MANAGEMENT SCIENCE
informs
Vol. 53, No. 7, July 2007, pp. 1102–1112 issn 0025-1909 eissn 1526-5501 07 5307 1102
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doi 10.1287/mnsc.1060.0621 © 2007 INFORMS
Call-Center Labor Cross-Training: It’s a Small World After All Seyed M. R. Iravani, Bora Kolfal
Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208 {
[email protected],
[email protected]}
Mark P. Van Oyen
Industrial and Operations Engineering, University of Michigan, Ann Arbor, Michigan 48109,
[email protected]
I
t is well known that flexibility can be created in manufacturing and service operations by using multipurpose production sources such as cross-trained labor, flexible machines, or flexible factories. We focus on flexible service centers, such as inbound call centers with cross-trained agents, and model them as parallel queueing systems with flexible servers. We propose a new approach to analyzing flexibility arising from the multifunctionality of sources of production. We create a work sharing (WS) network model for which its average shortest path length (APL) metric can predict the more effective of two alternative cross-training structures in terms of customer waiting times. We show that the APL metric of small world network (SWN) theory is one simple deterministic solution approach to the complex stochastic problem of designing effective workforce cross-training structures in call centers. Key words: cross-training; small world networks; average path length; call-center labor management; queueing; operational flexibility History: Accepted by Brian Uzzi and Luis Amaral, special issue editors; received November 6, 2004. This paper was with the authors 2 months for 2 revisions.
1.
Introduction
have expanded the cross-training of customer sales representatives (CSRs) in inbound call centers. That is to say, some or all of the agents are (cross-)trained to provide two or more types of service during a work shift. For example, a CSR may be trained for sales, and also for customer service, repairs, complaints, etc. Among the possible reasons for cross-training (e.g., motivation, providing a career path, improving the probability of service resolution on the first call, reducing the number of agents with whom a customer must speak, etc.), we focus on the operational benefits of cross-training in reducing the average number of customers in queue (and thus, by Little’s law, the average customer waiting time). In a nutshell, cross-training allows labor capacity to be dynamically reallocated to the services required by customers as call volumes shift and the mix across service types changes. Even when trends in the environment are absent, cross-training reduces the frequency with which agents starve for lack of calls due to intrinsic variability in the demand process and service times. The effect of cross-trained agents can result in, for example, callers experiencing a shorter wait to reach an agent and offering the same quality of service with a smaller workforce. These operational benefits alone may be sufficient to justify agent cross-training. It is obvious that full cross-training of every agent for every call type is very costly, and sometimes
Over the past two decades, businesses worldwide have vigorously worked to implement new operational approaches to deal with the difficult demands of the global economy. This has resulted in a rethinking of labor management practices. For a growing number of companies, this has meant a shift from workers trained only for one task to workforces trained for multiple tasks and, in some cases, dynamic worksharing (see Hopp and Van Oyen 2004). A vivid illustration of this change can be found in call centers, a large service industry employing roughly three to four million Americans, according to Datamonitor. Critical emergency services such as the police, ambulance services, and fire services depend on inbound call centers (such as 911 in North America) for dispatching; therefore, much more than convenience and profit are at stake. Call centers have found that careful attention to the management of the workforce (staffing, rostering, training, performance measurement, skills-based routing, etc.) can help avoid lost calls and reduce long waiting times. Competitive marketplace pressures (including increased pressure to outsource operations to nations with low-wage labor markets), increasing customer service level expectations, and the recent advent of skills-based routing technologies 1102
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impossible (e.g., call centers that serve clients in several languages). Therefore, an important question becomes: which type(s) of calls should each call-center agent be (cross-)trained to handle? In many applications, there will be a range of choices available in determining worker skill sets, and there is no easy way to tell which choices are better than others. We illustrate this with the following simple example to describe the critical issue of workforce cross-training in call centers. Consider a call center that receives 12 different types of calls, namely, call types A, B, C L. Calls of type i arrive randomly with a rate specified by the ith element of the demand rate vector = 05 0.5, 0.5, 0.75, 0.75, 0.5, 1.167, 0.667, 0.667, 0.333, 0.333, 0.333). For simplicity, suppose that the call center has seven agents. We assume that call-handling/service times are stochastic with an average of 0.9 units of time (which corresponds to a system utilization of 90% under an aggregate arrival rate of seven calls per unit time). Associated