Revenue Management in the Context of Dynamic Oligopolistic Competition

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, mar409}@psu.edu Presented at INFORMS Revenue Management and Pricing Section Conference, MIT

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

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. • We assume Cournot-Nash-Bertrand (CNB) non-cooperative behavior. • We use equilibrium dynamics that enforce flow conservation.

4. Firms' Decisions (Supply-Production-Distribution) • location and scale of activity • mix of input factors • timing of input factor deliveries • inventory and backorder levels • prices • output levels • timing of shipments • shipping/distribution patterns

5. Space of square integrable functions for real interval [t0,tf] Sobolov Space for real interval [t0,tf] Notation CONTINUED • Controls • States • Time

6. Taken as exogenous data by firm f Notation CONTINUED • Functions • Non-own allocations of demands are viewed as fixed by firm f

7. Total In-Flow Total Out-Flow Inventory Dynamics • Inventory Dynamics are equilibrium dynamics, namely differential flow conservation equations:

8. Variable Production Cost Gross Revenue Total Distribution Cost Inventory Cost Firm’s objective • Net Present Value of Profit of each Cournot-Nash firmfF:

9. Other constraints These reflect bounds on terminal inventories/backorders, as well as restrictions on output and consumption and shipment variables (controls).

10. Summary of Constraints The constraints are : • Shipment Dynamics • Inventory Dynamics • Inventory / Backorder Initial and Terminal Time Constraints • Upper and lower bounds on the controls: output, consumption and shipments

11. Optimal Control Problem for Each Firm • For each firm fF: This is a continuous time Optimal Control Problem

12. Hamiltonian formed by the OCP for each firm f F Cournot – Nash Equilibria • The solutions of the below DVI are Cournot – Nash Equilibria:

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. • In particular paper by Friesz et al (2004) generalizes Pontryagin’s maximum principle from optimal control theory to the DVI setting.

14. Revenue Management for Oligopolistic Competition in the Service Sector

15. The Pure RM Decision Environment • Abstract service providers • No variable costs • Fixed capacity environments • No concept of Inventory/Backorder • Faces variable demand • Low product variety

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

17. Our RM Competitive Environment CONTINUED • 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.

18. Notation

19. States CONTINUED • Price variables : where

20. Market Demand • Market demand is known for each of the services iS and instant of time t [t0,tf] therefore,

21. Controls • Demand allocation variables: where • Non-own demand for firm f  F (exogenous)

22. Controls CONTINUED • Rates of service provision: • where • Industry rate of provision of service iS

23. Excess demand Price Dynamics • Price of the service i S changes based on an excess demand

24. Nominal discount rate NPV of Revenue NPV of Fixed Cost Each firm’s objective • Each firm f  Fmaximizes Net Present Value (NPV) of profit (revenue) f(uf,vf, u-f,v-f ,t)

25. Constraints • Each firm has a finite upper bound on each type of service they provide; • We define

26. Constraints CONTINUED • Logical as well as capacity constraints of each firm f  Fare:

27. Feasible Control Sets • Set of feasible controls for firmf  F

28. Firm’s optimal control problem • Each firm f  Fseeks to solve the following problem with u-f ,v-fas exogenous inputs :

29. Differential Variational Inequality (DVI) • 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 :

30. Numerical Example

31. Market 2 Firm 2 a4 a1 Firm 1 Firm 4 a3 Market 1 Market 4 a5 a2 Firm 3 Market 3 5 arc 4 node network

32. Node 2 a4 a1 Node 1 Node 4 a3 a5 a2 Node 3 5 arc 4 node network

33. Summary of Controls and States • 29 controls : • 10 states :

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

35. Computational Resultsfor Spatial Oligopoly

36. Inventory Dynamics

37. Production output rates

38. Flow between O-D pair

39. Allocation of output for consumption

40. Consumption at Different Markets

41. NPV Profit of Firms

42. Summary • 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. • Extensions to a dynamic network service environment in progress. • Numerical experiments based on discrete time approximation of very large problem is underway

43. Summary CONTINUED • 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 • 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.