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abstract: 'We have derived new abundances of the rare-earth elements Pr, Dy, Tm, Yb, and Lu for the solar photosphere and for five very metal-poor, neutron-capture -rich giant stars. The photospheric values for all five elements are in good agreement with meteoritic abundances. For the low metallicity sample, these abundances have been combined with new Ce abundances from a companion paper, and reconsideration of a few other elements in individual stars, to produce internally-consistent Ba, rare-earth, and Hf (56 $\leq$ Z $\leq$ 72) element distributions. These have been used in a critical comparison between stellar and solar abundance mixes.'
author:
- |
Christopher Sneden, James E. Lawler, John J. Cowan,\
Inese I. Ivans, Elizabeth A. Den Hartog
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title: 'New Rare Earth Element Abundance Distributions for the Sun and Five $r$-Process-Rich Very Metal-Poor Stars'
---
INTRODUCTION\[intro\]
=====================
Early Galactic nucleosynthesis studies have been invigorated over the last decade by the discovery of many low metallicity halo stars with abundance distributions that depart significantly from that of our Solar System. The neutron-capture elements (Z $>$ 30, hereafter ) as a group exhibit particularly large star-to-star abundance variations with respect to Fe-peak elements. For example, data from a number of surveys collected in Sneden, Cowan, & Gallino (2008) show an abundance range in the rare-earth element Eu of at least $-$0.5 $\lesssim$ \[Eu/Fe\] $\lesssim$ +2.0 at metallicities \[Fe/H\] $\lesssim$ $-$2.5[^1]; see their Figure 14.
The abundances in the solar system and in most metal-rich Galactic disk stars arise from the combined effects of prior rapid and slow synthesis events (the “” and “”, respectively). The abundance patterns in low metallicity stars, however, vary widely. Examples have been found with element distributions that are consistent with the , the , and a variety of mixes in between these two extremes. These stars are thus natural test cases for nucleosynthesis predictions.
Rigorous tests of and theories require very accurate abundances in metal-poor stars. Good abundance determinations result from effort on all fronts: acquisition of very high resolution, low noise spectra of the stars; construction of realistic model stellar atmospheres; analysis of the spectra with few limiting simplifications; and improvement in basic atomic and molecular data. We have taken up the last consideration in the present series of papers: Lawler, Bonvallet, & Sneden (2001a), Lawler (2001b), Lawler (2001c), Den Hartog (2003), Lawler, Sneden, & Cowan (2004), Den Hartog (2005), Lawler (2006), Den Hartog (2006), Lawler (2007), Sobeck, Lawler, & Sneden (2007), Lawler (2008b), and Lawler (2009). We have concentrated most of our efforts on: (a) improving the basic laboratory data for (mostly) rare-earth ionized species that are detectable in metal-poor stars; (b) applying these data to derive new solar spectroscopic abundances and comparing these photospheric values to solar-system meteoritic data (Lodders 2003); and (c) extending the abundance analyses to a few well-studied low metallicity giants that are enriched in the products of the . Our most recent study (Lawler 2009) reports improved transition probabilities for 921 lines of . The present paper culminates this series with new solar and stellar analyses of Pr, Dy, Tm, Yb, and Lu. These elements all have good laboratory studies of their first ions in the literature, but have not been systematically subjected to solar/stellar analyses in the same manner as have other rare earths.
In this paper we expand the standard definition of the rare-earth elements from the lanthanides (57 $\leq$ Z $\leq$ 71) to include two adjacent elements Ba (Z = 56) and Hf (Z = 72), and adopt the collective shorthand notation “RE” for them. This broad definition covers a contiguous set of elements that have similar properties for stellar spectroscopy. In particular, these elements have relatively low first ionization potentials, 5.2 eV $\leq$ $IP$ $\leq$ 6.8 eV, and thus are almost completely ionized in the solar photosphere and in the atmospheres of low-metallicity giant stars. Their only detectable spectral features arise from their first ionized species. Element groups in the Periodic Table immediately preceding the REs (, I, Xe, Cs) and following them (, Ta, W, Re) have very different atomic properties. For various reasons traceable to very low abundances, Saha/Boltzmann energy level population effects, and/or lack of accessible transition wavelengths, these elements just outside the RE group are inaccessible to most stellar spectroscopic detection efforts.
