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Title: ISE 195 Introduction to Industrial Engineering


1
ISE 195 Introduction to Industrial Engineering
2
Lecture 3 Mathematical Optimization (Topics in
ISE 470 Deterministic Operations Research Models)
3
What is OR?
  • Operations Research Study of Mathematical
    Optimization
  • OR is short for Operations Research
  • Home professional society the Institute for
    Operations Research and the Management Sciences
    (INFORMS)
  • Researchers in OR focus on how to improve the
    theory (mathematical) and algorithm
    (computational) aspects of formulating and
    solving mathematical optimization problems
  • Basic OR tools are helpful when a decision
    problem has many variables and constraints that
    can be described with a linear function
  • Web Site, http//www.informs.org

4
INFORMS
5
ISE and OR
  • Industrial Systems Engineering Branch of
    Engineering Concerned with Integrating and
    Improving Systems
  • ISEs can use OR tools to do this, usually with
    the help of a computer
  • ISEs focus on problems in Logistics, Scheduling,
    Healthcare, etc. that have an optimization focus
    and that have a scale large enough to utilize
    OR tools
  • ISEs use OR to formulate design problems and
    generate solutions

6
Why the Comparison?
  • Pure Operations Research has a heavy mathematical
    and computational orientation
  • There are many mathematical details to
    formulating problems successfully
  • There are many computational (computer
    programming, algorithmic) details to successfully
    finding optimal solutions to a stated problem
  • ISE applications of OR do not have as high a
    theoretical mathematical or algorithmic content
  • ISEs try to use the correct technique to improve
    the integrated system under investigation,
    including OR when appropriate

7
Model Formulation and Solution
  • Mathematical optimization model formulation and
    solution
  • Represent the system or phenomena in some set of
    algebraic structures
  • Uses the decision-makers view, usually
    different from the real-world view
  • Simulation models have a closer mapping to real
    world details
  • Encode the resulting model in a computer via some
    modeling language
  • GAMS, X-Press, Excel
  • Find a solution to the model (hopefully
    optimal)
  • Solution algorithms vary for linear, nonlinear
    and integer decision variables
  • Solutions generated suggest new designs for a
    system
  • A prescriptive decision technique
  • Trying to find a best solution with which to
    prescribe how to make the best use of limited
    resources

8
Mathematical Modeling
  • Describe system with set of algebraic equations
  • Capture key relationships within the system
  • Capture key behaviors in system
  • Decisions for which insight needed are decision
    variables
  • Goal embedded within the objective function
  • Limitations/restrictions in constraints
  • Physical constraints
  • Logical constraints

9
General Form of Math Model
10
General Parametric Form
MAX (or MIN) c1X1 c2X2 cnXn Subject
to a11X1 a12X2 a1nXn lt
b1 ak1X1 ak2X2 aknXn lt bk
am1X1 am2X2 amnXn bm
11
A Simple Example
Blue Ridge Hot Tubs produces two types of hot
tubs Aqua-Spas Hydro-Luxes.
There are 200 pumps, 1566 hours of labor, and
2880 feet of tubing available. How many of
each type should be produced to maximize profits?
12
Model Formulation Process
1. Understand the problem 2. Identify the
decision variables X1number of Aqua-Spas to
produce X2number of Hydro-Luxes to
produce 3. State the objective function as a
linear combination of the decision
variables MAX 350X1 300X2
13
Model Formulation Process
4. State the constraints as linear combinations
of the decision variables 1X1 1X2 lt 200
pumps 9X1 6X2 lt 1566 labor 12X1 16X2
lt 2880 tubing 5. Identify any upper or lower
bounds on the decision variables X1 gt 0
X2 gt 0
14
Model Formulation Process
MAX (Objective function) 350X1 300X2 S.T.
(Constraint set) 1X1 1X2 lt 200 9X1 6X2
lt 1566 12X1 16X2 lt 2880 Non-negativity
X1 gt 0 X2 gt 0
15
Formulation in Excel Solver
16
2nd Simple Example Model
  • Objective Determine production mix that
    maximizes the profit under the raw material
    constraint and other production requirements
    (detailed next).
  • Maximize 50D 30C 6 M Subject to 7D 3C
    1.5M lt 2000 (raw steel) D gt 100 (contract
    ) C lt 500 (cushions available) D, C, M
    gt 0 (Non-negativity) D and C are integers

17
What We are Looking For
  • Want to find the best solution
  • Solution must satisfy each of the constraints
  • Constraints must be satisfied simultaneously
  • Common area satisfying all the constraints is
    called the feasible region
  • ANY point in the feasible region is a possible
    solution to the problem
  • What we want is that feasible solution that
    provides the largest value of the objective
    function (the optimal solution)

18
Feasible Region of an LP
X2
1000
Profit 4360
700
500
X1
500
19
Linear Programming
  • Assumptions of the linear programming model
  • The parameter values are known with certainty.
  • The objective function and constraints exhibit
    constant returns to scale.
  • There are no interactions between the decision
    variables (the additivity assumption).
  • The Continuity assumption Variables can take on
    any value within a given feasible range.

