Capability Flexibility: A Decision Support Methodology for Parallel Service and Manufacturing Systems with Flexible Servers Seyed M. Iravani1 • Bora Kolfal2 • Mark P. Van Oyen3 1
Department of Industrial Engineering and Management Sciences Northwestern University, Evanston, IL 60208, USA 2 Department of Finance and Management Science University of Alberta, Edmonton, AB T6G 0T1, CAN 3 Industrial and Operations Engineering University of Michigan, Ann Arbor, MI 48103, USA
[email protected] •
[email protected] •
[email protected] To obtain improved performance, many firms pursue operational flexibility by endowing their production operations with multi-skilled workers, flexible machines, and/or flexible plants. We focus on the problem of ranking (according to long run average wait in queue) alternative production system designs. We employ open, parallel queueing networks with partially flexible servers as our modeling paradigm. Prior literature introduced the concept of Structural Flexibility (SF) and methodologies to approximate the relative flexibility of production operations with multifunctional resources via a computationally tractable algorithm. The SF approach supported the robust design of flexible systems in a strategic context with very little information. Thus, the SF method did not model resources with differences in their capacities/speeds. In contrast, this paper extends the SF method to treat models with sources possessing heterogeneous capacity and multiple capabilities. We call this generalized approach the Capability Flexibility (CF) methodology. Simulation of a test suite indicates that the CF methodology captures fundamental relationships and insights for achieving flexibility through the interacting effects of capacity and multifunctionality of the production sources. Keywords: Operational Flexibility; Maxflow Algorithm; Parallel Queueing Systems, Cross-training.
1.
Introduction
The concept of the “capability flexibility” of a production operation (service or manufacturing) is introduced in this paper, and a method for quantifying it is developed and tested. As our context, we focus on service operations or make-to-order manufacturing environments where inventory is not held and short lead times are desirable. For such operations, the 1
negative consequences of variability in demand arrival rate or service times can be mitigated, among other means, by increasing production capacity (e.g., more workers, machines, plants, trucks, etc.) or increasing the operational flexibility of available capacity. Flexibility is a complex phenomenon with many facets. Among the flexibility mechanisms surveyed in Sethi and Sethi (1990), design for multifunctionality (e.g., cross-training labor or investing in operations/plants with multiple capabilities) is a very basic and powerful way of increasing operational flexibility (i.e., the variability-buffering effectiveness of the system’s capacity). Our scope targets decision support for how firms should endow their production operations with multiple capabilities (e.g., which skills to provide to each agent in a service center). Decision support in planning for multifunctionality is important for lead time reduction at the plant/operation level, which is in turn critical to improving overall supply chain performance. Thus, the tactical level decisions about how to create flexibility through multi-capability sources of capacity is coupled to strategic level considerations of supply chain performance. Our motivation is the need to better understand flexibility as a competitive strategy and design methods for the tactical implementation of flexibility. This motivation is derived not only from the needs of an individual production cell, but it also relates to macroeconomic phenomena. In the turmoil of the current world economic climate. almost every nation faces the challenge of whether or not they will be able to sufficiently develop and manage its organizations at the plant/office level to achieve a sufficiently well trained, equipped, and organized workforce to compete effectively in the new environment. Those workers who now enjoy relatively high wages can only continue to justify their high wages in those organizations in which they are able to effectively use their knowledge and skills to provide greater value than lower-paid workers. It is beyond our scope to survey the long history of approaches to human resources management, but increasing attention is being paid to skill develop, cross-training, and organizational function (see Boudreau et al. (2003) and Hopp and Van Oyen (2004). Sumukadas and Sawney (2004) have shown that within the wide range of employee involvement practices, power sharing practices are the most potent in advancing workforce agility. Further, power-sharing primarily means cross-training via job enrichment and/or job enlargement and self-managed teams. We agree with Mattoon (2003) that “It is clear that work force development remains a critical component of any economic development strategy ...” The authors have studied or worked with numerous organizations that are making un2
precedented changes in pursuit of higher levels of productivity and increased profitability through appropriate cross-training and worker coordination in operations. Call centers are good examples where technology, processes, and cross-training have been brought together to create service organizations that are highly flexible. Through flexible plants with mixed model automobile assembly, the authors have seen firsthand how multiple automakers have exploited the novel research of Jordan and Graves (1995) to build more and more assembly plants with an impressive ability to produce multiple vehicle types on the same line with seamless changes in product mix. Jordan et al. (2004) provides insight into how specific structures of cross-training for plant repair and maintenance workers (i.e., cross-training the categories of electrician, millright, pipe fitter, machine mechanic, and welding tool and die) can be very effective in an automotive manufacturing environment. Our goal is to provide high level decision support by developing a methodology for how to determine the most beneficial structure of cross-training of workers or to make the decision of which plant should be endowed the the ability to produce which products in a mixed-model mode of production. For generality, we employ a more abstract terminology and refer to production sources, demand types, and source capabilities. One might wonder why operational flexibility is so important. Simply put, operations (including supply chains) must be endowed with mechanisms to buffer against variability. Any production system will buffer variability (uncertainty) using some appropriate combination of (1) inventory, (2) capacity, or (3) (lead) time. Focusing on capacity, the effects of variability in demand can be reduced by either (1) increasing production capacity (e.g., more workers, machines, plants, trucks, etc.) or (2) increasing the operational flexibility of available capacity. Motivated by increasing competitive pressure to deliver both cost efficiency and responsiveness (as espoused by movements such as lean, time-based competition, and agile manufacturing), many firms are now trying to increase their operational flexibility. As mentioned above, cross-training labor and endowing equipment (or entire plants) with multiple capabilities are very basic and powerful ways of increasing the variability-buffering effectiveness of the system’s capacity. Without a good understanding of flexibility, it is easy for management to gravitate to the extremes: specialization (little to no flexibility) or generalization (full flexibility). The real question is how best to judiciously apply limited multifunctionality to achieve improved operational performance for the firm and for the customer. Once a budget/limit is set on the number of capabilities, we seek to inform the design of whom to cross-train for what 3
tasks in order to satisfy our constraint. For example, the intention may be to invest each worker with two skills. Our focus is on how best to select the two skills per worker from the many alternatives that may exist when many products/services are provided. We also note that our approach considers the effect of these choices on the entire group of workers, which exhibits complex network effects. For example, Hopp et al. (2004) provides insights into the special abilities of a “skill chain” (or even a partial chain) to form a structure that is particularly advantageous as a network. If an organization chooses to create production sources (workers, equipment, plants, etc.) with multiple capabilities (skills, functions, tooling, flexible processes, etc.) to respond to more than one demand type, the question becomes which capabilities to add to which sources in order to provide robust performance despite demand and service time variability. As an example, consider Figure 1, which shows three alternative design structures with multifunctional plants producing K = 4 demand/product types. The three structures differ in the manner in which capacity is allocated among the three or four plants and which products each plant will be capable of producing. Furthermore, one plant in Structure C has twice the capacity of all other plants in the structures. Production is modeled as a parallel queueing system with multi-capability servers following the (dynamic) longest queue policy. Processing times and interarrival times are generated from a Gamma distribution, which can accommodate any coefficient of variation (CV). Neither closed-form solutions nor good approximations are available to evaluate the performance measure (e.g., cycle times) of this class of flexible queueing systems under variability in demand and/or production process. The complexity of these systems is especially due to the flexible production sources (i.e., plants). The core issue is the following. How can one rank the performance of two alternative structures without resorting to a brute-force simulation of the alternative systems. We will benchmark a method by its ability to rank two designs, because the task of rank ordering a set of designs can be reduced to a sequence of pairwise comparisons. Our performance criterion is the long run time-average waiting time of customers in queue and not in service. Because our class of models is restricted to open queueing networks, by Little’s Law, our criterion is equivalent to the long run time-average number of jobs waiting in queue. On the face of it, the idea of attempting to develop a simple method to rank complex queueing system designs with partially flexible servers may seem naive. Certainly, lacking even good performance approximation methods for such queueing systems, a straightforward 4
approach is beyond the state of the art. Further, it is a fact that as the traffic/demand load applied varies, the set of designs will vary somewhat in terms of their rank ordering. Furthermore, it is beyond the state of the art to know how to control these systems according to an optimal policy (though numerical solutions are possible in the case of exponential service times for sufficiently small models). However, we will show that under a simple, implementable policy, we are able to develop a computationally based methodology to estimate the rank ordering of two given designs under the same demand processes. Moreover, we have been able to do so effectively without resorting to an approximation of the absolute performance of a design.