with every agent is a nonempty set, called a skill set, defining the type(s) of calls the agent is (cross-)trained to serve. Figure 1 shows two agent cross-training structures, in which an arrow from a call type to an agent indicates that the agent is trained to respond to that call type (i.e., this type of call is in the agent’s skill set). For example, in Structure 1, Call-Center Agent 5 is trained to respond to call types {G, H, I}, while in Structure 2, the skill set of that agent is {G, I, J}. Both structures are easily capable of handling demand vector because each worker can spend an equal fraction of effort on each skill in their skill set. All call types receive enough capacity, so the number of customers in the queues will not grow to infinity. However, the question is, which cross-training structure is more flexible, yielding a smaller average customer waiting time over a range of operating conditions? If there exists some property in a cross-training structure that impacts the relative performance of the corresponding stochastic queueing system over a broad operating range, then we are facing the interesting question of how to quantify this property with a computationally lightweight algorithm. A simple and intuitive way to capture the effectiveness of a cross-training structure is to simply use the total number of skills of a structure as an index. The number of arcs index of a system is defined to be the sum of the number of skills (arcs in the structure graph) of all agents. The intuition is clear: Every additional skill serving a call type provides more capacity that (as needed) can be used to serve that call type. In other words, every additional skill corresponding to a call type pools more capacity to serve that call type. As Figure 1 shows, the total number of skills in Structure 1 is 27, while the total number of skills in
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Structure 2 is only 25. Thus, the number of arcs index would select Structure 1 as a more effective crosstraining structure. To identify which structure yields the lowest average customer waiting time, we developed a discreteevent computer simulation program and estimated the average customer waiting times under each crosstraining structure. The system model is a queueing network with parallel, infinite-buffer queues. Call interarrival times as well as the call service times are modeled using gamma probability distributions (which covers a wide range of variability scenarios with coefficient of variation of less than, equal to, or greater than one). Over a range of variability levels in the call arrival and service processes as well as the utilization (load) of the system, simulation reveals that Structure 2 performs better than Structure 1 in minimizing the mean waiting times. In fact, for different variability and utilization scenarios, Structure 1 resulted in 15.6% to 24.6% (with an average of 20.2%) larger waiting times than Structure 2. As these results show, although Structure 1 has more skills to pool more capacity of different agents, the way that structure pools those capacities is not as effective as that in Structure 2. Thus, the number of arcs index can be misleading, as it is in this example. Furthermore, it is of not much help when ranking alternative structures that have the same number of skills (which is often the case for a given budget on total skills). While brute-force simulation is a useful tool for the performance analysis of problems that are not too large, it has limitations. It requires a large amount of computation and very detailed data to parameterize the system model (e.g., call arrival and process distributions, call-routing policies, etc.). Furthermore,
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it does not provide any insights about why some cross-training structures perform better than others. Because alternative designs for cross-training have a network structure, there is hope that the proper quantification of their properties using network theory may allow us to identify good designs without the use of brute-force simulation. A simple method that can efficiently predict the better of alternative structures would be very useful. Through our experiments, we have come to believe that if a cross-training structure is well designed, then it shares some properties of a small world network (SWN). The SWN literature reveals important characteristics of real-world networks (see Dorogovtsev and Mendes 2002 and Albert and Barabási 2002 for a review of SWNs and their evolution). For example, Uzzi and Spiro (2005) demonstrated that “smallworldliness” can be used as a predictor of performance, a concept fundamental to our work (see also Guimera et al. 2005). Watts and Strogatz (1998) and Watts (1999) have developed two network metrics that characterize SWN structure: the average shortest path length (APL) and the cluster coefficient. For our purposes, we have found the APL to have an obvious intuitive relationship to our application, while the cluster coefficient does not. Indeed, our experimentation has not shown the cluster coefficient to be very useful, and therefore we do not report on it here. The main contribution of this paper is the development of a work-sharing (WS) network model for which a revised version of the well-known APL metric has a strong ability to predict the more effective cross-training structure between two alternatives. Our work supports the conjecture that a call center with a well-designed and effective cross-training structure really is “a small world after all.”
