Introduction to Operation Research by gko95621

VIEWS: 103 PAGES: 54

Introduction to Operation Research document sample

More Info
									Continental
   Airlines




                  Introduction to Operations
                          Research
                      Statistical Concepts, Optimization,
                            Heuristics, Simulation,
                                and Forecasting

                                      Judy Pastor
                                      Steven Coy




2/22/2011 Operations Research Group
Continental
   Airlines                                             Sum and Product Notation

                             A sum of a row or column of n numbers
          n
           xi                        Ex. X = (1, 3, 4, 9)
           i                  4
                              xi          = 13  4  9 = 16
                              i


      n            m
                             An iterated sum: Sums a matrix of
      xi  y j                numbers having n rows and j columns
      i            j

                             A sequential product of n numbers
           n
           xi                        Ex. X = (1, 3, 4, 9)
               i              4
                              xi         = 13  4  9 = 108
                              i


2/22/2011 Operations Research Group                                                2
Continental
   Airlines                                              Convex and concave sets




                                      Convex Set
                                      The set of all the points that are bounded by this curve




                                      Concave Set
                                      The set of all points that are bound by this curve




2/22/2011 Operations Research Group                                                              3
Continental
   Airlines                                            Probability Concepts


       • Experiment
             – A repeatable procedure
             – Has a well defined set of possible outcomes
       • Sample outcome
             – Potential result of an experiment, denoted e
       • Sample space
             – The set of all possible outcomes, denoted S
       • Event
             – A subset of the sample space corresponding to the definition of the
               event, denoted E




2/22/2011 Operations Research Group                                                  4
Continental
   Airlines                                           Probability of an Event


        • Probability is the degree of chance or likelihood that
          and event will occur in an experiment
        • Calculating the probability for a discrete or countable
          problem
              1. Find the sum of possible outcomes that satisfy the definition of the
                 event
              2. Find the sum of the total number of possible outcomes
              3. Divide the result in 1 by the result in 2
        • In mathematical notation
              – P(E) = e  {E} /  e  {S}
              – P(E)  P(S) = 1




2/22/2011 Operations Research Group                                                     5
Continental
   Airlines                                                   Example


     Experiment: Pick a pax from the passenger database

     Sample Space: 5 million total pax; 1 million pax are
                   frequent fliers
     Event:                   Passenger is a frequent flier



                      P(FreqFlyer) = 1MM/5MM = .20 or 20%
                                       Note: P(S) = 1




2/22/2011 Operations Research Group                                     6
Continental
   Airlines                           Union and Intersection of Two Events

    • Intersection
          – The sum of the sample outcomes of two or more events that are common
            to all of these events
          – A B
          – Typically identified by the word “and” as in A and B

    • Union
          – The sum of the sample outcomes of two or more events
          – AB = A + B - A  B
          – Typically identified by the word “or” as in A or B

    • Probability of the Union of Two Events
          – P(AB) = P(A) + P( B) - P( A  B )



2/22/2011 Operations Research Group                                                7
Continental
   Airlines                                                             Example
    Experiment: Pick a pax at random from the passenger
                database

    Sample Space: 5 million total pax; 1 million pax are
                  frequent fliers; 2.7 million pax are female;
                 600 thousand frequent fliers are female

    Event:                  Passenger is a female or a frequent flier


                      P(Female) = 2.7MM/5MM = 54%
                             P(FreqFlyer) = 20%
                P(Female and FreqFlyer) = 0.6MM/5MM = 12%

             P(Female or FreqFlyer) = 54% + 20% - 12% = 62%
2/22/2011 Operations Research Group                                               8
Continental
   Airlines                              Conditional Probability


    • Conditional probability is the probability of an event, A,
      given that a related event, B, has already occurred
    • P(A|B) = P(A  B)/P(B)
    • Conditional probability effectively reduces the size of the
      sample space




2/22/2011 Operations Research Group                                 9
Continental
   Airlines                                                              Example

       Experiment: Pick a passenger at random from the passenger
                   database

       Sample Space: 5 million total pax; 1 million pax are
                     frequent fliers; 2.7 million pax are female;
                    600 thousand frequent fliers are female

       Event:                  Passenger is a frequent flier given that the
                               pax is female