In §\[obs\] we review the solar and stellar spectroscopic data and outline the abundance derivation methods. Results for individual elements are given in §\[newels\]. We summarize the total RE abundance sets for the solar photosphere and -rich metal-poor giant stars in §\[rareab\]. Finally, in §\[discuss\] we use the stellar RE abundance distributions in a critical examination of predictions.
SPECTROSCOPIC OBSERVATIONS, REDUCTIONS AND ANALYSES\[obs\]
==========================================================
For most of our stars, we analyzed the same high-resolution spectra that have been used in previous papers of this series. Additional descriptions of these stellar spectra can be found in their original studies: , Cowan (2002); , Sneden (2003); , Westin (2000); , Ivans (2006); see also Cowan (2005). The spectroscopic data sets employed in our analysis are summarized in Table \[tab1\]. For each of the instrumental setups listed, we report the useful wavelength range, and wavelength-dependent values of the signal-to-noise $S/N$, resolving power $R$, and quality factor per resolution element $F$ (sometimes also referred to as figure-of-merit), at selected wavelengths $\lambda_{app}$.
Data reduction for the Keck and McDonald data have been detailed in previous papers of this series and have largely relied on IRAF[^2] and FIGARO.[^3] For the recently acquired Magellan/MIKE data, we employed the MIKE Pipeline software[^4] (Kelson 2000, Kelson 2003). All of the data received final processing including continuum normalization and telluric feature removal using SPECTRE [@fit87]. Finally, for the solar analyses we employed the very high resolution ($R$ = $\ge$ 300,000), very high signal-to-noise (S/N $\ge$ 1000) center-of-disk photospheric spectrum of Delbouille, Roland, and Neven (1973)[^5].
The abundance analyses used the same methods that have been described at length in previous papers of this series. Here we summarize the main points; the reader should consult Lawler (2009) and references therein for details.
For each species, we begin with computations of relative strengths of all lines, in order to trim the often extensive laboratory line lists to a set that might produce detectable absorption in the solar photosphere and in our program stars. Line absorption coefficients are proportional to products of oscillator strengths and absorber number densities. In a standard LTE analysis, Boltzmann/Saha statistics describe the populations of atoms in various ionization stages and electronic levels. As discussed in §\[intro\], the REs have low ionization potentials, and thus exist almost completely as singly ionized species. Saha corrections for other ionization states can be neglected. Therefore the [*relative*]{} strength factors for ionized-species RE elements can be approximated by $\log(\epsilon gf) - \theta \chi$, where $\epsilon$ is the elemental abundance, $gf$ is the transition probability, $\theta$ = 5040/T is the reciprocal temperature, and $\chi$ is the excitation energy.
Almost all easily detectable RE lines are of low excitation, $\chi$ $\lesssim$ 1 eV, so the relative line strengths are not very sensitive to temperature. Choosing $\theta$ = 1.0 as a rough mean of the solar and stellar reciprocal temperatures, and adopting approximate solar abundance values for each element under consideration, we computed relative strength factors for , , , and lines, using the laboratory data that will be discussed in the appropriate subsections of §\[newels\]. We did not perform such computations for , as it has only two very strong lines of interest for abundance analyses (see §\[ybtext\]). The results of this exercise are displayed in Figure \[f1\]. With horizontal lines we mark the approximate minimum relative strength value for lines that can be considered “strong”. Such lines are those with evident saturation in their equivalent widths (EWs), which for the Sun empirically is log($EW/\lambda$) $\sim$ $-$5.3. We similarly mark the approximate strength value at which photospheric lines have log($EW/\lambda$) $\sim$ $-$6.5, too weak to be routinely used in abundance solar abundance analyses.
