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

Year Percentage of employees

1998 67.7%

1999 74.9%

2000 76.7%

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

ut (nt , dt) as the maximum expected revenue-to-go at stage tA = {ai j }, represents the firm’s resource flexibility structureD Demand Vectornt Resource availability

Because of the exponential state space, finding the optimal policy requiresEnormous 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?