PENN S TATE Department of Industrial Engineering 1 Revenue Management in the Context of Dynamic Oligopolistic Competition Terry L. Friesz Reetabrata Mookherjee Matthew A. Rigdon The Pennsylvania State University Industrial and Manufacturing Engineering {tfriesz, reeto, Presented at INFORMS Revenue Management and Pricing Section Conference, MIT

PENN S TATE Department of Industrial Engineering 2Outline Review the Dynamic Oligopolistic Network Competition model for non-service industries Review the Dynamic Oligopolistic Network Competition model for non-service industries Modify the above to treat Service/Revenue Management (RM) decision environment Modify the above to treat Service/Revenue Management (RM) decision environment The Modeling Perspective The Modeling Perspective Non-Cooperative Differential Oligopolistic Game among service providers Non-Cooperative Differential Oligopolistic Game among service providers Re-formulate as Differential Variational Inequlaity (DVI) and exploit available algorithms Re-formulate as Differential Variational Inequlaity (DVI) and exploit available algorithms Overview of One Particular Numerical Method Overview of One Particular Numerical Method

PENN S TATE Department of Industrial Engineering 3 Dynamic Oligopolistic Network Competition This is a foundation model upon which other models of dynamic network competition may be based: supply chains, telecomm, ecommerce, urban and intercity freight. This is a foundation model upon which other models of dynamic network competition may be based: supply chains, telecomm, ecommerce, urban and intercity freight. We assume Cournot-Nash-Bertrand (CNB) non-cooperative behavior. We assume Cournot-Nash-Bertrand (CNB) non-cooperative behavior. We use equilibrium dynamics that enforce flow conservation. We use equilibrium dynamics that enforce flow conservation.

PENN S TATE Department of Industrial Engineering 4 Firms' Decisions (Supply- Production-Distribution) location and scale of activity location and scale of activity mix of input factors mix of input factors timing of input factor deliveries timing of input factor deliveries inventory and backorder levels inventory and backorder levels prices prices output levels output levels timing of shipments timing of shipments shipping/distribution patterns shipping/distribution patterns

PENN S TATE Department of Industrial Engineering 5 Notation Controls Controls States States Time Time CONTINUED Space of square integrable functions for real interval [ t 0,t f ] Space of square integrable functions for real interval [ t 0,t f ] Sobolov Space for real interval [ t 0,t f ] Sobolov Space for real interval [ t 0,t f ]

PENN S TATE Department of Industrial Engineering 6Notation Functions Functions Non-own allocations of demands Non-own allocations of demands are viewed as fixed by firm f are viewed as fixed by firm f CONTINUED Taken as exogenous data by firm f

PENN S TATE Department of Industrial Engineering 7 Inventory Dynamics Inventory Dynamics are equilibrium dynamics, namely differential flow conservation equations: Inventory Dynamics are equilibrium dynamics, namely differential flow conservation equations: Total In-FlowTotal Out- Flow

PENN S TATE Department of Industrial Engineering 8 Firm’s objective Net Present Value of Profit of each Cournot- Nash firm f  F: Gross Revenue Variable Production Cost Total Distribution Cost Inventory Cost

PENN S TATE Department of Industrial Engineering 9 Other constraints These reflect bounds on terminal inventories/backorders, as well as restrictions on output and consumption and shipment variables (controls).

PENN S TATE Department of Industrial Engineering 10 Summary of Constraints The constraints are : 1. Shipment Dynamics 2. Inventory Dynamics 3. Inventory / Backorder Initial and Terminal Time Constraints 4. Upper and lower bounds on the controls: output, consumption and shipments

PENN S TATE Department of Industrial Engineering 11 Optimal Control Problem for Each Firm For each firm f  F: For each firm f  F: This is a continuous time Optimal Control Problem This is a continuous time Optimal Control Problem

