On the interaction between resource flexibility
and flexibility structures
Fikri Karaesmen, Zeynep Aksin,
Lerzan Ormeci
Koç University
Istanbul, 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
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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
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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
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Network flow problem with random demand
Framework of Jordan and Graves (1995)
Simplistic but captures basic characteristic of problem
Enables structural properties
• Dynamic
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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.
l1
l2
l3
demands
C1
C2
C3
capacities
No resource flexibility
l1
l2
l3
demands
C1
C2
C3
capacities
Partial resource flexibility
Definitions and Assumptions
• Demand l=(l1, l2,.. ln) 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
E[T1]
Obviously:
And less obviously:
E[T2]
E[T1 ]  E[T2 ]  E[T3 ]
E[T3 ]  E[T2 ]  E[T2 ]  E[T1 ]
E[T3]
More flexibility is better!
Diminishing returns to flexibility!
Some useful properties
E[T1]
E[T2]
E[T3]
E[T4 ]  E[T1 ]  E[T2 ]  E[T1 ]  E[T3 ]  E[T1 ]
Expected throughput is submodular in any two parallel arcs.
Parallel arcs are substitutes!
E[T4]
Some useful properties
E[T1]
E[T2]
If capacity is symmetric, then:
E[T1 ]  E[T2 ]
Balanced flexibility is better!
The right scale of flexibility
• Not all service representatives / workers have multiple
skills.
• Let a be the proportion of service representatives with
multiple skills
• What is the right level of a?
• What happens to the preceding properties as a
changes?
The right scale of flexibility
For any realization the following LP must be solved:
With the additional constraint:
The right scale of flexibility
E[T|a=0]
E[T|a=0.2]
E[T|a=0.4]
E[T | a  0.4]  E[T | a  0.2]  E[T | a  0.2]  E[T | a  0]
Expected throughput is concave in a.
Diminishing returns to scale!
Examples: effects of scale
E[T1]
E[T2]
E[T4]
E[T3]
Expected Throughput
0.8
0.75
0.7
Flex2
0.65
Flex3
Flex4
0.6
0.55
0.5
0
0.2
0.4
0.6
0.8
Proportion of Flexible Resources (a)
1
Examples: effects of scale
E[T1]
E[T2]
E[T4]
E[T3]
Expected Throughput
0.8
0.75
20%
0.7
40%
0.65
60%
80%
0.6
100%
0.55
0.5
1
2
Flexibility Structures
3
Example: scale, and variability of demand
E[T1]
E[T2]
E[T3]
E[T4]
Expected Throughput
0.9
0.85
0.8
Flex2 Low Var.
0.75
Flex3 Low Var.
0.7
Flex2 High Var.
0.65
Flex3 High Var.
0.6
0.55
0.5
0.2
0.4
0.6
0.8
1
Proportion of Flexible Resources (a)
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
li
• Processing times (talk times) are exponentially
distributed with rate m.
• Limited number of waiting spaces.
• Impatient customers abandon the queue: abandonment
times are exponentially distributed with rate q.
• 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
Expected Throughput
E[T1]
E[T2]
E[T3]
15
14.8
14.6
14.4
14.2
14
13.8
13.6
13.4
13.2
13
E[T4]
1
2
3
4
20%
40%
60%
80% 100%
a
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 crosstraining 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