An Improved Dynamic Programming Decomposition Approach for Network Revenue Management
|
|
- Leo Mason
- 7 years ago
- Views:
Transcription
1 An Improved Dynamic Programming Decomposition Approach for Network Revenue Management Dan Zhang Leeds School of Business University of Colorado at Boulder May 21, 2012
2 Outline Background Network revenue management formulation Classical dynamic programming decomposition An improved dynamic programming decomposition Numerical results Summary
3 Network RM Formulation (Gallego and van Ryzin, 1997; Gallego et. al., 2004; Liu and van Ryzin, 2008) m resources with capacity c (an m-vector) Capacity for resource i is c i. n products N = {1,..., n} Fare for product j is f j Product consumption matrix A = [a ij ] Finite time horizon with length τ In each period, there is one customer arrival with probability λ, and no customer arrival with probability 1 λ. Given a set of products S N, a customer chooses product j with probability P j (S). No-purchase probability P 0 (S) = 1 j S P j (S). Objective: Maximize expected total revenue
4 Applications Industry Resources Products Airlines Scheduled flights O-D itineraries at certain fare levels Hotels Room-days Single(multi)-day stays at certain rates Car rentals Car-days Single(multi)-day rentals at certain rates Air Cargo Scheduled flights (weight) O-D shipments at certain rates Scheduled flights (volume)
5 Dynamic Programming Formulation DP optimality equations: v t (x) = v τ+1 (x) = 0, max S N(x) { λ j S P j (S)(f j + v t+1 (x A j )) } + (λp 0 (S) + 1 λ)v t+1 (x), t, x, x. Notations v t (x): DP value function A j : resource incidence vector of product j N(x): {j N : x A j }
6 Dynamic Programming Formulation DP optimality equations: v t (x) = v τ+1 (x) = 0, max S N(x) { λ j S P j (S)(f j + v t+1 (x A j )) } + (λp 0 (S) + 1 λ)v t+1 (x), t, x, x. Notations v t (x): DP value function A j : resource incidence vector of product j N(x): {j N : x A j } Curse of dimensionality: state space grows exponentially with the number of resources
7 Choice-based Deterministic Linear Program (CDLP) z CDLP = max h S N λr(s)h(s) S N λq(s)h(s) c, h(s) τ, S N (Resource constraint) (Time constraint) h(s) 0, S N. (Non-negativity) Replace stochastic demand with deterministic fluid with rate λ Given offer set S N Total time S is offered: h(s) Revenue from unit demand: R(S) = j S f jp j (S) Consumption of resource i from unit demand: Q i (S) = j S a ijp j (S)
8 CDLP (Gallego et. al, 2004; Liu and van Ryzin, 2008) CDLP can by efficiently solved for certain class of choice models. The vector of dual values π associated with resource constraints can be used as bid-prices for resources z CDLP provides an upper bound on revenue Some recent references: Talluri (2010): Concave programming formulation Gallego, Ratliff, Shebalov (2011): Efficient reformulation
9 Classical Dynamic Programming Decomposition For each i, approximate the DP value function with v t(x) v t,i (x i ) + πk x k, } {{ } k i Value of the } {{ } i-th resource Value of all other resources t, x.
10 Classical Dynamic Programming Decomposition For each i, approximate the DP value function with v t(x) v t,i (x i ) + πk x k, } {{ } k i Value of the } {{ } i-th resource Value of all other resources t, x. Using the approximation in DP recursion leads to ( v t,i (x i ) = max λp j (S) f j ) a kj πk +v t+1,i (x i a ij ) S N(x i,c i ) j S k i } {{ } Fare proration + (λp 0(S) + 1 λ)v t+1(x i ), t, x i.
11 Classical Dynamic Programming Decomposition For each i, approximate the DP value function with v t(x) v t,i (x i ) + πk x k, } {{ } k i Value of the } {{ } i-th resource Value of all other resources t, x. Using the approximation in DP recursion leads to ( v t,i (x i ) = max λp j (S) f j ) a kj πk +v t+1,i (x i a ij ) S N(x i,c i ) j S k i } {{ } Fare proration + (λp 0(S) + 1 λ)v t+1(x i ), t, x i. Compute offer sets dynamically using the approximate value function v t(x) i v t,i (x i ), t, i.
12 Classical Dynamic Programming Decomposition A DP with m-dimensional state space is reduced to m one-dimensional DPs, one for each resource states (Assume 100 seats per flight) states
13 Classical Dynamic Programming Decomposition A DP with m-dimensional state space is reduced to m one-dimensional DPs, one for each resource states (Assume 100 seats per flight) states Variants of the approach are widely used in practice. Review: Talluri and van Ryzin (2004a)
14 DP Decomposition Bounds Proposition (Zhang and Adelman, 2009) The following relationships hold: (i) v t (x) min l=1,...,m {v t,l (x l ) + } k l π k x k v t,i (x i ) + k i π k x k, i, t, x; (ii) v 1 (c) v 1,i (c i ) + k i π k c k z CDLP, i. Decomposition value for each leg provides an upper bound on revenue Decomposition bounds are tighter than the bound from CDLP
15 Linear Programming Formulation of DP (Adelman, 2007) min v 1 (c) {v t( )} t v t (x) j S λp j (S)(f j + v t+1 (x A j )) + (λp 0 (S) + 1 λ)v t+1 (x), t, x, S N(x).