PENN S TATE Department of Industrial Engineering 12 Cournot – Nash Equilibria The solutions of the below DVI are Cournot – Nash Equilibria: The solutions of the below DVI are Cournot – Nash Equilibria: Hamiltonian formed by the OCP for each firm f  F

PENN S TATE Department of Industrial Engineering 13 Observations Regarding DVI Formulation The preceding re-statement of dynamic oligopolistic network competition as a differential variational inequality (DVI) allows powerful results on existence, computation and convergence to be applied. The preceding re-statement of dynamic oligopolistic network competition as a differential variational inequality (DVI) allows powerful results on existence, computation and convergence to be applied. In particular paper by Friesz et al (2004) generalizes Pontryagin’s maximum principle from optimal control theory to the DVI setting. In particular paper by Friesz et al (2004) generalizes Pontryagin’s maximum principle from optimal control theory to the DVI setting.

PENN S TATE Department of Industrial Engineering 14 Revenue Management for Oligopolistic Competition in the Service Sector

PENN S TATE Department of Industrial Engineering 15 The Pure RM Decision Environment Abstract service providers Abstract service providers No variable costs No variable costs Fixed capacity environments Fixed capacity environments No concept of Inventory/Backorder No concept of Inventory/Backorder Faces variable demand Faces variable demand Low product variety Low product variety

PENN S TATE Department of Industrial Engineering 16 Our RM Competitive Environment Service firms are involved in a dynamic oligopolistic competition Service firms are involved in a dynamic oligopolistic competition Firms compete to capture demands for services Firms compete to capture demands for services Price dynamics are a classical price- tatonnement model articulated at the market level. Price dynamics are a classical price- tatonnement model articulated at the market level.

PENN S TATE Department of Industrial Engineering 17 Our RM Competitive Environment The time scale we consider is, neither short nor long, rather of sufficient length that allows prices to reach equilibrium, but not long enough for firms to re-locate, open or close the business. The time scale we consider is, neither short nor long, rather of sufficient length that allows prices to reach equilibrium, but not long enough for firms to re-locate, open or close the business. CONTINUED

PENN S TATE Department of Industrial Engineering 18 Notation

PENN S TATE Department of Industrial Engineering 19 States Price variables : Price variables :where CONTINUED

PENN S TATE Department of Industrial Engineering 20 Market Demand Market demand is known for each of the services i  S and instant of time t  [ t 0, t f ] Market demand is known for each of the services i  S and instant of time t  [ t 0, t f ]therefore,

PENN S TATE Department of Industrial Engineering 21 Controls Demand allocation variables: Demand allocation variables:where  Non-own demand for firm f  F (exogenous)

PENN S TATE Department of Industrial Engineering 22 Controls Rates of service provision: Rates of service provision: where where Industry rate of provision of service i  S Industry rate of provision of service i  S CONTINUED

PENN S TATE Department of Industrial Engineering 23 Price Dynamics Price of the service i  S changes based on an excess demand Price of the service i  S changes based on an excess demand Excess demand Excess demand

PENN S TATE Department of Industrial Engineering 24 Each firm’s objective Each firm f  F maximizes Net Present Value (NPV) of profit (revenue)  f (u f,v f, u -f,v -f,t) Each firm f  F maximizes Net Present Value (NPV) of profit (revenue)  f (u f,v f, u -f,v -f,t) NPV of Revenue NPV of Fixed Cost Nominal discount rate

PENN S TATE Department of Industrial Engineering 25 Constraints Each firm has a finite upper bound on each type of service they provide; Each firm has a finite upper bound on each type of service they provide; We define We define

PENN S TATE Department of Industrial Engineering 26 Constraints Logical as well as capacity constraints of each firm f  F are: Logical as well as capacity constraints of each firm f  F are: CONTINUED

PENN S TATE Department of Industrial Engineering 27 Feasible Control Sets Set of feasible controls for firm f  F Set of feasible controls for firm f  F

