Mathematical Optimization is nearly everywhere

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Mathematical Optimization is nearly everywhere

• Finance
• Marketing
• E-business
• Telecommunications
• Games
• Operations Management
• Production Planning
• Transportation Planning
• System Design

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Home Runs in Management Science

• Merril Lynch
• 5 million customers
• 16,000 financial advisors
• Developed a model to design product features and
  pricing options to better reflect customer value
• Benefits (Integrated Choice Service)
• 80 million increase in annual revenue
• 22 billion increase in net assets

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Home Runs in Management Science

• NBC
• Must determine program schedules
• Schedules must meet advertisers demographic and
  cost requirements
• Developed optimization model to determine optimal
  timing and pricing of commercials
• Benefits
• 50 million increase in annual revenue

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Home Runs in Management Science

• Samsung Electronics
• Leading DRAM manufacturer
• Semiconductor facilities cost 2-3 billion
• High equipment utilization is key
• Developed comprehensive planning and scheduling
  system to control WIP (Work In Process)
• Benefits
• Cut cycle times in half
• 1 billion increase in annual revenue

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Linear Programming (our first tool, and probably
the most important one.)

• minimize or maximize a linear objective
• subject to linear equalities and inequalities
• maximize 3x 4y
• subject to 5x 8y ? 24
• x, y ? 0

A feasible solution satisfies all of the
constraints.
x 1, y 1 is feasible x 1, y 3 is
infeasible.
An optimal solution is the best feasible solution.
The optimal solution is x 4.8, y 0.
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Terminology

• Decision variables e.g., x and y.
• In general, there are quantities you can control
  to improve your objective which should completely
  describe the set of decisions to be made.
• Constraints e.g., 5x 8y ? 24 , x ? 0 , y ?
  0
• Limitations on the values of the decision
  variables.
• Objective Function. e.g., 3x 4y
• Value measure used to rank alternatives
• Seek to maximize or minimize this objective
• examples maximize NPV, minimize cost

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MSR Marketing Inc.adapted from Frontline Systems

• Need to choose ads to reach at least 1.5 million
  people
• Minimize Cost
• Upper bound on number of ads of each type

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Formulating as a math model

• Work with your partner
• The decisions are how many ads of each type to
  choose. Let x1 be the number of TV ads selected.
  Let x2, x3, x4 denote the number of radio, mail,
  and newspaper ads. These are the decision
  variables.
• What is the objective? Express the objective in
  terms of the decision variables.
• What are the constraints? Express these in terms
  of the decision variables.
• If you have time, try to find the best solution.

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The MSR Marketing Problem
Minimize
500 x1 200 x2 250 x3 125 x4
subject to
50 x1 25 x2 20 x3 15 x4 ? 1,500
0 ? x1 ? 20
0 ? x2 ? 15
0 ? x3 ? 10
0 ? x4 ? 15
MSR Marketing
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Characteristics of Models

• Models are usually simplified versions of the
  things they represent
• A valid model accurately represents the relevant
  characteristics of the object or decision being
  studied

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Benefits of Modeling

• Economy - it is often less costly to analyze
  decision problems using models.
• Timeliness - models often deliver needed
  information more quickly than their real-world
  counterparts.
• Feasibility - models can be used to do things
  that would be impossible.
• Models give us insight understanding that
  improves decision making.

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Gemstone Tool Company (Thanks to Rob Freund)

• Privately-held firm
• Consumer and industrial market for construction
  tools
• Headquartered in Seattle
• Manufacturing plants in the US, Canada, and
  Mexico.
• Simplifying assumptions, for purposes of
  illustration
• Winnipeg, Canada plant
• Wrenches and pliers.
• Made from steel
• Injection molding machine
• Assembly machine

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Data for the GTC Problem
We want to determine the number of wrenches and
pliers to produce given the available raw
materials, machine hours and demand.
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To do with your partner

