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.
Top Related