text: References Ak¸sin, Z., M. Armony, and V. Mehrotra (2007). The modern call center: A multi- disciplinary perspective on operations management research. Production and Operations Management 16 (6), 665–688. Ak¸sin, Z., B. Ata, S. M. Emadi, and C.-L. Su (2013). Structural estimation of callers’ delay sensitivity in call centers. Management Science 59 (12), 2727–2746. Aktekin, T. and R. Soyer (2012). Bayesian analysis of queues with impatient customers: Applications to call centers. Naval Research Logistics (NRL) 59 (6), 441–456. Aktekin, T. and R. Soyer (2014). Bayesian analysis of abandonment in call center opera- tions. Applied Stochastic Models in Business and Industry 30 (2), 141–156. Atlason, J., M. A. Epelman, and S. G. Henderson (2008). Optimizing call center staﬃng using simulation and analytic center cutting-plane methods. Management Science 54 (2), 295–309. Berzuini, C. and D. Clayton (1994). Bayesian analysis of survival on multiple time scales. Statistics in Medicine 13 (8), 823–838. Boyd, S., N. Parikh, E. Chu, B. Peleato, and J. Eckstein (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning 3 (1), 1–122. Brown, L., N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn, and L. Zhao (2005). Statistical analysis of a telephone call center: A queueing-science perspective. Journal of the American Statistical Association 100 (469), 36–50. 29

text: References

text: Ak¸sin,

text: Z.,

text: M.

text: Armony,

text: and

text: V.

text: Mehrotra

text: The

text: modern

text: call

text: center:

text: A

text: multi-

text: disciplinary

text: perspective

text: on

text: operations

text: management

text: research.

text: Production

text: and

text: Operations

text: Management

text: Ak¸sin,

text: Z.,

text: B.

text: Ata,

text: S.

text: M.

text: Emadi,

text: and

text: C.-L.

text: Su

text: Structural

text: estimation

text: of

text: callers’

text: delay

text: sensitivity

text: in

text: call

text: centers.

text: Management

text: Science

text: Aktekin,

text: T.

text: and

text: R.

text: Soyer

text: Bayesian

text: analysis

text: of

text: queues

text: with

text: impatient

text: customers:

text: Applications

text: to

text: call

text: centers.

text: Naval

text: Research

text: Logistics

text: (NRL)

text: Aktekin,

text: T.

text: and

text: R.

text: Soyer

text: Bayesian

text: analysis

text: of

text: abandonment

text: in

text: call

text: center

text: opera-

text: tions.

text: Applied

text: Stochastic

text: Models

text: in

text: Business

text: and

text: Industry

text: Atlason,

text: J.,

text: M.

text: A.

text: Epelman,

text: and

text: S.

text: G.

text: Henderson

text: Optimizing

text: call

text: center

text: staﬃng

text: using

text: simulation

text: and

text: analytic

text: center

text: cutting-plane

text: methods.

text: Management

text: Science

text: Berzuini,

text: C.

text: and

text: D.

text: Clayton

text: Bayesian

text: analysis

text: of

text: survival

text: on

text: multiple

text: time

text: scales.

text: Statistics

text: in

text: Medicine

text: Boyd,

text: S.,

text: N.

text: Parikh,

text: E.

text: Chu,

text: B.

text: Peleato,

text: and

text: J.

text: Eckstein

text: Distributed

text: optimization

text: and

text: statistical

text: learning

text: via

text: the

text: alternating

text: direction

text: method

text: of

text: multipliers.

text: Foundations

text: and

text: Trends

text: in

text: Machine

text: Learning

text: Brown,

text: L.,

text: N.

text: Gans,

text: A.

text: Mandelbaum,

text: A.

text: Sakov,

text: H.

text: Shen,

text: S.

text: Zeltyn,

text: and

text: L.

text: Zhao

text: Statistical

text: analysis

text: of

text: a

text: telephone

text: call

text: center:

text: A

text: queueing-science

text: perspective.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association


text: Appendix: Tuning Parameter Selection There are multiple penalty parameters in the optimization problem (5). Simultaneous selection of all the tuning parameters in a data-driven fashion can be a huge computational burden. As a remedy, we nest the selection of diﬀerent parameters inside the ADMM algorithm. We will discuss in more detail below. The roughness penalty parameters λ and λ control the smoothness of u and v when ui vi i i estimating U and V. Since we estimate U and V in (12) through tF-SVD, we adopt the adaptive tuning selection procedure as in Huang et al. (2009). In particular, the selection of λ and λ is embedded in the alternating algorithm of tF-SVD and use the generalized ui vi cross-validation (GCV) criteria. Closed-form GCV scores have been derived from leaving out rows and columns for fast computation (Huang et al., 2009). Another tuning parameter is ρ. Since ρ as a penalty parameter is left out in (12), it primarily serves as the step size in the dual update of (10). Although the step size in the ADMM algorithm can be ﬁxed as any positive value with little eﬀect on the ﬁnal solution theoretically, it does aﬀect convergence rate in practice (Boyd et al., 2011). To improve the convergence rate of the algorithm and alleviate the dependence on the initial choice of the penalty parameter, we follow He et al. (2000) and select ρ adaptively in each iteration. In particular, we ﬁrst deﬁne two residuals in each iteration as (cid:44) (cid:107) − − − (cid:107) Primal residual: z(l) H(l) U(l 1)V(l 1)T 2 , F (cid:44) (cid:107) − − (cid:107) Dual residual: s(l) H(l) H(l 1) 2 . F (cid:29) We remark that a small value of ρ leads to z(l) s(l), while a large value of ρ leads to (cid:28) z(l) s(l). To see this, we temporarily set Θ = 0, and examine (11). From the l’Hospital’s rule and the limit theory(cid:101), it is easy to(cid:98)see that (cid:101) d· lim h = jk = h , and lim h = uT v . ρ→0 jk t·jk jk ρ→∞ jk (j) (k) 34 ##LTLine##

text: Appendix:

label: paragraph

text: Tuning

text: Parameter

text: Selection

text: There

text: are

text: multiple

text: penalty

text: parameters

text: in

text: the

text: optimization

text: problem

text: Simultaneous

text: selection

text: of

text: all

text: the

text: tuning

text: parameters

text: in

text: a

text: data-driven

text: fashion

text: can

text: be

text: a

text: huge

text: computational

text: burden.

text: As

text: a

text: remedy,

text: we

text: nest

text: the

text: selection

text: of

text: diﬀerent

text: parameters

text: inside

text: the

text: ADMM

text: algorithm.

text: We

text: will

text: discuss

text: in

text: more

text: detail

text: below.

text: The

text: roughness

text: penalty

text: parameters

text: λ

text: and

text: λ

text: control

text: the

text: smoothness

text: of

text: u

text: and

text: v

text: when

text: ui

text: vi

text: i

text: i

text: estimating

text: U

text: and

text: V.

text: Since

text: we

text: estimate

text: U

text: and

text: V

text: in

text: through

text: tF-SVD,

text: we

text: adopt

text: the

text: adaptive

text: tuning

text: selection

text: procedure

text: as

text: in

text: Huang

text: et

text: al.

text: In

text: particular,

text: the

text: selection

text: of

text: λ

text: and

text: λ

text: is

text: embedded

text: in

text: the

text: alternating

text: algorithm

text: of

text: tF-SVD

text: and

text: use

text: the

text: generalized

text: ui

text: vi

text: cross-validation

text: (GCV)

text: criteria.

text: Closed-form

text: GCV

text: scores

text: have

text: been

text: derived

text: from

text: leaving

text: out

text: rows

text: and

text: columns

text: for

text: fast

text: computation

text: (Huang

text: et

text: al.,

text: Another

text: tuning

text: parameter

text: is

text: ρ.

