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OR IIOR IIGSLM 52800GSLM 52800
2
OutlineOutline
course outline
general OR approach
general forms of NLP
a list of NLP examples
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General General OROR Approach Approach
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Phases of Phases of OROR##
model construction (建模) model solution (解題)
model validity (驗證)
solution implementation (實施)
# Taha [2003] # Taha [2003] Operations ResearchOperations Research An Introduction, An Introduction, Prentice Hall, New JerseyPrentice Hall, New Jersey..
OR II
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Model ConstructionModel Construction (建模)
starting with defining xijk? No
starting with understanding the problem without any mathematics
who are the players in the system?
how do the players interact with each other?
what is (are) the objective(s)?
invoking mathematics only after understanding all the above
players = 持份者
相互作用
目標
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Model ConstructionModel Construction (建模)
to define a mathematical model variable xijk from players, functional relationships,
and logical relationships objective function(s) from the objective(s) constraints from functional relationships and
logical relationships equality constraints: gi(x) = bi
less than or equal to constraints: gi(x) bi
greater than or equal to constraints: gi(x) bi
目標函數
限制式
變量
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General Forms of General Forms of NLPNLP
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Non-Linear Programming (Non-Linear Programming (NLPNLP))
min f(x), s.t. gj(x) bj, j = 1, …, m, where
x = (x1, … , xn)T n: an n-dimensional vector f(x): the objective function g1(x), …, gm(x): functions of the constraints
f and gj: possibly non-linear, and assumed to be twice differentiable
bi: known constants
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2 2 and existj
i i
gf
x x
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Non-Linear Programming (Non-Linear Programming (NLPNLP))
for the above NLP how many decision variables are there? what is the value of m?
write out gj(x) for all j.
min f(x), s.t. gj(x) bj, j = 1, …, m,
2 3/2
2 2
max + ,
. .
+ 5,
+ 3,
0.
x y
s t
x y
x y
x
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Non-Linear Optimization (Non-Linear Optimization (NLPNLP))min f(x), s.t. gj(x) bj, j = 1, …, m,
2 3/2
2 2
max + ,
. .
+ 5,
+ 3,
0.
x y
s t
x y
x y
x
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Another Form Another Form of Non-Linear Optimization (of Non-Linear Optimization (NLPNLP))
min f(x),
s.t. gj(x) bj, j = 1, …, m,
xi 0, i = 1, …, n.
two forms being equivalent
no problem to model -constraints, =-constraints, and maximization
min f(x), s.t. gj(x) bj, j = 1, …, m,
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A List of ExamplesA List of Examples
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A List of ExamplesA List of Examples
1: 1: Non-Linear Profit
2: 2: Economic Order Quantity
3: 3: Non-linear Transportation Cost
4: 4: Portfolio Selection
5: 5: Location Selection
6: 6: Engineering Design
7: 7: System Reliability
8: 8: Routing in a Queueing Network Network
9: Line Fitting 9: Line Fitting
10: Electrical Circuit10: Electrical Circuit
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Example 1: Example 1: Non-Linear ProfitNon-Linear Profit
Suppose that the cost of making a unit is
and the demand for the unit
selling price p is p > 0. What is
the price to maximize the profit?
,2 p
,3 / p
Return
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Example 2: Example 2: Economic Order QuantityEconomic Order Quantity
Facing a demand of per unit time, a buyer places an
order of quantity Q every Q/ time units, and costs $K for
each order placed. Whenever a unit is kept in inventory,
the buyer spends $h per unit time. Find the best order
quantity for the buyer by balancing the long-run order
setup cost against the long-run inventory holding cost.
Assume that the replenishment lead time is zero and there
is no integer restriction on the order quantity.
Return
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Example 3: Example 3: Non-linear Transportation CostNon-linear Transportation Cost
Consider the context of Example 2 when K = $140, h = $1,
and = 70/unit time. Suppose that in addition there is a
transportation cost, which is $5/unit for the first 120 units,
$3/unit for the next 60 units, and $1/unit for quantity over
180 units; the transportation takes constant time.