Figure \[f1\] can be compared to similar plots for other RE elements in some of the previous papers of this series. Some general remarks apply to all RE ions. Most REs have complex energy structures, leading to large numbers of transitions. Their relative strength factors increase with decreasing wavelength; these usually are transitions from the lowest energy levels with the largest values. The most fertile regime for RE transitions is the near-UV domain, $\lambda$ $<$ 4000 Å. Unfortunately, the strong-line density of all species increases in this wavelength range, and many promising RE transitions are hopelessly blended with (usually) Fe-peak lines. Finally, as is evident in Figure \[f1\], very few RE ions have detectable transitions in the yellow-red ($\lambda$ $>$ 5000 Å) spectral region of the solar spectrum. Comments on the line strengths of individual species will be given in §\[newels\]. These same strength factors turn out to work reasonably well for the -rich giant stars. Their combination of cooler temperatures, more extended atmospheres, metal poverty, and enhanced abundances yields line strengths that are similar to or somewhat larger than those for the Sun.
We eliminated lines with relative strength factors that fell below the probable detection limits, and searched solar and stellar spectra for the remaining lines. In this effort we employed the large Kurucz (1998)[^6] line list, the solar line identifications of Moore, Minnaert, & Houtgast (1966), and the observed spectra described above. With these resources we were able to discard many additional lines that proved to be too weak and/or too blended to be of use either for the Sun or for the -rich stars.
We then constructed synthetic spectrum lists for small spectral regions (4–6 Å) surrounding each promising candidate line. These lists were built beginning with the Kurucz (1998) atomic line database. We updated the species transition probabilities with results from this series of papers, including the laboratory data cited below for Pr, Dy, Tm, Yb, and Lu. We also used recently published values for (Sobeck, Lawler, & Sneden 2007) and (Malcheva 2006). Lines missing from the Kurucz database but listed in the laboratory studies or in the Moore (1966) solar line atlas were added in. In spectral regions where molecular absorption is important, we used the Kurucz data for OH, NH, MgH, and CN, and Plez (private communication) data for CH.
We iterated the transition probabilities through repeated trial spectrum syntheses of the solar photosphere (and sometimes one of the -rich giant stars). For the Sun, as in previous papers of this series we adopted the Holweger & Müller (1974) empirical model photosphere, and computed the synthetic spectra with the current version of Sneden’s (1973) LTE, 1-dimensional (1D) line analysis code MOOG. In these trial syntheses, no alterations were made to the lines with good laboratory ’s. On occasion, obvious absorptions without plausible lab or solar identifications were arbitrarily defined to be lines with excitation energies $\chi$ = 3.5 eV and values to match the photospheric absorption. We discarded all candidate RE lines that proved to be seriously blended with unidentified contaminants.
Final solar abundances for each line were determined through matches between the Delbouille (1973) photospheric center-of-disk spectra and the empirically-smoothed synthetic spectra. The same procedures were applied to the observed stellar spectra (Table \[tab1\]) and synthetic spectra generated with the model atmospheres whose parameters and their sources are given in Table \[tab2\].
ABUNDANCES OF PR, DY, TM, YB, AND LU\[newels\]
==============================================
In this section we discuss our abundance determinations of elements Pr, Dy, Tm, Yb, and Lu in the Sun and the -rich stars. Tables \[tab3\], \[tab4\], and \[tab5\] contain the mean abundances in the solar photosphere and in the -rich low metallicity giants for these elements and for other REs that have been analyzed in previous papers of this series. The full suite of elements will be discussed in §\[rareab\]. Table \[tab3\] also gives estimates of abundance components in solar-system meteoritic material. These data will be discussed in more detail in §\[discuss\].
Praseodymium\[prtext\]
----------------------
Pr ($Z$ = 59) has one naturally-occurring isotope, [$^{\rm 141}$Pr]{}. The spectrum has been well studied in the laboratory, with transition probabilities reported by Ivarsson (2001; hereafter Iv01), Biémont (2003) and Li (2007; hereafter Li07), as well as numerous publications on its wide hyperfine structure. We will consider the hfs data in more detail in the Appendix.
