Mathematics for Industry

1
Mathematics for Industry

• Kathryn E. Stecke
• University of Texas at Dallas
• School of Management
• Richardson, Texas

2
(No Transcript)
3
Introduction

• Mathematics has been called the language of
  science (manthanein)
• Mathematics is used to solve many real-world
  problems in
• Industry
• Physical sciences
• Life sciences
• Economics
• Social and human sciences
• Engineering and technology

4
Mathematics and Ancient Wonders

• Mathematics was used to build many of the ancient
  wonders of the world, such as
• The pyramids of Egypt
• The Great Wall of China
• The hanging gardens of Babylon
• The Taj Mahal of Agra

5
Early Industrial Applications

• Developed by
• Taylor
• The Gilbreths
• Gantt

6
Industrial Applications

• Early mathematics (computations, statistics, and
  accounting) has been applied to operations
  problems, in
• Administration
• Managing technical activities

7
Types of ModelsSuri 1985

• Generative
• Evaluative

8
Generative Models

• Linear, integer, dynamic, and nonlinear
  programming (Kim et al. 2003)
• Differential equations
• Number theory
• Tabu search and genetic algorithms
• Fuzzy set theory
• Fluid dynamics
• Game theory

9
Evaluative Models

• Queueing network theory
• Petri nets
• Data envelope analysis
• Simulation
• Perturbation analysis
• Neural networks

10
Inventory Management
The Total Cost Curve is U Shaped
Annual Cost
Holding Cost
Ordering Cost
Order Quantity Q
EOQ
11
Inventory Control Extensions

• Quantity discounts for items or transportation
  costs or warehouse costs
• Consideration of the lead time
• Exact lead time is uncertain
• Newsboy problem
• (S,T) and (R,Q) policies

12
Network Flow Models

• 1974 (and 1975) Nobel Prize winners Tjalling
  Koopmans (and L.V. Kantorovich) were the first to
  propose network flow models
• Koopmans modeled the problem of moving people,
  supplies, and equipment from various U.S. bases
  to foreign bases
• The goal was to optimize one or more of
• Minimizing total transportation cost
• Minimizing total transit time
• Maximizing defensive effectiveness

13
Network Flow Models

• Kantorovich used network flow models to address
  some important problems in the Soviet economy
• He investigated the problems of allocating
  production levels among factories and
  distributing the resulting products among markets

14
Network Flow Application in Airline Industry

• The marketing department of a major airline
  company develops forecasts of the number of
  passengers taking different flights in each of
  several fare classes.
• Profits are affected by the number of available
  seats allocated to different fare classes on
  different flights.
• This can be formulated as a network flow problem
  based on the network of flights, fare classes,
  and seats.

15
Network Flow Application in Employee Scheduling

• Large grocery stores need to determine how to
  schedule their employees.
• I.e., in scheduling check-out clerks, a store
  manager must determine, each week, which and how
  many employees should be assigned to check out
  stations at each hour of the day.
• varying volumes of customers at different hours
• employees individual restrictions.
• A network flow model can be used based on the
  network formed by employees, customers, and
  working hours.

16
Network Flow Application in the Finance Industry

• Changing interest rates and opportunities for
  investment underscore the need for financial
  institutions to develop ways to manage their
  funds more effectively.
• Issues of liquidity and rate of return must be
  balanced to achieve proper relationships between
  inflows and outflows.
• A network flow model helps financial officers
  find the best composition and timing of
  investments.

17
Network Flow Applications

• The US Department of Transportation determines
  the best routes for proposed highways and other
  transportation channels by shortest path and
  assignment network methods
• NASA uses network flow models to determine
  characteristics of optimal space satellite
  orbits.
• In option trading, the calculation of the minimum
  deposit or margin that the broker must require of
  the investor can be found.