2.
Literature
The application of our model is related to past work done to analyze approaches to multifunctionality and cross-training. Studies by the operations management community often seek to quantify or guide the evaluation of additional skills or functions in production and service operations systems. Jordan and Graves (1995) introduced the concept of a “chain” strategy for multifunctionality in a single-stage manufacturing system with random demand and deterministic production. This work was extended to multistage manufacturing systems by Graves and Tomlin (2003). Sheikzadeh et al. (1998) analyzed the chain structure for achieving equipment flexibility. Gurumurthi and Benjaafar (2004) examined throughput in queueing systems under varying parameters, and provide evidence for the surprising effectiveness of chain structures. Hopp et al. (2004) defined the structure of
“D-skill chaining” for cross-training and used queueing models of flexible workers in serial production systems to show that the two-skill chaining structure dramatically mitigates the ill effects of variability. The graph structure called a D-skill chain by Hopp et al. (2004) had previously been important for the work of Watts and Strogatz (1998), which describes these networks as one-dimensional ring lattices with D connected neighbors. Their work emphasized the APL metric and the cluster coefficient, and they showed that random perturbations of the lattice generated SWNs. Of particular relevance to this paper, Aksin and Karaesmen (2002) and Iravani et al. (2005) model systems with cross-trained workers and try to predict their performance ranking. Aksin and Karaesmen (2002) performed analyses and provided insights into the characteristics of cross-training structures that may improve the system throughput. Iravani et al. (2005) provided a new methodology for assessing the flexibility of systems resulting from crosstraining or multifunctionality of their resources. They used the maximum-flow algorithm and developed an ordinal metric that can effectively rank structures by flexibility/effectiveness. Their perspective demonstrated a strong link between the deterministic structure of capabilities (e.g., cross-training structures) and the performance of the resulting complex stochastic queueing system. Aksin and Karaesmen (2002) and Iravani et al. (2005) are especially strong and general in their assertions that the underlying network structure of the cross-trained workforce has a lot to say about their efficiency and robustness. Graves and Tomlin (2003), Gurumurthi and Benjaafar (2004), Hopp et al. (2004), Hopp and Van Oyen (2004), Jordan and Graves (1995), Jordan et al. (2004), McClain et al. (2000), Sennott et al. (2006), Sheikzadeh et al. (1998), and Tekin et al. (2004) also contribute evidence toward this claim. By adapting concepts from SWN theory to the design of service operations, we introduce a new approach to a decision-support methodology for this important problem.
3.
Scope, Assumptions, and Approach
We focus on a call center that receives N different types of calls. Calls of type i arrive randomly with average rate i. The time that an agent takes to serve call type i (if the agent is trained for that call type) is a random variable that follows some probability distribution Fi s. We assume that agents who are trained for call type i follow the same servicetime distribution in their responses to call type i (i.e., Fi s is independent of the agents). One reason for this assumption is that planning models rarely try to
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model current employees with regard to the speed at which they work. Furthermore, it is reasonable to assume that all agents have received proper training, and therefore they all have almost the same speed in responding to a particular call type. Our call-center model is a queueing system with N queues and N multifunctional servers (and no customer rejection or abandonment/reneging). Within a queue, jobs are served first-come-first-served, but there is a longestqueue policy implemented in the skills-based routing technology that coordinates the dynamic allocation of the servers (agents) to the calls using a policy that keeps customer waits brief. Our investigation emphasizes scenarios in which there is a limited budget for cross-training. This creates a constraint on the number of skills, and it may arise from financial limits or practical considerations regarding training time, stress resulting from handling multiple call types, motivation/turnover, etc. For instance, suppose that we begin with four specialists (i.e., agents who can only respond to one call type) and a limited budget that provides training for, say, four additional skills. Then, many different training programs exist that will result in different cross-training structures. If we train each CSR for one additional skill, then which additional skill should each agent receive? In addition, we could consider allowing some CSRs to remain specialists while others receive two or more additional skills. In general, one must compare the performances of all possible cross-training choices (subject to constraints). The use of computer simulation and complete enumeration is one option for situations without too many alternatives; however, we will show that the cross-training structures can be converted to SWNs, and the APL metric can be used as a simple algorithm to choose the best-performing cross-training pattern. In the language of SWNs, the constraint on the number of skills Figure 2
(or average number of skills) is related to the “coordination number” of a network (see Newman 2000). In the sequel, §4 shows how a cross-training structure can be converted to a WS network. Section 5 uses simulation to evaluate the ability of the APL metric to predict which of two alternative cross-training structures will be the more effective. Finally, §6 concludes the paper.
