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OR IIOR IIGSLM 52800GSLM 52800

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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