text: Since

text: ρ

text: as

text: a

text: penalty

text: parameter

text: is

text: left

text: out

text: in

text: it

text: primarily

text: serves

text: as

text: the

text: step

text: size

text: in

text: the

text: dual

text: update

text: of

text: Although

text: the

text: step

text: size

text: in

text: the

text: ADMM

text: algorithm

text: can

text: be

text: ﬁxed

text: as

text: any

text: positive

text: value

text: with

text: little

text: eﬀect

text: on

text: the

text: ﬁnal

text: solution

text: theoretically,

text: it

text: does

text: aﬀect

text: convergence

text: rate

text: in

text: practice

text: (Boyd

text: et

text: al.,

text: To

text: improve

text: the

text: convergence

text: rate

text: of

text: the

text: algorithm

text: and

text: alleviate

text: the

text: dependence

text: on

text: the

text: initial

text: choice

text: of

text: the

text: penalty

text: parameter,

text: we

text: follow

text: He

text: et

text: al.

text: and

text: select

text: ρ

text: adaptively

text: in

text: each

text: iteration.

text: In

text: particular,

text: we

text: ﬁrst

text: deﬁne

text: two

text: residuals

text: in

text: each

text: iteration

text: as

text: Primal

text: residual:

text: z(l)

text: H(l)

text: U(l

text: F

text: Dual

text: residual:

text: s(l)

text: H(l)

text: H(l

text: F

text: We

text: remark

text: that

text: a

text: small

text: value

text: of

text: ρ

text: leads

text: to

text: z(l)

text: s(l),

text: while

text: a

text: large

text: value

text: of

text: ρ

text: leads

text: to

text: z(l)

text: s(l).

text: To

text: see

text: this,

text: we

text: temporarily

text: set

text: Θ

text: and

text: examine

text: From

text: the

text: l’Hospital’s

text: rule

text: and

text: the

text: limit

text: it

text: is

text: easy

text: that

text: d·

text: lim

text: h

text: jk

text: h

text: and

text: lim

text: h

text: uT

text: v

text: ρ→0

text: jk

text: t·jk

text: jk

text: ρ→∞

text: jk

text: (j)

text: (k)

text: ##LTLine##


text: Setting 1 Setting 2 5 1.5 E E 1 S S M M 0.5 0 MLE tFHazard MLE tFHazard Setting 3 Setting 4 30 5 E 20 E 4 S S M 10 M 3 0 2 MLE tFHazard MLE tFHazard Setting 5 Setting 6 50 E E 4 S S M M 0 3 MLE tFHazard MLE tFHazard Figure S.1: Simulation results. The box-plots show the Frobenius norm of the diﬀerence between the estimated and the true hazard rate matrices. Each plot corresponds to one simulation setting. 38 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Setting

font: default

text: Setting

text: E

text: E

text: S

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text: M

text: M

text: MLE

text: tFHazard

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text: Setting

text: Setting

text: E

text: E

text: S

text: S

text: M

text: M

text: MLE

text: tFHazard

text: MLE

text: tFHazard

text: Setting

text: Setting

text: E

text: E

text: S

text: S

text: M

text: M

text: MLE

text: tFHazard

text: MLE

text: tFHazard

text: Figure

text: Simulation

text: results.

text: The

text: box-plots

text: show

text: the

text: Frobenius

text: norm

text: of

text: the

text: diﬀerence

text: between

text: the

text: estimated

text: and

text: the

text: true

text: hazard

text: rate

text: matrices.

text: Each

text: plot

text: corresponds

text: to

text: one

text: simulation

text: setting.

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text: As far as we know, Brown et al. (2005), Mandelbaum and Zeltyn (2013), and Aktekin and Soyer (2014) are the only empirical papers studying patience patterns in tele-queues. In particular, Aktekin and Soyer (2014) proposed mixture models and piecewise linear models to estimate customer patience hazard rate. However, all three studies focused on one-way variation of the customer patience as a function of waiting time. In practice, customer patience varies across the time of day. To cope with it, one would like the patience hazard function to also depend on the time of day, i.e., it varies in two directions: how long a customer waits and what time s/he starts waiting. This paper makes a unique and important contribution by developing a two-way func- tional hazards model (i.e., tF-Hazards) to provide an eﬃcient estimator of the two-way patience hazard surface as inputs to call center staﬃng models. The tF-Hazards model stems from a simple yet ﬂexible two-way piecewise constant hazard model, and incor- porates low-rank and smooth structure into the discrete hazard matrix indexed by two timescales (in our application, the two timescales are waiting duration and time of day). The model can be viewed as a generalization of the two-way functional singular value de- composition (tF-SVD) approach for Gaussian data (Huang et al., 2009). Unlike tF-SVD though, our model accommodates censored time-to-event data and estimates the underly- ing two-way hazard surface. The model is estimated via a regularized likelihood framework, where the low-rank constraint and the roughness penalty are incorporated simultaneously. We exploit an alternating direction method of multipliers (ADMM) algorithm to solve the constrained optimization problem. We apply the proposed method to the US bank waiting times previously analyzed by Mandelbaum and Zeltyn (2013), and oﬀer novel insights into the customer waiting behavior across time of day, reﬂecting dynamics of customer patience. We also investigate the oﬀered wait patterns, which provides interesting ﬁndings reﬂecting the dynamic eﬀect of system protocol and congestion. Both the patience time and the 3

text: As

text: far

text: as

text: we

text: know,

text: Brown

text: et

text: al.

text: Mandelbaum

text: and

text: Zeltyn

text: and

text: Aktekin

text: and

text: Soyer

text: are

text: the

text: only

text: empirical

text: papers

text: studying

text: patience

text: patterns

text: in

text: tele-queues.

text: In

text: particular,

text: Aktekin

text: and

text: Soyer

text: proposed

text: mixture

text: models

text: and

text: piecewise

text: linear

text: models

text: to

text: estimate

text: customer

text: patience

text: hazard

text: rate.

text: However,

text: all

text: three

text: studies

text: focused

text: on

text: one-way

text: variation

text: of

text: the

text: customer

text: patience

text: as

text: a

text: function

text: of

text: waiting

text: time.

text: In

text: practice,

text: customer

text: patience

text: varies

text: across

text: the

text: time

text: of

text: day.

text: To

text: cope

text: with

text: it,

text: one

text: would

text: like

text: the

text: patience

text: hazard

text: function

text: to

text: also

text: depend

text: on

text: the

text: time

text: of

text: day,

text: i.e.,

text: it

text: varies

text: in

text: two

text: directions:

text: how

text: long

text: a

text: customer

text: waits

text: and

text: what

text: time

text: s/he

text: starts

text: waiting.

text: This

text: paper

text: makes

text: a

text: unique

text: and

text: important

text: contribution

text: by

text: developing

text: a

text: two-way

text: func-

text: tional

text: hazards

text: model

text: (i.e.,

text: tF-Hazards)

text: to

text: provide

text: an

text: eﬃcient

text: estimator

text: of

text: the

text: two-way

text: patience

text: hazard

text: surface

text: as

text: inputs

text: to

text: call

text: center

text: staﬃng

text: models.

text: The

text: tF-Hazards

text: model

text: stems

text: from

text: a

text: simple

text: yet

text: ﬂexible

text: two-way

text: piecewise

text: constant

text: hazard

text: model,

text: and

text: incor-

text: porates

text: low-rank

text: and

text: smooth

text: structure

text: into

text: the

text: discrete

text: hazard

text: matrix

text: indexed

text: by

text: two

text: timescales

text: (in

text: our

text: application,

text: the

text: two

text: timescales

text: are

text: waiting

text: duration

text: and

text: time

text: of

text: day).

text: The

text: model

text: can

text: be

text: viewed

text: as

text: a

text: generalization

text: of

text: the

text: two-way

text: functional

text: singular

text: value

text: de-

text: composition

text: (tF-SVD)

text: approach

text: for

text: Gaussian

text: data

text: (Huang

text: et

text: al.,

text: Unlike

text: tF-SVD

text: though,

text: our

text: model

text: accommodates

text: censored

text: time-to-event

text: data

text: and

text: estimates

text: the

text: underly-

text: ing

text: two-way

text: hazard

text: surface.

