Mathematics for Industry

1. Mathematics forIndustry Kathryn E. Stecke University of Texas at Dallas School of Management Richardson, Texas

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: ExamplesGlover, 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

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 Sanyo’s 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: PerceptronRosenblatt [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