = 450 dozen 8(450) + 5(100)
Zapper
= 100 dozen
Profit
= $4,100 per week
19
Management is seeking a
production schedule that will
increase the company’s profit.
20
A Linear Programming model can
provide an insight and an
intelligent solution to this problem.
21
Defining Control/Decision
Variables
22

Ask, “Does the decision maker have the authority to
decide the numerical value (amount) of the item?”

If the answer “yes” it is a control/decision variable.

By very precise in the units (and if appropriate, the
time frame) of each decision variable.
The Galaxy Linear Programming
Model

Decisions variables:
–X1 =
Weekly production level of Space Rays
–X2 =
Weekly production level of Zappers
(in dozens)
23
Objective Function
24

The objective of all optimization models, is to
figure out how to do the best you can with what
you’ve got.

“The best you can” implies maximizing something
(profit, efficiency...) or minimizing something (cost,
time...).
The Galaxy Linear Programming
Model

Objective Function:
Decisions variables:
X1 = Weekly production of Space Rays,
X2 = Weekly production of Zappers
–
Weekly profit, to be maximized
Max 8X1 + 5X2
25
Space Ray- $8/dozen
Zappers $5/dozen
Writing Constraints

Create a limiting condition in words in the following
manner:
(The amount of a resource required) (Has some
relation to) (The availability of the resource)

Make sure the units on the left side of the relation are
the same as those on the right side.

Translate the words into mathematical notation using
known or estimated values for the parameters and the
previously defined symbols for the decision variables.
26
Decisions variables X1 = Space Rays, X2 = Zappers
Space Rays uses 2 of
plastic and 3 min of
labor
Zappers uses 1 of
plastic and 4 min of
labor
There is 1000 of special plastic
and 40 hours (2,400 min) of
production time/week.
Total production  700,
Number Space Rays cannot
exceed number of dozens of
Zappers by more than 350,
27
Writing Constraints
2X1 + 1X2  1000
(Plastic)
3X1 + 4X2  2400
(Prod Time - Min)
X1 + X2  700
(Total production)
X1 - X2  350
(Mix)
Writing Constraints

Additional constraints
Non negativity constraint - X 0
Lower bound constraint - X  L
Upper bound constraint - X  U
Integer constraint - X = integer
Binary constraint - X = 0 or 1
28
The Galaxy Linear Programming
Model
Max 8X1 + 5X2 (Weekly profit)
subject to (the constraints)
2X1 + 1X2  1000
3X1 + 4X2  2400
X1 + X2  700
X1 - X2  350
(Plastic)
(Production Time - Min)
(Total production)
Is there Additional
(Mix)
Xj  0, j = 1,2
29
Non negativity constraint
Lower bound constraint Upper bound constraint Integer constraint
Binary constraint
Constraints?
(Nonnegativity)
Integers??
The Graphical Analysis of Linear
Programming
The set of all points that satisfy all the
constraints of the model is called a
FEASIBLE REGION
30
Using a graphical presentation we can
represent:
 All
31
the constraints
 The
objective function
 The
three types of feasible points.
Graphical Analysis – the Feasible
Region
X2
The non-negativity constraints
X1
32
Graphical Analysis – the Feasible
Region
X2
The Plastic constraint
2X1+X2  1000
1000
Total production constraint: X1+X2  700 (redundant)
700
500
Infeasible
Production
Time
3X1+4X2  2400
Feasible
500
33
700
X1
Graphical Analysis – the Feasible Region
X2
1000
The Plastic constraint
2X1+X2 1000
Total production constraint: X1+X2 700 (redundant)
700
500
Production
Time
3X1+4X22400
Infeasible
Production mix
constraint:
X1-X2  350
Feasible
500
700
X1
Interior points. Boundary points. Extreme points (5 Vertices).
34
• There are three types of feasible points
Max 8X1 + 5X2
The search for an optimal solution
Start at some arbitrary profit, say profit = $2,000...
X2
Then increase the profit, if possible...
...and continue until it becomes infeasible
1000
Optimal Profit =$4,360
and optimal solution:
Space Rays = 320 dozen
Zappers
= 360 dozen
Current solution:
Space Rays = 450, Zapper
= 100 and Profit = $4,100
700
600
8X1 + 5X2 = 3,000
8X1 + 5X2 = 2,000
400
X1
35
250
500
36
Simulation
37
37
Overview of Simulation
–
When do we prefer to develop simulation model over an
analytic model?
 When
not all the underlying assumptions set for analytic model
are valid.
 When mathematical complexity makes it hard to provide useful
results.
 When “good” solutions (not necessarily optimal) are
satisfactory (In general it is the interest of the Enterprises).
- A simulation develops a model to numerically evaluate a
system over some time period.
38
- By estimating characteristics of the system, the best
alternative from a set of alternatives under consideration
(sceneries) can be selected.
38
Overview of Simulation
–
Continuous simulation systems monitor the
system each time a change in its state takes
place.
- Discrete simulation systems monitor changes in a
state of a system at discrete points in time.
Simulation of most practical problems requires the
use of a computer program.
–
39
39
Overview of Simulation
–
Approaches to developing a simulation model
 Using add-ins to Excel such as @Risk or Crystal Ball
 Using general purpose programming languages such
as: FORTRAN, PL/1, Pascal, Basic.
 Using simulation languages such as GPSS, SIMAN,
SLAM.
 Using a simulator software program (ARENA,
SIMUL8, PROMODEL).
- Modeling and programming skills, as well as knowledge
of statistics are required when implementing the
simulation approach.
40
40
Monte Carlo Simulation
41