text: The

text: model

text: is

text: estimated

text: via

text: a

text: regularized

text: likelihood

text: framework,

text: where

text: the

text: low-rank

text: constraint

text: and

text: the

text: roughness

text: penalty

text: are

text: incorporated

text: simultaneously.

text: We

text: exploit

text: an

text: alternating

text: direction

text: method

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text: multipliers

text: (ADMM)

text: algorithm

text: to

text: solve

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text: constrained

text: optimization

text: problem.

text: We

text: apply

text: the

text: proposed

text: method

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text: the

text: US

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text: Mandelbaum

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text: and

text: oﬀer

text: novel

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text: customer

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text: the piecewise constant model due to the low variance; when the sample size is large enough (Settings 4 and 6), the MLE is more accurate than the tF-Hazards estimate. We also note that Settings 5 and 6 represent situations where the low-rank assumption is violated, which suggest that tF-Hazards remains useful when the sample size is insuﬃcient. B Customer Patience Study with Short Waiting Calls We apply the tF-Hazards method to estimate the underlying hazard rate surface for the Quick&Reilly group with and without those calls with waiting times less than 2 seconds. The results are shown in Figure S.2. Panels (a) and (b) plot the intra-day pattern as proportion of the baseline hazard under the two scenarios. Panels (c) and (d) plot the ≥ baseline hazard rate as a function of waiting duration 2 seconds, while Panel (e) shows ≥ the baseline hazard function for waiting time 0 second. We can make the following observations: • Panels (a) vs. (b): Including the short-wait calls has no obvious eﬀect on the intra-day pattern. • Panels (c)-(e): Again, the two baseline hazard functions in Panel (c) and Panel (d) look similar in general, with periodic peaks every minute or so. The new estimate (with the short-waiting data) in Panel (e) has a sharp spike in the ﬁrst couple of sec- onds (due to the high-answering-rate) and contains more wiggles (Panel (d)). This is because our method uses a single tuning parameter to control the degree of smooth- ness in the loading estimation. The inclusion of the short-waiting data leads to the selection of a smaller roughness penalty tuning parameter and thus results in a more wiggly estimate, without losing the general shape of the baseline function. 39

text: the

text: piecewise

text: constant

text: model

text: due

text: to

text: the

text: low

text: variance;

text: when

text: the

text: sample

text: size

text: is

text: large

text: enough

text: (Settings

text: and

text: the

text: MLE

text: is

text: more

text: accurate

text: than

text: the

text: tF-Hazards

text: estimate.

text: We

text: also

text: note

text: that

text: Settings

text: and

text: represent

text: situations

text: where

text: the

text: low-rank

text: assumption

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text: violated,

text: which

text: suggest

text: that

text: tF-Hazards

text: remains

text: useful

text: when

text: the

text: sample

text: size

text: is

text: insuﬃcient.

text: B

label: section

text: Customer

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text: Short

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text: the

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text: The

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text: plot

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text: no

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text: intra-day

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text: Panels

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text: Panel

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text: and

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text: look

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text: 2.5#104 Histogram of Waiting Time (>=2s) 10000 Arrivals of Calls with Waiting >=2s 9000 2 8000 7000 Calls1.5 Calls 6000 er of er of 5000 mb 1 mb 4000 u u N N 3000 0.5 2000 1000 0 0 2 30 60 90 120 150 180 0 2 4 6 8 10 12 14 16 18 20 22 24 Waiting Time (1s interval) Time of Day (15min interval) Figure S.4: Retail service group. Left: the histogram of observed waiting times (in seconds) for calls with waiting time between 2 and 200 seconds. Right: the histogram of arrival rates ≥ (in 15-minute intervals) of calls with waiting time 2 seconds. Histogram of Waiting Time (>=2s) Arrivals of Calls with Waiting >=2s 8000 2000 7000 1800 1600 6000 1400 Calls5000 Calls1200 er of 4000 er of 1000 umb3000 umb 800 N N 600 2000 400 1000 200 0 0 2 30 60 90 120 150 180 210 240 270 300 0 2 4 6 8 10 12 14 16 18 20 22 24 Waiting Time (1s interval) Time of Day (15min interval) Figure S.5: CCO service group. Left: the histogram of observed waiting times (in seconds) for calls with waiting time between 2 and 300 seconds. Right: the histogram of arrival ≥ rates (in 15-minute intervals) of calls with waiting time 2 seconds. 44 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Histogram

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text: D Table 1: A transaction dataset . 1 Items I I I Trans. 1 2 3 T t B C G H K 1 1 t A D G K 2 T t A C D H 3 2 t A E F 4 T t B E F 5 3 t B E F G K 6 D Figure 1: Queries on dataset . dataset. Figure 1 graphically illustrates this. The itemsets that are solution for Q (i.e., D 1 P , P and P ) are extracted from D = . 1 2 3 1 An example of such a query where user’s constraints are expressed only on itemsets is the query Q asking for FCIs: 1 ≡ ∧ Q (D, P) freqent(D, P, θ) closed(D, P) 1 where frequent(D, P, θ) and closed(D, P) are predicates expressing user’s constraints on the frequency (with a minimum frequency θ) and the closure of an itemset P in D, D D where D is in this case. The query Q on the dataset of Table 1 with a minimum ≥ 1 1 frequency θ 50% returns A, B, EF and GK as FCIs. (cid:48) As a second example of such a query on itemsets the user can ask a query Q where 1 the extracted itemsets are FCIs and the items are taken from at least lb and at most ub categories: (cid:48) ≡ ∧ ∧ Q (D, P) Q (D, P) atLeast(P, lb) atMost(P, ub) 1 1 where atLeast(P, lb) and atMost(P, ub) are user’s constraints ensuring that the item- (cid:48) set P overlaps between lb and ub categories of items. The query Q on the dataset 1 4 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: D

label: caption

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text: A

text: transaction

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text: I

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text: D of Table 1 with lb = ub = 2 and minimum frequency θ = 50% only returns EF. 1 It does not return A, B, and GK because each of these itemsets belongs to a single category. 3.2 User’s constraints on items In addition to constraints on itemsets, the user may want to put constraints on the items themselves. Such constraints are constraints on the dataset. They specify on which items/columns the mining will occur. In Figure 1, constraints on items lead the query, Q , to mine on the sub-dataset D satisfying constraints on items. 2 2 As an example, the user can ask a query Q , where the extracted itemsets are FCIs 2 of sub-datasets containing at least lb categories of items and at most ub categories: I I ≡ ∧ ∧ Q (D, P) Q (D, P) atLeastI(D, lb ) atMostI(D, ub ) 2 1 I I where atLeastI(D, lb ) and atMostI(D, ub ) are user’s constraints ensuring that the I I (cid:48) dataset D contains between lb and ub categories of items. As opposed to Q and Q , I I 1 1 Q seeks itemsets in sub-datasets satisfying a property on their items. The query Q 2 D 2 on the dataset of Table 1 with lb = ub = 2 and minimum frequency θ = 50% 1 I I I I I I I I returns A, B and E on + , A, B, F and GK on + , EF and GK on + . 1 2 1 3 2 3 3.3 User’s constraints on transactions The user may also want to put constraints on transactions. Such constraints de- termine on which transactions/rows the mining will occur. In Figure 1, constraints on transactions lead the query, Q , to mine on the subset D of transactions. 3 3 As an example, the user can ask a query Q , where the extracted itemsets are FCIs 3 on at least lb and at most ub categories: T T ≡ ∧ ∧ Q (D, P) Q (D, P) atLeastT(D, lb ) atMostT(D, ub ) 3 1 T T where atLeastT(D, lb ) and atMostT(D, ub ) are user’s constraints ensuring that T T the dataset D contains between lb and ub categories of transactions. The query Q D T T 3 on the dataset of Table 1 with lb = ub = 2 and minimum frequency θ = 50% 1 T T T T T T returns: A, AD, CH and GK on + , B, BEF, BGK and GK on + , A, T T 1 2 1 3 BEF and EF on + . 2 3 3.4 User’s constraints on items and transactions Finally, the user may want to put constraints on both items and transactions. In (cid:48) Figure 1, such constraints lead the query, Q , to mine on D and D . 4 4 4 The user can ask a query Q , where the extracted itemsets are FCIs on at least lb 4 I and at most ub categories of items and on at least lb and at most ub categories of I T T transactions: ≡ ∧ Q (D, P) Q (D, P) Q (D, P) 4 2 3 D The query Q on the dataset of Table 1 with lb = ub = lb = ub = 2 and 4 1 I I T T minimum frequency θ = 50% will have to explore nine possible sub-datasets in which to look for frequent closed itemsets: 3.5 A simple illustration: Where Ferrari cars are frequently bought? Consider a dataset of cars purchases in France, where each transaction/purchase contains items representing the city, the department, and the region where the purchase 5

text: D

text: of

text: Table

text: with

text: lb

text: ub

text: and

text: minimum

text: frequency

text: θ

text: only

text: returns

text: EF.