                P(Female and FreqFlyer) = 0.6MM/5MM = 12%
                              P(Female) = 54%

                      P(FreqFlyer| Female) = 0.12/0.54 = 22.2%

2/22/2011 Operations Research Group                                                10
Continental
   Airlines                                 Calculating Expected Value

    • Expectation is a long-run weighted average
    • Example
          – What is the expected return from running a revenue integrity process?
          – The process, which searches for duplicate bookings and expired TTLs,
            costs $0.1 for every record that the process scans.
          – For each duplicate reservation found, we receive $100 in incremental
            revenue and for each expired TTL, we receive $25.
          – We know that the long-run probability that a reservation will have a dupe
            is P(D) = 0.15% and that a TTL is expired is P(TE) = 0.1%
          – If we use the process to scan 100 K reservations, what is the expected
            return?
    • This requires an expected value computation

                   E(R) = P(D) * $100 + P(TE) * $25 - $0.10
                E(R) = 0.15% * $100 + 0.1% * 25 - $0.10 = $0.175
                      E(R/100 K) = $0.05 * 100 K = $5000
2/22/2011 Operations Research Group                                                     11
Continental
   Airlines                                              Random Variables


   • Random Variables (RV) are characteristics or outcomes that
     vary from observation to observation
   • Independence of two RVs
         – two RVs are independent if the outcome of one does not effect the
           outcome of another => P(A|B) = P(A)
   • Correlation of two random variables
         – Two RVs are correlated if the knowledge of the outcome of one gives us
           an indication of the outcome of the other
         – Positive: X moves with Y
         – Negative: as X increases, Y decreases




2/22/2011 Operations Research Group                                                 12
Continental
   Airlines                                             Probability Distribution of a RV


      Consider the unconstrained demand for a leisure class ticket on Flt 102:


 Let’s compile the
 demand for each
 departure of Flt 102
 for a full year and
 create a frequency
 histogram.

                                                                                                         Notice that the
                                                                                                         histogram is
                                                                                                         mound-shaped and
                                                                                                         approximates a
                                                                                                         familiar bell-
                                                                                                         shaped curve.
                                   11

                                        13

                                             16

                                                  19

                                                       22

                                                            25

                                                                 27

                                                                      30

                                                                           33

                                                                                36

                                                                                     39

                                                                                          41

                                                                                               44

                                                                                                    47
                               8




2/22/2011 Operations Research Group                                                                                         13
Continental
   Airlines                Probability Distribution of a Continuous RV

   • The bell-shaped curve that we saw on the last slide is a
     normal density curve
   • Using this chart, we could argue that demand for this flight is
     normally distributed
   • Probability calculations with a normal distribution
         – Example: What is the probability that demand will be less than or equal to
           35 pax?
         – First, we standardize the curve--transform it so that the area under the
           curve is equal to 1 (we use a z-transform)
         – We then find the area under the curve that satisfies the definition of our
           event (the interval 0 to 35)
         – The area under the curve from 0 to 35 = P(D  35)  .84




2/22/2011 Operations Research Group                                                     14
Continental
   Airlines                                          Calculating a Normal Probability
 To find the probability, we find the interval on
 the horizontal axis and calculate the area under
 the curve corrsponding to that interval

                      0.50
                      0.45
                      0.40
                      0.35
                      0.30
              f (x)




                      0.25
                      0.20
                      0.15
                      0.10
                      0.05                           P(D<X)=A
                      0.00
                             -5

                                    -4

                                           -3

                                                  -2

                                                         -1

                                                                    0.

                                                                          1.

                                                                                2.

                                                                                      3.

                                                                                            4.

                                                                                                  5.
                               .0

                                      .0

                                             .0

                                                    .0

                                                           .0

                                                                      0

                                                                            0

                                                                                  0

                                                                                        0

                                                                                              0

                                                                                                    0
                                                                x




2/22/2011 Operations Research Group                                                                     15
Continental
   Airlines                                                                   More About Distributions

    • Cumulative distribution
          – In our demand example, we found a probability for a single value of x
          – A cumulative distribution gives us the probability of D  x for any
            value of x



                                               1
                                             0.9
                                             0.8
                                             0.7
                        probability D <= x




                                             0.6
                                             0.5
                                             0.4
                                             0.3
                                             0.2
                                             0.1
                                               0                                                        Cumulative
                                                                                                        Normal
                                                   0