16 Linear Programming Formulation of DP (Adelman, 2007) min v 1 (c) {v t( )} t v t (x) j S λp j (S)(f j + v t+1 (x A j )) + (λp 0 (S) + 1 λ)v t+1 (x), t, x, S N(x). Huge number of decision variables and constraints
17 Linear Programming Formulation of DP (Adelman, 2007) min v 1 (c) {v t( )} t v t (x) j S λp j (S)(f j + v t+1 (x A j )) + (λp 0 (S) + 1 λ)v t+1 (x), t, x, S N(x). Huge number of decision variables and constraints Functional approximation idea: use a parameterized representation of the value function to reduce the number of decision variables
18 The Affine Functional Approximation (Zhang and Adelman, 2009) Affine approximation is given by v t (x) θ t + i V t,i x i, t, x. (1)
19 The Affine Functional Approximation (Zhang and Adelman, 2009) Affine approximation is given by v t (x) θ t + i V t,i x i, t, x. (1) Using (1) in the linear programming formulation leads to min θ,v θ1 + i V 1,i c i ( θ t + V t,i x i λp j (S) f j + θ t+1 + V t+1,i (x i a ij ) i j S i ( + (λp 0(S) + 1 λ) θ t+1 + ) V t+1,i x i, t, x, S N(x). i )
20 The Affine Functional Approximation The dual program is given by z P1 = max λp j (S)f j Y t,x,s Y t,x,s N(x) j S { c i, if t = 1, x i Y t,x,s = x,s N(x) (x i ) j S λp j (S)a ij Y t 1,x,S, t = 2,..., τ x,s N(x) { Y t,x,s = 1, if t = 1, x,s N(x) Y t 1,x,S, t = 2,..., τ. x,s N(x) Y 0. i, t,
21 The Affine Functional Approximation The dual program is given by z P1 = max λp j (S)f j Y t,x,s Y t,x,s N(x) j S { c i, if t = 1, x i Y t,x,s = x,s N(x) (x i ) j S λp j (S)a ij Y t 1,x,S, t = 2,..., τ x,s N(x) { Y t,x,s = 1, if t = 1, x,s N(x) Y t 1,x,S, t = 2,..., τ. x,s N(x) Y 0. i, t, Due to the large number of columns, solving the linear program above still requires considerable computational effort.
22 Functional Approximation Approaches for Network RM Citation Choice Model Functional approximation Solution strategy Adelman (2007) Independent demand Affine Column generation Zhang and Adelman (2009) MNLD Affine Column generation Zhang (2011) MNLD Nonlinear non-separable CDLP+Simultaneous DP Liu and van Ryzin (2008) MNLD Separable (fare proration) CDLP+DP Decomposition Miranda Bront et. al. (2009) MNLO Separable (fare proration) CDLP+DP Decomposition Farias and Van Roy (2008) Independent demand Separable concave Constraint sampling Meissner and Strauss (2012) MNLD Separable concave Column generation Kunnumkal and MNLD Separable (fare proration) Convex programming Topaloglu (2011) +DP Decomposition Tong and Topaloglu (2011) Independent demand Affine Reduction + Constraint generation Vossen and Zhang Independent demand Affine Reduction + MNLD + Dynamic disaggregation MNLD: Multinomial logit model with disjoint consideration sets MNLO: Multinomial logit model with overlapping consideration sets
23 Research Questions Computational cost: ADP (affine or separable concave approximation) classical DP decomposition
24 Research Questions Computational cost: ADP (affine or separable concave approximation) classical DP decomposition How can we balance solution quality with solution time? Can we improve the classical DP decomposition?
25 A Strong Functional Approximation (Zhang, 2011) v t (x) min ˆv t,i(x i ) + πk x k, t, x. i k i Nonlinear and non-separable functional approximation Each value v t (x) is approximated by a single value across legs Motivated by the decomposition bounds (Zhang and Adelman, 2009)
26 A Nonlinear Optimization Problem Using the new functional approximation leads to z NLP = min min ˆv 1,i(c i ) + πk c k ˆv t,i ( ) t,i i k i min ˆv t,i(x i ) + πk x k i k i λp j (S) f j + min ˆv t+1,l(x l a lj ) + πk (x k a kj ) l j S k l + (λp 0(S) + 1 λ)min ˆv t+1,l(x l ) + πk x k l, t, x, S N(x). k l The problem is a nonlinear optimization problem with a huge number of nonlinear constraints.
27 A Restricted Optimization Problem Step 1: Writing each constraint as m equivalent constraints Step 2: Restricting the constraints so that each constraint only involves one resource The restricted problem provides a relaxed bound: Proposition The objective value of the restricted program, z NLP, is bigger than z NLP.