text: It

text: does

text: not

text: return

text: A,

text: B,

text: and

text: GK

text: because

text: each

text: of

text: these

text: itemsets

text: belongs

text: to

text: a

text: single

text: category.

text: User’s

text: constraints

text: on

text: items

text: In

text: addition

text: to

text: constraints

text: on

text: itemsets,

text: the

text: user

text: may

text: want

text: to

text: put

text: constraints

text: on

text: the

text: items

text: themselves.

text: Such

text: constraints

text: are

text: constraints

text: on

text: the

text: dataset.

text: They

text: specify

text: on

text: which

text: items/columns

text: the

text: mining

text: will

text: occur.

text: In

text: Figure

text: constraints

text: on

text: items

text: lead

text: the

text: query,

text: Q

text: to

text: mine

text: on

text: the

text: sub-dataset

text: D

text: satisfying

text: constraints

text: on

text: items.

text: As

text: an

text: example,

text: the

text: user

text: can

text: ask

text: a

text: query

text: Q

text: where

text: the

text: extracted

text: itemsets

text: are

text: FCIs

text: of

text: sub-datasets

text: containing

text: at

text: least

text: lb

text: categories

text: of

text: items

text: and

text: at

text: most

text: ub

text: categories:

text: I

text: I

text: Q

text: (D,

text: P)

text: Q

text: (D,

text: P)

text: atLeastI(D,

text: lb

text: atMostI(D,

text: ub

text: I

text: I

text: where

text: atLeastI(D,

text: lb

text: and

text: atMostI(D,

text: ub

text: are

text: user’s

text: constraints

text: ensuring

text: that

text: the

text: I

text: I

text: dataset

text: D

text: contains

text: between

text: lb

text: and

text: ub

text: categories

text: of

text: items.

text: As

text: opposed

text: to

text: Q

text: and

text: Q

text: I

text: I

text: Q

text: seeks

text: itemsets

text: in

text: sub-datasets

text: satisfying

text: a

text: property

text: on

text: their

text: items.

text: The

text: query

text: Q

text: D

text: on

text: the

text: dataset

text: of

text: Table

text: with

text: lb

text: ub

text: and

text: minimum

text: frequency

text: θ

text: I

text: I

text: I

text: I

text: I

text: I

text: I

text: I

text: returns

text: A,

text: B

text: and

text: E

text: on

text: A,

text: B,

text: F

text: and

text: GK

text: on

text: EF

text: and

text: GK

text: on

text: User’s

text: constraints

text: on

text: transactions

text: The

text: user

text: may

text: also

text: want

text: to

text: put

text: constraints

text: on

text: transactions.

text: Such

text: constraints

text: de-

text: termine

text: on

text: which

text: transactions/rows

text: the

text: mining

text: will

text: occur.

text: In

text: Figure

text: constraints

text: on

text: transactions

text: lead

text: the

text: query,

text: Q

text: to

text: mine

text: on

text: the

text: subset

text: D

text: of

text: transactions.

text: As

text: an

text: example,

text: the

text: user

text: can

text: ask

text: a

text: query

text: Q

text: where

text: the

text: extracted

text: itemsets

text: are

text: FCIs

text: on

text: at

text: least

text: lb

text: and

text: at

text: most

text: ub

text: categories:

text: T

text: T

text: Q

text: (D,

text: P)

text: Q

text: (D,

text: P)

text: atLeastT(D,

text: lb

text: atMostT(D,

text: ub

text: T

text: T

text: where

text: atLeastT(D,

text: lb

text: and

text: atMostT(D,

text: ub

text: are

text: user’s

text: constraints

text: ensuring

text: that

text: T

text: T

text: the

text: dataset

text: D

text: contains

text: between

text: lb

text: and

text: ub

text: categories

text: of

text: transactions.

text: The

text: query

text: Q

text: D

text: T

text: T

text: on

text: the

text: dataset

text: of

text: Table

text: with

text: lb

text: ub

text: and

text: minimum

text: frequency

text: θ

text: T

text: T

text: T

text: T

text: T

text: T

text: returns:

text: A,

text: AD,

text: CH

text: and

text: GK

text: on

text: B,

text: BEF,

text: BGK

text: and

text: GK

text: on

text: A,

text: T

text: T

text: BEF

text: and

text: EF

text: on

text: User’s

text: constraints

text: on

text: items

text: and

text: transactions

text: Finally,

text: the

text: user

text: may

text: want

text: to

text: put

text: constraints

text: on

text: both

text: items

text: and

text: transactions.

text: In

text: Figure

text: such

text: constraints

text: lead

text: the

text: query,

text: Q

text: to

text: mine

text: on

text: D

text: and

text: D

text: The

text: user

text: can

text: ask

text: a

text: query

text: Q

text: where

text: the

text: extracted

text: itemsets

text: are

text: FCIs

text: on

text: at

text: least

text: lb

text: I

text: and

text: at

text: most

text: ub

text: categories

text: of

text: items

text: and

text: on

text: at

text: least

text: lb

text: and

text: at

text: most

text: ub

text: categories

text: of

text: I

text: T

text: T

text: transactions:

text: Q

text: (D,

text: P)

text: Q

text: (D,

text: P)

text: Q

text: (D,

text: P)

text: D

text: The

text: query

text: Q

text: on

text: the

text: dataset

text: of

text: Table

text: with

text: lb

text: ub

text: lb

text: ub

text: and

text: I

text: I

text: T

text: T

text: minimum

text: frequency

text: θ

text: will

text: have

text: to

text: explore

text: nine

text: possible

text: sub-datasets

text: in

text: which

text: to

text: look

text: for

text: frequent

text: closed

text: itemsets:

text: A

text: simple

text: illustration:

text: Where

text: Ferrari

text: cars

text: are

text: frequently

text: bought?

