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Transcript of Introduction Modeling
8/2/2019 Introduction Modeling
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OPERATIONS RESEARCH
An Intro
By
Farizal, PhD
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2
Operations Research, OR
What is It?
Operations research (also known as
management science, MS) is a
collection of techniques based onmathematics and other scientific
approaches a problem within a
system to yield the optimal
solution.
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History of O.R.
• World War II-research on military operation.
• 1947-simplex method by George Dantzig.
•
1950-LP , DP , Queueing Theory , andInventory Theory.
• Computer revolution.
•
1980s-software package.
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Optimisation = Efficiency + Savings
• Kellogg’s – The largest cereal producer in the world. – LP-based operational planning (production, inventory, distribution) system
saved $4.5 million in 1995.
• Procter and Gamble – A large worldwide consumer goods company. – Utilised integer programming and network optimization worked in concert
with Geographical Information System (GIS) to re-engineering productsourcing and distribution system for North America.
– Saved over $200 million in cost per year.
• Hewlett-Packard – Robust supply chain design based on advanced inventory optimization
techniques. – Realized savings of over $130 million in 2004
Source: Interfaces
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Properties of O.R.
• O.R. is concerned with OPTIMAL decision
making in, and modeling of, deterministic &
probabilistic systems that originate from Real
life.
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Properties of O.R.
• Creative scientific research into the
fundamental properties of operations.
• Search for optimality.
• Team approach-involving the backgrounds of
mathematics, statistics & probability theory,
economics, business administration, electronic
computing, engineering & physics, and
behavior sciences etc.
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Distinct Nature of OR
• Simultaneously analyzes all variables
• Seeks global, balanced solutions definedby:
– Multiple criteria – Multiple, conflicting objectives
• Helps mitigate risk and reduces uncertainty
by modeling different scenarios• Goes beyond single-issue management
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Contribution of O.R.
• The structuring of the real life situation into a
mathematical model, abstracting the essential elements
so that a solution relevant to the decision maker’s
objectives can be sought. This involves looking at the
problem in the context of the ENTER SYSTEM.
• Exploring the STRUCTURE of such solutions & developing
systematic procedures for obtaining them.
•Developing a solution, including the mathematical theory,if necessary that yields an optimal measure of
DESIRABILITY.
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What in O.R.?
Deterministic
Problem
LP , DP , NLP ,
IP ,Inventory ,Network
Scheduling,
PERT/CPM
Stochastic
Problem
D.S.
Simulation.
Queueing ,Game Theory
Forecasting ,
Decision
Analysis,Markov Chain
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Deterministic versus Stochastic• Two broad categories of optimization models exist
– deterministic
• parameters/data known with certainty
– stochastic
• parameters/data know with uncertainty
• Deterministic models are easier to solve. we
pretend we know the parameter/input with
certainty).
• Stochastic model are difficult to solve. In reality, we
know a distribution about our demand. We get
around this in real life by re-optimizing.
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Deterministic versus Stochastic
• Deterministic optimization ignores risk of beingwrong about parameter/data estimates.
• No commercial software packages are currently
available to do generalized, stochasticoptimization.
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Linear Programs
A Major Tool of OR
• Linear Programs (LPs) are a special type of mathematical model where all relationshipsbetween parts of the system being modeled can be
represented linearly (a straight line).• Not always realistic, but we know how to solve LPs.
• May need to approximate a relationship that is
slightly non-linear with a linear one.• When to use: if a problem has too many
dimensions and alternative solutions to evaluate allmanually, use an LP to evaluate.
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General Optimization Model
Problem(1):
Min f ( x )
s.t. g( x
)
0 --------(1) x 0
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Linear Programs
• LPs can evaluate thousands, millions, etc. of different alternatives to find the one that best
meets the objective of the business problem.