2.
Literature Survey
To identify the importance of the judicious choice of the cross-training structure, Jordan and Graves (1995) explored how a “chain” structure with each plant (source) capable of producing only two product types, as illustrated in Structure B of Figure 1, is a particularly powerful structure that achieves most of the potential benefits from flexibility while keeping costs relatively low. In the context of labor cross-training, having a limited number of capabilities per worker (e.g., two skills per worker in a chain structure) can also help improve quality (see Pinker and Shumsky 2000). We refer the reader to Gurumurthi and Benjaafar (2001), Hopp et al. (2004), Iravani and Krishnamurthy (2005), and Sheikhzadeh et al. (1997) for subsequent work on the value of chains in queueing systems. Note that Structure B of Figure 1 is an example of a “partial chain” as defined and studied in Hopp et al. (2004), while adding the obvious connecting capability to the specialized plant would make it a standard chain as defined in Jordan and Graves (1995). Gurumurthi and Benjaafar (2001) relate flexibility and throughput under varying parameters, congruent with the observations made in Hopp et al. (2004). Aksin and Karaesmen (2002) took a graph-theoretic approach to determining the maximum throughput achievable in symmetric, connected queueing network loss models and obtained some analytical results and useful insights on the importance of balance in the structure. If the design problem is limited only to information on the approximate arrival rates for the various demand types, one can assume standard (unit capacity) production sources and equal service times for the four products. In such “strategic” settings, the simple deterministic Structural Flexibility (SF) method of Iravani et al. (2005) established that it is 5
1
1
1
1
1
1
1
1
2 1
1
Structure A
Structure B
Structure C
Figure 1: An example of flexible plants possible to assign a SF index to design structures of multifunctionality and use those indices to rank structures by performance. Structures A and B (but not C) of Figure 1 both have plants/sources with unit capacity (as denoted by the number 1 beside the plant) and fall within the scope of systems treated by the SF method. As developed in Iravani et al. (2007), a small world network model can be constructed for which the Average Path Length (APL) metric is about as effective as the maxflow based computational method developed in Iravani et al. (2005). The key to the SF method was to focus on a graph-theoretic analysis of the structure of a design without explicitly modeling capacity, arrival rate or service rate information. As a result, the method required nearly homogeneous capacity servers to be accurate. Furthermore, the SF methodology assumes that every server allocates its effort roughly equally across all of its capabilities. A nice feature of the SF approach is that, for structures within its scope, it provides a clear understanding of the intrinsic flexibility of a structure, as well as a simple yet powerful method to distinguish effective structures among alternatives. The index was obtained as the dominant eigenvalue of a SF matrix formed by computing the number of nonoverlapping paths through the structure from every demand type to every other. While it causes complications for the SF method, quantifying the source capacities unique to each source is often of great practical importance for modeling fidelity, especially at the tactical level. Even in a strategic setting, it is not uncommon for a firm to recognize that, for example, 80% of its production effort may be expended on 20% of its product lines. Some plants may have much greater capacity than others as well, as is the case in Structure C of Figure 1, where the plant three (serving classes 1, 3, and 4) has twice the capacity of the first two plants. To rank two structures with heterogeneous capacity (such as comparing C 6
to A, C to B, or A to B), we develop the Capability Flexibility (CF) methodology, which incorporates source/server speeds and mean service times in addition to demand arrival rates. The CF method uses a linear program to assess and rank alternate cross-training designs with respect to their flexibility, i.e., their effectiveness in minimizing the long run average (equivalently, expected) waiting time of the customers in the queueing network disregarding time in service. It improves upon the accuracy of the SF method and greatly expands the class of systems that can be ranked according to flexibility. For example, it allows the comparison of any pair of structures in Figure 1, and thus selection of the most flexible design. Simulation of a large test suite indicates that the CF methodology captures fundamental insights into how to achieve flexibility through a mix of source capacity and source multifunctionality. As one might expect, the CF method is more nuanced than the SF method, but it is still dramatically more intuitive and computationally tractable than the queueing system it approximates. Our approach also provides additional insights into the behavior of parallel queueing networks (even those without exponential process times). The first order necessary conditions for stability are well-understood for our class of parallel queueing networks with multifunctional servers (see Andradottir et al. (2003)). This paper breaks ground in providing a heuristic, deterministic, second-order approximation of a capability-design’s relative flexibility/performance. The success of our index-based method for ranking designs, however, does not resolve the problem of how to approximate the absolute performance of this class of queueing systems. In the sequel, Section 3 presents the Capability Demand/Source (CDS) graph as a system model. In Section 4, we relate the CDS graph of a system to its flexibility via a linear program and introduce the CF vector and obtain the CF index from it. In Section 5, we numerically evaluate the accuracy of the predictions made by the CF index.
3.
The Capability Demand/Source (CDS) Graph
We use the terms “source” for a worker, machine, or plant, “capability” for a worker skill or a machine/plant production capability, and “demand type” for a particular task, job, or product that must be serviced by the sources. Our method is based on a parsimonious, deterministic model called the CDS graph, which we illustrate for the Structure C in Figure 1 in Figure 2.Left. The CDS graph contains a set of source nodes S = {s1 , s2 , . . . , sN }, where 7
N is the number of sources in the structure, a set of demand nodes D = {d1 , d2 , . . . , dK }, where K is the number of demand types in the structure, and A, which is the set of all arcs (s, d) in the structure, where s ∈ S and d ∈ D. A undirected arc connecting a source to a demand node indicates the capability of that source to service the connected demand type. We define the graph defined by these arcs as the “system structure.” The service rate vector, µ = (µ1 , µ2 , . . . , µK ), represents the service rates of demand types d1 to dK when processed by a unit-capacity server, whereas the demand vector, λ = (λ1 , λ2 , . . . , λK ), represents the demand arrival rates. The source speed vector ν = (ν1 , ν2 , . . . , νN ) represents the speeds of the source nodes s1 to sN . A source with a capacity/speed of νs is modeled as a single server for which a service time realization that would take τ time units under a unit speed server takes τ /νs time units. We assume that every server follows the longest queue policy within their capability set, because it is robustly effective, commonly used, and easy to implement. As a mild necessary condition for the stability of the queueing system, we assume that source capacities “fit” demand in the following sense. A structure with source speed vector ν and service and arrival rate vectors µ and λ “fits” an environment if it is possible to allocate source capacities in such a way that all average demands are met (an LP can be easily constructed to examine this). As an example, the structure in Figure 2.Left fits when the source speed vector is ν = (1, 1, 2), but not when ν = (2, 1, 1), since the average demand for type 4 cannot be satisfied, and therefore that queue will grow to infinity. Note that connected systems are often more flexible than disconnected systems, since they have the potential to directly or indirectly shift capacity from any source to any demand type. Thus, we focus on connected structures, where there is at least one path through the system structure between any demand node pair di and dj (or any source pair si and sj ).