20
How Do We Solve an LP?
  • Solving an LP Finding the best solution
    possible Finding the optimal solution
  • In ISE 470, you will learn the Simplex Method
    of solving an LP
  • Uses ideas from matrix algebra (i.e. pay
    attention in MTH 235)
  • Can be performed by hand using matrix operations
    (pivots)
  • The simplex algorithm is available in many forms
    in software
  • Excel Solver Tool
  • Many commercial solver packages (CPLEX, XPRESS,
    etc)

21
Integer Programming
  • Many real life problems call for at least one
    integer decision variable.
  • There are three types of Integer models
  • Pure integer (AILP)
  • Mixed integer (MILP)
  • Binary (BILP) (zero-one variables, on-off)
  • Unfortunately, these get quite hard to solve
  • Real-world problems can have hundreds or
    thousands of variables and constraints
  • Some problems are theoretically hard, even when
    they seem to have small numbers of elements
  • Real benefit of these types of models is that we
    can use binary variables to represent a host of
    logical conditions within the mathematical
    formulation

22
Logical Contraints (1)
1. Three of six projects must be selected.
Y1 Y2 Y3 Y4 Y5 Y6 gt 3
2. No more than three of six projects can be
selected.
Y1 Y2 Y3 Y4 Y5 Y6 lt 3
23
Logical Contraints (2)
 3. Either Project 2 or Project 4 or Project 6
must be selected
Y2 Y4 Y6 1
4. Production level for Product 1 cannot exceed
the production level for Product 5.
X1 lt X5
24
Logical Constraint (3)
5. If Product 4 is produced, then at least 120
units of Product 4 must be produced.
X4 lt MY4 X4 gt 120Y4
6. Four projects are numbered in ascending
order. If any project selected, all lower
numbered projects must also be selected.
Y2 lt Y1 Y3 lt Y2 Y4 lt Y3
25
Logical Constraint (4)
7. If Project 3 selected, either Project 4 or
Project 5 must be selected.
Y3 lt Y4 Y5
8. If Product 6 produced, production must be 50
or 100 units.
X6 50Y61 100Y62 Y61 Y62 lt 1
26
Basic Solution Methods
  • Linear models Simplex algorithm
  • Fast in practice
  • Lots of good sensitivity analysis information
  • Integer models Branch and bound
  • Can take very long time
  • No sensitivity information
  • Can be sped up with specific knowledge
  • Nonlinear models
  • Variety of solution methods, usually based on the
    derivative of the objective function
  • Use Hill Climbing and other methods

27
Applications of Mathematical Optimization
  • Budgeting, capital budgeting
  • Resource allocation, how much to make
  • How much to make, how much to contract
  • Assignment of personnel to jobs
  • Any other kind of assignment application
  • Funding allocation, investment planning
  • Inventory planning
  • Facility layout and planning

28
Applications of Mathematical Optimization
  • Routing
  • Ever use Google Maps?
  • Pick-up and delivery of items
  • Selection of items for loading onto delivery
    trucks
  • Scheduling
  • Jobs onto machines
  • Patients to doctors
  • Doctors to shifts, workers to shifts
  • Packing problems, cutting problems
  • How to cut patterns from material
  • And the list goes on and on

29
Other Related Areas
  • Non-linear optimization
  • These involve non-linear functions within the
    formulations
  • Most applicable solution methods are
    gradient-based local search procedures
  • Modern heuristic optimization
  • These involve integer programming models that are
    particularly difficult to solve
  • Also involve non-linear models with forms for
    which the existing non-linear solution codes have
    a particularly hard time solving
  • These are essentially search methods with some
    pretty strange analogies to nature
  • Genetic Algorithms
  • Ant-Colony Algorithms
  • Simulated Annealing Algorithms

30
Large-Scale Applications
  • Airline Crew Scheduling
  • Scenario A late-day storm has canceled many
    flights for American Airlines in and out of
    Chicago OHare Airport. Many planes scheduled to
    come to Chicago and leave Chicago have not made
    it to their planned destination for the end of
    the day
  • Problem How should we reassign crews and planes
    to the routes for the next day, to optimize our
    use of people, aircraft and other resources?
  • Companies such as American Airlines, IBM,
    Hewlett-Packard, UPS, FedEx, Toyota have
    large-scale problems that are well suited to
    mathematical optimization

31
  • Questions?
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