We adopted Li07 as our primary transition probability source. This is the most recent and largest set, 260 lines, of purely experimental measurements (Li07 combined their own branching fractions with previously published lifetimes). Iv01 also conducted a smaller lab study, reporting values for 31 lines. However, their list includes four lines not published by Li07. Therefore we considered both Li07 and Iv01 data sets in our abundance determinations. In Figure \[f2\] we plot the differences between individual Iv01 and Li07 values, using different symbols to distinguish those lines employed in our abundance analyses from those that proved to be unsuitably weak or blended. There is generally good agreement: ignoring the five obviously discrepant lines that are labeled by wavelength in the figure, the mean difference is $<\loggf_{Iv01} - \loggf_{Li07}>$ = +0.03 $\pm$ 0.01 ($\sigma$ = 0.06, 23 lines). Comments on individual lines in common are given below. Biémont (2003) also published ’s for 150 lines. However, their values were determined by combining experimental lifetimes and theoretical branching fractions, which are very difficult to compute for the complex RE atomic structures (, Lawler (2008a).
Moore (1966) give 21 identifications for the solar spectrum. However, most of them are very weak and/or blended. An early study by Biémont (1979) has a good discussion of the benefits and disadvantages of many of these lines for photospheric abundance work. They used nine lines to derive $_\odot$ = 0.71 $\pm$ 0.08,[^7] with individual lines contributing to the average with different weights. Only three of these lines were considered to be high-quality ones. More recently, Ivarsson, Wahlgren, & Ludwig (2003) employed synthetic/observed spectral matches to suggest $_\odot$ = 0.4 $\pm$ 0.1, more than a factor of two smaller than the meteoritic value of $_{met}$ = 0.78 $\pm$ 0.03.
We searched for useful lines in the solar spectrum by first identifying them in , which is the most extreme -rich metal-poor star of our sample: \[Fe/H\] = –2.9, \[Eu/Fe\] = +1.6 (Hill 2002). This star’s low metallicity and large \[/Fe-peak\] abundance ratios combine to yield many strong (and often essentially unblended) candidate transitions. Inspection of the spectrum yielded 43 lines from Li07 and an additional 3 lines from Iv01 that merited abundance consideration. Preliminary synthetic spectrum calculations suggested that 13 of these candidate lines were either too weak or too blended in both and the Sun. The wide hyperfine structure of all prominent lines made this exercise much easier than it would be in searches for lines with no hfs. In Figure \[f3\] we illustrate this point with synthetic and observed spectra of the strong 4408.8 and 4179.4 Å transitions. Visual inspection of the profiles suggests that their full-width-half-maxima are FWHM $\simeq$ 0.4 Å, while observed and synthetic profiles of single lines (, 4178.86 Å and 4179.59 Å ) have FWHM $\simeq$ 0.25 Å Wavelengths of the remaining useful lines are given in Table \[tab6\], along with their excitation energies and the Li07 and Iv01 transition probabilities. In Figure \[f2\] one sees five lines with large discrepancies between these studies. Three of the lines were not involved for our abundance studies and so we cannot comment further on them. Li07 caution that 5219.1 Å is blended on their spectra. We adopted the Iv01 value for this line. Finally, the difference between Iv01 and Li07 for 5322.8 Å is 0.2 dex, but abundances derived with the Li07 proved to be consistent with those from other Pr lines.