28 An Equivalent Simultaneous Dynamic Program ˆv t,i(x i ) = max S N(x i,c i ) { min l i ( { λp j (S) f j + min ˆv t+1,i(x i a ij ) πk a kj, j S k i max [ˆv t+1,l(y l ) y l πl ] } }) a kj πk + πi x i 0 y l c l a lj k + (λp 0(S) + 1 λ) min i, t, x i. { ˆv t+1,i(x i ), min l i { max 0 y l c l [ˆv t+1,l(y l ) π l y l ] + π i x i DP recursion for resource i involves values from all other resources }}
29 An Equivalent Simultaneous Dynamic Program ˆv t,i(x i ) = max S N(x i,c i ) { min l i ( { λp j (S) f j + min ˆv t+1,i(x i a ij ) πk a kj, j S k i max [ˆv t+1,l(y l ) y l πl ] } }) a kj πk + πi x i 0 y l c l a lj k + (λp 0(S) + 1 λ) min i, t, x i. { ˆv t+1,i(x i ), min l i { max 0 y l c l [ˆv t+1,l(y l ) π l y l ] + π i x i DP recursion for resource i involves values from all other resources The dynamic program is equivalent to the restricted nonlinear program can be solved efficiently via a simultaneous dynamic programming algorithm leads to tighter revenue bounds }}
30 New Bounds Proposition (Zhang, 2011) Let {ˆv t,i ( )} t,i,x i be the optimal solution from the simultaneous dynamic program. The following results hold: (i) ˆv t,i (x i) v t,i (x i ), i, x i ; (ii) v 1 (c) z NLP z NLP = min i {ˆv 1,i (c i) + } k i π k c k min i {v 1,i (c i ) + } k i π k c k z CDLP. The simultaneous dynamic program provides tighter bounds on revenue than the classical decomposition.
31 Recap High dimensional dynamic program
32 Recap High dimensional dynamic program Large scale linear program
33 Recap High dimensional dynamic program Large scale linear program Large scale nonlinear program with nonlinear constraints
34 Recap High dimensional dynamic program Large scale linear program Large scale nonlinear program with nonlinear constraints Restricted nonlinear program with nonlinear constraints
35 Recap High dimensional dynamic program Large scale linear program Large scale nonlinear program with nonlinear constraints Restricted nonlinear program with nonlinear constraints Simultaneous dynamic program
36 Comparison: Classical vs. Improved Approaches Classical dynamic programming decomposition: Solve m single-leg DPs Prorated fares Fare proration Static bid-prices Solve CDLP
37 Comparison: Classical vs. Improved Approaches Classical dynamic programming decomposition: Solve m single-leg DPs Prorated fares Fare proration Static bid-prices Solve CDLP Network effects only captured through fare proration
38 Comparison: Classical vs. Improved Approaches Classical dynamic programming decomposition: Solve m single-leg DPs Prorated fares Fare proration Static bid-prices Solve CDLP Network effects only captured through fare proration Improved dynamic programming decomposition: Solve one simultaneous DP Static bid-prices Solve CDLP
39 Comparison: Classical vs. Improved Approaches Classical dynamic programming decomposition: Solve m single-leg DPs Prorated fares Fare proration Static bid-prices Solve CDLP Network effects only captured through fare proration Improved dynamic programming decomposition: Solve one simultaneous DP Static bid-prices Solve CDLP Network effects captured during DP recursion!
40 Computational Study: Problem Instances Randomly generated hub-and-spoke instances Number of non-hub locations (flights) in the set {4, 8, 16, 24} Number of periods in the set {100, 200, 400, 800} Two products for each possible itinerary Multinomial Logit Choice Model with Disjoint Consideration Sets (MNLD) Largest problem instance: 24 non-hub locations (flights), 336 products, 800 periods
41 Numerical Study: Policies DCOMP1: the new decomposition approach where the approximation m v t (x) ˆv t,i(x i ), t, x i=1 is used to compute control policies. DCOMP: the classical dynamic programming decomposition CDLP: static bid-price policy based on the dual values of resource constraints in CDLP CDLP10: A version of CDLP that resolves 10 times with equally spaced resolving intervals Each policy is simulated times