4.
The Capability Flexibility (CF)
Within the context of flexibility of parallel queueing models of production systems, we begin with an intuitive definition of the phenomenon we refer to as capability flexibility. Definition: Capability flexibility refers to the relative effectiveness of a queueing system (with average traffic arrival rates specified for each demand type and with multi-capability sources that are quantified by capacity) in achieving short waiting times via the reallocation of production in response to variability (e.g., fluctuations in the demand process, work content, product mix, etc.).
8
1 = 1.25 d1 1 = 1
100% =1 1
=1 s 1 1
d2
d2
= 1 s2 2
= 1.25
2
= 0.5
40% s2
60% m3 = 1.35
3 = 0.5 =1 3
12.5% =2 3
= 2 s3 3
d4
1
12.5%
2 = 0.5 = 1.25 2
=1 2
d3
s1
d1
3
= 0.5
4
= 1.25
25% s3 50%
4 = 1.25 = 1.25 4
d3
d4
Figure 2: Left: The CDS graph of Structure C; Right: the computation of m3 Our method generates an approximate measure of this type of operational flexibility in a scalar index called the Capability Flexibility (CF) index by generalizing the method used by Iravani et al. (2005). We evaluate the CF of a system by computing an index from the CF row vector, M . Element mj of M is intended to capture the operational flexibility of the system to allocate capacity to demand type dj while satisfying the other demand types. Let us illustrate the method for Structure C in Figure 1 for which the CDS graph is presented in Figure 2.Left. This system has a demand vector of λ = (1.25, 0.5, 0.5, 1.25), a service rate vector of µ = (1, 1.25, 1, 1.25), and a source speed vector of ν = (1, 1, 2). Element m3 of CF vector M shows how much capacity the structure can provide for demand type d3 while not compromising the other demand types d1 , d2 , and d4 . Figure 2.Right shows how element m3 is obtained. In order to find m3 , we compute an assignment of the source capacities to the demand types to satisfy the demand vector λ and maximize the total source capacity that can be directed towards demand d3 . Figure 2.Right presents such an allocation. The solid lines in Figure 2.Right represent the percentage of source capacities allocated to meet the demands given by the demand arrival rate vector λ. For example, to satisfy the demand rate of 1.25 for demand d1 , all the capacity of source s1 and 12.5% of capacity of source s3 is dedicated to the demand d1 , i.e., λ1 = 1.25 = (100%)µ1 ν1 + (12.5%)µ1 ν3 . The dashed lines represent the remaining unused capacities of the source nodes (as a percentage
9
of their total capacities), that can be directed to demand type d3 . Lastly, m3 is the sum of all of the capacities that can be directed to demand type d3 , hence m3 = (60%ν2 +37.5%ν3 )µ3 = (0.6 + 0.75) = 1.35. Clearly, m3 is the maximum demand rate that can be met for demand d3 , without creating an unstable system for a given demand vector λ. As Figure 2.Right shows, finding m3 is similar to solving a maxflow problem. Let variable ysd denote the percentage of source s ∈ S capacity allocated to demand d ∈ D. Also, let rs denote the remaining percentage of source s ∈ S capacity after all demands are satisfied on average, and zsj denote the idle capacity as percentage of source s ∈ S capacity that can be directed to demand j ∈ D via arc (s, j) ∈ A, if needed. The following LP computes mj , the maximum flow rate out of the demand type j ∈ D, as Z ∗ , the maximized objective function: X
max Z =
νs (zsj + ysj )µj
{s:(s,j)∈A}
X
s.t. ysd + rs ≤ 1
∀s∈S
(1)
νs ysd µd ≥ λd
∀d∈D
(2)
∀ s ∈ {s : (s, j) ∈ A}
(3)
{d:(s,d)∈A}
X
{s:(s,d)∈A}
zsj ≤ rs zsd ≥ 0,
ysd ≥ 0
rs ≥ 0
∀ (s, d) ∈ A
∀ s ∈ S.
The first set of constraints, (1), deduces for each source s ∈ S the fraction, rs , of remaining capacity of each source given the effort commitments captured in ysd . The second set of constraints, (2), ensures that each demand d ∈ D is satisfied on average under ysd . Constraint (3) relates idle capacity that can be directed to the demand j ∈ D to the skill set of source s ∈ S (the cross-training structure). Constraint (3) ensures that the idle capacity of only the sources that are capable of serving demand j are considered for generating flow out of demand j. Note that in our setting, mj is the maximum amount of demand j that can be met while the structure also meets all other demand types.
4.1
The CF Index
We obtain the elements of the CF row vector M by solving the linear program presented in this section for each demand type dj , where j = 1, 2, . . . , K. Therefore, the CF vector 10
requires solving K linear programs. For example, the CF row vector for the system for which the CDS graph is presented in Figure 2.Left is M = (2.10, 1.25, 1.35, 2.31). It is interesting to note that the SF matrix developed in Iravani et al. (2005) focused on the flow through the sources (all pairs). This required N (N − 1)/2 maxflow linear programs. The approach here focuses instead on the ability to meet demand, defining a new approach suitable to our richer model, which has capacity and demand rate information explicitly modeled. The result is only K linear programs. In many applications such as call centers and service centers, there are far more sources than demand types, so our approach makes the computation tractable, and we will show that it is also effective for the class of models we consider. The next step of our CF approach computes the ρj ratio for each demand type from the element mj of the CF vector M as follows: ρj =
λj mj
∀ dj ∈ D.
(4)
For a demand type dj , ρj is the ratio of its arrival rate λj to mj . It should be noted that ρ defined in (4) is an adaptation of the “traffic intensity” from the queueing theory. In our setting, we utilize ρ to capture the relative size of the total capacity that can be directed towards a demand type compared to the arrival rate of the same demand type. As we only focus on stable systems, the ρ values will be smaller than one. The row vector of ρ values for the system for which the CDS graph is presented in Figure 2.Left is ρ = (0.60, 0.40, 0.37, 0.54). We expect a more flexible structure to be able to direct more total capacity to demand types, resulting in larger CF vector M elements and hence, smaller ρ values. Because we expect a more flexible system to have smaller ρ values, we take the mean of the ρ values presented in (4) as the CF index, ICF : ICF = (
X
ρj )/K .