We calculated solar photospheric synthetic spectra for all the lines of Table \[tab6\]. We found, as have the previous studies cited above, that there are few useful solar Pr abundance indicators. Our final value was based on five lines (Table \[tab6\]). We show the synthetic/observed photospheric spectrum matches for four of these lines in left-hand panels (a), (c), (e), and (g) of Figure \[f4\], contrasting their appearance in right-hand panels (b), (d), (f), and (h) for . We do not include the 5219.1 Å line in Figure \[f4\] because it was too weak in the spectrum of to analyze in that star. Note that Li07 transition probabilities were used for the 4222.9, 4510.1, and 5322.8 Å lines and Iv01 values for the 5219.1 and 5259.7 Å lines. However, consistent abundances from all five lines were derived: the mean value (Table \[tab3\]) is $_\odot$ = 0.76 $\pm$ 0.02 (sigma = 0.04). Our new photospheric abundance is in good agreement with the meteoritic and the Biémont (1979) photospheric abundances that were quoted above.
For the -rich low metallicity stars we derived abundances from 10–27 lines (Table \[tab6\]). We plot the individual line abundances for these stars and the Sun as functions of wavelength in Figure \[f5\], with their summary abundance statistics in the panel legends. In each case the line-to-line scatter was small, $\sigma$ $\simeq$ 0.06, and we found no significant abundance trends with wavelength, excitation energy (the range in this quantity is only $\simeq$1 dex), or .
Dysprosium\[dytext\]
--------------------
Dy ($Z$ = 66) has seven naturally-occurring isotopes, five of which contribute substantially to its solar-system abundance: [$^{\rm 156,158}$Dy]{}, $\ll$1%, [$^{\rm 160}$Dy]{}, 2.34%; [$^{\rm 161}$Dy]{}, 18.91%; [$^{\rm 162}$Dy]{}, 25.51%; [$^{\rm 163}$Dy]{}, 24.9%; and [$^{\rm 164}$Dy]{}, 28.19% (Lodders 2003). The atomic structure of is complex, leading to a rich spectrum of transitions arising from low-excitation energy levels. This species has been well-studied in the laboratory recently, with published transition probabilities by Kusz (1992), Biémont & Lowe (1993), and Wickliffe, Lawler, and Nave (2000). The Wickliffe study contains a detailed comparison of their transition probabilities with those of Kusz and Biémont & Lowe (as well as earlier investigations), and will not be repeated here.
We adopted the Wickliffe (2000) values, as in our earlier analyses of the -rich stars. Those studies (, Ivans 2006 for , and Sneden 2003 for ) performed extensive searches for promising lines. However, the Dy abundances reported in those papers were derived from both EW matches and synthetic spectrum calculations. Therefore, to be internally consistent in our new analyses, we began afresh with new solar identifications and new synthesis line lists for each chosen feature. In principle lines should have both isotopic wavelength splitting and (for [$^{\rm 161,163}$Dy]{}) hyperfine substructure. We inspected the profiles of many of the strongest lines appearing in National Solar Observatory (NSO) Fourier Transform Spectrometer (FTS) laboratory spectra. Some line substructure is present in each line. However, the components that are shifted away from the line centers are always very weak ($\lesssim$10% of central intensities), and the full widths near profile baselines are $\sim$0.05 Å. For all lines, FWHM $\sim$ 0.02 Å in the lab spectra. These widths are substantially smaller than the measured solar and stellar spectrum line widths. Therefore we treated all lines as single features.
There are many candidate lines, as indicated by their relative strength values shown in panel (b) of Figure \[f1\]. Solar Dy abundances could be determined from 13 of these transitions. The resulting mean photospheric abundance (Table \[tab3\]) is $_\odot$ = +1.13 $\pm$ 0.02 ($\sigma$ = 0.06). This value is in excellent agreement with the meteoritic abundance, $_{met}$ = +1.13 $\pm$ 0.04 and with the Kusz (1992) photospheric abundance, $_\odot$ = +1.14 $\pm$ 0.08. It is also in reasonable accord with the Biémont & Lowe (1993) value, $_\odot$ = +1.20 $\pm$ 0.06.
Synthetic spectra of 24–35 lines were used in the Dy abundance derivations for the -rich low-metallicity giants (Table \[tab7\]). The analyses were straightforward, as many lines in each star’s spectrum were strong and unblended. This led to very well-determined mean abundances (Tables \[tab4\] and \[tab5\]).