–
Fleet Assignment Model - assign aircraft to flightlegs to minimize cost and maximize revenue
– Revenue Management - set bid prices to maximize
revenue and/or minimize spill
– Crew Scheduling - schedule crew members to
minimize number of crew needed and maximize
utilization
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Other Types of Linear Optimizations
• MIP (Mixed Integer Programming)
– is similar to LP but at least one decision variable is
required to be a integer value
– violates the LP rule that decision variables be
continuous – is solved by “branch and bound” - solving a series of
LPs that fix the integer decision variables to various
integer values and comparing the resulting objective
function values
– is done in a smart way to avoid enumerating all
possibilities
–is useful, since you can not have .3 of an aircraft
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Other Types of Linear Optimization
• Network problem
– is a special form of LP which turns out to be
“naturally integer”
– can be solved faster than an LP, using a special
network optimization algorithm – is very restrictive on types of constraints that can be
present in the problem
•
Shortest Path – finds the shortest path from the source (start) to
sink (end) nodes, along connecting arcs, each having
a cost associated with them
– is used in many applications
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Other Optimization Models
• Quadratic Program
– has a quadratic objective function with linear
constraints
– can be applied to revenue management, because itallows fare to rise with demand within a problem
• price(OD) = 50 + [5*numpax(OD)]
• max revenue = price * numpax
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Other Optimization Models
• Non-linear Program (NLP)
– can have either non-linear objective function or
non-linear constraints or both
– feasible region is generally not convex
– much more difficult to solve
– but it is worth our time to learn to solve them since
world is actually non-linear most of the time – some non-linear programs can be solved with LPs or
MIPs using piecewise linear functions
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Applications of O.R.
• Inventory & Production Problem
• Maximization Problem
•
Minimization Problem• Work-Force Planning Problem
• Waiting Line Problem
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Application Details
• Accounting:
– Cash flow planning
– Credit policy
–Strategy planning
• Facilities Planning
– Location & size
– Logistics systems
– Transportation Planning
– Hospital planning
• Manufacturing
– Production scheduling
– Production-marketing
balance• Organization Behavior
– Employee recruiting
– Skills balancing
– Training programs
scheduling
– Manpower justification \
planning
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OR/MS Successes
Best cases from the annual INFORMS
Edelman Competition
2002: Continental Airlines Survives 9/112001: Merrill Lynch Integrated Choice
2001: NBC’s Optimization of Ad Sales
2000: Ford Motor Prototype Vehicle Testing1996: Procter & Gamble Supply Chain
1991: American Airlines Revolutionizes Pricing
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Case 4: Ford Motor Prototype Vehicle
Testing
• Business Problem: Developing prototypes
for new cars and modified products is
enormously expensive. Ford sought toreduce costs on these unique, first-of-a-
kind creations.
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Ford Motor (con’t)
• Model Structure: Ford and a team fromWayne State University developed a
Prototype Optimization Model (POM) to
reduce the number of prototype vehicles.The model determines an optimal set of
vehicles that can be shared and used to
satisfy all testing needs.• Project Value: Ford reduced annual
prototype costs by $250 million.
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Case 5: Procter & Gamble Supply
Chain
• Business Problem: To ensure smart growth,
P&G needed to improve its supply chain,
streamline work processes, drive out non-value-added costs, and eliminate
duplication.
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P&G Supply Chain (con’t)
•
Model Structure: The P&G operations researchdepartment and the University of Cincinnati created
decision-making models and software. They
followed a modeling strategy of solving two easier-
to-handle subproblems:
– Distribution/location
– Product sourcing
• Project Value: The overall Strengthening GlobalEffectiveness (SGE) effort saved $200 million a year
before tax and allowed P&G to write off $1 billion of
assets and transition costs.
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P&G Supply Chain (con’t)
• Project Value: The overall Strengthening
Global Effectiveness (SGE) effort saved
$200 million a year before tax and allowedP&G to write off $1 billion of assets and
transition costs.
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Case 6: American Airlines
Revolutionizes Pricing
• Business Problem: To compete effectively in
a fierce market, the company needed to“sell the right seats to the right customers
at the right prices.”
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American Airlines (con’t)
• Model Structure: The team developed yield
management, also known as revenue management
and dynamic pricing. The model broke down the
problem into three subproblems: – Overbooking
– Discount allocation
– Traffic management
The model was adapted to American Airlinescomputers.