(5)
dj ∈D
We conjecture that structures with smaller ICF are more flexible and therefore result in a lower average waiting times in queue only. Our next section will test this conjecture. For a given structure, let λmax be the arrival rate vector with the largest elements that the structure can meet on average, i.e., any increase in an element of λmax makes the LP presented in this section infeasible. If we set the arrival rate vector to uλmax , with u ∈ [0, 1], u values close to 1 correspond to high system utilization levels. In such cases, the excess capacity that can be redirected via the cross-training structure becomes very limited and the 11
value of flexibility reduces. As a separate case, if we keep adding source skills to a system, the cross-training structure will get closer to that of total flexibility where every source is capable to serve any demand type. For structures with total system capacity levels that are close to each other, the chances of getting a tie with ICF increases if the utilization is too high or if the structures are “saturated” with too many skills. If two systems have the same ICF for an arrival rate vector λ, we recalculate the ICF indices for those systems for a lower arrival rate vector uλ, where u ∈ [0, 1), until the tie is broken. When needed, we started with u = 0.9 and picked u values as multiples of 0.1. In our numerical studies (presented in Section 5), we did not have to use a secondary tie-breaking rule, however we now present further tie-breaking rules for the sake of completeness. As our second tie-breaking rule, we pick the more flexible system to be the one with larger total P number of capabilities weighted by the source speeds (i.e., use the index ICC = s∈S νs cs , where cs is the number of capabilities of source s). If there is still a tie, we choose the one P with larger total number of capabilities (i.e., use IC = s∈S cs ). And lastly, we randomly choose among the structures if we still cannot break the tie.
5.
Testing the Capability Flexibility Method
We used simulation to evaluate the accuracy of the CF index over an extensive set of stable, open, parallel queueing environments. These environments represent systems such as service/call centers or parallel flexible machines or factories. In our study a structure is considered to be better than another, if it results in a smaller overall long run average (equivalently, expected) waiting time in queue only.
5.1
Structures and Environments Under Study
Our study includes two parameter sets for six test suites, each designed for a particular number of demand types and a demand arrival rate vector. The test suites 1, 2, 4, and 5 are based on the parallel system test suites 2–5 used in Iravani et al. (2005). The structures of all of the six test suites are shown in Figures 3 to 8, where the numbers to the right of the demand type nodes are the demand rates. To avoid clutter, the service rates for the test suites for each parameter set is presented separately in Table 1. Similarly, the source speeds for structures in each test suite are presented in the Appendix in Figures A4 and A8, for sets 1 and 2, respectively. The arrival rate vectors for the six test suites are also provided below: 12
λ(1) = (1.5, 1, 0.5, 0.5, 1, 1.5) λ(2) = (1.5, 1.5, 1.5, 0.5, 0.5, 0.5) λ(3) = (2, 2, 0.5, 0.5, 0.5, 0.5)
λ(4) = (0.5, 0.5, 1, 1, 2, 2, 1, 1, 0.5, 0.5) λ(5) = (1, 1, 1, 1, 1, 1, 1, 1) λ(6) = (1.12, 0.94, 1, 0.88, 1.06, 1)
1
a
1.5
1
a
1.5
1
a
1.5
1
a
1.5
1
a
1.5
2
b
1
2
b
1
2
b
1
2
b
1
2
b
1
3
c
0.5
3
c
0.5
3
c
0.5
3
c
0.5
3
c
0.5
4
d
0.5
4
d
0.5
4
d
0.5
4
d
0.5
4
d
0.5
5
e
1
5
e
1
5
e
1
5
e
1
5
e
1
6
f
1.5
6
f
1.5
6
f
1.5
6
f
1.5
6
f
1.5
Structure 1-1
Structure 1-2
Structure 1-3
Structure 1-4
Structure 1−5
Figure 3: Structures for demand arrival rate λ(1) Note that the last two suites have almost uniformly distributed demand, while suites 3 and 4 have some demand rates four times larger than others. Our numerical study includes two different set of parameters for service rates as follows: Table 1: Service Rate Vectors Service rates for parameter set 1 µ(1) = (1.25, 1.1, 1, 1, 1.1, 1.25) µ(2) = (1.1, 1.2, 1.1, 1.2, 1, 1) µ(3) = (1.25, 1.2, 1.1, 1, 1, 1)
µ(4) = (1, 1, 1.1, 1.2, 1.25, 1.25, 1.2, 1.1, 1, 1) µ(5) = (1.25, 1.2, 1.2, 1.15, 1.15, 1.1, 1.1, 1) µ(6) = (1.1, 1.2, 1, 1, 1.2, 1.1)
Service rates for parameter set 2 µ(1) = (1, 1.5, 0.5, 0.5, 1.5, 1) µ(4) = (0.8, 0.6, 1.2, 0.8, 1, 1, 0.8, 1.2, 0.6, 0.8) µ(2) = (1.5, 1.5, 1, 0.75, 0.5, 0.5) µ(5) = (0.75, 0.75, 0.8, 1.25, 1.25, 0.8, 0.75, 0.75) µ(3) = (1.5, 1.5, 0.75, 0.75, 0.5, 0.5) µ(6) = (1.5, 0.75, 1, 1, 0.75, 1.5) As Table 1 shows, in parameter set 1 the service rates vary up to 25% within each test suite, while set 2 has some cases with service rates up to 300% faster. The speed vectors for each structure within the test suites are given in Figures A4 and A8 in Appendix. As these figures show for parameter set 1, the sources in suites 1 and 5 have the same speed, while some sources in suite 2 are four times faster than other sources in the same structure. For set 2, some sources are three times faster than others in suites 3 and 6. Structures in suites 1, 4, and 5 have the same number of sources, whereas suites 2, 3, and 6 contain structures with fewer or more sources compared to the other structures in the same test suite. The CF indices and the rank of the structures obtained via the CF index after tie-breaking rule is applied, if required, are found in Appendix Figures A1 and A2 for set 1 and Figures 13
1
a
1.5
1
a
1.5
1
a
1.5
2
b
1.5
2
b
1.5
2
b
1.5
3
c
1.5
3
c
1.5
3
c
1.5
4
d
0.5
4
d
0.5
4
d
0.5
5
e
0.5
5
e
0.5
5
e
0.5
6
f
0.5
6
f
0.5
6
f
0.5
a
1.5
b
1.5
c
1.5
d
0.5
e
0.5
f
0.5
Structure 2-1
Structure 2-3
Structure 2-2 a
1.5
a
1.5
1
1 b
1
1.5
b
1.5
2
2 c
2
1.5
c
1.5
3
3 d
3
0.5
d
0.5
4
4 e
4
0.5
e
0.5
5
5 f
5
0.5
f
0.5
Structure 2-5
Structure 2-4
Structure 2-6
Figure 4: Structures for demand arrival rate λ(2)
1
a
2
1
a
2
1
a
2
2
b
2
2
b
2
2
b
2
3
c
0.5
3
c
0.5
3
c
0.5
4
d
0.5
4
d
0.5
4
d
0.5
5
e
0.5
5
e
0.5
5
e
0.5
f
0.5
6
f
0.5
6
f
0.5
a
2
b
2
c
0.5
d
0.5
6
1
Structure 3-3
Structure 3-2
Structure 3-1 a
2
a
2
b
2
1 2
b
2 2
3
c
1 c
0.5
0.5
3 4
d
2 d
0.5
0.5
4 5
e
3
0.5
e
0.5
e
0.5
f
0.5
f
0.5
5 6
f
Structure 3-4
0.5
Structure 3-5
Structure 3-6
Figure 5: Structures for demand arrival rate λ(3)
14
1
a
0.5
1
a
0.5
1
a
0.5
2
b
0.5
2
b
0.5
2
b
0.5
3
c
1
3
c
1
3
c
1
4
d
1
4
d
1
4
d
1
5
e
2
5
e
2
5
e
2
6
f
2
6
f
2
6
f
2
7
g
1
7
g
1
7
g
1
8
h
1
8
h
1
8
h
1
9
i
0.5
9
i
0.5
9
i
0.5
j
0.5
10
j
0.5
10
j
0.5
10
Structure 4-1
Structure 4-2
Structure 4-3
1
a
0.5
1
a
0.5
1
a
0.5
2
b
0.5
2
b
0.5
2
b
0.5
3
c
1
3
c
1
3
c
1
4
d
1
4
d
1
4
d
1
5
e
2
5
e
2
5
e
2
6
f
2
6
f
2
6
f
2
7
g
1
7
g
1
7
g
1
8
h
1
8
h
1
8
h
1
9
i
0.5
9
i
0.5
9
i
0.5
10
j
0.5
10
j
0.5
10
j
0.5
Structure 4-4
Structure 4-5
Structure 4-6
1
a
0.5
1
a
0.5
2
b
0.5
2
b
0.5
3
c
1
3
c
1
4
d
1
4
d
1
5
e
2
5
e
2
6
f
2
6
f
2
7
g
1
7
g
1
8
h
1
8
h
1
9
i
0.5
9
i
0.5
j
0.5
10
j
0.5
10
Structure 4-7
Structure 4-8
Figure 6: Structures for demand arrival rate λ(4)
15
1
a
1
1
a
1
1
a
1
2
b
1
2
b
1
2
b
1
3