42 Computational Time Case # Parameters Capacity Load CPU seconds DCOMP1 DCOMP per leg factor CDLP DCOMP DCOMP1 DCOMP A1 (100,4,4,16) % A2 (200,4,4,16) % A3 (400,4,4,16) % A4 (800,4,4,16) % A5 (100,8,8,48) % A6 (200,8,8,48) % A7 (400,8,8,48) % A8 (800,8,8,48) % A9 (100,16,16,160) % A10 (200,16,16,160) % A11 (400,16,16,160) % A12 (800,16,16,160) % A13 (100,24,24,336) % A14 (200,24,24,336) % A15 (400,24,24,336) % A16 (800,24,24,336) %
43 Bound Performance Case # CDLP DCOMP DCOMP1 Bound improvement %-difference across legs bound bound bound %-CDLP %-DCOMP DCOMP DCOMP1 A % 4.74% 4.46% 0.00% A % 1.55% 1.87% 0.36% A % 0.98% 2.36% 0.00% A % 0.36% 0.58% 0.00% A % 0.99% 3.18% 0.05% A % 0.21% 2.18% 0.22% A % 0.18% 1.09% 0.00% A % 0.03% 1.49% 0.00% A % 0.76% 4.69% 0.00% A % 0.64% 2.95% 0.03% A % 0.30% 1.52% 0.01% A % 0.15% 0.94% 0.00% A % 14.05% 10.37% 0.00% A % 2.27% 5.50% 0.00% A % 0.66% 2.80% 0.03% A % 0.24% 1.44% 0.00%
44 Bounds from Individual Legs Bounds from individual Legs DCOMP DCOMP Leg
45 Bounds from Individual Legs Bounds from individual Legs DCOMP DCOMP Leg DCOMP1 bounds are more homogeneous across legs
46 A Hub-and-spoke Network with 2 Non-Hub Locations Case # τ Load Capacity DCOMP1 DCOMP1 Revenue Gains OPT-GAP factor per leg REV %-CDLP %-CDLP10 %-DCOMP B % % 8.56% 2.74% B % % 5.74% 4.11% B % -1.00% -0.33% 0.03% B % % 4.41% 7.17% B % 48.88% 3.00% 0.09% B % 13.42% 3.62% 5.77% B % % 4.78% 7.97% B % 42.16% -4.49% 0.00% B % 13.27% 1.33% 1.94% B % 15.05% 0.98% 4.65% B % 15.46% 1.70% 8.14% B % 15.47% 2.45% 5.52% B % 2.89% 0.47% 0.34% B % 4.84% 1.14% 0.11% B % 22.11% 2.22% 0.04% B % 7.92% 2.14% 0.01% B % 23.82% 1.46% 0.18% B % 2.61% 1.24% 0.04% B % 30.16% 2.48% 0.01% B % 31.46% 2.60% 0.00%
47 DCOMP1 Percentage Revenue Gain vs. Load Factor DCOMP1 percentage revenue gain % CDLP10 % DCOMP Load factor
48 DCOMP1 Percentage Revenue Gain vs. Load Factor DCOMP1 percentage revenue gain % CDLP10 % DCOMP Load factor Higher load factor Higher revenue gains
49 DCOMP1 Percentage Revenue Gain vs. Number of Periods DCOMP1 percentage revenue gain % CDLP10 % DCOMP Number of periods
50 DCOMP1 Percentage Revenue Gain vs. Number of Periods DCOMP1 percentage revenue gain % CDLP10 % DCOMP Number of periods Significant revenue gains for problems with long selling horizons!
51 A Hub-and-spoke Network with 4 Non-Hub Locations Case # τ Load Capacity DCOMP1 DCOMP1 Revenue Gains OPT-GAP factor per leg REV %-CDLP %-CDLP10 %-DCOMP C % 14.60% 0.46% 0.10% C % 52.79% 1.95% 1.32% C % 12.35% -0.11% 0.03% C % 52.31% 1.36% 0.66% C % 44.57% 1.71% 1.96% C % 14.18% 1.72% 2.67% C % 29.01% -2.27% -0.81% C % 1.02% 0.49% 0.18% C % 4.98% 1.46% 2.67% C % 4.18% 1.61% 4.11% C % 3.44% 1.51% 4.43% C % 57.07% 1.92% 3.08% C % 1.30% -0.90% 0.21% C % 2.18% 0.21% 0.08% C % 6.38% 0.74% 0.00% C % 2.66% 0.89% -0.06% C % 2.50% -0.81% 0.44% C % 3.05% -0.03% 0.20% C % 3.24% 0.26% 0.05% C % 3.21% 0.41% 0.02%
52 DCOMP1 Percentage Revenue Gain vs. Load Factor DCOMP1 percentage revenue gain % CDLP10 % DCOMP Load factor
53 DCOMP1 Percentage Revenue Gain vs. Number of Periods DCOMP1 percentage revenue gain % CDLP10 % DCOMP Number of periods
54 Summary and Future Directions Functional approximation approach is promising for solving large scale stochastic dynamic programs. However, implementations of the approach often require very high computational cost. The first nonlinear non-separable functional approximation for network RM problem Novel approximation architecture Better revenue bounds Improved heuristic policies Moderate computational cost Current work Exploiting special structures of the LP formulations of dynamic programs in value function approximation (Vossen and Zhang, 2012) Applications with real data (Zhang and Weatherford, 2012)
