Scheduling of Flexible Resources in Professional Service Firms Arun Singh CS 537- Dr. G.S. Young Dept. of Computer Science Cal Poly Pomona

Problem

Solution

Flexibility Flexibility is the ability of an organization to effectively cope with uncertainties and changes in the market by employing resources that can process different types of jobs

Workplace Learning Industry YearPercentage of employees % % % Percentage of employees who received training Comparisons Report of the American Society for Training Development (ASTD)

Problem Definition  Offers m different types of training programs  Employs l different types of instructors  Specialized  Flexible  Versatile  Index the program type by i = 1, 2,...,m  πi denote the return or revenue  Client requests for training on this day arrive randomly, starting T periods before this date

WLS Manager Firm accumulates client requests during the period, say, one day and allocates resources at the end of the period. Goal  Maximum expected total revenue Major decisions  Acceptance  Resource assignment Stochastic dynamic program

Maximum expected revenue-to-go u t (n t, d t ) as the maximum expected revenue-to-go at stage t A = {ai j }, represents the firm’s resource flexibility structure D Demand Vector nt Resource availability Because of the exponential state space, finding the optimal policy requires Enormous computational effort even for problems with moderate number of resources

Special Case: Two Job Types with Unit Job Arrivals  Two types of jobs, types 1 and 2  Three types of resources, types 1, 2, and 3  Type-1 and type-2 resources are specialized resources  Type-3 resources are flexible  In each period t, we assume that no more than one job arrives

Special Case: Two Job Types with Unit Job Arrivals Cont.. The following observations are intuitively true and can also be proved formally using interchange arguments.  The optimal online policy must accept as many type-1 jobs as possible, utilizing the type-1 resources before the type-3 resources.  The optimal online policy must accept a type-2 job whenever type-2 resources are available. The problem is more challenging when a type-2 job arrives and only type-1 and type-3 resources are available It would make sense to reserve the flexible resources for jobs arriving toward the end of the decision horizon

Theorem for the optimal online policy Suppose a type-2 job arrives in period t, and let (n1, 0, n3; e2) be the state of the system in this period. 1. There exists a non increasing function Ft (n1) such that the optimal online policy accepts the type-2 job and assigns a type-3 resource to it if and only if n3 ≥ Ft (n1). 2. Ft (n1) is a non decreasing function of t for any given n1.

Cont.. if p1 + p2= 1 (i.e. a type-1 or type-2 job always arrives in each period and hence the total demand for the two job types is known), then the threshold function takes the simple form Ft (n1) = t − n1, t = the number of periods remaining until the end of the decision horizon, is also the total remaining demand. Thus, the optimal online policy accepts an incoming type-2 job and uses a flexible resource only if n3 ≥ Ft (n1) = t - n1

Basic threshold policy 1. Accept a type-i job if and only if  The total number of resources that can only be assigned to either the type-i jobs or any job type of lesser value (job types with indices larger than i ) is greater than zero, or  The total number of resources that can be assigned to job types that are more valuable than type-i jobs (i.e., jobs with indices smaller than i ) exceeds the total expected demand for these job types over the remaining time horizon. 2. If the type-i job is accepted in step 1, assign a resource which is expected to generate the smallest return among all resources that can be assigned to the type-i job.

Capacity reservation policy 1. Accept a type-i job if and only if (a) the total number of resources that can only be assigned to either type-i jobs or any job of lesser value (jobs with indices larger than i ) is greater than zero, or (b) the probability that the overflow demand of job types that are more valuable than the type-I job is greater than or equal to the total number of flexible resources that can process type-i or more valuable jobs is less than the threshold value of πi/πi. 2. If the type-i job is accepted in the first step, assign a resource which is expected to generate the smallest return among all resources that can be assigned to the type-i job.

Rollout policy 1. Accept a type-i job if and only if its reward πi is greater than or equal to the minimum opportunity cost ωj t among the available resources that can process type-i jobs. 2. If the type-i job is accepted, then assign to it the available resource with the minimum opportunity cost. ωjt = opportunity cost

Concluding Remarks  Little flexibility is sufficient to achieve the maximum reward for the systems with moderate capacity  More flexibility is required for the system with tighter capacity  Our methods significantly out perform the naive benchmark method (first-come first-served), and provide near-optimal solutions in most problem instances  Capacity reservation policy consistently dominates the other solution approaches, since it incorporates more problem parameters (rewards, demand distributions) into its decision process.

Questions?