                                                       5

                                                           10

                                                                15

                                                                     20

                                                                          25

                                                                               30

                                                                                    35

                                                                                         40

                                                                                              45

                                                                                                   50
                                                                          x                             distribution


2/22/2011 Operations Research Group                                                                                    16
Continental
   Airlines                                              Truncated Normal Distribution


       Demand processes that are described by a normal
       distribution are truncated at 0--demand cannot be negative

                      0.50
                      0.45
                      0.40
                      0.35
                      0.30
                      0.25
              f (x)




                      0.20
                      0.15
                      0.10
                      0.05
                      0.00
                             -5

                                    -4

                                           -3

                                                  -2

                                                         -1

                                                                0.

                                                                      1.

                                                                            2.

                                                                                  3.

                                                                                        4.

                                                                                              5.
                                .

                                       .

                                              .

                                                     .

                                                            .

                                                                  0

                                                                        0

                                                                              0

                                                                                    0

                                                                                          0

                                                                                                0
                               0

                                      0

                                             0

                                                    0

                                                           0




                                                                x



2/22/2011 Operations Research Group                                                                 17
Continental
   Airlines                               Some Common Distributions

    • Poisson: used to describe the arrival pattern (or process) of
      people to a system
          – For example, 5 people request a certain fair class product per hour
          – Mean = variance
    • Exponential: used to describe service times
          – If a process generates poisson arrivals, the generating process is
            exponential
          – Has special properties that makes it a favorite for queueing systems
            (waiting lines) and reliability
    • Uniform (a favorite, because it is easy): the probability is
      the same throughout
          – ExampleU(10, 20): P(x > 15 ) = P(10 < x < 15)




2/22/2011 Operations Research Group                                                18
Continental
   Airlines                                     Statistical Measures

    • Central tendency
          – Mean = average
          – Median = middle value in a sorted list
          – Mode = largest value or highest portion of a probability
            density curve
    • Error measures
          – e = x - prediction of x
          – MAD(MAE): Mean absolute deviation (error):  |e| / n
          – MAPE: Mean absolute percentage error  |e| /x / n




2/22/2011 Operations Research Group                                    19
Continental
   Airlines
                                                         More Measures

       • Variance: Measures the dispersion (spread) of the
         observations
       • Standard deviation: The square root of the variance
       • Coefficient of variation: The standard deviation divided
         by the mean--stated as a percentage
       • Skew: Measures the asymmetry of a distribution
             – mean > median: Positive or right-skewed

             – mean < median: Negative or left-skewed
       • Correlation: Statistical measure of the relationship
         between two variables from -1, perfect negative
         correlation to 1, perfect positive correlation


2/22/2011 Operations Research Group                                      20
Continental
   Airlines                                           Truncated Normal Distribution

    • In RM, the tails of the normal demand curve typically
      extend beyond the capacity of the plane
    • This is why we use unconstraining (detruncation)
      algorithms to approximate the tail of the curve

                            0.02




                            0.01
                    f (x)




                            0.00                                                                         Cap = 125
                                   0




                                                               10

                                                                      12

                                                                             14

                                                                                    16

                                                                                           18

                                                                                                  20
                                       20

                                            40

                                                 60

                                                      80


                                                                  0

                                                                         0

                                                                                0

                                                                                       0

                                                                                              0

                                                                                                     0
                                                           x



2/22/2011 Operations Research Group                                                                                  21
Continental
   Airlines                                           Operations Research

    • “application of mathematical techniques, models, and tools to
      a problem within a system to yield the optimal solution”
    • Phases of an OR Project
          –   formulate the problem
          –   develop math model to represent the system
          –   solve and derive solution from model
          –   test/validate model and solution
          –   establish controls over the solution
          –   put the solution to work




2/22/2011 Operations Research Group                                         22
Continental
   Airlines                           Linear Programs – A Major Tool of OR

    • Linear Programs (LPs) are a special type of mathematical
      model where all relationships between parts of the system
      being modeled can be represented linearly (a straight line).
    • Not always realistic, but we know how to solve LPs.
    • May need to approximate a relationship that is slightly non-
      linear with a linear one.
    • When to use: if a problem has too many dimensions and
      alternative solutions to evaluate all manually, use an LP to
      evaluate.