text: Consider

text: a

text: dataset

text: of

text: cars

text: purchases

text: in

text: France,

text: where

text: each

text: transaction/purchase

text: contains

text: items

text: representing

text: the

text: city,

text: the

text: department,

text: and

text: the

text: region

text: where

text: the

text: purchase


text: For item based collaborative ﬁltering to work, similarity among items has to be ﬁgured. In this case, requirements are the items and similarity among these requirements are estimated using correlation analysis. There are three correlation techniques which were considered for ﬁnding correlation among requirements. The techniques - Pearson Coeﬃcient Correlation, Cosine Sim- ilarity and Jaccard Distance. All of these techniques are implemented on the user-requirement rela- tion matrix to detect similarities among requirements. Best results are (p,q) produced by cosine similarity and Pearson coeﬃcient correlation as both of these approaches are invariant to scaling. This means similarities among ele- ments are invariant even if all elements are multiplied by a nonzero constant. However, cosine similarity is not invariant when any constant is added to all elements. But Pearson correlation is also invariant to adding any constant to all elements. For example, if there are two vectors X1 and X2, and Pearson correlation function is called pearson(), pearson(X1, X2) == pearson(X1, 2 * X2 + 3). This property is really important as we are looking for similarity patterns among items. The items do not need to be exactly identical to be aﬃrmed similar by our approach. Hence, Pearson Coeﬃcient Correlation is used to determine similarity among items and used to predict ratings of the stakeholders. The equation of Pearson Coeﬃcient Correlation is stated in (cid:80) Equation 5. (cid:113) (cid:113) (cid:80) (cid:80) − − ∈ (R R¯ )(R R¯ ) sim(i, j) = u U (u,i) i (u,j) j (5) − − ∈ (R R¯ )2) ∈ (R R¯ )2) u U (u,i) i u U (u,j) j To ensure accuracy of the correlation computation, we must ﬁrst isolate the co-rated cases (i.e., cases where the users rated both i and j items). Let the set of users who both rated i and j are denoted by U then the correlation similarity is given by Here R denotes the rating of user u on item i, R is u,i i the average rating of the i-th item. Hereafter, using this similarity function an Requirement-to-Requirement similarity matrix, as presented in Table 2, will be generated. 3.4.2 Stakeholder Selection Using the similarity matrix obtained from the previous step, stakeholders likely to rate a requirement can be predicted based on the commonly used (cid:80) Equation 6. (cid:80) ∗ (S R ) P = allsimilaritems,N i,N| |u,N (6) (u,i) S allsimilaritems,N i,N 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: For

text: item

text: based

text: collaborative

text: ﬁltering

text: to

text: work,

text: similarity

text: among

text: items

text: has

text: to

text: be

text: ﬁgured.

text: In

text: this

text: case,

text: requirements

text: are

text: the

text: items

text: and

text: similarity

text: among

text: these

text: requirements

text: are

text: estimated

text: using

text: correlation

text: analysis.

text: There

text: are

text: three

text: correlation

text: techniques

text: which

text: were

text: considered

text: for

text: ﬁnding

text: correlation

text: among

text: requirements.

text: The

text: techniques

text: Pearson

text: Coeﬃcient

text: Correlation,

text: Cosine

text: Sim-

text: ilarity

text: and

text: Jaccard

text: Distance.

text: All

text: of

text: these

text: techniques

text: are

text: implemented

text: on

text: the

text: user-requirement

text: rela-

text: tion

text: matrix

text: to

text: detect

text: similarities

text: among

text: requirements.

text: Best

text: results

text: are

text: (p,q)

text: produced

text: by

text: cosine

text: similarity

text: and

text: Pearson

text: coeﬃcient

text: correlation

text: as

text: both

text: of

text: these

text: approaches

text: are

text: invariant

text: to

text: scaling.

text: This

text: means

text: similarities

text: among

text: ele-

text: ments

text: are

text: invariant

text: even

text: if

text: all

text: elements

text: are

text: multiplied

text: by

text: a

text: nonzero

text: constant.

text: However,

text: cosine

text: similarity

text: is

text: not

text: invariant

text: when

text: any

text: constant

text: is

text: added

text: to

text: all

text: elements.

text: But

text: Pearson

text: correlation

text: is

text: also

text: invariant

text: to

text: adding

text: any

text: constant

text: to

text: all

text: elements.

text: For

text: example,

text: if

text: there

text: are

text: two

text: vectors

text: and

text: and

text: Pearson

text: correlation

text: function

text: is

text: called

text: pearson(),

text: This

text: property

text: is

text: really

text: important

text: as

text: we

text: are

text: looking

text: for

text: similarity

text: patterns

text: among

text: items.

text: The

text: items

text: do

text: not

text: need

text: to

text: be

text: exactly

text: identical

text: to

text: be

text: aﬃrmed

text: similar

text: by

text: our

text: approach.

text: Hence,

text: Pearson

text: Coeﬃcient

text: Correlation

text: is

text: used

text: to

text: determine

text: similarity

text: among

text: items

text: and

text: used

text: to

text: predict

text: ratings

text: of

text: the

text: stakeholders.

text: The

text: equation

text: of

text: Pearson

text: Coeﬃcient

text: Correlation

text: is

text: stated

text: in

text: Equation

text: (R

text: R¯

text: )(R

text: R¯

text: sim(i,

text: j)

text: u

text: U

text: (u,i)

text: i

text: (u,j)

text: j

text: (R

text: R¯

text: (R

text: R¯

text: u

text: U

text: (u,i)

text: i

text: u

text: U

text: (u,j)

text: j

text: To

text: ensure

text: accuracy

text: of

text: the

text: correlation

text: computation,

text: we

text: must

text: ﬁrst

text: isolate

text: the

text: co-rated

text: cases

text: (i.e.,

text: cases

text: where

text: the

text: users

text: rated

text: both

text: i

text: and

text: j

text: items).

text: Let

text: the

text: set

text: of

text: users

text: who

text: both

text: rated

text: i

text: and

text: j

text: are

text: denoted

text: by

text: U

text: then

text: the

text: correlation

text: similarity

text: is

text: given

text: by

text: Here

text: R

text: denotes

text: the

text: rating

text: of

text: user

text: u

text: on

text: item

text: i,

text: R

text: is

text: u,i

text: i

text: the

text: average

text: rating

text: of

text: the

text: i-th

text: item.

text: Hereafter,

text: using

text: this

text: similarity

text: function

text: an

text: Requirement-to-Requirement

text: similarity

text: matrix,

text: as

text: presented

text: in

text: Table

text: will

text: be

text: generated.

text: Stakeholder

text: Selection

text: Using

text: the

text: similarity

text: matrix

text: obtained

text: from

text: the

text: previous

text: step,

text: stakeholders

text: likely

text: to

text: rate

text: a

text: requirement

text: can

text: be

text: predicted

text: based

text: on

text: the

text: commonly

text: used

text: Equation

text: (S

text: R

text: P

text: allsimilaritems,N

text: i,N|

text: |u,N

text: (u,i)

text: S

text: allsimilaritems,N

text: i,N

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: stakeholders. All the requirements must be mutually independent. This pro- posed approach deals with rank reversal and dependency issues. But the method of collecting requirement weight did not consider budget and time constraints. K-means algorithm is used in [8] to resolve rank reversal problem of large scale software prioritization. Multiple criteria are used to form clusters. The clusters were prioritized based on weights. However, the approach did not prioritizes stakeholders, used ambiguous methods to gain weights of require- ments and did not handle dependencies in requirement prioritization. Lim et al. [5] prioritized stakeholders using ‘StakeNet’- a social network- ing tool. This tool obtained recommendation of stakeholders from each stake- holder in the system through interviews. They extended this work to ‘Stake- Source2.0’ [14], which prioritizes requirements and stakeholders by means of social networking and collaborative ﬁltering. Their work also highlighted stakeholders conﬂict and proposed recommending requirements to applicable stakeholders. However, rank reversal was not considered in either of these approaches. The authors of [14] also proposed ‘StakeRare’ [15] which used social net- works and collaborative ﬁltering for large scale requirements. The paper addressed three problems for large scale projects: information overload, in- adequate stakeholder input, and biased prioritization of requirements. The authors collected stakeholder list by eliciting requirements and deriving inﬂu- ence of stakeholders using interviews and the importance of each requirement was determined. From that, requirements were analyzed and a list of pri- oritized requirements were generated. Although the method performed well compared to other existing methods, it did not cover rank reversal problem. Review of state of the art framework for requirements prioritization illus- trates that issues regarding rank reversal are not fully addressed in most of the approaches. Moreover, the approaches considering rank reversal suﬀers from several problems such as lack of stakeholder prioritization and compu- tational complexity. In our knowledge, current approaches did not emphasize on reducing human interactions necessary for prioritizing new requirements. Depending solely on the feedback from stakeholders for prioritization, will increase time and cost needed for the process and introduce scalability issues. 3 The SAFFRON Framework This paper proposes a framework named SAFFRON which reduces hu- 5

text: stakeholders.

text: All

text: the

text: requirements

text: must

text: be

text: mutually

text: independent.