18
Network Flow Models Applications Glover,
Klingman, and Phillips 1992

• Electrical circuit board design
• Telecommunications
• Water management
• Design of transportation systems
• Metalworking
• Chemical processing
• Aircraft design
• Fluid dynamic analysis

19
Network Flow Models Applications

• Computer job processing
• Production
• Marketing
• Distribution
• Financial planning
• Project selection
• Facility location
• Accounting

20
Network Flow Models More Applications Glover,
Klingman, and Phillips 1992

• Airline revenue management
• Employee shift scheduling
• Best use of energy resources

21
Network Flow Models Examples Glover, Klingman,
and Phillips 1992

• A car manufacturer
• Chemical products companies
• An international pharmaceutical company
• Lumber company
• The U.S. Air Force
• An automobile components manufacturer
• Oil company
• The Tennessee Valley Authority
• Texas

22
Network Flow ModelsGlover, Klingman, and
Phillips 1992

• They provide references that detail most of these
  applications
• Methods to solve these problems, both discrete
  and continuous, are also given

23
Fuzzy Logic Zadeh 1965

• Fuzzy set theory, originally introduced by Zadeh,
  resembles human reasoning in its use of
  approximate information and uncertainty to
  generate decisions
• As complexity rises,
• precise statements lose meaning
• and meaningful statements lose precision.
• --Lotfi Zadeh

24
(No Transcript)
25
Fuzzy Logic Concept Dubois and Prade 1980

• Specifically designed to mathematically represent
  uncertainty and vagueness
• Using fuzzy sets, many engineering and decision
  problems can be greatly simplified

26
Fuzzy Set Theory

• Implements classes of groupings of data with
  boundaries that are not sharply defined
• Any methodology or theory implementing crisp
  definitions such as classical set theory,
  arithmetic, and programming, may be fuzzified

27
Fuzzy Logic Critical Aspects

• Linguistic variables are used, where general
  terms such as large, medium, and small are used
  to capture a range of numerical values
• Allows these stratified sets to overlap
• A 160 pound man may be classified in both the
  large and medium categories, with varying
  degrees of belonging or membership to each group

28
Fuzzy Logic and Boolean Logic

• Implements soft linguistic variables on a
  continuous range of truth values which allows
  intermediate values to be defined between
  conventional binary
• Differs from the classical two-valued sets and
  logic in that it uses soft linguistic variables
  rather than strict binary variables (large, tall,
  cold) (true or false)

29
Fuzzy Logic and Boolean Logic

• It can be considered a superset of Boolean or
  crisp logic, in the way fuzzy set theory is a
  superset of conventional set theory
• Can handle approximate information in a
  systematic way
• Formally, fuzzy logic is a structured, model-free
  estimation that approximates a function through
  linguistic input/output associations

30
Fuzzy Logic Control Applications

• Chemical process control
• Consumer products as washing machines, video
  cameras, and automobiles
• Robotics and automation
• Laundry washing machines (Matsushita and Hitachi)
• The Japanese bullet trains (Hitachi)
• Intelligent cruise control, anti-lock brake
  systems, automatic transmission control, adaptive
  traffic signal control, mobile robots, and
  baggage handling at the Denver airport

31
Fuzzy Logic Other Industrial Applications

• Automatic control of dam gates for the
  hydroelectric power plants of Tokio Electric
  Power
• Robot control (Hirota, Fuji Electric, Toshiba,
  and Omron)
• Preventing unwanted temperature fluctuations in
  air conditioning systems (Mitsubishi and Sharp)
• Efficient and stable control of car engines
  (Nissan)
• Cruise control for automobiles (Nissan and Subaru)

32
Fuzzy Logic Other Applications

• Improved efficiency of their industrial control
  applications (Aptronix, Omron, Meiden, Sha,
  Micom, Mitsubishi, Nisshin-Denki, and
  Oku-Electronics)
• Positioning of wafer steppers in the production
  of semiconductors (Canon)
• Optimized planning of bus time tables (Toshiba,
  Nippon-System, and Keihan-Express)
• Automatic motor-control for vacuum cleaners while
  recognizing the surface conditions and degree of
  soil (Matsushita)

33
Fuzzy Logic Other Applications

• Back light control for Sanyos camcorders
• Software design for industrial processes
  (Aptronix, Harima, and Ishikawajima-OC
  Engineering)
• Controlling machine speed and temperature for the
  steel works (Kawasaki Steel, Nippon Steel, and
  NKK)
• Improved fuel consumption for automobiles (NOK
  and Nippon Denki Tools)
• Improved sensitivity and efficiency for elevator
  controls (Fujitec, Hitachi, and Toshiba)
• Improved safety of nuclear reactors (Hitachi,
  Bernard, and Nuclear Fuel Division)