2/22/2011 Operations Research Group                                          23
Continental
   Airlines                           Linear Programs – A Major Tool of OR

    • LPs can evaluate thousands, millions, etc. of different
      alternatives to find the one that best meets the objective of
      the business problem.
          – Fleet Assignment Model - assign aircraft to flight legs to minimize cost
            and maximize revenue
          – Revenue Management - set bid prices to maximize revenue and/or
            minimize spill
          – Crew Scheduling - schedule crew members to minimize number of crew
            needed and maximize utilization




2/22/2011 Operations Research Group                                                    24
Continental
   Airlines                           Linear Program Formulation

    • Understand the system and environment to which the
      problem belongs
    • Understand the problem and the objective to be achieved
    • State the model - clear idea of problem and what can and can
      not be included in the model
    • Collect Data - get data/parameters/constraints and
      boundaries of system and interrelationships
    • Determine decisions - define decision variables - what do we
      need the model to tell us?
    • Formulate and solve model




2/22/2011 Operations Research Group                                  25
Continental
   Airlines                           Example: RM Network LP


    • Problem - how many passengers of each itinerary and fare
      class should be accepted on each flight to achieve the
      maximum revenue for the flight network?
    • Statement - the model should tell us the above
    • Data - demand by itinerary/fare class, aircraft capacity,
      overbooking levels, expected revenue by itinerary/fare class
    • Decisions - how many passengers of each itinerary/fare class
      to accept on each flight leg




2/22/2011 Operations Research Group                                  26
Continental
   Airlines                Example: RM Network LP Data Collection

    •    Two Flights: SFO-IAH, IAH-AUS
    •    Two fare classes: Y-high fare, Q-low fare
    •    Three itineraries: SFOIAH, IAHAUS, SFOAUS
    •    Six fares:
                                             Fares
                              Market    Y            Q
                             SFOIAH    400           300
                             IAHAUS    250           100
                             SFOAUS    450           320


    • Flight capacity: SFO-IAH 124, IAH-AUS 94
    • No overbooking

2/22/2011 Operations Research Group                                 27
Continental
   Airlines                Example: RM Network LP Data Collection




                                           Demand
                           Market     Y             Q
                          SFOIAH      30            90
                         IAHAUS       50            30
                         SFOAUS       20            50




2/22/2011 Operations Research Group                                 28
Continental
   Airlines                           Example: RM Network LP Formulation Model

    • Data Definition:
          – F               set of flights = {SFOIAH, IAHAUS}
                • f                     index of F (1,2)
          –   CAPf          capacity of flightf = {124, 94 }
          –   I             set of itineraries {SFOIAH, IAHAUS, SFOAUS}
          –   i             index of I (1,2,3)
          –   IFf           set of itineraries over flight f
                • IF1={SFOIAH,SFOAUS} IF2={IAHAUS,SFOAUS}
          – Cset of classes {Y, Q}
                • c                     index of C (1,2)
          – DMDi,c          demand for itinerary i and class c
          – FAREi,c         fare for itinerary i and class c




2/22/2011 Operations Research Group                                              29
Continental
   Airlines                           Example: RM Network LP Formulation Model

    • Now that we have defined all the data that we know about
      the model, we now must define what we want the model to
      tell us.
    • Problem - how many passengers of each itinerary and fare
      class should be accepted on each flight to achieve the
      maximum revenue for the flight network?
    • Define decision variables:
          – Xi,c            # pax accepted for itinerary i and class c
    • There are 3 itineraries and 2 classes so there are a total of 6
      decision variables.




2/22/2011 Operations Research Group                                              30
Continental
   Airlines                           Example: RM Network LP Formulation Model

    • Many sets of values (collectively called solutions) for the six
      Xi,c variables exist which could satisfy the constraints
      (formulation coming) of aircraft capacity and maximum
      demand. These are “feasible” solutions.
    • Which solution do we want?
    • Problem - how many passengers of each itinerary and fare
      class should be accepted on each flight to achieve the
      maximum revenue for the flight network?
    • The feasible solution for this is “optimal”.