c
1
3
c
1
3
c
1
4
d
1
4
d
1
4
d
1
5
e
1
5
e
1
5
e
1
6
f
1
6
f
1
6
f
1
7
g
1
7
g
1
7
g
1
h
1
8
h
1
8
h
1
8
Structure 5-2
Structure 5-1
Structure 5-3
1
a
1
1
a
1
1
a
1
2
b
1
2
b
1
2
b
1
3
c
1
3
c
1
3
c
1
4
d
1
4
d
1
4
d
1
5
e
1
5
e
1
5
e
1
6
f
1
6
f
1
6
f
1
7
g
1
7
g
1
7
g
1
h
1
8
h
1
8
h
1
8
Structure 5-4
Structure 5-6
Structure 5-5
1
a
1
1
a
1
1
a
1
2
b
1
2
b
1
2
b
1
3
c
1
3
c
1
3
c
1
4
d
1
4
d
1
4
d
1
5
e
1
5
e
1
5
e
1
6
f
1
6
f
1
6
f
1
7
g
1
7
g
1
7
g
1
h
1
8
h
1
8
h
1
8
Structure 5-7
Structure 5-8
Structure 5-9
Figure 7: Structures for demand arrival rate λ(5)
16
1
a
1.12
1
b
0.94
1
2
c
1
d
0.88
3
d
0.88
e
1.06
4
e
1.06
f
1
f
1
a
1.12
1
b
0.94
1
2
c
d
0.88
3
e
1.06
4
f
1
a
1.12
b
0.94
c 2
3
Structure 6−2
Structure 6-1
Structure 6−3 a
1.12
1
a
1.12
1
b
0.94
2
b
0.94
1
2
c
1
3
c
1
d
0.88
3
d
0.88
4
d
0.88
e
1.06
4
e
1.06
5
e
1.06
f
1
f
1
6
f
1
a
1.12
1
b
0.94
2
c
3
4
Structure 6−4
Structure 6−5
Structure 6−6
Figure 8: Structures for demand arrival rate λ(6) A5 and A6 for set 2, respectively. Even though we did not need to use the tie-breaking indices ICC (number of speed-weighted capabilities) and IC (number of capabilities) for our test suite, they are provided in Figure A3 for set 1 and in Figure A7 for set 2 in Appendix. Our test cases were chosen to satisfy two conditions. First, the utilization of every source must be at least 50% so as to avoid excessive waste of capacity that would usually be unacceptable in practice. Second, the difference between the highest and the lowest source utilization in every structure is at most 25%. This feature tends to create a fair workload balance, but still allows significant variation. For example, in a call center environment, managers avoid having an agent with 25% more idle time than another. More generally, good management will use judgement to consider the effects of learning and forgetting, and therefore will avoid cases where skills are used infrequently and therefore get “rusty.”
5.2
Evaluation Process
Our evaluation process is based on pairwise performance comparisons of every possible pair of structures within the test suites for sets 1 and 2. For every pair of structures in the same test suite, we predict that the structure with the smaller CF index has a lower average waiting time in queue only. We take this approach because it is the most fundamental mechanism for evaluating the performance of designs. All other approaches such as determining a ranked 17
ordering are based on this pairwise comparison. Further, it is consistent with the previous approach that was used in Iravani et al. (2005) and Iravani et al. (2007). We then used simulation to obtain the average waiting time (in queue only) for each structure. In our simulation when a source completes service, it is assigned to the longest queue (with random selection for tie-breaking). Within a queue, service is nonpreemptive and first in first out (FIFO). If a demand can be served by multiple idle sources, the source with longest idle time is chosen to serve the demand. Our simulation was written in the C++ language. Runs typically ended after 100,000 jobs exited the system, in addition to a warm-up period of 25,000 jobs. Depending on the demand rate vector, each run was replicated either 1000 or 1500 times. For variance reduction purposes, common random number streams are used: one dedicated to each demand arrival and source service times. To test each structure under a reasonable variety of operating conditions, we used the Gamma distribution for service times and demand interarrival times since it can generate CV’s of 1 (the exponential case) and 2. Relatively higher variability system are the ones for which flexibility is most needed. To investigate variation in the degree of congestion, we use three scalings of the demand arrival rate vector: uλ(i) with u = 1, 0.9, and 0.8 for i = 1, 2, . . . , 6. So, our experiments have 6 combinations of scaling factor u and CV for each demand vector λ(i). We consider structure which results in a smaller expected waiting time to be a more flexible structure. If, and only if, the simulation contradicts the CF index and the simulation results differ more than 0.1 percent (at the 95% confidence level), we count that as a wrong prediction and calculate the percentage relative error, ∆. In such cases, the percentage relative error is defined as ∆=
WCF − WSim × 100%, WSim
where WCF is the simulated performance (i.e., expected waiting time) of the structure chosen by the CF index, while WSim denotes the truly better performance among the pair revealed by simulation. We then calculate the number of wrong predictions among the total number of pairwise comparisons as a measure for evaluating the prediction power of the CF index. Tables 2 and 3 summarize the number of comparisons made (that differ more than 0.1% performancewise), the number of wrong predictions made by the CF index, and the percentage of correct predictions for set 1 and set 2, respectively. Given that the CF index’s prediction is wrong,
18
it also presents the average and maximum percent error over such cases. Although the CF index is only a deterministic metric obtained by using a fairly simple LP, Tables 2 and 3 present very high prediction success rates of 98.60% and 98.04% for sets 1 and 2, respectively. Over the small set of cases with an error, for set 1 (set 2) the average error percentage is 1.47% (1.85%) and the maximum error percentage is 2.86% (4.45%). Demand Vector λ(1) λ(2) λ(3) λ(4) λ(5) λ(6) Total
Table 2: Performance evaluation of the CF index for set 1. # of # of Pairwise # of Wrong % of Correct Error, ∆ Struct. Comparisons Predictions Predictions Average Max. 5 60 1 98.33% 2.86% 2.86% 6 90 2 97.78% 0.88% 1.43% 6 90 0 100% 0% 0% 8 166 7 95.78% 1.44% 2.35% 9 216 0 100% 0% 0% 6 90 0 100% 0% 0% 40 712 10 98.60% 1.47% 2.86%
Demand Vector λ(1) λ(2) λ(3) λ(4) λ(5) λ(6) Total
Table 3: Performance evaluation of the CF index for set 2. # of # of Pairwise # of Wrong % of Correct Error, ∆ Struct. Comparisons Predictions Predictions Average Max. 5 60 0 100% 0% 0% 6 90 3 96.67% 3.06% 4.45% 6 90 0 100% 0% 0% 8 168 6 96.43% 2.27% 4.10% 9 215 4 98.14% 0.72% 1.12% 6 90 1 98.89% 0.12% 0.12% 40 713 14 98.04% 1.85% 4.45%
It should be noted that the error percentages are small compared to the average and maximum spreads (56.68% and 348.92%, respectively for set 1, and 48.83% and 271.15%, respectively for set 2), where the spread for two structures is defined as the percentage average waiting time difference of those structures with respect to the smaller of the two. Detailed results for set 1 and set 2 are presented in Table 4 below, showing excellent performance for the CF index. Table 4: Performance spread for sets 1 and 2 Demand # of Wait Spread, Set 1 Wait Spread, Set 2 Vector Struct. Average Max. Average Max. λ(1) 5 51.61% 189.53% 50.24% 185.16% λ(2) 6 33.91% 125.68% 35.41% 132.12% λ(3) 6 55.09% 149.98% 65.77% 188.63% λ(4) 8 75.33% 348.92% 45.19% 152.09% λ(5) 9 56.38% 285.05% 57.30% 271.15% λ(6) 6 50.30% 177.15% 30.80% 101.50% Total 40 56.68% 348.92% 48.83% 271.15%
19
6.