Thulium\[tmtext\]
-----------------
Tm ($Z$ = 69) has one naturally-occurring isotope, [$^{\rm 169}$Tm]{}. This element is one of the least abundant of the REs: $_{met}$ = 0.11 $\pm$ 0.06 (Lodders 2003). Therefore transitions in solar and stellar spectra are weak, and relatively few can be employed in abundance analyses. Moore (1966) list only 10 identifications in their solar line compendium; all of these lie at wavelengths $\lambda$ $<$ 4300 Å.
We considered the 146 lines investigated by Wickliffe & Lawler (1997). That study reported laboratory experimental transition probabilities derived from their branching fractions and the radiative lifetimes of Anderson, Den Hartog, and Lawler (1996). The relative strengths of these lines are displayed in panel (c) of Figure \[f1\]. Inspection of this plot suggests that few detectable lines will be found redward of 4000 Å, in accord with the Moore (1966) identifications.
As in the case of Pr (§\[prtext\]), we began our search for suitable transitions with , since they should stand out most clearly among the weaker Fe-peak contaminants in this star’s spectrum. Only nine lines were sufficiently strong and unblended to warrant further investigation. We computed synthetic spectra for each of these candidate features. Although Tm is an odd-$Z$, odd-$A$ atom with a non-zero nuclear spin ($I$ = $\frac{1}{2}$), inspection of the chosen lines in very high-resolution NSO FTS spectra showed that hyperfine splitting is very small, and could be safely ignored in the calculations.
Our synthetic spectra of lines for the solar photosphere showed that only three of them could be used for abundance analysis. The synthetic/observed spectrum matches for these lines in the solar photosphere are displayed in Figure \[f6\], along with those for . It is clear that each of these lines is weak and blended in the photospheric spectrum, while being much stronger and cleaner in the -rich low metallicity giant star.
These lines and their photospheric abundances are listed in Table \[tab8\]. We derive a formal mean abundance (Table \[tab3\]) of $_\odot$ = +0.14 $\pm$ 0.02 ($\sigma$ = 0.04). Caution obviously is warranted here. Probably the $\sigma$ value is a truer estimate of the abundance uncertainty than the standard deviation of the mean. However, this photospheric abundance is in reasonable agreement with the meteoritic value, $_{met}$ = +0.11 $\pm$ 0.06.
More features could be employed in the abundance determinations for the -rich low-metallicity giants (Table \[tab8\]). Their mean values (Tables \[tab4\] and \[tab5\]) were based on 5–7 lines per star. For stars analyzed previously by our group, the new Tm abundances agree with the published values to within the uncertainty estimates. The Tm abundance for will be discussed along with this star’s other REs in §\[rrichstars\].
Ytterbium\[ybtext\]
-------------------
Yb ($Z$ = 70) has seven naturally-occurring isotopes, six of which are major components of its solar-system abundance: [$^{\rm 168}$Yb]{}, $\ll$1%; [$^{\rm 170}$Yb]{}, 3.04%; [$^{\rm 171}$Yb]{}, 14.28%; [$^{\rm 172}$Yb]{}, 21.83%; [$^{\rm 173}$Yb]{}, 16.13%; [$^{\rm 174}$Yb]{}, 31.83%; and [$^{\rm 176}$Yb]{}, 12.76% (Lodders 2003). The atomic structure of is similar to that of , with a $^2$S ground state and first excited state more than 2.5 eV above the ground state. Therefore this species has very strong resonance lines at 3289.4 and 3694.2 Å as the only obvious spectral signatures of this element. All other lines are expected to be extremely weak.