34
Neural Networks Origin

• Emulations of biological neurons, the most
  sophisticated collection of which is the human
  brain
• Crude electronic networks of neurons based on the
  neural structure of the brain
• Can be viewed as a massive distributed processor
  that has a natural propensity for storing
  experimental knowledge and making it available
  for use

35
Basic Element of a Neural Network Perceptron
Rosenblatt 1958

• The perceptron, built in hardware, is the oldest
  neural network still in use today
• A single-layer perceptron was found to be useful
  in classifying a continuous-valued set of inputs
  into one of two possible classes
• The perceptron computes a weighted sum of the
  inputs, subtracts a threshold, and passes one of
  two possible values as the result

36
Biological Neurons

• Biological neurons can learn from experience,
  detect subtle relationships between various
  inputs, and adapt to changing and uncertain
  circumstances
• What has been attained is the development of
  simple systems that exhibit the kind of
  generalized processing and adaptive properties
  inherent in biological neural networks

37
Neural Networks Advantages

• Resilient against distortions in input data and
  their capability of learning through training
• Ability to derive meaning from complicated or
  imprecise data
• Used to extract patterns and detect trends that
  are too complex to be noticed
• For example, law enforcement agents have looked
  for travel patterns that might indicate drug
  smuggling activities

38
Neural Networks Applications

• Navigation and vision or pattern recognition in
  robotics
• Expert systems
• Decision analysis
• Control systems
• Signal processing
• Character and speech recognition
• Control robot arm tracking movements

39
Tabu Search Glover and Laguna 1998

• Meta-heuristic that guides a local heuristic
  search procedure to explore a solution space
  beyond local optimality
• Iterative improvement search technique

40
Tabu Search How Does it Work

• Adaptive memory is used to provide a flexible
  search behavior
• It avoids getting trapped in local optima by
  using a limited memory of past moves
• A tabu list contains a memory of only some of the
  iterations
• Old memory is updated by new learning as the
  iterations proceed

41
Tabu Search The Role of Memory

• The memory of past iterations helps tabu search
  to continue exploration without becoming
  confounded by the absence of an improving move
• A simple form of tabu search methodology
  constrains a search by classifying certain moves
  as forbidden

42
Tabu Search Industrial Applications

• Job shop, flow shop, flexible flow line, and
  audit scheduling
• Resource allocation of a single and multiple
  plants
• Production planning with workforce learning
• Process plan optimization
• Determining the location of hub facilities in the
  design of communication networks

43
Tabu Search Industrial Applications

• Transportation, routing, and network design
• Vehicle routing
• VLSI systems with learning
• Task assignment for balancing assembly lines
• Facility layout in manufacturing

44
Genetic Algorithms Concept Holland 1992

• GAs are
• Adaptive heuristic search algorithms that use
  evolutionary ideas of natural selection and
  genetics
• Random directed search to seek optimal solutions
• Artificial intelligence, meta-heuristic technique

45
Genetic Algorithms Ecological Setting

• Natural selection and random variations determine
  the attributes of an individual
• Variations in generations are brought about by
  crossover and mutation of chromosomes

46
Genetic Algorithms for Problem Solving

• A GA views solutions as chromosomes, which are
  members of a population
• Fitness of a chromosome determines its chances of
  procreating progenies
• A fitness function is the objective function to
  be optimized

47
Advantage of Genetic Algorithms

• Ability to deal with many types of constraints
  and objective functions
• GA has been implemented for machine learning
• These can be thought of as living programs that
  learn from their environment and evolve over time

48
Genetic Algorithms Applications

• Industrial scheduling problems
• Automated design
• Composite material design
• Multi-objective design of automotive components
  for crashworthiness, weight savings, and other
  desirable characteristics
• Mobile communication infrastructure optimization
• Plant flow layout problems
• Aircraft design
• Robot trajectory generation

49
Simulated Annealingvan Laarhoven and Aarts 1987

• Annealing
• Is the process of heating a solid and cooling it
  slowly to remove strain and crystal imperfections
• During the process the free energy of the solid
  is minimized
• Initial heating is necessary to avoid becoming
  trapped in a local minimum (resulting in an
  imperfect crystal)

50
Simulated Annealing for Problem Solving

• Mathematically imitating the actual annealing
  process
• Objective functions can be viewed as the free
  energy
• Imitating how nature reaches a minimum yields
  optimization algorithms