CF Index versus Alternative Indices
In the prior section, our large test suite provided a good basis on which to empirically assess the effectiveness of the CF index. The CF method is still a relatively simple computational procedure, but the model for capability flexibility has much greater parametric detail. It is very significant that the the CF index is very effective. In fact, the prediction rate is so high that there is little room left for improvement. In addition to the above, the fact that the CF index is a heuristic makes it quite important to carefully benchmark it relative to the best known alternatives. We have test these two alternatives: (1) the SF index developed in Iravani et al. (2005), and (2) the Jordan-Graves index, which represents our best effort to extend the index concept proposed in Jordan and Graves (1995)to our modeling framework. As is presented in the Online Appendix of Iravani et al. (2005), the flexibility metric conceptualized in Jordan and Graves (1995) for periodic inventory models does not apply directly to our models, and we adapted it to unit-capacity queueing network models. Thus, we must state clearly that any shortcoming in the performance do not imply that the JordanGraves index is not appropriate in other settings. For the sake of clarity, we detail the development the Jordan-Graves index. Jordan and Graves (1995) originally considered periodic inventory models with demand loss in which source capacities are deterministic and demand for different product types are stochastic. Structures are evaluated in terms of their expected demand loss in Jordan and Graves (1995). They show that, for a realization of demands di ∈ D, the minimum demand loss is equivalent to SH = maxP ∈P
³X
X
di −
i∈P
´ cj
(6)
j∈S −1 (P )
where SH denotes the shortfall for the structure, cj is the capacity of the source j, P is the power set of the demand set D, S −1 (P ) is the set of source nodes that can serve at least one demand type in the set P . Hence, S −1 (P ) is the subset of sources that can serve the demand subset P . The expected shortfall of the structure is obtained by taking the expectation of (6) over demand distribution. Hence, " E[SH] = E maxP ∈P
³X i∈P
Di −
X
´ cj
# ,
(7)
j∈S −1 (P )
where Di is the random variable for demand of type i. Note that expected shortfall for total 20
flexibility (where every source is pooled to serve any demand type) can be written as " # ³ X X ´ E[SHtf ] = E max 0, Di − cj . i∈D
(8)
j∈S
If the expected demand loss decreases with increased flexibility, as assumed in Jordan and Graves (1995), then expected shortfall, E[SH], of structures with higher flexibilities should be smaller and closer to expected shortfall of structures with total flexibility, E[SHtf ]. As comparing (7) and (8) might be very difficult, the probability of the shortfall of a structure exceeding the shortfall of total flexibility is proposed instead. This probability can be expressed as follows: " IJG = P r maxP ∈P
³X i∈P
Di −
X
´ cj
# ³ X X ´ > max 0, Di − cj .
j∈S −1 (P )
i∈D
(9)
j∈S
For any two structures, the one with higher flexibility is expected to have a lower probability, IJG as given in (9), which we call Jordan-Graves index. A simulation program is used to compute the Jordan-Graves index for the test cases in Iravani et al. (2005) (see the Online Appendix of Iravani et al. (2005) for a detailed explanation). As noted in Section 5, the test suites 1, 2, 4, and 5 are extensions of the parallel system test suites used in Iravani et al. (2005). However, even for the structures that are common in both papers, we can not directly compare the performance of the CF index to the performance of the SF index, as SF index is not applicable to situations where demand types have different service rates and sources have different speeds. Therefore, in order to make a comparison, we applied the CF index to the test suite given in Iravani et al. (2005) with the setting used in that paper, i.e., unit demand service rates for all of the demand types and unit speeds for all of the sources. Table 5 summarizes the predictive performance of both the CF and the SF indices for the demand vectors D1 to D5 as defined in Iravani et al. (2005). Detailed information on the performance of the SF and the Jordan-Graves indices for the structural flexibility test suite was provided in Iravani et al. (2005), and for that test suite, the CF Index has a correct prediction percentage of 96.7%, outperforming the correct prediction percentage of both the SF index, which is 93.4%. It also outperforms the Jordan-Graves index, which is 95.5%. The average and maximum error percentages of the CF index are 1.86% and 3.81%, respectively, whereas the corresponding percentages for the SF index (Jordan-Graves index) are 2.31% and 6.32% (2.33% and 7.05%). Thus, the study suggests that the CF index outperforms 21
both the SF index and the Jordan-Graves index for the test cases presented in Iravani et al. (2005). Table 5: Performance evaluation of the CF and the SF indices for the structural flexibility test suite. CF Index SF Index Demand Number of % of Correct Error, ∆ % of Correct Error, ∆ Vector Structures Predictions Average Max. Predictions Average Max. D1 4 100% 0% 0% 100% 0% 0% D2 3 100% 0% 0% 100% 0% 0% D3 5 90% 0.62% 1.12% 100% 0% 0% D4 4 100% 0% 0% 100% 0% 0% D5 9 97.22% 3.10% 3.81% 88.89% 2.31% 6.32% Total 25 96.72% 1.86% 3.81% 93.4% 2.31% 6.32%
7.