2/22/2011 Operations Research Group                                              31
Continental
   Airlines                    Example: RM Network LP Objective Function




                     MAX  fare i ,c  X i ,c
                                      i   c




2/22/2011 Operations Research Group                                   32
Continental
   Airlines                    Example: RM Network LP Obj. Function & Constraints


    • The Objective Function is an expression that defines the
      optimal solution, out of the many feasible solutions. We can
      either
          – MAXimize - usually used with revenue or profit or
          – MINimize - usually used with costs
    • Feasible solutions must satisfy the constraints of the problem.
      LPs are used to allocate scarce resources in the best possible
      manner. Constraints define the scarcity.
    • The scarcity in this problem involves a fixed number of seats
      and scarce high paying customers.




2/22/2011 Operations Research Group                                            33
Continental
   Airlines                                  Example: RM Network LP Constraints


         Capacity Constraints :
               X
                 c iIFf
                                      i ,c   CAP f for each f in F

          Demand Constraints :
              X i ,c  DMDi ,c for each itinerary i,
                                                 fare class c

2/22/2011 Operations Research Group                                          34
Continental
   Airlines                           Example: RM Network LP Constraints

    • Rules for Constraints
          – must be a linear expression
          – decision variables can be summed together but not multiplied or divided
            by each other
          – have relational operators of =, <=, or >=
          – must be continuous
    • Constraints define the “feasible region” - all points within the
      feasible region satisfy the constraints.
    • The feasible region is convex.
    • The optimal solution lies at an extreme point of the feasible
      region.




2/22/2011 Operations Research Group                                                   35
Continental
   Airlines                       Example: RM Network LP Cplex Input File

   MAX
      400 X_SFOIAH_Y + 300 X_SFOIAH_Q +
      250 X_IAHAUS_Y + 100 X_IAHAUS_Q +
      450 X_SFOAUS_Y + 320 X_SFOAUS_Q
   ST
   CAPY_SFOIAH: X_SFOIAH_Y + X_SFOIAH_Q + X_SFOAUS_Y + X_SFOAUS_Q <= 124
   CAPY_IAHAUS: X_SFOAUS_Y + X_SFOAUS_Q + X_IAHAUS_Y + X_IAHAUS_Q <=94
   DMD_SFOIAH_Y: X_SFOIAH_Y <= 30
   DMD_SFOIAH_Q: X_SFOIAH_Q <= 90
   DMD_IAHAUS_Y: X_IAHAUS_Y <= 50
   DMD_IAHAUS_Q: X_IAHAUS_Q <= 30
   DMD_SFOAUS_Y: X_SFOAUS_Y <= 20
   DMD_SFOAUS_Q: X_SFOAUS_Q <= 50
   END




2/22/2011 Operations Research Group                                         36
Continental
   Airlines                  Example: RM Network LPSolution - Constraints
    SECTION 1 - ROWS

     NUMBER   ......ROW....... AT   ...ACTIVITY...   SLACK ACTIVITY   ..LOWER LIMIT.   ..UPPER LIMIT.   .DUAL ACTIVITY

          1   obj              BS            58100           -58100             NONE             NONE                1
          2   CAPY_SFOIAH      UL              124                0             NONE              124             -300
          3   CAPY_IAHAUS      UL               94                0             NONE               94             -100
          4   DMD_SFOIAH_Y     UL               30                0             NONE               30             -100
          5   DMD_SFOIAH_Q     BS               74               16             NONE               90                0
          6   DMD_IAHAUS_Y     UL               50                0             NONE               50             -150
          7   DMD_IAHAUS_Q     BS               24                6             NONE               30                0
          8   DMD_SFOAUS_Y     UL               20                0             NONE               20              -50
          9   DMD_SFOAUS_Q     BS                0               50             NONE               50                0


   obj is the objective function value - total revenue from the small network of flights
   CAPY_SFOIAH and CAPY_IAHAUS are the capacity constraints. Both are at UL -
       upper limit with activities of 124 and 94, respectively (i.e. both flight legs are full).
   Dual Activity on each capacity constraint is also known as the Shadow Price of the
       flight. The SP of SFOIAH is 300 and the SP of IAHAUS is 100. In RM terms, this
       means that the value of one more seat on SFOIAH is 300 and the value of one
       more seat on IAHAUS is 100. Alternately, 300 and 100 also define the lowest fare
       that should be accepted on each leg.
   DMD_{SFOIAH,IAHAUS}_{Y,Q} are the demand constraints. SFOIAH_Y,
       IAHAUS_Y, and SFOAUS_Y are at upper level (i.e. accept all Y passengers).
       Reject some/all of Q.