51
Simulated Annealing Applicationsvan Laarhoven
and Aarts 1987

• Computer-aided design of integrated circuits
• Layout
• Code design
• Neural network theory
• Cutting patterns
• Vehicle routing

52
Decision Analysis

• Structures multiple objective problems
• Multi-attribute utility theory
• Analytic hierarchy process (uses pairwise
  comparison of alternatives)
• Useful when a group needs to make a decision
• To reach a good or best choice

53
Decision AnalysisKeefer et al. 2004

• Provides references to many applications of
  decision analysis methods such as multi-attribute
  utility theory to
• Real options
• Purchasing equipment
• New product strategies
• Telecommunication applications

54
Decision AnalysisUlvila and Gaffney 2004

• Use decision analysis to evaluate computer
  intrusion detection systems. Their method can be
  used to
• Decide the optimal operating point on an
  intrusion detector
• Choose the best intrusion detection system
• Compare the value of one intrusion detection
  system to another
• Determine the value of an intrusion detection
  system over no detector
• Determine how to adjust the operation of an
  intrusion detector to respond to changes in its
  environment

55
Decision Analysis Applications Keeney and
Raiffa 1976

• Airport location
• Heroin addiction treatment
• Business problems (profit versus ethics)
• Hospital blood bank inventory control
• Air pollution control
• Fire department operations
• Safety of landing aircraft
• Strategic and operational policy concerning
  frozen blood
• Sewage sludge disposal in the metropolitan Boston
  area
• Selecting a job or profession
• Transporting hazardous substances
• Development of water quality indices
• Airport development for Mexico City

56
Petri Nets Peterson 1981

• Used to describe and study information processing
  systems that can be characterized as being
• Concurrent
• Asynchronous
• Distributed
• Parallel
• Nondeterministic and/or stochastic

57
Petri Nets

• State and algebraic equations can be set up to
  describe the behavior of a system
• Decision-free timed Petri nets are equivalent to
  linear state equations in a max,-based algebra

58
Petri Nets Applications

• Flexible manufacturing
• Performance evaluation
• Industrial control systems
• Discrete event dynamic systems
• Fault tolerant systems
• Programmable logic and VLSI arrays
• Local-area networks
• Neural networks
• Decision models

59
Petri Nets More Applications

• Software design
• Workflow management
• Data analysis
• Concurrent programming
• Reliability engineering
• Diagnosis for finding the original error in the
  line of error -gt error state -gt visible error

60
Queueing Models

• Queueing models have been used to investigate
  industrial problems for many years
• In the 1940s, queueing models were used to solve
  a variety of machine interference problems
• How many repairpersons to assign
• to maintain a system (Stecke 1992 )
• How many telephone operators to
• handle traffic calls (Palm 1943

61
Queueing ModelsBuzacott and Shanthikumar 1993
Tempelmeier and Kuhn 1993

• Analyze tradeoffs concerning the number of
  servers versus the waiting time of customers
• Determining the appropriate number of service
  facilities to cover expected demand, as well as
  determining the efficiency of servers and the
  number of servers of different types at the
  service facilities (Hillier and Lieberman
  2002)
• Suri 1998 suggests using queueing theory to
  provide quick solutions to industrial problems

62
Queueing Models Applied in Retail Services
Berman and Larson 2004

• Many retail service facilities have both front
  and back room operations.
• The front room deals with serving customers.
• The back room focuses typically on restocking
  shelves and sorting and/or processing paperwork.
• Workers can be cross-trained to do both jobs.
• A manager brings a worker from the back to the
  front when the customer checkout queue becomes
  "too long".
• A reverse assignment occurs when the number of
  customers is small.
• Queueing theory is used to find the minimum
  number of workers to staff the facility subject
  to the performance constraints.

63
Queueing Models Applied in Call Centers

• One important application is in staffing call
  centers.
• A common target is to have 80 of the customers
  wait less than 20 seconds.
• Queueing theory is used to find the minimum
  number of workers to staff a call center subject
  to the performance constraint.

64
Queueing Models Applications

• Banks
• Airports - runway layout, luggage collection,
  shops, passport control ...
• Supermarkets
• Restaurants
• Manufacturing processes
• Hospital appointment bookings
• Ticket counters

65
(No Transcript)
66
Future Industrial Applications

• Developing and operating reconfigurable
  manufacturing systems Koren (2002)
• Homeland security (aka WWII)
• Schmitt and Stecke (2003)