On the interaction between resource flexibility and flexibility structures

Fikri Karaesmen, Zeynep Aksin, Lerzan Ormeci

Koç UniversityIstanbul, Turkey

Sponsored by a KUMPEM research grant

FIFTH INTERNATIONAL CONFERENCE ON "Analysis of Manufacturing Systems –Production Management"May 20-25, 2005 - Zakynthos Island, Greece

Outline

• Motivation• The methodology• Some structural results• Numerical examples• Work-in-Progress

Resource flexibility in practice:multilingual call (contact) centers

• Compaq’s call centers in Ireland: supports nine European languages

• Toshiba call center in Istanbul: eight European languages

• Similar centers for Dell, Gateway, IBM, DHL, Intel, etc.• Language and cultural know-how mix.• Language and technical skills mix.

• Excellent example of multi-skill service structure

Resource flexibility

• Part of a general framework that encompasses manufacturing and services– Flexible manufacturing capacity: assigning demand types to

flexible plants– FMS: routing parts to the right flexible machine

– Human resources: cross-training of workers or service

representatives

Emerging questions

• What is the value of cross-training?• What can be expected out of a good dynamic routing

system?• What is the right scale of flexibility?

– is everyone x-trained?– if only some, how many?

• What is the right scope of flexibility? – can x-trained personnel deal with all calls?– if not, what is the right skills mix?

Related literature

• Process Flexibility– Jordan and Graves (1995): manufacturing flexibility, demand-plant assignments

(motivated by a GM case)

– Graves and Tomlin (2003)

– Iravani, Van Oyen and Sims (2005)

– Aksin and Karaesmen (2004)

• Flexible servers in queueing systems– Van Oyen, Senturk-Gel, Hopp (2001)

– Pinker and Shumsky (2000)

– Chevalier, Shumsky, Tabordon (2004)

– Aksin and Karaesmen (2002)

– Hopp, Tekin, Van Oyen (2004)

• Review papers– Sethi and Sethi (1990)

– Hopp, Van Oyen (2004)

Methodological issues

• Static– Network flow problem with random demand– Framework of Jordan and Graves (1995)– Simplistic but captures basic characteristic of problem– Enables structural properties

• Dynamic– Can take into account queueing, abandonments, blocking– Difficult to decouple staffing question from call routing– Stochastic dynamic optimization problem– Very difficult problem in general

The Network Flow Model

• The system is represented by a graph.• An arc between demand i and resource j implies that

demand i can be treated by resource j.• Without loss of generality, each demand type has a main

corresponding department.

demands capacities

1

2

3

C1

C2

C3

No resource flexibility

demands capacities

1

2

3

C1

C2

C3

Partial resource flexibility

Definitions and Assumptions

• Demand =(1, 2,.. n) is a random vector.• Capacities and flexibility structure are given.• The allocation (routing) takes place after the realization

of the demand. • Plausible objective: maximization of expected throughput

(flow)• Solve max-flow problem for each possible realization and

take expectations (over the random demand vector).

• Easy to simulate, difficult to establish structural results.

Some useful properties

1 2 3[ ] [ ] [ ]E T E T E T

E[T1] E[T2] E[T3]

Obviously:

And less obviously:

3 2 2 1[ ] [ ] [ ] [ ]E T E T E T E T

More flexibility is better!

Diminishing returns to flexibility!

Some useful properties

E[T1] E[T4]E[T3]E[T2]

4 1 2 1 3 1[ ] [ ] [ ] [ ] [ ] [ ]E T E T E T E T E T E T

Expected throughput is submodular in any two parallel arcs.

Parallel arcs are substitutes!

Some useful properties

E[T1] E[T2]

If capacity is symmetric, then:

1 2[ ] [ ]E T E T

Balanced flexibility is better!

The right scale of flexibility

• Not all service representatives / workers have multiple skills.

• Let be the proportion of service representatives with multiple skills

• What is the right level of ?• What happens to the preceding properties as

changes?

The right scale of flexibility

With the additional constraint:

For any realization the following LP must be solved:

The right scale of flexibility

E[T|=0] E[T|=0.2] E[T|=0.4]

[ | 0.4] [ | 0.2] [ | 0.2] [ | 0]E T E T E T E T

Expected throughput is concave in .

Diminishing returns to scale!

Examples: effects of scale

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0 0.2 0.4 0.6 0.8 1

Proportion of Flexible Resources ()

Exp

ecte

d T

hro

ug

hp

ut

Flex2

Flex3

Flex4

E[T1] E[T2] E[T3] E[T4]

Examples: effects of scale

E[T1] E[T2] E[T3] E[T4]

0.5

0.55

0.6

0.65

0.7

0.75

0.8

1 2 3

Flexibility Structures

Exp

ecte

d T

hro

ug

hp

ut

20%

40%

60%

80%

100%

Example: scale, and variability of demand

E[T1] E[T2] E[T3] E[T4]

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.2 0.4 0.6 0.8 1

Proportion of Flexible Resources ()

Exp

ecte

d T

hro

ug

hp

ut

Flex2 Low Var.

Flex3 Low Var.

Flex2 High Var.

Flex3 High Var.

Robustness of the results: comparison with a call center model

• A call center with N customer classes and departments• Arrivals occur according to Poisson processes with rates

i

• Processing times (talk times) are exponentially distributed with rate .

• Limited number of waiting spaces.• Impatient customers abandon the queue: abandonment

times are exponentially distributed with rate .• C servers per department.

Methodology

• Call routing policies have an effect on the performance. • Difficult stochastic dynamic control problem in multiple

dimensions• We extend a bound/approximation by Kelly by reducing

the problem to N single dimensional Markov Decision Processes

• Combine the solutions of the MDPs in a concave optimization problem (an LP).

• Solve the LP: the result is a bound on the expected throughput per unit time which is fairly tight.

A numerical example: the symmetric case

• A three class call center• All parameters symmetric (call volumes, service rates,

abandonment parameters)• Five servers, twenty five phone lines for each class• Vary scale: 0-5 x-trained servers• Vary flexibility structure

Results

1313.213.413.613.814

14.214.414.614.815

20% 40% 60% 80% 100%

1

2

3

4

Exp

ecte

d T

hrou

ghpu

t

E[T1] E[T2] E[T3] E[T4]

Flexibility Insights

• Obvious result: more flexibility is better• Balanced skill sets are better

– spread out flexibility rather than exclusive flexibility

• High scale is desireable but..– diminishing returns to scale

– marginal value of scale increases with better scope for low levels of scale

– scale and scope decisions interact

– good skill-set design is essential for optimal cross-training practice

Managerial Implications

• Start with skill-set design; determining the right scale should follow this design decision: what type of flexibility followed by how much

• If the call center deals with calls that share similar parameters (symmetric) prefer a low scope strategy at high scale to a high scope strategy at low scale.

• For large call centers, even low scope and low scale should be sufficient (20% flexible capacity?)

• For smaller call centers higher scope is desirable.

Future and ongoing work

• On network flow models– More structural results on scale effects– A complete numerical study– Flexibility/capacity interactions

• On queueing models– Call routing policies– Capacity design

• Some information available at: http://call.ku.edu.tr