text: This

text: pro-

text: posed

text: approach

text: deals

text: with

text: rank

text: reversal

text: and

text: dependency

text: issues.

text: But

text: the

text: method

text: of

text: collecting

text: requirement

text: weight

text: did

text: not

text: consider

text: budget

text: and

text: time

text: constraints.

text: K-means

text: algorithm

text: is

text: used

text: in

text: to

text: resolve

text: rank

text: reversal

text: problem

text: of

text: large

text: scale

text: software

text: prioritization.

text: Multiple

text: criteria

text: are

text: used

text: to

text: form

text: clusters.

text: The

text: clusters

text: were

text: prioritized

text: based

text: on

text: weights.

text: However,

text: the

text: approach

text: did

text: not

text: prioritizes

text: stakeholders,

text: used

text: ambiguous

text: methods

text: to

text: gain

text: weights

text: of

text: require-

text: ments

text: and

text: did

text: not

text: handle

text: dependencies

text: in

text: requirement

text: prioritization.

text: Lim

text: et

text: al.

text: prioritized

text: stakeholders

text: using

text: ‘StakeNet’-

text: a

text: social

text: network-

text: ing

text: tool.

text: This

text: tool

text: obtained

text: recommendation

text: of

text: stakeholders

text: from

text: each

text: stake-

text: holder

text: in

text: the

text: system

text: through

text: interviews.

text: They

text: extended

text: this

text: work

text: to

text: ‘Stake-

text: which

text: prioritizes

text: requirements

text: and

text: stakeholders

text: by

text: means

text: of

text: social

text: networking

text: and

text: collaborative

text: ﬁltering.

text: Their

text: work

text: also

text: highlighted

text: stakeholders

text: conﬂict

text: and

text: proposed

text: recommending

text: requirements

text: to

text: applicable

text: stakeholders.

text: However,

text: rank

text: reversal

text: was

text: not

text: considered

text: in

text: either

text: of

text: these

text: approaches.

text: The

text: authors

text: of

text: also

text: proposed

text: ‘StakeRare’

text: which

text: used

text: social

text: net-

text: works

text: and

text: collaborative

text: ﬁltering

text: for

text: large

text: scale

text: requirements.

text: The

text: paper

text: addressed

text: three

text: problems

text: for

text: large

text: scale

text: projects:

text: information

text: overload,

text: in-

text: adequate

text: stakeholder

text: input,

text: and

text: biased

text: prioritization

text: of

text: requirements.

text: The

text: authors

text: collected

text: stakeholder

text: list

text: by

text: eliciting

text: requirements

text: and

text: deriving

text: inﬂu-

text: ence

text: of

text: stakeholders

text: using

text: interviews

text: and

text: the

text: importance

text: of

text: each

text: requirement

text: was

text: determined.

text: From

text: that,

text: requirements

text: were

text: analyzed

text: and

text: a

text: list

text: of

text: pri-

text: oritized

text: requirements

text: were

text: generated.

text: Although

text: the

text: method

text: performed

text: well

text: compared

text: to

text: other

text: existing

text: methods,

text: it

text: did

text: not

text: cover

text: rank

text: reversal

text: problem.

text: Review

text: of

text: state

text: of

text: the

text: art

text: framework

text: for

text: requirements

text: prioritization

text: illus-

text: trates

text: that

text: issues

text: regarding

text: rank

text: reversal

text: are

text: not

text: fully

text: addressed

text: in

text: most

text: of

text: the

text: approaches.

text: Moreover,

text: the

text: approaches

text: considering

text: rank

text: reversal

text: suﬀers

text: from

text: several

text: problems

text: such

text: as

text: lack

text: of

text: stakeholder

text: prioritization

text: and

text: compu-

text: tational

text: complexity.

text: In

text: our

text: knowledge,

text: current

text: approaches

text: did

text: not

text: emphasize

text: on

text: reducing

text: human

text: interactions

text: necessary

text: for

text: prioritizing

text: new

text: requirements.

text: Depending

text: solely

text: on

text: the

text: feedback

text: from

text: stakeholders

text: for

text: prioritization,

text: will

text: increase

text: time

text: and

text: cost

text: needed

text: for

text: the

text: process

text: and

text: introduce

text: scalability

text: issues.

text: The

text: SAFFRON

text: Framework

text: This

text: paper

text: proposes

text: a

text: framework

text: named

text: SAFFRON

text: which

text: reduces

text: hu-


text: elicited requirements follows these steps: • At ﬁrst the requirements relevant to the project and its ratings will be elicited from the stakeholders using human interaction. • Then the approach described in StakeRare [24] will be applied. The stakeholders will be prioritized using the ratings provided by other stakeholders. • After that each stakeholders’ inﬂuence on the project will be calculated. • After all these computations, the requirements will be prioritized using the ratings provided by the stakeholders and project inﬂuences calcu- lated from role and stakeholder inﬂuence. To solve the classic rank reversal problem, the following steps using prediction techniques are used. These steps are stated as follows: • When new requirements arrive, ratings given by a portion of stakehold- ers are elicited for each requirement. • Item-to-Item collaborative ﬁltering is then used to ﬁnd similarity among already elicited requirements and the new requirements. Although col- laborative ﬁltering technique was used in [15], they used it to ﬁnd simi- lar stakeholders instead of requirements. However, in this scenario it is more reasonable to ﬁnd similar requirements ﬁrst and then determine which stakeholders are more likely to rate those. Thus item-to-item technique was used. • Model Based Collaborative Filtering using latent factors - learning pa- rameter of users and feature vector of requirements are ﬁnally used for actual prediction of the values for the determined users from the previ- ous step. This step also uses merged rating matrix and corresponding user-requirement relation matrix. • Finally, StakeRare [15] is applied to the updated requirements list and new prioritized requirements list is attained from the approach. A portion of the ratings are predicted rather than collecting all the ratings from stakeholders. Collecting all ratings from stakeholders for new require- ments is time consuming and has scalability issues. Also large scale project 7

text: elicited

text: requirements

text: follows

text: these

text: steps:

label: list

text: At

text: ﬁrst

text: the

text: requirements

text: relevant

text: to

text: the

text: project

text: and

text: its

text: ratings

text: will

text: be

text: elicited

text: from

text: the

text: stakeholders

text: using

text: human

text: interaction.

text: Then

text: the

text: approach

text: described

text: in

text: StakeRare

text: will

text: be

text: applied.

text: The

text: stakeholders

text: will

text: be

text: prioritized

text: using

text: the

text: ratings

text: provided

text: by

text: other

text: stakeholders.

text: After

text: that

text: each

text: stakeholders’

text: inﬂuence

text: on

text: the

text: project

text: will

text: be

text: calculated.

text: After

text: all

text: these

text: computations,

text: the

text: requirements

text: will

text: be

text: prioritized

text: using

text: the

text: ratings

text: provided

text: by

text: the

text: stakeholders

text: and

text: project

text: inﬂuences

text: calcu-

text: lated

text: from

text: role

text: and

text: stakeholder

text: inﬂuence.

text: To

text: solve

text: the

text: classic

text: rank

text: reversal

text: problem,

text: the

text: following

text: steps

text: using

text: prediction

text: techniques

text: are

text: used.

text: These

text: steps

text: are

text: stated

text: as

text: follows:

text: When

text: new

text: requirements

text: arrive,

text: ratings

text: given

text: by

text: a

text: portion

text: of

text: stakehold-

text: ers

text: are

text: elicited

text: for

text: each

text: requirement.

text: Item-to-Item

text: collaborative

text: ﬁltering

text: is

text: then

text: used

text: to

text: ﬁnd

text: similarity

text: among

text: already

text: elicited

text: requirements

text: and

text: the

text: new

text: requirements.

text: Although

text: col-

text: laborative

text: ﬁltering

text: technique

text: was

text: used

text: in

text: they

text: used

text: it

text: to

text: ﬁnd

text: simi-

text: lar

text: stakeholders

text: instead

text: of

text: requirements.