Conclusions
The purpose of this work is to support strategic and tactical design of source multi-functionality to enhance performance through what we term Capability Flexibility (CF). We introduced the CF method as a relatively tractable yet powerful extension of the SF method introduced in Iravani et al. (2005). For test suites with open, parallel queueing models and flexible servers, performance analysis via simulation showed this new CF method to be effective in ranking alternative system designs according to waiting time in queue only (or, by Little’s Law, average length of the waiting line). Furthermore, the CF methodology is an improvement even in the special cases handled by the SF method. The CF methodology not only offers insight into how the interaction of capacity and flexibility can improve performance, but also offers a fast numerical algorithm for decision support in place of system simulation. For example, our methodology assists in the decision of whom to cross-train for what tasks in service centers with a budget/limit on the amount of multi-functionality by creating designs yielding robust performance despite demand uncertainty. Furthermore, it can also be incorporated into shift scheduling software. Each work shift schedule with cross-trained labor defines a distinct structure. The number of potential structures is extremely large. Attrition, vacations, leaves, demand fluctuations and the like force an organization to adapt to changes in the availability of skills and workers that make shift scheduling difficult. Given N workers, two shifts, and a list of each worker’s skill (capability) set, different partitions of the workforce between the two shifts result in a huge number of pairs of capability structures. Search based on simulation is computationally 22
intensive and complex to implement, so the speed and simplicity of the CF method allows larger problems to be addressed. Note that the CF methodology considers the effect of these design choices on the ensemble of workers, which exhibits complex queueing network effects. It thereby provides strategic support for creating designs yielding robust performance despite demand uncertainty.
Acknowledgments This work was supported by NSF under Grants No. DMI-0542063, 0423048 & 0500479.
References Aksin, O.Z. and F. Karaesmen. 2002. Designing flexibility: Characterizing the value of cross-training practices, Working Paper, INSEAD, Fontainebleau Cedex, France. http://home.ku.edu.tr/fkaraesmen/pdfs/flex2102.pdf Andradottir, S., H. Ayhan, and D.G. Down. 2003. Dynamic server allocation for queueing networks with flexible servers. Operations Research, 51, 952–968. Gurumurthi, S. and S. Benjaafar. 2001. Modeling and analysis of flexible queueing systems. Working Paper, Department of Mechanical Engineering, University of Minnesota, Minneapolis. http://www.me.umn.edu/labs/logistics/flexq.pdf Hopp, W.J., E. Tekin, and M.P. Van Oyen. 2004. Benefits of skill-chaining in production lines with cross-trained workers. Management Science, 50, 83–98. Hopp, W.J. and M.P. Van Oyen. 2004. Agile workforce evaluation: A framework for crosstraining and coordination. IIE Transactions, 36, 919–940. Iravani, S.M.R., B. Kolfal, and M.P. Van Oyen, 2007. Call center labor cross-training: It’s a small world after all. Management Science, 53, 1102–1112. Iravani, S.M.R. and V. Krishnamurthy. 2005. Workforce agility in repair and maintenance environments. Working Paper, Department of Industrial Engineering and Management sciences, Northwestern University, Evanston, IL. Iravani, S.M.R., M.P. Van Oyen, and K.T. Sims. 2005. Structural flexibility: A new perspective on the design of manufacturing and service operations. Management Science, 51, 151–166. 23
Jordan, W.C. and S.C. Graves. 1995. Principles on the benefits of manufacturing process flexibility. Management Science, 41, 577–594. Jordan, W.C., R.R. Inman, and D.E. Blumenfeld. 2004. Chained cross-training of workers for robust performance. IIE Transactions, 36, 953-967. Mattoon, R. 2003. Strategies for improving the Midwest work force. Chicago Fed Letter, 186, Feb. Pinker, E. and R. Shumsky. 2000. The efficiency-quality tradeoff of crosstrained workers. Manufacturing and Service Operations Management (M&SOM), 2, 32–48. Sethi, A.K. and S.P. Sethi. 1990. Flexibility in Manufacturing: A survey. The International Journal of Flexible Manufacturing Systems, 2, 289–328. Sheikhzadeh, M., S. Benjaafar, and D. Gupta. Machine Sharing in Manufacturing System: Total flexibility versus chaining. Working paper, Dept. of Mech. Engr., University of Minnesota, 1997. Sumukadas, R. and R. Sawhney. 2004. Workforce agility through employee involvement. IIE Transactions, 36, 1011–1021.
24
APPENDIX Demand Vector λ(1) λ(2) λ(3) λ(4) λ(5) λ(6)
Demand Scale 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8
1 3.354 2.743 2.332 3.475 2.845 2.435 3.309 2.820 2.399 5.248 4.554 3.939 3.785 3.161 2.810 3.358 2.353 1.713
2 3.423 2.862 2.398 3.475 2.805 2.347 3.309 2.850 2.486 3.975 3.236 2.818 3.785 2.867 2.387 3.358 2.353 1.713
3 3.285 2.624 2.266 3.475 2.969 2.563 2.896 2.071 1.601 3.834 3.157 2.734 3.785 2.810 2.356 3.358 2.353 1.806
4 3.066 2.191 1.659 3.808 3.033 2.492 2.905 2.208 1.801 5.696 4.609 3.906 3.785 2.558 1.873 2.568 1.899 1.433
Structures 5 3.286 2.689 2.341 3.475 2.947 2.529 3.171 2.481 2.041 5.143 4.483 3.913 3.785 2.625 2.022 3.358 2.353 1.795
6
3.370 2.341 1.707 2.515 1.857 1.456 6.444 5.521 4.775 3.785 2.724 2.207 3.358 2.406 2.104
7
8
9
4.173 3.489 2.977 3.785 2.558 1.821
4.507 3.844 3.368 3.785 2.558 1.821
3.785 2.558 1.821
Figure A1: CF indices of the test structures for the first part of the numerical study
Demand Vector λ(1) λ(2) λ(3) λ(4) λ(5) λ(6)
Demand Scale 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8
1 4 4 3 3 3 3 5 5 5 6 6 7 9 9 9 3 3 3
2 5 5 5 2 2 2 6 6 6 2 2 2 8 8 8 2 2 2
3 2 2 2 5 5 6 2 2 2 1 1 1 7 7 7 5 5 5
4 1 1 1 6 6 4 3 3 3 7 7 5 4 4 4 1 1 1
Structures 5 3 3 4 4 4 5 4 4 4 5 5 6 5 5 5 4 4 4
6
1 1 1 1 1 1 8 8 8 6 6 6 6 6 6
7
8
9
3 3 3 2 2 2
4 4 4 3 3 3
1 1 1
Figure A2: Ranks (ascending order) of the structures within each test suite in set 1, obtained via the CF index (with tie-breaking rule).