Monte Carlo simulation generates random events.

Random events in a simulation model are needed when the
input data includes random variables.

To reflect the relative frequencies of the random variables,
the random number mapping method is used.
41
JEWEL VENDING COMPANY –
an example for the random mapping
technique
42

Jewel Vending Company (JVC) installs and stocks
vending machines.

Bill, the owner of JVC, considers the installation of a
certain product (“Super Sucker” jaw breaker) in a
vending machine located at a new supermarket.
42
Bill would like to estimate the
JEWEL VENDING COMPANY
expected number of days it takes
for a filled machine to become half empty.

Data
– The vending machine holds 80 units of the product.
– The machine should be filled when it becomes half empty.
Daily demand distribution is estimated from similar
vending machine placements.
P(Daily demand = 0 jaw breakers) = 0.10
P(Daily demand = 1 jaw breakers) = 0.15
P(Daily demand = 2 jaw breakers) = 0.20
P(Daily demand = 3 jaw breakers) = 0.30
P(Daily demand = 4 jaw breakers) = 0.20
P(Daily demand = 5 jaw breakers) = 0.05
–
43
43
Random number mapping –
The Probability function Approach
Random number mapping uses the probability function
to generate random demand.
A number between
00 and 99 is selected
randomly.
0.10
00-09
44
0
3434
3434
0.20
34
0.15 34
34
3434
The daily demand is
determined by the mapping
demonstrated below.
0.30
0.20
10-25
26-44
45-74
75-94
0.05
95-99
1
2
3
4
5
26-44
Demand
44
Random number mapping –
The Cumulative Distribution Approach
F(X)
1.00
Daily demand
X is determined by
the random number Y between
0 and 1, such that X is the
smallest value for which F(X) Y.
34
1.00
0.95
0.75
0.45
Y = 0.34
0.34
F(1) = .25 < .34
F(2) = .45 > .34
0.25
0.10
45
0.00
0
1
2
3
4
5
45
X
Simulation of the JVC Problem
46

A random demand can be generated by hand (for
small problems) from a table of pseudo random
numbers.

Using Excel a random number can be generated by
– The RAND() function
– The random number generation option
(Tools>Data Analysis)
46
Simulation of the JVC Problem

An illustration of generating a daily random demand.

Since we have two digit probabilities, we use the first two
digits of each random number.
Random
Number
6506
7761
6170
8800
4211
7452
Day
1
2
3
4
5
6
00-09
47
0
Two First
Digits
65
77
61
88
42
74
Demand
3
4
3
4
2
3
10-25
26-44
45-74
75-94
1
2
3
4
Total Demand
to Date
3
7
10
14
16
19
95-99
5
47
Simulation of the JVC Problem
Simulation is repeated and stops once
total demand reaches 40 or more.
Day
1
2
3
4
5
6
Random
Number
6506
7761
6170
8800
4211
7452
Two First
Digits
65
77
61
88
42
74
Demand
3
4
3
4
2
3
Total Demand
to Date
3
7
10
14
16
19
The number of “simulated” days required for the
total demand to reach 40 or more is recorded.
48
48
Simulation Results and Hypothesis Tests
49
–
The purpose of performing the simulation runs
is to find the average number of days required
to sell 40 jaw breakers.
–
Each simulation run ends up with (possibly) a
different number of days.
Hypothesis test is conducted to test
whether or not m = 16.
Null hypothesis H0 : m = 16
Alternative hypothesis HA : m > 16
49