55 Thank you! Questions? Comments?
Role of Stochastic Optimization in Revenue Management. Huseyin Topaloglu School of Operations Research and Information Engineering Cornell University
Role of Stochastic Optimization in Revenue Management Huseyin Topaloglu School of Operations Research and Information Engineering Cornell University Revenue Management Revenue management involves making
More informationRandomization Approaches for Network Revenue Management with Customer Choice Behavior
Randomization Approaches for Network Revenue Management with Customer Choice Behavior Sumit Kunnumkal Indian School of Business, Gachibowli, Hyderabad, 500032, India sumit kunnumkal@isb.edu March 9, 2011
More informationA Randomized Linear Programming Method for Network Revenue Management with Product-Specific No-Shows
A Randomized Linear Programming Method for Network Revenue Management with Product-Specific No-Shows Sumit Kunnumkal Indian School of Business, Gachibowli, Hyderabad, 500032, India sumit kunnumkal@isb.edu
More informationUpgrades, Upsells and Pricing in Revenue Management
Submitted to Management Science manuscript Upgrades, Upsells and Pricing in Revenue Management Guillermo Gallego IEOR Department, Columbia University, New York, NY 10027, gmg2@columbia.edu Catalina Stefanescu
More informationRe-Solving Stochastic Programming Models for Airline Revenue Management
Re-Solving Stochastic Programming Models for Airline Revenue Management Lijian Chen Department of Industrial, Welding and Systems Engineering The Ohio State University Columbus, OH 43210 chen.855@osu.edu
More informationAppointment Scheduling under Patient Preference and No-Show Behavior
Appointment Scheduling under Patient Preference and No-Show Behavior Jacob Feldman School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853 jbf232@cornell.edu Nan
More informationRevenue Management for Transportation Problems
Revenue Management for Transportation Problems Francesca Guerriero Giovanna Miglionico Filomena Olivito Department of Electronic Informatics and Systems, University of Calabria Via P. Bucci, 87036 Rende
More informationA Lagrangian relaxation approach for network inventory control of stochastic revenue management with perishable commodities
Journal of the Operational Research Society (2006), 1--9 2006 Operational Research Society Ltd. All rights reserved. 0160-5682/06 $30.00 www.palgrave-journals.com/jors A Lagrangian relaxation approach
More informationA Statistical Modeling Approach to Airline Revenue. Management
A Statistical Modeling Approach to Airline Revenue Management Sheela Siddappa 1, Dirk Günther 2, Jay M. Rosenberger 1, Victoria C. P. Chen 1, 1 Department of Industrial and Manufacturing Systems Engineering
More informationCargo Capacity Management with Allotments and Spot Market Demand
Submitted to Operations Research manuscript OPRE-2008-08-420.R3 Cargo Capacity Management with Allotments and Spot Market Demand Yuri Levin and Mikhail Nediak School of Business, Queen s University, Kingston,
More informationA central problem in network revenue management
A Randomized Linear Programming Method for Computing Network Bid Prices KALYAN TALLURI Universitat Pompeu Fabra, Barcelona, Spain GARRETT VAN RYZIN Columbia University, New York, New York We analyze a
More informationPrinciples of demand management Airline yield management Determining the booking limits. » A simple problem» Stochastic gradients for general problems
Demand Management Principles of demand management Airline yield management Determining the booking limits» A simple problem» Stochastic gradients for general problems Principles of demand management Issues:»
More informationAirline Revenue Management: An Overview of OR Techniques 1982-2001
Airline Revenue Management: An Overview of OR Techniques 1982-2001 Kevin Pak * Nanda Piersma January, 2002 Econometric Institute Report EI 2002-03 Abstract With the increasing interest in decision support
More informationDiscrete Optimization
Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using
More informationA General Attraction Model and an Efficient Formulation for the Network Revenue Management Problem *** Working Paper ***
A General Attraction Model and an Efficient Formulation for the Network Revenue Management Problem *** Working Paper *** Guillermo Gallego Richard Ratliff Sergey Shebalov updated June 30, 2011 Abstract
More informationOnline Network Revenue Management using Thompson Sampling
Online Network Revenue Management using Thompson Sampling Kris Johnson Ferreira David Simchi-Levi He Wang Working Paper 16-031 Online Network Revenue Management using Thompson Sampling Kris Johnson Ferreira
More informationDynamics of Network Programming Decomposition
econstor www.econstor.eu Der Open-Access-Publikationsserver der ZBW Leibniz-Informationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Gönsch,
More informationTwo-Stage Stochastic Linear Programs
Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic
More informationRevenue Management Through Dynamic Cross Selling in E-Commerce Retailing
OPERATIONS RESEARCH Vol. 54, No. 5, September October 006, pp. 893 913 issn 0030-364X eissn 156-5463 06 5405 0893 informs doi 10.187/opre.1060.096 006 INFORMS Revenue Management Through Dynamic Cross Selling
More informationCyber-Security Analysis of State Estimators in Power Systems
Cyber-Security Analysis of State Estimators in Electric Power Systems André Teixeira 1, Saurabh Amin 2, Henrik Sandberg 1, Karl H. Johansson 1, and Shankar Sastry 2 ACCESS Linnaeus Centre, KTH-Royal Institute
More informationEfficient and Robust Allocation Algorithms in Clouds under Memory Constraints
Efficient and Robust Allocation Algorithms in Clouds under Memory Constraints Olivier Beaumont,, Paul Renaud-Goud Inria & University of Bordeaux Bordeaux, France 9th Scheduling for Large Scale Systems
More informationDecentralization and Private Information with Mutual Organizations
Decentralization and Private Information with Mutual Organizations Edward C. Prescott and Adam Blandin Arizona State University 09 April 2014 1 Motivation Invisible hand works in standard environments
More informationOptimization of Mixed Fare Structures: Theory and Applications Received (in revised form): 7th April 2009
Original Article Optimization of Mixed Fare Structures: Theory and Applications Received (in revised form): 7th April 29 Thomas Fiig is Chief Scientist in the Revenue Management Development department
More informationLecture 10 Scheduling 1
Lecture 10 Scheduling 1 Transportation Models -1- large variety of models due to the many modes of transportation roads railroad shipping airlines as a consequence different type of equipment and resources
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no
More informationLinear Programming in Matrix Form
Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,
More informationSimulation in Revenue Management. Christine Currie christine.currie@soton.ac.uk
Simulation in Revenue Management Christine Currie christine.currie@soton.ac.uk Introduction Associate Professor of Operational Research at the University of Southampton ~ 10 years experience in RM and
More informationLecture 11: 0-1 Quadratic Program and Lower Bounds
Lecture : - Quadratic Program and Lower Bounds (3 units) Outline Problem formulations Reformulation: Linearization & continuous relaxation Branch & Bound Method framework Simple bounds, LP bound and semidefinite
More informationHETEROGENEOUS AGENTS AND AGGREGATE UNCERTAINTY. Daniel Harenberg daniel.harenberg@gmx.de. University of Mannheim. Econ 714, 28.11.