4.
WS Networks: The SWN Representation of Cross-Training Structures
The key difficulty in our approach is to design a method for converting cross-training structures into a particular network model for which the APL criterion will be a useful indicator of flexibility (i.e., the effectiveness of the cross-training structure). For this, it is crucial to understand how an additional skill contributes to the performance of a cross-training structure. Consider the four cross-training structures in Figure 2. In Figure 2(I), there is no cross-training, and therefore each agent can only answer to one particular call type. Suppose that there is high variability in call interarrival times and call service times. This may result in situations where the queue of a call type, say Call Type A, becomes empty, while the queue of some other call type, say Call Type B, is very long, and therefore customers in that queue experience a long waiting time. Because Agent 1 is not cross-trained for Call Type B, her available unused capacity (during her idle time) cannot be used to help Agent 2. Note that in the presence of variability, some workers will occasionally be starved for work while others are overwhelmed, and this may cause long queue lengths. Additional cross-training (i.e., partial server pooling) can alleviate congestion. In Systems (II) and (III),
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Agent 1 is cross-trained to serve calls of Type B. By using this skill appropriately (for example, at times when Queue A is empty and Queue B has two or more calls), then customers of Type B will experience less waiting time in the line. Both Structures (II) and (III) have the same total number of skills. Furthermore, in both (II) and (III), all agents are cross-trained for two skills, and every call type can be answered by two agents. However, it has been shown (see Jordan and Graves 1995) that in the presence of variability, Structure (III) is more effective than Structure (II). One reason is that in (II), Agents 1 and 2 cannot help Agents 3 and 4, while in (III), Agents 1 and 2 can (directly or indirectly) help Agents 3 and 4. Figure 2(IV) shows another example in which all agents are cross-trained for all call types, and therefore each agent can help every other agent in responding to any call type. This structure is known as complete server pooling or full cross-training, and it is well understood that pooling usually reduces congestion (depending on the service policy). (See Mandelbaum and Reiman 1998 for elaboration on a system for which pooling inflates the total average waiting time.) Note that while (IV) is the best-performing cross-training structure, it is also the one with the highest training and/or wage costs. The above example shows how and why the addition of a skill can improve system performance, provided the control or coordination of the workers is effective. The main role of the additional skill is to give an agent the capability (flexibility) to help another agent to serve a particular call type when needed. This example also suggests that the more ways agents can help each other, the more effective the cross-training structure is in improving the system performance (something we have found to usually be the case in our study of systems with cross-trained labor). We now present our methodology based on the APL metric for converting the design of a crosstraining structure into a useful SWN representation, the WS network. There are three rules that define the construction of a WS network. We begin with the first two: Rule 1. Every agent i is represented by node i, where i = 1 2 N . Rule 2. An undirected arc is placed between nodes i and j if agents i and j both share at least one call type in common in their respective skill sets. To clarify the second rule, consider the structures in Figure 2 and their corresponding WS networks in Figure 3. In Figure 3(II), there is an undirected arc connecting nodes i = 1 and j = 2 because Agents 1 and 2 in Figure 2(II) can both serve (i.e., help each other in serving) at least one common call type (A or B). In the WS network in Figure 3(III), there is also
Management Science 53(7), pp. 1102–1112, © 2007 INFORMS
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an undirected arc connecting node i = 1 to node j = 2. The reason is that, as Figure 2(III) shows, Agents 1 and 2 can both help each other in serving as least one (i.e., in this case only one) common Call Type B. Note that Figure 3(I) has no link between any nodes because the agents are not cross-trained and therefore hold no skills in common. Although Systems (I) and (II) have illustrative value, our methodology is intended for structures such as (III) and (IV), that are connected, a typical SWN assumption (see Watts and Strogatz 1998). Figure 3(IV) has every pair connected, and this represents the fully cross-trained case in which every agent can help each other. A novel aspect of our work is that we assign a length to each arc of the WS network, as specified in the third rule: Rule 3. The length of an arc connecting two nodes, i and j, is the reciprocal of the number of call types that can be served by both agents i and j. Given variability in the demand and/or service processes, it is clear that the more call types agents can help each other with (i.e., the greater number of shared call types), the more effective the crosstraining structure would be. We set the length of the arc between i and j, arci j, to be the reciprocal of the number of call types that both agents i and j can serve. Therefore, if agents i and j help each other in more call types, the length of the arci j becomes smaller. Consequently, the WS structures with smaller lengths between their nodes (and thus smaller APL) will represent cross-training structures in which agents can help each other in more call types.