The resonance lines have complex hyperfine and isotopic substructures that broaden their absorption profiles by 0.06 Åand must be included in synthetic spectrum computations. In the Appendix we discuss the literature sources for the resonance lines and tabulate their substructures in a form useful for stellar spectroscopists. Moore (1966) identified major , , and contaminants to the 3289.4 Å line, and our synthetic spectra confirmed that is a small contributor to the total feature. From our synthetic spectra of the 3694.2 Å line we derived $_\odot$ = +0.86 $\pm$ 0.10 (Table \[tab3\]), in reasonable agreement with $_{met}$ = +0.94 $\pm$ 0.03. The large uncertainty attached to our photospheric abundance arises from a variety of sources: (a) reliance on a single line; (b) its large absorption strength, which increases the dependence on adopted microturbulent velocity; (c) the contaminating presence of the strong 3694.0 Å line; and (b) closeness of this spectral region to the Balmer discontinuity.
We then synthesized the 3289 and 3694 Å lines in the stellar spectra. However, these are -rich stars, and the isotopic mix in a pure nucleosynthetic mix is different than that of the solar system ( and ) combination. For our computations we adopted (see Sneden 2008): [$^{\rm 168,170}$Yb]{}, 0.0%; [$^{\rm 171}$Yb]{}, 17.8%; [$^{\rm 172}$Yb]{}, 22.1%; [$^{\rm 173}$Yb]{}, 19.0%; [$^{\rm 174}$Yb]{}, 22.7%; and [$^{\rm 176}$Yb]{}, 18.4%.
The Yb contribution to the 3289 Å feature is very large in the -rich stars. In the most favorable case, , Yb accounts for roughly 75% of the total blend. Unfortunately, the contributions of the contaminants (mostly ) cannot be assessed accurately enough for this line to be a reliable Yb abundance indicator. The synthetic/observed spectral matches of the 3694 Å line provide the new Yb abundances listed in Tables \[tab4\] and \[tab5\]. These values are consistent with the ones reported in the original papers on these stars. However, while the absorption dominates that of the possible metal-line contaminants, the Balmer lines in this spectral region are substantially stronger in these low-pressure giant stars than they are in the solar photospheric spectrum. In particular, lines at 3691.6 and 3697.2 Å significantly depress the local continuum at the wavelength. Caution is warranted in the interpretation of these Yb abundances.
Lutetium\[lutext\]
------------------
Lu ($Z$ = 71), has two naturally-occurring isotopes: [$^{\rm 175}$Lu]{}, 97.416%; and [$^{\rm 176}$Lu]{}, 2.584% (Lodders 2003). It is the least abundant RE: $_{met}$ = 0.09 $\pm$ 0.06 (Lodders). has a relatively simple structure, with a $^1$S ground state. It has no other very low-energy states; the first excited level lies 1.5 eV above the ground state. This ion with only two valence electrons has relatively few strong lines in the visible and near UV connected to low E.P. levels, although most of the prominent lines have well-determined experimental transition probabilities.
We considered only the transitions of Quinet (1999), using their experimental branching fractions and lifetime measurements by Fedchak (2000) to determine transition probabilities. These are are listed, along with wavelengths and excitation energies, in Table 12 of Lawler (2009). The combination of a small solar-system Lu abundance and the (unfavorable) atomic parameters produces very small relative strength factors for these lines, as shown in panel (d) of Figure \[f1\]. No line even rises to our defined “weak-line” threshold of usefulness. Moore (1966) lists only 3077.6, 3397.1, and 3472.5 Å identifications in their solar line compendium, and all of these lines appear to be blended.
We made a fresh search for detectable lines of , and succeeded mainly in confirming the results of a previous investigation by Bord, Cowley, & Mirijanian (1998). Those authors argued that all of the lines identified by Moore (1966) are unsuitable for solar Lu abundance work. They quickly dismissed the 3077.6 and 3472.5 Å lines and performed an extended analysis of 3397.1 Å. Synthetic spectrum computations around this feature (see their Figures 2 and 3) convinced them that molecular NH dominates the absorption at the wavelength. Our own trials produced the same outcome.
