Stochastic Models of Resource Allocation for Services

1. Stochastic Models of Resource Allocation for Services Ralph D. Badinelli Virginia Tech

2. Motivation Manufacturing Service Service design PSS design Capacity acquisition Revenue process design Location/layout/IT design Resource planning Resource allocation Resource dispatching Quality control • Product design • Process design • Capacity acquisition • Location/layout • Revenue management • Aggregate planning • P&IC • Shop floor control • Quality control

3. INFORMS Service Science Section • Formed in February 2007 • Meetings sponsored/co-sponsored • National INFORMS 2007 (Seattle) • 2008 Logic of Service Science (Hawaii) • Service, Operations, Logistics, Informatics SOLI 2008 (Beijing) • 2008 Frontiers in Service (Washington) • National INFORMS 2008 (Washington, DC) • International Conference on Service Science (Hong Kong) • National INFORMS 2009 (San Diego) • November, 2008 - New Quarterly Journal • Service Science • http://www.sersci.com/ServiceScience/ • 2010 – First on-line INFORMS SIG conference • Vice Chair/Chair-Elect = Ralph D. Badinelli

4. Purpose • We develop a resource allocation model with general forms of service technology functions • We describe the relationship between inputs and outputs of a process of co-creation of value by a service provider and a service recipient.  • Model development is directed at providing useful policy prescription for service providers

5. Contributions • A useful optimization model for resource allocation and dispatch • Some basic guidelines for optimal resource allocation/dispatching, for client involvement and adaptation of resource management to process learning • A modeling framework for service processes that can serve as a foundation for further model development

6. Service Process Definition: A service process is a coordinated set of activities which transforms a set of tangible and intangible resources (inputs), which include the contributions from the service recipient and the service provider, into another set of tangible and intangible resources (outputs). • E.g., agile software development, IT consulting, higher education

7. Technology functions • A technology function for a service encounter is a function that effectively maps inputs to outputs according to the capabilities of the service participants to transform inputs into outputs. • We construct this functional relationship by considering the inputs and outputs of a process to be functions of the volume, or number of service “cycles”, of the process which are simultaneously executed. Athanossopoulus (1998)

8. Assumptions • The set of inputs of a service process is comprised of two sets of inputs • provider inputs • client inputs • Resource constraints • Awareness – the client/provider may not have full knowledge of the technology function. • Objective function - maximization of utility of the service participants.

9. Efficiency & Returns to scale

10. Technology functions • The general nonlinear (VRS) technology function: • The linear VRS technology function

11. The linear CRS technology function Benchmark technological coefficient of inputi of process p Benchmark technological coefficient of inputi of process p = benchmark usage rate of resource ipercycle of process p benchmark generation of resource j per cycle of process p number of cycles of process that are executed

12. Basic I/O relationships (Benchmark PSS)

13. Real PSS – performance and uncertainty

14. Problem P1 Resource allocation problem subject to: for all p a vector of target outputs for process the distribution of , a function of the resource allocations vector of capacities of available resources

15. Loss function Lemma 1: The loss function increases with inefficiency Lemma 2: Loss is increasing in the targets, Lemma 4: Loss is decreasing and convex in volume

16. Process uncertainty Self adjusting assumption:after the process inputs are allocated, the process usage rates are dispatched by the service provider and the service recipient in such a way that they mutually adjust to values that support a certain volume and which are consistent with the inefficiency of the bottleneck input.

17. Problem re-statement Define, subject to: for all p

18. Optimality conditions First-order KKT conditions imply: where,

19. Optimal resource dispatch , Theorem 2: Processes that have lower usage rates will be allocated higher proportions of available input resources and achieve higher volumes under an optimal policy.

20. Optimal effort vs. performance

21. The cost of poor performance

22. Optimal effort vs. uncertainty

23. The cost of uncertainty

24. General outcomes The need for model-based resource planning Optimal allocation of input resources across processes that are different in terms of their efficiencies, uncertainties and/or output targets is quite complex and, in some cases counter-intuitive Conflict resolution Service providers and service recipients should make every attempt to educate themselves jointly about the nature of a service process before they engage in dispatching resources to it.

25. Outcome adaptive policies The transition law is simple

26. Estimation adaptive policies • Update estimates of the parameters of the process pdf with each service period. • Consider improvements in efficiency as well as random variation. • ARIMA (0,1,1) forecasting model? • Non parametric updating • Must use approximate DP due to dimensionality of estimation state

27. Conclusions • We began the modeling of stochastic, multi-period resource allocation problems • Service models can borrow much mathematical structure from manufacturing models • The multi-dimensionality of service processes introduces new mathematical features to planning models • In our lifetimes, a cure will be found!