Determine the new economic order quantity.
Return
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Example 4: Example 4: Portfolio SelectionPortfolio Selection
In investment, one would like to maximize his
(expected) profit and minimize his risk, subject
to his budget constraint. The modern portfolio
theory says that the profits of assets are
interrelated, and the risk of investment can be
measured by the variation of a portfolio.
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Example 4: Example 4: Portfolio Selection Portfolio Selection
Consider a collection of n assets. Let pj be the price
of asset j; j and jj be the mean and the variance of
return on a unit of asset j, respectively; ij be the
covariance of return on one unit of asset i and asset
j. How should we invest if we want to minimize the
risk with $B for investment, aiming at earning at
least $L?Return
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Example 5: Example 5: Location SelectionLocation Selection
Retail outlets A, B, and C are located at (2, 2), (3, 4), and
(6, 2), respectively. The annual quantities of goods
transported from a depot to outlets A, B, and C are 3, 2,
and 5 units, respectively. (a). Determine the location of
the depot that minimizes the total distance between the
depot and the outlets. (b). Determine the location of the
depot that minimizes the total goods-distance between
the depot and the outlets. Return
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Example 6: Example 6: Engineering DesignEngineering Design
(a). Determine the dimensions of a rectangular box
of volume 1,000 cm3 such that its total surface area
is minimized.
(b). Suppose that costs of the top and the bottom
plates of a rectangular box are three times of the
side plates. Determine the dimensions of the box
that minimize the total cost of the box.
Return
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Example 7: Example 7: System ReliabilitySystem Reliability
We are going to decide the most reliable configuration of
a system, where the reliability of a system (a component)
is the probability that the system (the component) works.
The system puts three types of components in series
such that each type can have a number of backup units
to increase the reliability provided by the type of
component, and hence the overall reliability of the
system.
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Example 7: Example 7: System ReliabilitySystem Reliability
The cost of the system is no more than $1,000K and its weight no
more than 300 g. The details of each type of components are
given below. Assume that the conditions of working or not of
components are independent.
Component Cost/unit Weight/unit Reliability
1 $50K 20 g 0.9
2 $20K 40 g 0.8
3 $100K 15 g 0.85
Return
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Example 8: Example 8: Routing in a Queueing NetworkRouting in a Queueing Network
For an M/M/1 station of arrival rate and service rate
(> ), the (stationary) expected number in station =
/(1), where 0 < = / < 1. For a stable Jackson
network of M/M/1 stations, the expression for the
stationary number in system holds for the stations, as
long as the total arrival rate to the station remains
lower than the service rate of the station. These
relationships can be used to determine the optimal
routing in such a stable Jackson network.
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Example 8: Example 8: Routing in a Queueing NetworkRouting in a Queueing Network
Consider the above stable Jackson network formed by 4
M/M/1 stations. Suppose that we can control the routing
of parts in the system. Determine the optimal values of
B, C, and D that minimize the expected total number
in system. Return
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Example 9: Line FittingExample 9: Line Fitting
The relationship between 3 independent variables x1, x2, x3 and
the dependent variable y should be linear in nature, i.e., y = b0 +
b1x1 + b2x2 + b3x3 for some unknown parameters. Suppose we
have the following set of 8 data points. Define the deviation dj of
the jth point by yj – b1x1j – b2x2j – b3x3j; e.g., d1 =
10923b12b219b3. Find the best fitted line if the objective
function is to minimize the sum of the pth power of the
deviations.
Return
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Example 10: Electrical CircuitExample 10: Electrical Circuit
Suppose that the electrical current in the LHS circuit
is given by 1000 = I(20+R). As electrical current I
passes through the cell and the resistors, certain
substances are generated, of quantities 1000I for the
cell, and 20I2 and I2R for the 20 and R resistors,
respectively. Set R to minimize the total amount of
substance generated.
Return