COMPUTING EQUILIBRIUM WITH HETEROGENEOUS AGENTS AND AGGREGATE UNCERTAINTY (BASED ON KRUEGER AND KUBLER, 2004) Daniel Harenberg daniel.harenberg@gmx.de University of Mannheim Econ 714, 28.11.06 Daniel Harenberg
More informationRevenue Management & Insurance Cycle J-B Crozet, FIA, CFA
J-B Crozet, FIA, CFA Abstract: This paper investigates how an insurer s pricing strategy can be adapted to respond to market conditions, and in particular the insurance cycle. For this purpose, we explore
More informationHow To Optimize Online Ads
Yield Optimization of Display Advertising with Ad Exchange Santiago R. Balseiro Fuqua School of Business, Duke University, 100 Fuqua Drive, NC 27708, srb43@duke.edu Jon Feldman, Vahab Mirrokni, S. Muthukrishnan
More informationMathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
More informationA Log-Robust Optimization Approach to Portfolio Management
A Log-Robust Optimization Approach to Portfolio Management Dr. Aurélie Thiele Lehigh University Joint work with Ban Kawas Research partially supported by the National Science Foundation Grant CMMI-0757983
More informationDynamic Capacity Management with General Upgrading
Submitted to Operations Research manuscript (Please, provide the mansucript number!) Dynamic Capacity Management with General Upgrading Yueshan Yu Olin Business School, Washington University in St. Louis,
More informationOptimization under uncertainty: modeling and solution methods
Optimization under uncertainty: modeling and solution methods Paolo Brandimarte Dipartimento di Scienze Matematiche Politecnico di Torino e-mail: paolo.brandimarte@polito.it URL: http://staff.polito.it/paolo.brandimarte
More informationTOWARDS AN EFFICIENT DECISION POLICY FOR CLOUD SERVICE PROVIDERS
Association for Information Systems AIS Electronic Library (AISeL) ICIS 2010 Proceedings International Conference on Information Systems (ICIS) 1-1-2010 TOWARDS AN EFFICIENT DECISION POLICY FOR CLOUD SERVICE
More informationDecentralized control of stochastic multi-agent service system. J. George Shanthikumar Purdue University
Decentralized control of stochastic multi-agent service system J. George Shanthikumar Purdue University Joint work with Huaning Cai, Peng Li and Andrew Lim 1 Many problems can be thought of as stochastic
More informationPreliminary Draft. January 2006. Abstract
Assortment Planning and Inventory Management Under Dynamic Stockout-based Substitution Preliminary Draft Dorothée Honhon, Vishal Gaur, Sridhar Seshadri January 2006 Abstract We consider the problem of
More informationScheduling Home Health Care with Separating Benders Cuts in Decision Diagrams
Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams André Ciré University of Toronto John Hooker Carnegie Mellon University INFORMS 2014 Home Health Care Home health care delivery
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More informationPurposeful underestimation of demands for the airline seat allocation with incomplete information
34 Int. J. Revenue Management, Vol. 8, No. 1, 2014 Purposeful underestimation of demands for the airline seat allocation with incomplete information Lijian Chen* School of Business Administration, Department
More informationRobust Airline Schedule Planning: Minimizing Propagated Delay in an Integrated Routing and Crewing Framework
Robust Airline Schedule Planning: Minimizing Propagated Delay in an Integrated Routing and Crewing Framework Michelle Dunbar, Gary Froyland School of Mathematics and Statistics, University of New South
More informationCHAPTER 9. Integer Programming
CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral
More informationIEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2
IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3
More informationModels in Transportation. Tim Nieberg
Models in Transportation Tim Nieberg Transportation Models large variety of models due to the many modes of transportation roads railroad shipping airlines as a consequence different type of equipment
More informationREVENUE MANAGEMENT: MODELS AND METHODS
Proceedings of the 2008 Winter Simulation Conference S. J. Mason, R. R. Hill, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. REVENUE MANAGEMENT: MODELS AND METHODS Kalyan T. Talluri ICREA and Universitat
More informationNonlinear Optimization: Algorithms 3: Interior-point methods
Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,
More information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
More informationBranch-and-Price Approach to the Vehicle Routing Problem with Time Windows
TECHNISCHE UNIVERSITEIT EINDHOVEN Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows Lloyd A. Fasting May 2014 Supervisors: dr. M. Firat dr.ir. M.A.A. Boon J. van Twist MSc. Contents
More informationSimulating the Multiple Time-Period Arrival in Yield Management
Simulating the Multiple Time-Period Arrival in Yield Management P.K.Suri #1, Rakesh Kumar #2, Pardeep Kumar Mittal #3 #1 Dean(R&D), Chairman & Professor(CSE/IT/MCA), H.C.T.M., Kaithal(Haryana), India #2