• Work with your partner to formulate the GTC
  problem as a linear program.
• Let P number of pliers made
• Let W number of wrenches made

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Formulating the GTC Problem

• Step 1 Determine Decision Variables
• W number of wrenches manufactured
• P number of pliers manufactured
• Step 2 Determine Objective Function
• Maximize Profit

.4 W .3 P
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The Formulation Continued

• Step 3 Determine Constraints

Steel
1.5 W P 15,000
Molding
W P 12,000
0.4 W 0.5 P 5,000
Assembly
0 W 8,000
Wrench Demand
Plier Demand
0 P 10,000
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Another example
A Modeling Example Eli Daisy produces the drug
Wozac in huge batches by heating a chemical
mixture in a pressurized container. Each time a
batch is produced, a different amount of Wozac is
produced. The amount produced is the process
yield (measured in pounds). Daisy is
interested in understanding the factors that
influence the yield of Wozac production process.
The solution on subsequent slides describes a
model building process for this situation.
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Another example

• Daisy is interested in determining the factors
  that influence the process yield. This would be
  referred to as a descriptive model since it
  describes the behavior of the actual yield as a
  function of various factors.
• Daisy might determine the following factors
  influence yield
• Container volume in liters (V)
• Container pressure in milliliters (P)
• Container pressure in degrees centigrade (T)
• Chemical composition of the processed mixture

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Another example
Letting A, B, and C be the percentage of the
mixture made up of chemical A, B, and C, then
Daisy might find , for example, that
Yield 300 0.8V 0.01P 0.06T 0.001TP -
0.01T2 0.001P2 11.7A 9.4B 16.4C 19AB
11.4AC 9.6BC
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Another example
To determine this relationship, the yield of the
process would have to measured for many different
combinations of the previously listed factors.
Knowledge of this equation would enable Daisy to
describe the yield of the production process once
volume, pressure, temperature, and chemical
composition were known.
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Another example

• Prescriptive or Optimization Models
• Prescriptive models prescribes behavior for an
  organization that will enables it to best meet
  its goals. Components of this model include
• objective function(s)
• decision variables
• constraints
• An optimization model seeks to find values of the
  decision variables that optimize (maximize or
  minimize) an objective function among the set of
  all values for the decision variables that
  satisfy the given constraints.

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The Objective Function The Daisy example seeks
maximize the yield for the production process.
In most models, there will be a function we wish
to maximize or minimize. This function is called
the models objective function. To maximize the
process yield we need to find the values of V, P,
T, A, B, and C that make the yield equation
(below) as large as possible.
Yield 300 0.8V 0.01P 0.06T 0.001TP -
0.01T2 0.001P2 11.7A 9.4B 16.4C 19AB
11.4AC 9.6BC
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The Decision Variables Variables whose values are
under our control and influence system
performance are called decision variables. In the
Daisy example, V, P, T, A, B, and C are decision
variables.
Constraints In most situations, only certain
values of the decision variables are possible.
For example, certain volume, pressure, and
temperature conditions might be unsafe. Also, A,
B, and C must be nonnegative numbers that sum to
one. These restrictions on the decision variable
values are called constraints.
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• Suppose the Daisy example has the following
  constraints
• Volume must be between 1 and 5 liters
• Pressure must be between 200 and 400 milliliters
• Temperature must be between 100 and 200 degrees
  centigrade
• Mixture must be made up entirely of A, B, and C
• For the drug to perform properly, only half the
  mixture at most can be product A.

