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OR II GSLM 52800. Outline. course outline general OR approach general forms of NLP a list of NLP examples. General OR Approach. Phases of OR #. model construction (建模) model solution (解題) model validity ( 驗證 ) solution implementation (實施). OR II.

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outline
Outline
  • course outline
  • general OR approach
  • general forms of NLP
  • a list of NLP examples

2

phases of or
Phases of OR#
  • model construction (建模)
  • model solution (解題)
  • model validity (驗證)
  • solution implementation (實施)

OR II

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# Taha [2003] Operations Research An Introduction, Prentice Hall, New Jersey.

model construction
Model Construction(建模)
  • starting with defining xijk? No
  • starting with understanding the problem withoutany 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 construction1
Model Construction(建模)
  • to define a mathematical model
    • variable xijkfrom 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

變量

目標函數

限制式

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non linear programming nlp
Non-Linear Programming (NLP)
  • 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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non linear programming nlp1
Non-Linear Programming (NLP)

min f(x),

s.t. gj(x) bj, j = 1, …, m,

  • for the above NLP
    • how many decision variables are there?
    • what is the value of m?
    • write out gj(x) for all j.

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non linear optimization nlp
Non-Linear Optimization (NLP)

min f(x),

s.t. gj(x) bj, j = 1, …, m,

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another form of non linear optimization nlp
Another Form of Non-Linear Optimization (NLP)
  • 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 examples1
A List of Examples
  • 1: Non-Linear Profit
  • 2: Economic Order Quantity
  • 3: Non-linear Transportation Cost
  • 4: Portfolio Selection
  • 5: Location Selection
  • 6: Engineering Design
  • 7: System Reliability
  • 8: Routing in a Queueing Network
  • 9: Line Fitting
  • 10: Electrical Circuit

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example 1 non linear profit
Example 1: Non-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?

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example 2 economic order quantity
Example 2: Economic 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.

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example 3 non linear transportation cost
Example 3: Non-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.

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example 4 portfolio selection
Example 4: Portfolio 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 portfolio selection1
Example 4: 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?

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example 5 location selection
Example 5: Location 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.

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example 6 engineering design
Example 6: Engineering 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.

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example 7 system reliability
Example 7: System 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 system reliability1
Example 7: System 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.

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example 8 routing in a queueing network
Example 8: Routing 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 routing in a queueing network1
Example 8: Routing 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.

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example 9 line fitting
Example 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 = 10923b12b219b3. Find the best fitted line if the objective function is to minimize the sum of the pth power of the deviations.

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example 10 electrical circuit
Example 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.

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