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To clarify Rule 3, again consider Structures (II) and (III) in Figure 2 and their corresponding WS networks in Figure 3. In Figure 2(II), Agents 1 and 2 can help each other to serve two Call Types A and B. Therefore, the length of the arc connecting Nodes 1 and 2 in the WS network in Figure 3(II) is reciprocal of 2, i.e., 0.5. On the other hand, in Figure 2(III), Agents 1 and 2 can help each other in serving only one call type, namely, Type B. Hence, the length of the arc connecting Nodes 1 and 2 in the WS network in Figure 3(III) is one (i.e., the reciprocal of one). Similarly, in Figure 2(IV), there are four call types (Call Types A, B, C, and D) that Agents 1 and 2 can both serve. Therefore, in Figure 3(IV) the length of the arc connecting Nodes 1 and 2 is 0.25, the reciprocal of four. For a graph with N nodes, computation of the APL metric requires the calculation of the minimum distance between every possible pair i j, denoted as min Li j . Because the path between nodes i and i has no meaning in our WS network, and because the shortest path from node i to node j is the same as the shortmin min est path from node j to node i (i.e., Li j = Lj i ), we only need to calculate N N − 1/2 shortest paths. The APL of a WS network with N nodes is therefore the average length of these N N − 1/2 paths, which can be calculated as APL =
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Note that our SW network is defined such that a smaller APL number generally corresponds to a crosstraining structure in which agents have greater versatility in helping each other. Thus, the smaller the APL, the more flexible and more effective the crosstraining structure is in reducing the customer average waiting times. We call our methodology based on the WS network and the APL metric the WS-APL method. We now go back to our example in Figure 1 and present the WS networks of those structures in Figure 4. Although Structure 1 in Figure 1 has a larger total number of skills than Structure 2, the WS network of Structure 2 has a smaller APL compared to that of Structure 1 (i.e., APL2 = 121 < APL1 = 153), predicting that the cross-training Structure 2 should be more effective than Structure 1. This is exactly what our simulation confirms. As this example shows, our WS-APL method was not misled by the larger total number of skills of Structure 1.
5.
Evaluating the Performance of the WS-APL Method
In this section, using a set of five rigorous numerical test suites, we evaluate the ability of the WS-APL method to distinguish effective cross-training structures between alternative structures.
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5.1. Design of the Test Suites Many considerations went into the generation of the test suites, and here we briefly explain the rationale. An important consideration was the ability to compare the WS-APL method to alternatives. The test suites developed in Iravani et al. (2005) were used for the benchmarking of several good methods, so we chose them for benchmarking the performance evaluation of the WS-APL method. The test suites model call centers with six, eight, and ten different call types and the same number of agents. We use discreteevent simulation to compute the waiting-time performance of each structure. The nature of these queueing models requires fairly lengthy simulation run lengths as well as multiple replications to achieve good performance estimates. The majority of SWN theory is applied in contexts with very large networks. Call centers range in size from several agents to over a thousand. The examples in this paper have a small size, primarily as a consequence of the computational burden of our system models, which are relatively complex. Our experience with theoretical and computational analysis of queueing networks suggests that large systems should exhibit the same characteristics and effects as the small ones that are computationally
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their capability set, thereby allowing them to maintain and use the skills for which they have been trained. This condition is consistent with the approach taken in Aksin and Karaesmen (2002). However, our shock models, which we will describe later, are designed to systematically violate this condition in a measured and meaningful way as a robustness check. To create difficult test suites and to reflect the notion that there is a budget on the total number of Figure 6
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tractable. The network models of many industrial service operations usually have few nodes to begin with. Thus, one perspective on this work is to show that “small-worldliness” still has meaning even when the “world” is small to begin with.