2/22/2011 Operations Research Group                                                                                      37
Continental
   Airlines                   Example: RM Network LP Solution - Decision Variables
 SECTION 2 - COLUMNS

   NUMBER   .....COLUMN..... AT   ...ACTIVITY...   ..INPUT COST..   ..LOWER LIMIT.   ..UPPER LIMIT.   .REDUCED COST.

       10   X_SFOIAH_Y       BS               30              400                0             NONE                0
       11   X_SFOIAH_Q       BS               74              300                0             NONE                0
       12   X_IAHAUS_Y       BS               50              250                0             NONE                0
       13   X_IAHAUS_Q       BS               24              100                0             NONE                0
       14   X_SFOAUS_Y       BS               20              450                0             NONE                0
       15   X_SFOAUS_Q       LL                0              320                0             NONE              -80



 This section of the solution report shows the values for the decision variables at
   the optimal solution.
 The LP tells us to accept 30 SFOIAH Y, 74 SFOIAH Q (reject 16), accept 50
   IAHAUS Y, accept 24 IAHAUS Q (reject 6), accept 20 SFOAUS Y, and
   accept no SFOAUS Q.
 Note that the LP cut off all SFOAUS Q booking requests because their fare of
   320 is less than the sum of the shadow prices of the two flights (300+100 =
   400 > 320).




2/22/2011 Operations Research Group                                                                                    38
Continental
   Airlines                           Example: RM Network LP Solving LPs

    • A problem that sounds small, like our example, can balloon
      out into many decision variables and constraints.
    • Computer software is available to solve linear programs.
    • Cost of programs depends on size of problems to be solved.
    • Excel has an Add-in to solve small LPs.
    • CPLEX is state of the art, but more expensive.
    • LPs with 100,000s row and columns can be solved.




2/22/2011 Operations Research Group                                        39
Continental
   Airlines                           Example: RM Network LP Solving LPs

    • The first method of solving LPs was invented during WWII
      by George Dantzig. The algorithm is called SIMPLEX. It is
      based on convexity theory and that the optimal solution will
      occur at an extreme point of the solution space
    • Newer state of the art algorithms are based on steepest
      descent gradient methods and are called “interior point”
      methods
    • Interior point methods can be extremely fast (much faster
      than SIMPLEX) for certain structures of problems




2/22/2011 Operations Research Group                                        40
Continental
   Airlines                                         Degeneracy

    • When an LP has more than one unique way to reach an
      optimal objective function value, we say that the problem is
      “degenerate”
    • LP solvers can detect degeneracy but only report one solution
    • It would be nice to see all possible solutions
    • Different solvers can “land” on different solutions of a
      degenerate problem, depending on solution strategy
    • The RM Network problem is usually degenerate




2/22/2011 Operations Research Group                                   41
Continental
   Airlines                           Other Types of Linear Optimizations

    • MIP (Mixed Integer Programming)
          – is similar to LP but at least one decision variable is required to be a integer
            value
          – violates the LP rule that decision variables be continuous
          – is solved by “branch and bound” - solving a series of LPs that fix the
            integer decision variables to various integer values and comparing the
            resulting objective function values
          – is done in a smart way to avoid enumerating all possibilities
          – is useful, since you can not have .3 of an aircraft




2/22/2011 Operations Research Group                                                           42
Continental
   Airlines                           Other Types of Linear Optimization

    • Network problem
          – is a special form of LP which turns out to be “naturally integer”
          – can be solved faster than an LP, using a special network optimization
            algorithm
          – is very restrictive on types of constraints that can be present in the problem
    • Shortest Path
          – finds the shortest path from the source (start) to sink (end) nodes, along
            connecting arcs, each having a cost associated with them
          – is used in many applications




2/22/2011 Operations Research Group                                                          43
Continental
   Airlines                                   Other Optimization Models

    • Quadratic Program
          – has a quadratic objective function with linear constraints
          – can be applied to revenue management, because it allows fare to rise with
            demand within a problem
                • price(OD) = 50 + [5*numpax(OD)]
                • max revenue = price * numpax