text: However,

text: in

text: this

text: scenario

text: it

text: is

text: more

text: reasonable

text: to

text: ﬁnd

text: similar

text: requirements

text: ﬁrst

text: and

text: then

text: determine

text: which

text: stakeholders

text: are

text: more

text: likely

text: to

text: rate

text: those.

text: Thus

text: item-to-item

text: technique

text: was

text: used.

text: Model

text: Based

text: Collaborative

text: Filtering

text: using

text: latent

text: factors

text: learning

text: pa-

text: rameter

text: of

text: users

text: and

text: feature

text: vector

text: of

text: requirements

text: are

text: ﬁnally

text: used

text: for

text: actual

text: prediction

text: of

text: the

text: values

text: for

text: the

text: determined

text: users

text: from

text: the

text: previ-

text: ous

text: step.

text: This

text: step

text: also

text: uses

text: merged

text: rating

text: matrix

text: and

text: corresponding

text: user-requirement

text: relation

text: matrix.

text: Finally,

text: StakeRare

text: is

text: applied

text: to

text: the

text: updated

text: requirements

text: list

text: and

text: new

text: prioritized

text: requirements

text: list

text: is

text: attained

text: from

text: the

text: approach.

text: A

text: portion

text: of

text: the

text: ratings

text: are

text: predicted

text: rather

text: than

text: collecting

text: all

text: the

text: ratings

text: from

text: stakeholders.

text: Collecting

text: all

text: ratings

text: from

text: stakeholders

text: for

text: new

text: require-

text: ments

text: is

text: time

text: consuming

text: and

text: has

text: scalability

text: issues.

text: Also

text: large

text: scale

text: project


text: Merging respective matrices is concerned with: merging of previous rat- ing matrix and new rating matrix, merging of previous user-requirement relation matrix and user-requirement relation matrix for new requirements. User-requirement relation matrix is used in the next step to determine the probability of stakeholders to provide rating to lately considered require- ments. Merged matrix is used to conduct actual prediction of the ratings. Figure 2 illustrates the merging process. User-requirement relation matrix consists of binary values: 1 and 0. If user-requirement relation matrix = 1, it denotes that for i-th require- (i,j) ment speciﬁed j-th stakeholder has provided a rating and If user-requirement relation matrix = 0, it denotes that for i-th requirement speciﬁed j-th (i,j) stakeholder has not provided any rating. This matrix aids to ﬁnd the pat- tern of stakeholders giving ratings to particular requirements. A sample of an user-requirement relation matrix is presented in Table 1. Table 1: A Sample User-Requirement Relation Matrix d.5.1. view and modify a.3.1. combine ID a.3.2. combine access rights, time Users card and library card of access, online, without session card card being present Aaron Toms 1 1 1 Adrian Bank 1 0 1 Alison Crane 1 0 ? Andy Hicks 1 1 ? Andrew Dawn 0 0 1 3.4 Determining users for prediction Ratings of which users are to be predicted has to be determined ﬁrst to implement actual prediction. To determine, similarity among new and previ- ously elicited requirements can be used. Item to item collaborative ﬁltering can be used for ﬁnding similarities among requirements. This approach pre- dicts probability of users to give ratings to new requirements based on this similarities. It learns if any speciﬁed user tends to rate the new requirement based on his/her rating on similar previous requirements. This approach can be divided into two steps. The steps are stated as following: 3.4.1 Correlation Computation 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Merging

text: respective

text: matrices

text: is

text: concerned

text: with:

text: merging

text: of

text: previous

text: rat-

text: ing

text: matrix

text: and

text: new

text: rating

text: matrix,

text: merging

text: of

text: previous

text: user-requirement

text: relation

text: matrix

text: and

text: user-requirement

text: relation

text: matrix

text: for

text: new

text: requirements.

text: User-requirement

text: relation

text: matrix

text: is

text: used

text: in

text: the

text: next

text: step

text: to

text: determine

text: the

text: probability

text: of

text: stakeholders

text: to

text: provide

text: rating

text: to

text: lately

text: considered

text: require-

text: ments.

text: Merged

text: matrix

text: is

text: used

text: to

text: conduct

text: actual

text: prediction

text: of

text: the

text: ratings.

text: Figure

text: illustrates

text: the

text: merging

text: process.

text: User-requirement

text: relation

text: matrix

text: consists

text: of

text: binary

text: values:

text: and

text: If

text: user-requirement

text: relation

text: matrix

text: it

text: denotes

text: that

text: for

text: i-th

text: require-

text: (i,j)

text: ment

text: speciﬁed

text: j-th

text: stakeholder

text: has

text: provided

text: a

text: rating

text: and

text: If

text: user-requirement

text: relation

text: matrix

text: it

text: denotes

text: that

text: for

text: i-th

text: requirement

text: speciﬁed

text: j-th

text: (i,j)

text: stakeholder

text: has

text: not

text: provided

text: any

text: rating.

text: This

text: matrix

text: aids

text: to

text: ﬁnd

text: the

text: pat-

text: tern

text: of

text: stakeholders

text: giving

text: ratings

text: to

text: particular

text: requirements.

text: A

text: sample

text: of

text: an

text: user-requirement

text: relation

text: matrix

text: is

text: presented

text: in

text: Table

text: Table

text: A

text: Sample

text: User-Requirement

text: Relation

text: Matrix

text: view

text: and

text: modify

text: combine

text: ID

text: combine

text: access

text: rights,

text: time

text: Users

text: card

text: and

text: library

text: card

text: of

text: access,

text: online,

text: without

text: session

text: card

text: card

text: being

text: present

text: Aaron

text: Toms

text: Adrian

text: Bank

text: Alison

text: Crane

text: Andy

text: Hicks

text: Andrew

text: Dawn

text: Determining

text: users

text: for

text: prediction

text: Ratings

text: of

text: which

text: users

text: are

text: to

text: be

text: predicted

text: has

text: to

text: be

text: determined

text: ﬁrst

text: to

text: implement

text: actual

text: prediction.

text: To

text: determine,

text: similarity

text: among

text: new

text: and

text: previ-

text: ously

text: elicited

text: requirements

text: can

text: be

text: used.

text: Item

text: to

text: item

text: collaborative

text: ﬁltering

text: can

text: be

text: used

text: for

text: ﬁnding

text: similarities

text: among

text: requirements.

text: This

text: approach

text: pre-

text: dicts

text: probability

text: of

text: users

text: to

text: give

text: ratings

text: to

text: new

text: requirements

text: based

text: on

text: this

text: similarities.

text: It

text: learns

text: if

text: any

text: speciﬁed

text: user

text: tends

text: to

text: rate

text: the

text: new

text: requirement

text: based

text: on

text: his/her

text: rating

text: on

text: similar

text: previous

text: requirements.

text: This

text: approach

text: can

text: be

text: divided

text: into

text: two

text: steps.