25
Demand Vector λ(1) λ(2) λ(3) λ(4) λ(5) λ(6)
Index IC ICC IC ICC IC ICC IC ICC IC ICC IC ICC
1 2 12 12 12 12 12 12 12 12 12 11 12 11.25 20 26 20 29 16 20 16 20 12 16 25 25
Structures 3 4 5 6 12 24 12 12 24 12 11 12 10 16 11 16 12 22.5 18 16 12 12 19.5 17.25 14.75 25 28 20 20 20 29 20 20 20 24 24 26 28 24 24 26 28 16 17 16 12 25 29 25 12.5
7
8
9
26 26 32 32
24 24 36 36
42 42
Figure A3: ICC (number of speed-weighted capabilities) and IC (number of capabilities) indices for the first part (set 1)
26
Demand Vector λ(1)
λ(2)
λ(3)
λ(4)
λ(5)
λ(6)
4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.50 1.00 0.75 1.00 1.00 0.75 1.50 1.25
Source 5 6 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 0.75 0.75 1.00 1.25 0.75
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.75 0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.75 0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2.00 2.00 1.75 1.50 1.00
1.00
1.25
Structure 1-1 1-2 1-3 1-4 1-5 2-1 2-2 2-3 2-4 2-5 2-6 3-1 3-2 3-3 3-4 3-5 3-6 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 6-1 6-2 6-3 6-4 6-5 6-6
1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.50 1.75 1.00 0.75 1.25 0.75 1.25 2.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.50 1.50 1.00
2 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.50 1.75 1.00 1.25 1.25 0.75 1.25 2.00 1.00 1.25 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.25 1.25 1.25 1.50 1.50 1.00
3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.75 1.00 0.75 1.00 1.00 1.25 0.75 1.50 2.25 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 2.00 2.00 2.00 1.75 1.00
7
8
9
10
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.25 1.25 1.00 1.00 1.00 1.00 1.00
1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00
Figure A4: Source speeds of the test structures for the first part (set 1) of the numerical study
27
Demand Vector λ(1) λ(2) λ(3) λ(4) λ(5) λ(6)
Demand Scale 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8
1 3.885 3.099 2.598 3.626 3.121 2.668 3.543 3.033 2.651 5.507 4.716 4.083 4.159 3.208 2.852 3.368 2.355 1.715
2 3.718 2.969 2.551 3.600 2.965 2.589 3.745 3.267 2.850 4.262 3.500 3.028 4.159 3.078 2.519 2.944 2.120 1.642
3 3.718 2.870 2.489 3.784 3.248 2.810 2.943 2.163 1.761 4.202 3.292 2.843 4.680 3.119 2.410 3.368 2.355 1.792
4 3.585 2.469 1.838 3.512 2.770 2.319 3.014 2.416 1.995 5.914 4.722 4.070 4.159 2.741 1.948 3.368 2.355 1.715
Structures 5 3.718 2.962 2.550 3.710 3.183 2.728 3.110 2.501 2.059 5.396 4.660 4.060 4.159 2.919 2.225 3.368 2.355 1.780
6
3.269 2.307 1.689 2.617 1.964 1.565 6.094 5.248 4.569 4.159 2.959 2.371 3.368 2.461 2.123
7
8
9
4.044 3.276 2.770 4.159 2.741 1.923
4.856 3.984 3.403 4.159 2.741 1.923
4.159 2.741 1.923
Figure A5: CF indices of the test structures for the second part of the numerical study
Demand Vector λ(1) λ(2) λ(3) λ(4) λ(5) λ(6)
Demand Scale 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8 1 0.9 0.8
1 5 5 5 4 4 4 5 5 5 6 6 7 8 9 9 3 3 3
2 4 4 4 3 3 3 6 6 6 3 3 3 7 7 8 1 1 1
3 2 2 2 6 6 6 2 2 2 2 2 2 9 8 7 5 5 5
4 1 1 1 2 2 2 3 3 3 7 7 6 4 4 4 2 2 2
Structures 5 3 3 3 5 5 5 4 4 4 5 5 5 5 5 5 4 4 4
6
1 1 1 1 1 1 8 8 8 6 6 6 6 6 6
7
8
9
1 1 1 2 2 2
4 4 4 3 3 3
1 1 1
Figure A6: Ranks (ascending order) of the structures within each test suite in set 2, obtained via the CF index. Tie-breaking rule is applied when required.
28
Demand Vector λ(1) λ(2) λ(3) λ(4) λ(5) λ(6)
Index IC ICC IC ICC IC ICC IC ICC IC ICC IC ICC
Structures 1 2 3 4 5 6 12 12 12 24 12 14 14 14 28 14 12 12 11 12 10 16 14 14 13.25 17.25 14 24.5 12 11 18 16 12 12 14 13.25 22 20.5 17.25 29 20 26 28 20 20 20 25 35.5 37 25 25 25 16 20 24 24 26 28 21 26.5 30.5 31.5 34.5 37 12 16 16 17 16 12 27 28 27 29 27 13.5
7
8
9
26 24 35 32 32 36 42 42 48.75 56.25
Figure A7: ICC (number of speed-weighted capabilities) and IC (number of capabilities) indices for the second part (set 2)
29
Demand Vector λ(1)
λ(2)
λ(3)
λ(4)
λ(5)
λ(6)
4 1.25 1.50 1.00 1.00 1.50 1.25 1.50 1.00 1.00 1.50 1.00 1.25 1.25 1.00 1.75 1.50
Source 5 6 1.25 1.50 1.00 1.50 1.00 1.50 1.25 1.25 1.50 1.50 0.75 1.50 0.75 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.00 1.00 1.25 1.75 1.50
1.50 1.25 1.25 1.00 1.25 1.50 1.00 1.00 1.00 1.25 1.25 1.00 1.25 1.25 1.25 1.50 1.25
1.00 0.75 0.75 1.25 1.00 1.25 1.75 1.25 1.25 1.50 1.25 1.25 1.25 1.25 1.25 1.50 1.50
1.00 0.75 0.75 1.50 1.25 1.00 1.75 2.25 1.25 1.25 1.25 1.50 1.25 1.25 1.50 1.50 1.50
2.00 1.75 1.75 2.00 1.25
1.00
1.00
Structure 1-1 1-2 1-3 1-4 1-5 2-1 2-2 2-3 2-4 2-5 2-6 3-1 3-2 3-3 3-4 3-5 3-6 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 6-1 6-2 6-3 6-4 6-5 6-6
1 1.00 1.00 1.00 1.25 0.75 1.00 1.00 0.75 1.50 1.00 1.75 0.75 0.75 1.25 0.75 1.00 2.25 1.00 1.50 1.50 1.50 1.25 1.00 1.50 1.25 1.25 1.50 1.50 1.50 1.25 1.25 1.25 1.25 1.25 3.00 2.00 1.50 1.50 1.75 1.25
2 1.00 1.00 1.50 1.25 0.75 1.50 1.00 1.00 1.75 1.75 1.75 0.75 1.00 1.25 0.75 1.25 2.50 1.50 1.50 1.50 1.50 1.75 1.25 1.00 1.00 1.50 1.25 1.25 1.25 1.25 1.50 1.25 1.25 1.25 1.75 1.50 1.75 1.50 1.50 1.25
3 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.25 1.25 1.00 1.25 1.00 1.50 0.75 1.75 2.50 1.25 1.25 1.25 1.00 1.25 1.50 1.00 1.25 1.25 1.25 1.25 1.25 1.50 1.25 1.50 1.25 1.50 2.00 1.50 1.75 2.00 1.50 1.00
7
8
9
10
1.25 1.25 1.25 1.00 1.00 1.25 1.25 1.00 1.50 1.25 1.25 1.25 1.50 1.50 1.25 1.25 1.25
1.00 1.25 1.25 1.00 1.25 1.50 1.25 1.25 1.50 1.25 1.25 1.50 1.25 1.25 1.25 1.00 1.00
1.75 1.25 1.25 1.50 1.50 1.25 1.00 1.00
1.25 1.75 1.75 1.25 1.00 1.00 1.25 1.25
Figure A8: Source speeds of the test structures for the second part (set 2) of the numerical study
30