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American Airlines (con’t)
• Project Value: In 1991, American Airlines
estimated a benefit of $1.4 billion over the
previous three years. Since then, yieldmanagement was adopted by other
airlines, and spread to hotels, car rentals,
and cruises, resulting in added profits going
into billions of dollars.
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Phases of an OR Project
– Define the problem
– Develop math model to represent the
system
– Solve and derive solution from model
– Test/validate model and solution
–
Establish controls over the solution – Put the solution to work
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Mathematics in Operation
Mathematical Solution Method (Algorithm)
Real Practical Problem
Mathematical (Optimization) Problem x2
Computer Algorithm
Human Decision-Maker
Decision Support Software System
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Problem Solving Stages
Mathematical Solution Method (Algorithm)
Real Practical Problem
Mathematical (Optimization) Problem
Computer Algorithm
Human Decision-Maker
Decision Support Software System
Staff Rostering at
Childcare Centre
Mathematical
Programming
CPLEX
XpressMP
LINGO
Excel with VBA
Childcare Centre
Manager
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General Optimization Model
Problem(1):
Min f ( x )
s.t. g( x )0 --------(1)
x 0
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• The Objective Function is an expression that
defines the optimal solution, out of the many
feasible solutions. We can either – MAXimize - usually used with revenue or profit or
– MINimize - usually used with costs
• Feasible solutions must satisfy the constraints of the problem. LPs are used to allocate scarce
resources in the best possible manner.
Constraints define the scarcity.• The scarcity in this problem involves a fixed
number of seats and scarce high paying
customers.
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• Rules for Constraints
– must be a linear expression
– decision variables can be summed together but notmultiplied or divided by each other
– have relational operators of =, <=, or >=
–
must be continuous• Constraints define the “feasible region” - all
points within the feasible region satisfy the
constraints.• The feasible region is convex.
• The optimal solution lies at an extreme point of
the feasible region
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ExampleA company with three plants produces two products.
First plant that operates for 4 hours produces only
product 1. Second plant operates for 12 hours but
produces only product 2. The last plant operates
for 18 hours and produces both products. It takesone hour to produce product 1 at plant 1 and 3
hours at plant 3 while product 2 needs 2 hour to be
produced at available facilities. If the selling price
for product 1 and 2 is $3,000 and $5,000,respectively. Find how many product 1 and product
2 should be made to maximize the profit? How
much the profit?
S l ti
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Solution:
Problem Articulation
Plant Product Product time
available1 2
1
2
3
1
0
3
0
2
2
4 hours
12 hours
18 hours
Profit $3,000 $5,000
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Linear Programming Formulation
Define:
: number of batches of product i produce per week
: total profit per week form purchasing this two
products
Objective function: to maximize
Max 3 X1 + 5 X2
Constraints: production time available
X1 ≤ 4
i x
z
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Complete LP Model
Max z =
s.t21 53 x x
0,0
1823122
4
21
21
2
1
x x
x x
x
x
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Example-ExtendedA company with three plants produces two products.
First plant that operates for 4 hours produces onlyproduct 1. Second plant operates for 12 hours but
produces only product 2. The last plant operates for
18 hours and produces both products. It takes one
hour to produce product 1 at plant 1 and 3 hours atplant 3 while product 2 needs 2 hour to be produced
at available facilities. If the selling price for product 1
and 2 is $3,000 and $5,000, respectively, and if
product 1 in order to be economical should be
produced at least 3, find how many product 1 and
product 2 should be made to maximize the profit?
How much the profit?
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Complete LP Model
Max z =
s.t21 53 x x
0,0
1823
122
3
4
21
21
2
1
1
x x
x x
x
x
x
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Example-ExtendedA company with three plants produces two products .
First plant that operates for 4 hours produces only
product 1. Second plant operates for 12 hours but
produces only product 2. The last plant operates for
18 hours and produces both products. It takes one
hour to produce product 1 at plant 1 and 3 hours at
plant 3 while product 2 needs 2 hour to be produced
at available facilities. If the selling price for product 1
and 2 is $3,000 and $5,000, respectively. Find howmany product 1 and product 2 should be made to
maximize the profit. How much the profit?
Determine which plants produce the products?