More informationManaging Revenue from Television Advertising Sales
Managing Revenue from Television Advertising ales Dana G. Popescu Department of Technology and Operations Management, INEAD ridhar eshadri Department of Information, Risk and Operations Management, University
More informationBig Data Optimization at SAS
Big Data Optimization at SAS Imre Pólik et al. SAS Institute Cary, NC, USA Edinburgh, 2013 Outline 1 Optimization at SAS 2 Big Data Optimization at SAS The SAS HPA architecture Support vector machines
More informationDantzig-Wolfe bound and Dantzig-Wolfe cookbook
Dantzig-Wolfe bound and Dantzig-Wolfe cookbook thst@man.dtu.dk DTU-Management Technical University of Denmark 1 Outline LP strength of the Dantzig-Wolfe The exercise from last week... The Dantzig-Wolfe
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More information36106 Managerial Decision Modeling Revenue Management
36106 Managerial Decision Modeling Revenue Management Kipp Martin University of Chicago Booth School of Business October 5, 2015 Reading and Excel Files 2 Reading (Powell and Baker): Section 9.5 Appendix
More informationSpatial Decomposition/Coordination Methods for Stochastic Optimal Control Problems. Practical aspects and theoretical questions
Spatial Decomposition/Coordination Methods for Stochastic Optimal Control Problems Practical aspects and theoretical questions P. Carpentier, J-Ph. Chancelier, M. De Lara, V. Leclère École des Ponts ParisTech
More informationOptimization Modeling for Mining Engineers
Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2
More information6.231 Dynamic Programming and Stochastic Control Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 6.231 Dynamic Programming and Stochastic Control Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 6.231
More informationUnit 1. Today I am going to discuss about Transportation problem. First question that comes in our mind is what is a transportation problem?
Unit 1 Lesson 14: Transportation Models Learning Objective : What is a Transportation Problem? How can we convert a transportation problem into a linear programming problem? How to form a Transportation
More informationKeywords: Beta distribution, Genetic algorithm, Normal distribution, Uniform distribution, Yield management.
Volume 3, Issue 9, September 2013 ISSN: 2277 128X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com Simulating
More informationSIMULATING CANCELLATIONS AND OVERBOOKING IN YIELD MANAGEMENT
CHAPTER 8 SIMULATING CANCELLATIONS AND OVERBOOKING IN YIELD MANAGEMENT In YM, one of the major problems in maximizing revenue is the number of cancellations. In industries implementing YM this is very
More informationMonte Carlo Simulation
1 Monte Carlo Simulation Stefan Weber Leibniz Universität Hannover email: sweber@stochastik.uni-hannover.de web: www.stochastik.uni-hannover.de/ sweber Monte Carlo Simulation 2 Quantifying and Hedging
More informationTransportation Polytopes: a Twenty year Update
Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,
More informationOffline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
More informationMaximum Utility Product Pricing Models and Algorithms Based on Reservation Prices
Maximum Utility Product Pricing Models and Algorithms Based on Reservation Prices R. Shioda L. Tunçel T. G. J. Myklebust April 15, 2007 Abstract We consider a revenue management model for pricing a product
More informationDemand Forecasting in a Railway Revenue Management System
Powered by TCPDF (www.tcpdf.org) Demand Forecasting in a Railway Revenue Management System Economics Master's thesis Valtteri Helve 2015 Department of Economics Aalto University School of Business Aalto
More informationMIT ICAT. Airline Revenue Management: Flight Leg and Network Optimization. 1.201 Transportation Systems Analysis: Demand & Economics
M I T I n t e r n a t i o n a l C e n t e r f o r A i r T r a n s p o r t a t i o n Airline Revenue Management: Flight Leg and Network Optimization 1.201 Transportation Systems Analysis: Demand & Economics
More informationAPPLICATIONS OF REVENUE MANAGEMENT IN HEALTHCARE. Alia Stanciu. BBA, Romanian Academy for Economic Studies, 1999. MBA, James Madison University, 2002
APPLICATIONS OF REVENUE MANAGEMENT IN HEALTHCARE by Alia Stanciu BBA, Romanian Academy for Economic Studies, 1999 MBA, James Madison University, 00 Submitted to the Graduate Faculty of Joseph M. Katz Graduate
More informationOptimizing Replenishment Intervals for Two-Echelon Distribution Systems with Fixed Order Costs
Optimizing Replenishment Intervals for Two-Echelon Distribution Systems with Fixed Order Costs Kevin H. Shang Sean X. Zhou Fuqua School of Business, Duke University, Durham, North Carolina 27708, USA Systems
More informationIntroduction to Support Vector Machines. Colin Campbell, Bristol University
Introduction to Support Vector Machines Colin Campbell, Bristol University 1 Outline of talk. Part 1. An Introduction to SVMs 1.1. SVMs for binary classification. 1.2. Soft margins and multi-class classification.