text: The

text: steps

text: are

text: stated

text: as

text: following:

text: Correlation

text: Computation

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The next step is to observe that the set CD(a, b) of all morphisms between a to × b, has the structure of a G Gop-module according to: b a · (g , g ) m = g mg . 1 2 1 2 Note also that there is an isomorphism of algebras C(G ×Gop) (cid:39) CG ⊗CGop (cid:39) CG ⊗ (CG )op. Now we can give a description of theb Caratan matribx in (cid:91)tearms b a of the representation theory of the endomorphism groups. D Proposition 3.9. Let be a ﬁnite and skeletal EI-category. Let I = I a ∈ a D0 be a set of indices for the primitive idempotents up to equivalence as described ∈ ∈ above. Take i I and j I . The number of times that S(j) appears as a a b ⊗ Jordan-Hölder factor in P(i) is the number of times that Sj D(Si) appears as an irreducible constituent in the G × Gop-module CD(a, b). b a Proof. The number of times that S(j) appears as a Jordan-Hölder factor in P(i), i.e., the entry (j, i) of the Cartan matrix of CD equals the dimension CD dim e e . j i ∈ D ∈ D Given m 1, it is clear that e me = 0 unless m (a, b) hence we have j i CD CD dim e e . = dim e (a, b)e j i j i where we consider CD(a, b) as a CG ⊗ CGop-module as explained above. Fol- b a lowing this point of view we can write CD ⊗ · CD e (a, b)e = (e e ) (a, b) j i j i It is easy to see that e ⊗ e is a primitive idempotent of CG ⊗(CG )op which j i b a corresponds to the simple module C ⊗ C ⊗ (cid:39) C ⊗ C (cid:39) C ⊗ C ( G ( G )op) (e e ) G e ( G )ope G e e G . b a j i b j a i b j i a For any G-module M and a primitive idempotent e ∈ CG we know that dim eM is the number of times that CGe appears as an irreducible constituent in M. Therefore, CD dim e (a, b)e j i is the number of times that the simple module CG e ⊗e CG appears as an irreducible constituent in CD(a, b). As CG e (cid:39) Sj anb djbyiLemama 3.8 we know that e CG (cid:39) D(Si), so we have the desirebdjresult. i a We will make another step in order to avoid explicit use of the opposite group. If M is an Gop-module, we can also regard M as a G-module with new action deﬁned by ∗ − · g m = g 1 m (where on the right hand side we use the Gop-module action). 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: next

text: step

text: is

text: to

text: observe

text: that

text: the

text: set

text: CD(a,

text: b)

text: of

text: all

text: morphisms

text: between

text: a

text: to

text: b,

text: has

text: the

text: structure

text: of

text: a

text: G

text: Gop-module

text: according

text: to:

text: b

text: a

text: (g

text: g

text: m

text: g

text: mg

text: Note

text: also

text: that

text: there

text: is

text: an

text: isomorphism

text: of

text: algebras

text: C(G

text: ×Gop)

text: CG

text: ⊗CGop

text: CG

text: (CG

text: )op.

text: Now

text: we

text: can

text: give

text: a

text: description

text: of

text: theb

text: Caratan

text: matribx

text: in

text: b

text: a

text: of

text: the

text: representation

text: theory

text: of

text: the

text: endomorphism

text: groups.

text: D

text: Proposition

text: Let

text: be

text: a

text: ﬁnite

text: and

text: skeletal

text: EI-category.

text: Let

text: I

text: I

text: a

text: a

text: be

text: a

text: set

text: of

text: indices

text: for

text: the

text: primitive

text: idempotents

text: up

text: to

text: equivalence

text: as

text: described

text: above.

text: Take

text: i

text: I

text: and

text: j

text: I

text: The

text: number

text: of

text: times

text: that

text: S(j)

text: appears

text: as

text: a

text: a

text: b

text: Jordan-Hölder

text: factor

text: in

text: P(i)

text: is

text: the

text: number

text: of

text: times

text: that

text: Sj

text: D(Si)

text: appears

text: as

text: an

text: irreducible

text: constituent

text: in

text: the

text: G

text: Gop-module

text: CD(a,

text: b).

text: b

text: a

text: Proof.

text: The

text: number

text: of

text: times

text: that

text: S(j)

text: appears

text: as

text: a

text: Jordan-Hölder

text: factor

text: in

text: P(i),

text: i.e.,

text: the

text: entry

text: (j,

text: i)

text: of

text: the

text: Cartan

text: matrix

text: of

text: CD

text: equals

text: the

text: dimension

text: CD

text: dim

text: e

text: e

text: j

text: i

text: D

text: D

text: Given

text: m

text: it

text: is

text: clear

text: that

text: e

text: me

text: unless

text: m

text: (a,

text: b)

text: hence

text: we

text: have

text: j

text: i

text: CD

text: CD

text: dim

text: e

text: e

text: dim

text: e

text: (a,

text: b)e

text: j

text: i

text: j

text: i

text: where

text: we

text: consider

text: CD(a,

text: b)

text: as

text: a

text: CG

text: CGop-module

text: as

text: explained

text: above.

text: Fol-

text: b

text: a

text: lowing

text: this

text: point

text: of

text: view

text: we

text: can

text: write

text: CD

text: CD

text: e

text: (a,

text: b)e

text: (e

text: e

text: (a,

text: b)

text: j

text: i

text: j

text: i

text: It

text: is

text: easy

text: to

text: see

text: that

text: e

text: e

text: is

text: a

text: primitive

text: idempotent

text: of

text: CG

text: ⊗(CG

text: )op

text: which

text: j

text: i

text: b

text: a

text: corresponds

text: to

text: the

text: simple

text: module

text: C

text: C

text: C

text: C

text: C

text: C

text: G

text: G

text: )op)

text: (e

text: e

text: G

text: e

text: G

text: )ope

text: G

text: e

text: e

text: G

text: b

text: a

text: j

text: i

text: b

text: j

text: a

text: i

text: b

text: j

text: i

text: a

text: For

text: any

text: G-module

text: M

text: and

text: a

text: primitive

text: idempotent

text: e

text: CG

text: we

text: know

text: that

text: dim

text: eM

text: is

text: the

text: number

text: of

text: times

text: that

text: CGe

text: appears

text: as

text: an

text: irreducible

text: constituent

text: in

text: M.

text: Therefore,

text: CD

text: dim

text: e

text: (a,

text: b)e

text: j

text: i

text: is

text: the

text: number

text: of

text: times

text: that

text: the

text: simple

text: module

text: CG

text: e

text: ⊗e

text: CG

text: appears

text: as

text: an

text: irreducible

text: constituent

text: in

text: CD(a,

text: b).

text: As

text: CG

text: e

text: Sj

text: anb

text: djbyiLemama

text: we

text: know

text: that

text: e

text: CG

text: D(Si),

text: so

text: we

text: have

text: the

text: desirebdjresult.

text: i

text: a

text: We

text: will

text: make

text: another

text: step

text: in

text: order

text: to

text: avoid

text: explicit

text: use

text: of

text: the

text: opposite

text: group.

text: If

text: M

text: is

text: an

text: Gop-module,

text: we

text: can

text: also

text: regard

text: M

text: as

text: a

text: G-module

text: with

text: new

text: action

text: deﬁned

text: by

text: g

text: m

text: g

text: m

text: (where

text: on

text: the

text: right

text: hand

text: side

text: we

text: use

text: the

text: Gop-module

text: action).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 1: A sketch of the expected Columbia Plot, indicating the order of the chiral phase transition as a function of light and strange quark masses at zero chemical po- tential. The crossover region is separated from regions of ﬁrst-order phase transitions by second-order phase transition lines. and extend this holographic analysis to ﬁnite chemical potential. 2 Soft-Wall Model To consider the thermodynam(cid:18)ics of AdS/QCD, we use(cid:19)an asymptotically anti-de Sitter 5-D black hole metric L2 − dz2 ds2 = f(z)dt2 + dx2 + . (1) z2 i f(z) Following the procedure established in our previous work and in [22, 23, 24], we model ﬁnite temperature and chemical potential with a charged black hole described (cid:18) (cid:19) (cid:18) (cid:19) by the 5D AdS–Reissner-Nordstro¨m metric − z 4 z 6 f(z) = 1 (1 + Q2) + Q2 , (2) z z h h 3 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: A

text: sketch

text: of

text: the

text: expected

text: Columbia

text: Plot,

text: indicating

text: the

text: order

text: of

text: the

text: chiral

text: phase

text: transition

text: as

text: a

text: function

text: of

text: light

text: and

text: strange

text: quark

text: masses

text: at

text: zero

text: chemical

text: po-

text: tential.

text: The

text: crossover

text: region

text: is

text: separated

text: from

text: regions

text: of

text: ﬁrst-order

text: phase

text: transitions

text: by

text: second-order

text: phase

text: transition

text: lines.

text: and

text: extend

text: this

text: holographic

text: analysis

text: to

text: ﬁnite

text: chemical

text: potential.

text: Soft-Wall

text: Model

text: To

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