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skills in the design, Test Suites 1, 2, and 3 have almost the same number of total skills and almost the same number of skills per worker. 5.1.2. Variability in Call Arrivals and Call Service Times. Call service times and interarrival times were generated from gamma distributions, which can accommodate any coefficient of variation (CV), but reduces to an exponential distribution when CV = 1. Using CV levels of 1.0 and 2.0 for interarrival and service times, there were four variability scenarios per case. Given a demand arrival rate vector, i , we adjusted the mean process times to 0.7 and 0.9 for all demand types to test system utilizations of 70% and 90%, respectively. Thus, we tested eight regular (nonshock) operating conditions plus the same eight scenarios under the shock model (see the description of “peak hours” below). This creates a total of 16 different scenarios per test comparison. 5.1.3. Peak Hours. Call centers often experience a high volume of calls around peak hours (e.g., 8:00 a.m. to 9:00 a.m.), and we incorporated this by repeating every experiment under a shock model. The shock models sequentially provide a shock to every possible demand pair i j. A shock of type i j
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with i = j boosts type j call volume (i.e., call arrival rate) with an absolute rate increase of 0225 in the 70% utilization case (0.075 in the 90% utilization case) for a period of 5,000 calls, while also dropping type i by 0.225. Shocks of type i i simply boost the demand for type i to increase agent utilization 32% (8.3% in the 90% utilization case). Each shock is preceded by an equilibrium period of 5,000 calls. 5.1.4. Call Routing. In our simulation program, we used the longest-queue policy to assign calls to available agents. According to a longest-queue policy, when an agent finishes a call, the call routed to him or her is the next call in the queue that is longest. We selected the longest-queue policy for several reasons. Among them, the longest-queue policy is intuitive, easily implemented, and widely used in industry (see Iravani et al. 2005 for further details). Furthermore, in our simulation, when a call arrives and two agents are free, the agent that has been idle the longest is selected. 5.2. Evaluation by Simulation The APL metric for every WS network is presented in Table 1. Our evaluation process is based on all possible pairwise performance rankings of the test
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Figure 8
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suite structures for a particular demand vector i , i = 1 2 3 4 5. We evaluate the performance of the WS-APL method based on the following two metrics: (i) the percent of correct predictions, and (ii) the percent relative error, , of the wrong predictions. For every possible pairwise comparison, we checked the ranking predicted by the APL metric against the true ranking as determined by the simulation of long-run average waiting times. If the simulation model shows that the APL metric picked the less-effective structure, we count that as a failure. In those cases for which the
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APL metric failed, we calculated the relative percentage error, (the precent increase in customer average waiting time in the inferior structure relative to the superior structure), and reported the average and maximum percent errors. Table 2 summarizes the predictive ability of the APL in detecting the more-effective cross-training structures between alternative structures. Note that the average and maximum percent errors are calculated conditioned upon an error outcome. To eliminate comparisons with insignificant differences or
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Table 1 Demand vector 1 2 3 4 5
Average Path Length (APL) Index for Cross-Training Structures
1
2
3
4
3.18 1.67 1.63 2.06 2.29
2.53 1.57 1.77 2.07 1.72
2.29 1.40 1.60 1.88 1.35
2.33 1.60 1.61 1.14
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0.82
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Error (%)
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8911 8034 9090 9546
2.10 4.54 1.26 2.33
680 1552 632 705
0.36
Performance Evaluation of the APL Metric
Error (%) Variability Number of pairwise Predicted scenario comparisons correctly (%) Average Max.