More informationFinal Report. to the. Center for Multimodal Solutions for Congestion Mitigation (CMS) CMS Project Number: 2010-018
Final Report to the Center for Multimodal Solutions for Congestion Mitigation (CMS) CMS Project Number: 2010-018 CMS Project Title: Impacts of Efficient Transportation Capacity Utilization via Multi-Product
More informationLogistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression
Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max
More informationBranch and Cut for TSP
Branch and Cut for TSP jla,jc@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark 1 Branch-and-Cut for TSP Branch-and-Cut is a general technique applicable e.g. to solve symmetric
More informationA Service Design Problem for a Railway Network
A Service Design Problem for a Railway Network Alberto Caprara Enrico Malaguti Paolo Toth Dipartimento di Elettronica, Informatica e Sistemistica, University of Bologna Viale Risorgimento, 2-40136 - Bologna
More informationRevenue Management and Capacity Planning
Revenue Management and Capacity Planning Douglas R. Bish, Ebru K. Bish, Bacel Maddah 3 INTRODUCTION Broadly defined, revenue management (RM) 4 is the process of maximizing revenue from a fixed amount of
More informationBig Data - Lecture 1 Optimization reminders
Big Data - Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Big Data - Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Schedule Introduction Major issues Examples Mathematics
More informationA Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem
A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationLecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
More informationProximal mapping via network optimization
L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
More informationLECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005
LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005 DAVID L. BERNICK dbernick@soe.ucsc.edu 1. Overview Typical Linear Programming problems Standard form and converting
More informationA stochastic programming approach for supply chain network design under uncertainty
A stochastic programming approach for supply chain network design under uncertainty Tjendera Santoso, Shabbir Ahmed, Marc Goetschalckx, Alexander Shapiro School of Industrial & Systems Engineering, Georgia
More informationMODELS AND ALGORITHMS FOR WORKFORCE ALLOCATION AND UTILIZATION
MODELS AND ALGORITHMS FOR WORKFORCE ALLOCATION AND UTILIZATION by Ada Yetunde Barlatt A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Industrial
More informationCredit Risk Models: An Overview
Credit Risk Models: An Overview Paul Embrechts, Rüdiger Frey, Alexander McNeil ETH Zürich c 2003 (Embrechts, Frey, McNeil) A. Multivariate Models for Portfolio Credit Risk 1. Modelling Dependent Defaults:
More informationRevenue Management with Correlated Demand Forecasting
Revenue Management with Correlated Demand Forecasting Catalina Stefanescu Victor DeMiguel Kristin Fridgeirsdottir Stefanos Zenios 1 Introduction Many airlines are struggling to survive in today's economy.
More informationRecovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach
MASTER S THESIS Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach PAULINE ALDENVIK MIRJAM SCHIERSCHER Department of Mathematical
More informationCustomers with positive demand lead times place orders in advance of their needs. A
Replenishment Strategies for Distribution Systems Under Advance Demand Information Özalp Özer Department of Management Science and Engineering, Stanford University, Stanford, California 94305 ozalp@stanford.edu
More informationImproved Forecast Accuracy in Airline Revenue Management by Unconstraining Demand Estimates from Censored Data by Richard H. Zeni
Improved Forecast Accuracy in Airline Revenue Management by Unconstraining Demand Estimates from Censored Data by Richard H. Zeni ISBN: 1-58112-141-5 DISSERTATION.COM USA 2001 Improved Forecast Accuracy
More informationSensitivity analysis of utility based prices and risk-tolerance wealth processes
Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More information2007/26. A tighter continuous time formulation for the cyclic scheduling of a mixed plant
CORE DISCUSSION PAPER 2007/26 A tighter continuous time formulation for the cyclic scheduling of a mixed plant Yves Pochet 1, François Warichet 2 March 2007 Abstract In this paper, based on the cyclic
More informationIntegrated Multi-Echelon Supply Chain Design with Inventories under Uncertainty: MINLP Models, Computational Strategies
Integrated Multi-Echelon Supply Chain Design with Inventories under Uncertainty: MINLP Models, Computational Strategies Fengqi You, Ignacio E. Grossmann * Department of Chemical Engineering, Carnegie Mellon
More informationDistributionally robust workforce scheduling in call centers with uncertain arrival rates
Distributionally robust workforce scheduling in call centers with uncertain arrival rates S. Liao 1, C. van Delft 2, J.-P. Vial 3,4 1 Ecole Centrale, Paris, France 2 HEC. Paris, France 3 Prof. Emeritus,
More informationSTRATEGIC CAPACITY PLANNING USING STOCK CONTROL MODEL
Session 6. Applications of Mathematical Methods to Logistics and Business Proceedings of the 9th International Conference Reliability and Statistics in Transportation and Communication (RelStat 09), 21
More informationDuality in Linear Programming
Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow
More informationLoad Balancing and Switch Scheduling
EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load
More informationA Review of alexible Multi-product Inventory Model
Dynamic Capacity Management with Substitution Robert A. Shumsky Simon School of Business University of Rochester Rochester, NY 14627 shumsky@simon.rochester.edu Fuqiang Zhang UC Irvine Graduate School
More informationCASH FLOW MATCHING PROBLEM WITH CVaR CONSTRAINTS: A CASE STUDY WITH PORTFOLIO SAFEGUARD. Danjue Shang and Stan Uryasev
CASH FLOW MATCHING PROBLEM WITH CVaR CONSTRAINTS: A CASE STUDY WITH PORTFOLIO SAFEGUARD Danjue Shang and Stan Uryasev PROJECT REPORT #2011-1 Risk Management and Financial Engineering Lab Department of
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More information