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• Mathematically, these constraints can be
  expressed

V 5 V 1 P 400 P 200 T 200 T 100
A 0 B 0 C 0 A B C 1.0 A 0.5
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The Complete Daisy Optimization Model
Letting z represent the value of the objection
function (the yield), the entire optimization
model may be written as
maximize z 300 0.8V 0.01P 0.06T 0.001TP
- 0.01T2 0.001P2 11.7A 9.4B 16.4C
19AB 11.4AC 9.6BC
V 5 V 1 P 400 P 200
T 200 T 100 A B C 1.0 A 0.5
A 0 B 0 C 0
Subject to (s.t.)
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Daisy problem formulation in LINGO 7.0
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Solution extract of Daisy example (shown without
slack, surplus, or dual prices) using LINGO 7.0
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Static and Dynamic Models A static model is one
in which the decision variables do not involve
sequences of decisions over multiple periods. A
dynamic model is a model in which the decision
variables do involve sequences of decisions over
multiple periods. In a static model, we solve a
one shot problem whose solutions are prescribe
optimal values of the decision variables at all
points in time. The Daisy problem is an example
of a static model.
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Static and Dynamic Models For a dynamic model,
consider a company (SailCo) that must determine
how to minimize the cost of meeting (on-time) the
demand for sail boats it produces during the next
year. SailCo must determine the number of sail
boats to produce during each of the next four
quarters. SailCos decisions must be made over
multiple periods and thus posses a dynamic model.
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Linear and Nonlinear models Suppose that when
ever decision variables appear in the objective
function and in the constraints of an
optimization model the decision variables are
always multiplied by constants and then added
together. Such a model is a linear model. The
Daisy example is a nonlinear model. While the
decision variables in the constraints are linear,
the objective function is nonlinear since the
objective function terms 0.001TP, - 0.01T2,
0.001P2, 19AB, 11.4AC, and 9.6BC are
nonlinear. In general, nonlinear models are much
harder to solve.
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Integer and Noninteger Models If one or more of
the decision variables must be integer, then we
say that an optimization model is an integer
model. If all the decision variables are free to
assume fractional values, then an optimization
model is a noninteger model. The Daisy example
is a noninteger example since volume, pressure,
temperature, and percentage composition are all
decision variables which may assume fractional
values. If decision variables in a model
represent the number of workers starting during
each shift, then clearly we have a integer model.
Integer models are much harder to solve then
noninteger models.
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Deterministic and Stochastic Models Suppose that
for any value of the decision variables the value
of the objective function and whether or not the
constraints are satisfied is known with
certainty. We then have a deterministic model.
If this is not the case, then we have a
stochastic model. If we view the Daisy example
as a deterministic model, then we are making the
assumption that for given values of V, P, T, A,
B, and C the process yield will always be the
same. Since this is unlikely, the objective
function can be viewed as the average yield of
the process for given decision variable values.
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Addressing managerial problems A management
science framework

• 1. Determine the problem to be solved
• 2. Observe the system and gather data
• 3. Formulate a mathematical model of the problem
  and any important subproblems
• 4. Verify the model and use the model for
  prediction or analysis
• 5. Select a suitable alternative
• 6. Present the results to the organization
• 7. Implement and evaluate

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Some Success Stories

• Optimal crew scheduling saves American Airlines
  20 million/yr.
• Improved shipment routing saves Yellow Freight
  over 17.3 million/yr.
• Improved truck dispatching at Reynolds Metals
  improves on-time delivery and reduces freight
  cost by 7 million/yr.
• GTE local capacity expansion saves 30 million/yr.

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Other Success Stories (cont.)

• Optimizing global supply chains saves Digital
  Equipment over 300 million.
• Restructuring North America Operations, Proctor
  and Gamble reduces plants by 20, saving 200
  million/yr.
• Optimal traffic control of Hanshin Expressway in
  Osaka saves 17 million driver hours/yr.
• Better scheduling of hydro and thermal generating
  units saves southern company 140 million.

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Success Stories (cont.)

• Improved production planning at Sadia (Brazil)
  saves 50 million over three years.
• Production Optimization at Harris Corporation
  improves on-time deliveries from 75 to 90.
• Tata Steel (India) optimizes response to power
  shortage contributing 73 million.
• Optimizing police patrol officer scheduling saves
  police department 11 million/yr.
• Gasoline blending at Texaco results in saving of
  over 30 million/yr.