2/22/2011 Operations Research Group                                                     44
Continental
   Airlines                                   Other Optimization Models

   • Non-linear Program (NLP)
         – can have either non-linear objective function or non-linear constraints or
           both
         – feasible region is generally not convex
         – much more difficult to solve
         – but it is worth our time to learn to solve them since world is actually non-
           linear most of the time
         – some non-linear programs can be solved with LPs or MIPs using
           piecewise linear functions




2/22/2011 Operations Research Group                                                       45
Continental
   Airlines                             Deterministic versus Stochastic

    • Two broad categories of optimization models exist
          – deterministic
                • parameters/data known with certainty
          – stochastic
                • parameters/data know with uncertainty
    • Deterministic models are easier to solve. Our RM LP is
      deterministic (we pretend we know the demand with
      certainty).
    • Stochastic model are difficult to solve. In reality, we know a
      distribution about our demand. We get around this in real
      life by re-optimizing.




2/22/2011 Operations Research Group                                       46
Continental
   Airlines                           Deterministic versus Stochastic

    • Deterministic optimization ignores risk of being wrong about
      parameter/data estimates.
    • No commercial software packages are currently available to
      do generalized, stochastic optimization.




2/22/2011 Operations Research Group                                     47
Continental
   Airlines                                             Heuristics

    • Definition - educated guess
    • When you use a heuristic to solve a problem, you have a gut
      feeling that it is a pretty good solution, but can not prove it
      mathematically
    • You can not prove that there is not a better solution out there
    • To qualify as an “optimal” solution, there must be a
      mathematical proof to say that no better solution exists




2/22/2011 Operations Research Group                                     48
Continental
   Airlines                                Types of Heuristics

    • “Greedy Algorithms” - also called “myopic” - nearsighted
      solutions.
    • Example: in our RM Network LP, the greedy solution would
      be to take the highest fare passengers possible on each leg,
      without looking at the consequences of doing so on the
      connecting leg. So the greedy solution is to take the SFOAUS
      Q passengers at a fare of $320. But the optimal solution
      looks at displacement and says do not take any SFOAUS Q
      passengers.




2/22/2011 Operations Research Group                                  49
Continental
   Airlines                                             Heuristics

    • Combinatorial problems can grow exponentially when the
      number of decisions needed to be made grows linearly.
      Heuristics can be used in these cases to get a good solution in
      a reasonable amount of time.
    • TSP - Travelling Salesman Problem is a good example of this.
    • EMSR is a heuristic. It is provably optimal for two fare
      classes, but not more. However, it gives a good answer in a
      finite amount of time and takes probability into account.




2/22/2011 Operations Research Group                                     50
Continental
   Airlines                                             Simulation

    • When a problem is too complicated to be put into an LP or a
      solvable non-linear optimization, one way to study the
      problem is to simulate it under different conditions.
    • PODS (Passenger O & D Simulation) is one example.
    • Simulation can tell us something about a set of parameters
      (i.e. total revenue, load factor), but does not point us in the
      direction of an improvement.




2/22/2011 Operations Research Group                                     51
Continental
   Airlines                                           Forecasting

    • Types of forecasting techniques used in RM:
          – pick-up
          – booking regression
          – exponential smoothing
    • Pick-up - adds average future bookings from historical
      observations to bookings on hand.
    • Booking Regression - computes best fit for history of
      bookings on hand (independent) to final booked (dependent)
          – final booked = a + b*(bookings on hand)




2/22/2011 Operations Research Group                                 52
Continental
   Airlines                                                                  Forecasting

    • Exponential Smoothing - similar to pick-up except the
      average is weighted. The most recent historical observations
      are weighted most heavily, decreasing for earlier
      observations.
          – recursive relationship
                                                        2
                • Avg Pick Up = a * (pick upt-1) + (1-a) * (pick upt-2) + ...
                • boils down to AvgPU 
                    – Avg Pick Up = a * (pick upt-1) + (1-a) * (last fcst pick up)
    • Problem is how to estimate a. 0<=a<=1
    • How much weight should be on most recent observation?




2/22/2011 Operations Research Group                                                        53
Continental
   Airlines
                                      Questions?




2/22/2011 Operations Research Group                54

								
To top