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48 48
8333 8333
1.46 1.46
2.28 1.91
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24 24
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77 78
8896 8974
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1.13 0.98
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8333 8333
4.73 4.75
6.80 6.59
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8993 9021
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Patterns
overlapping confidence intervals, we threw out any comparisons for which the performance outcomes (customer average waiting times) were less than 0.1% different. Overall, our experiment generated 969 valid pairwise comparisons and threw out only seven. We highlight the fact that the APL metric has an impressive overall predictive ability of 89.1% to rank the more-effective cross-training structure between two alternatives. When the APL metric chooses wrongly, the average and maximum relative performance between the less-effective structure chosen by the APL, and the more-effective structure, are 2.1% and 6.8%. This supports our assertion that, in general, the smaller the world of the WS network i.e., the smaller the APL, the better the performance of the corresponding cross-training structure. This also provides a very positive indication of the usefulness of the small world approach to these problems, suggesting that further investigation may be fruitful. The WS-APL method is a very good one in light of two criteria that we have set for a methodology: (1) the method should achieve a good rate of correct predictions over a reasonable test suite and have reasonably small errors, and (2) the ranking comparison should be simple to compute. This latter objective is achieved by solving a deterministic WS model to compute the APL, thereby avoiding the complex analysis of the underlying parallel queueing network model, which is intractable. Table 2
Table 3
Several metrics for the effectiveness of a design have been developed from an operations research perspective. Iravani et al. (2005) tested the most relevant ones: the number of arcs, the eigenvalue structural flexibility (SF) index, and the loss probability of Jordan and Graves (1995). The performance results across the entire test suite are reported for reference in Table 3. We handle the case of equal (tied) indices by tossing a fair coin, resulting in a 50% chance of success. The number of arcs method resulted in a relatively poor correct prediction rate of 80.34% and generated a maximum error of 15.52%. On the other hand, it is the simplest method computationally. The results strongly support our intuition that it is often better to have more skills. On the other hand, the metric presented in Jordan and Graves (1995) has only a 5% greater correct prediction rate than APL and tends to generate slightly larger errors. However, it is important to note that this improved performance was obtained at the expense of vastly greater computational complexity as well as limited applicability. First, it required a search over the power set of the network nodes and also required simulation to compute a loss probability for each candidate of the search (and thus it did not meet our criterion of a deterministic method for assessing flexibility). Second, the need for a detailed probabilistic model to compute these loss probabilities makes it an open research problem to determine how to compute them under our shock models. Hence, the ensemble results for the Jordan and Graves (1995) metric exclude the shock models (half of our test cases). The interesting result of the benchmark test is that the WS-APL method is roughly equivalent to the SF index, which is based on a maxflow linear program. The SF index was computed as the dominant eigenvalue of an SF matrix. Each element i j of the SF matrix was in turn calculated by solving a linear programming model (i.e., a maxflow model) that finds the number of distinct paths through the cross-training graph by which demand i can be connected to demand j. Note that this model emphasizes
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a view of the network based on the demand nodes. The WS-APL method, in contrast, is based on a network model of the production sources (agents) and uses shortest paths to summarize the flexibility of the structure.
6.
Conclusion
7.
Electronic Companion
In this paper, we have shown how a cross-training structure for call-center agents can be converted to a WS network. Using standard shortest path algorithms, the APL index of the WS network can be computed easily. Over a large test suite, we show that this WS-APL method can be used effectively as an approximate method to distinguish the better-performing cross-training structure between two alternatives. That is to say, the WS-APL methodology is an effective decision-support tool for assessing flexibility. This deterministic index correctly identifies designs with short average waiting times, thereby suggesting that this type of graph-theoretic approach captures the key elements of good system design using limited cross-training (multifunctionality). Interestingly, this method harnesses small world theory to make predictive or normative statements about the flexible design of service operations. We have emphasized the application of call-center workforce cross-training, but our models have application to many service or manufacturing operations.
An electronic companion to this paper is available as part of the online version that can be found at http://mansci.journal.informs.org/. Acknowledgments
This work was supported in part by the National Science Foundation under Grants DMI-0099821, DMI-0423048, and DMI-0542063.
References Aksin, O. Z., F. Karaesmen. 2002. Designing flexibility characterizing the value of cross-training practices. Working paper, INSEAD, Fontainebleau Cedex, France.
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