1 OR II GSLM 52800

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

3 General OR Approach

4 Phases of OR #  model construction （建模）  model solution （解題）  model validity （驗證）  solution implementation （實施） # Taha [2003] Operations Research  An Introduction, Prentice Hall, New Jersey. OR II

5 Model Construction Model Construction （建模）  starting with defining x ijk ? 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

6 Model Construction Model Construction （建模）  to define a mathematical model  variable x ijk from players, functional relationships, and logical relationships  objective function(s) from the objective(s)  constraints from functional relationships and logical relationships  equality constraints: g i (x) = b i  less than or equal to constraints: g i (x)  b i  greater than or equal to constraints: g i (x)  b i 目標函數 限制式 變量

7 General Forms of NLP

8 Non-Linear Programming (NLP)  min f(x),  s.t.g j (x)  b j, j = 1, …, m,  where  x = (x 1, …, x n ) T   n : an n-dimensional vector  f(x): the objective function  g 1 (x), …, g m (x): functions of the constraints  f and g j : possibly non-linear, and assumed to be twice differentiable  b i : known constants

9 Non-Linear Programming (NLP)  for the above NLP  how many decision variables are there?  what is the value of m?  write out g j (x) for all j. min f(x), s.t. g j (x)  b j, j = 1, …, m,

10 Non-Linear Optimization (NLP) min f(x), s.t. g j (x)  b j, j = 1, …, m,

11 Another Form of Non-Linear Optimization (NLP)  min f(x),  s.t.g j (x)  b j, j = 1, …, m,  x i  0,i = 1, …, n.  two forms being equivalent  no problem to model  -constraints, =-constraints, and maximization min f(x), s.t. g j (x)  b j, j = 1, …, m,

12 A List of Examples

13 A List of Examples  1: Non-Linear Profit Non-Linear ProfitNon-Linear Profit  2: Economic Order Quantity Economic Order QuantityEconomic Order Quantity  3: Non-linear Transportation Cost Non-linear Transportation CostNon-linear Transportation Cost  4: Portfolio Selection Portfolio SelectionPortfolio Selection  5: Location Selection Location SelectionLocation Selection  6: Engineering Design Engineering DesignEngineering Design  7: System Reliability System ReliabilitySystem Reliability  8: Routing in a Queueing Network Routing in a Queueing NetworkRouting in a Queueing Network  9: Line Fitting Line FittingLine Fitting  10: Electrical Circuit Electrical CircuitElectrical Circuit

14 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? Return

15 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. Return

16 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. Return

17 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.

18 Example 4: Portfolio Selection  Consider a collection of n assets. Let p j 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

19 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. Return

20 Example 6: Engineering Design  (a). Determine the dimensions of a rectangular box of volume 1,000 cm 3 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

21 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.

22 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. ComponentCost/unitWeight/unitReliability 1$50K20 g0.9 2$20K40 g0.8 3$100K15 g0.85 Return

23 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.

24 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. Return

25 Example 9: Line Fitting  The relationship between 3 independent variables x 1, x 2, x 3 and the dependent variable y should be linear in nature, i.e., y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 for some unknown parameters. Suppose we have the following set of 8 data points. Define the deviation d j of the jth point by y j – b 1 x 1j – b 2 x 2j – b 3 x 3j ; e.g., d 1 = 109  23b 1  2b 2  19b 3. Find the best fitted line if the objective function is to minimize the sum of the pth power of the deviations. Return

26 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 20I 2 and I 2 R for the 20  and R  resistors, respectively. Set R to minimize the total amount of substance generated. Return