PENN S TATE Department of Industrial Engineering 28 Firm’s optimal control problem Each firm f  F seeks to solve the following problem with u -f,v -f as exogenous inputs : Each firm f  F seeks to solve the following problem with u -f,v -f as exogenous inputs :

PENN S TATE Department of Industrial Engineering 29 Differential Variational Inequality (DVI) Cournot-Nash-Bertrand differential (i.e. dynamic) games are a specific realization of the DVI problem Cournot-Nash-Bertrand differential (i.e. dynamic) games are a specific realization of the DVI problem Solutions of the following DVI are the Nash equilibria : Solutions of the following DVI are the Nash equilibria :

PENN S TATE Department of Industrial Engineering 30 Numerical Example

PENN S TATE Department of Industrial Engineering 31 5 arc 4 node network Firm2 Firm1Firm4 Firm3 a1a1 a5a5 a4a4 a3a3 a2a2 Market 1Market 4 Market 2 Market 3

PENN S TATE Department of Industrial Engineering 32 5 arc 4 node network Path Arc sequence P1P1P1P1 a1a1a1a1 P2P2P2P2 a2a2a2a2 P3P3P3P3 a 1, a 3 P4P4P4P4 a 1, a 4 P5P5P5P5 a 1, a 3, a 5 P6P6P6P6 a 2, a 5 P7P7P7P7 a3a3a3a3 P8P8P8P8 a4a4a4a4 P9P9P9P9 a 3, a 5 P 10 a5a5a5a5 Node2 Node1Node4 Node3 a1a1 a5a5 a4a4 a3a3 a2a2

PENN S TATE Department of Industrial Engineering 33 Summary of Controls and States 29 controls : 10 states : 10 states :

PENN S TATE Department of Industrial Engineering 34 Other Information linear demand linear demand quadratic variable cost quadratic variable cost quadratic inventory cost quadratic inventory cost N = 20 (time steps) N = 20 (time steps) L = 20 (planning horizon) L = 20 (planning horizon) Step size,  =1 Step size,  =1 Bounds on Controls : 0 and 75 Bounds on Controls : 0 and 75 These choices lead to nearly 700 variables. These choices lead to nearly 700 variables.

PENN S TATE Department of Industrial Engineering 35 Computational Results for Spatial Oligopoly

PENN S TATE Department of Industrial Engineering 36 Inventory Dynamics

PENN S TATE Department of Industrial Engineering 37 Production output rates

PENN S TATE Department of Industrial Engineering 38 Flow between O-D pair

PENN S TATE Department of Industrial Engineering 39 Allocation of output for consumption

PENN S TATE Department of Industrial Engineering 40 Consumption at Different Markets

PENN S TATE Department of Industrial Engineering 41 NPV Profit of Firms

PENN S TATE Department of Industrial Engineering 42Summary Theoretical framework and computational experimentation of the traditional production – distribution system in a dynamic network completed Theoretical framework and computational experimentation of the traditional production – distribution system in a dynamic network completed Theoretical framework supporting extensions to non-network dynamic service sector environment completed. Theoretical framework supporting extensions to non-network dynamic service sector environment completed. Extensions to a dynamic network service environment in progress. Extensions to a dynamic network service environment in progress. Numerical experiments based on discrete time approximation of very large problem is underway Numerical experiments based on discrete time approximation of very large problem is underway

PENN S TATE Department of Industrial Engineering 43 Summary Continuous time algorithms for descent in Hilbert space without time discretization have been designed and analyzed qualitatively Continuous time algorithms for descent in Hilbert space without time discretization have been designed and analyzed qualitatively Preliminary tests of continuous time algorithms are promising Preliminary tests of continuous time algorithms are promising We have shown treatment of dynamics with explicit time lags is possible using continuous time algorithms. This opens the door to consideration of explicit service response delays – a previously unstudied topic. We have shown treatment of dynamics with explicit time lags is possible using continuous time algorithms. This opens the door to consideration of explicit service response delays – a previously unstudied topic. CONTINUED