Bord (1998) detected 6221.9 Å in the Delbouille (1973) photospheric spectrum. This line is extremely weak, EW $\sim$ 1 mÅ, and its hyperfine substructure spreads the absorption over $\sim$0.5 Å. The complex absorption profile of this line (see their Figure 4) actually increases one’s confidence in its identification in the photospheric spectrum. Bord reported $_\odot$ = +0.06 with an estimated $\pm$0.10 uncertainty from this line.
We repeated their analysis, using the hyperfine substructure pattern given in Table 13 of Lawler (2009), and derived $_\odot$ = +0.12 $\pm$ 0.08 (Table \[tab3\]), where the error reflects uncertainties in matching synthetic and observed feature profiles. This photospheric abundance is consistent with the Bord (1998) value and with the meteoritic abundance quoted above, given the uncertainties attached to each of these estimates. Our lack of success in identifying other Lu abundance indicators in the solar photospheric spectrum suggests that prospects are poor for reducing its error bar substantially in the future.
We also attempted to study the 3397 and 6621 Å lines in our sample of -rich low metallicity giants. Absorption by at 3397.1 Å is certainly present in the spectra of at least and . Unfortunately, the lower resolutions of our stellar spectra compared to that of the solar spectrum creates more severe blending of the Lu transition with neighboring lines, and NH contamination of the total feature still creates substantial abundance ambiguities. The 6221.9 Å line should be present, albeit very weak, in these stars. However, our spectra (when they extend to this wavelength range) lack the S/N to allow meaningful detections. We therefore cannot report Lu abundances for these -rich stars.
RARE EARTH ABUNDANCE DISTRIBUTIONS IN THE SUN AND R-PROCESS-RICH STARS\[rareab\]
================================================================================
The Sun and Solar System\[solarab\]
-----------------------------------
With new analyses of Pr, Dy, Tm, Yb, and Lu we now have determined abundances for the entire suite of REs in the solar photosphere. In Table \[tab3\] we merge the results of this and our previous papers. Missing from the list is of course Pm (Z = 61), whose longest-lived isotope, [$^{\rm 145}$Pm]{}, is only 17.7 years (Magill, Pfennig, and Galy 2006). We also chose not to include a photospheric value for Ba, whose few transitions are so strong that their solar absorptions cannot be reliably modeled in the sort of standard photospheric abundance analysis that we have performed.
The photospheric abundance uncertainties quoted in Table \[tab3\] are combinations of internal “scatter” factors (mainly continuum placement, observed/synthetic matching, and line blending problems) and external “scale” factors (predominantly solar model atmosphere choices). These issues are discussed Lawler (2009) and in previous papers of this series. We remind the reader that our abundance computations have been performed with the traditional assumptions of LTE and 1D static atmosphere geometry. Very little has been done to date to explore the effects of these computational limitations for RE species in the solar atmosphere. Mashonkina & Gehren (2000) have performed non-LTE abundance analyses of Ba and Eu, but their photospheric abundances are not substantially different from LTE results. There have been efforts to model the solar spectrum with more realistic 3-dimensional (3D) hydrodynamic models; see the summary in Grevesse, Asplund, & Sauval (2007), and references therein. These studies so far have reported new solar abundances only for the lighter elements (CNO, Na$-$Ca, and Fe). Generally the 3D non-LTE line modeling efforts yield lower abundances: comparing the photospheric values in Grevesse to those of the older standard compilation of Anders & Grevesse (1989), $<\delta$log $\epsilon>$ = $-$0.12 $\pm$ 0.03 ($\sigma$ = 0.09, for 11 elements that can be studied with photospheric spectra). We thus expect that any RE abundance shifts with 3D modeling would be similar from element to element, leaving their abundance ratios essentially unchanged. Future studies to explore these effects in detail will be welcome.
In Figure \[f7\] we compare RE photospheric abundances to their meteoritic values. In the top panel the “OLD” values are best estimates by Anders & Grevesse (1989). While the average agreement is good, significant discrepancies between individual abundances are evident, particularly at the low-abundance end. Formally, a simple mean is $<\log \epsilon_{\odot-AG89} - \log \epsilon_{met-AG89}>$ = 0.00 $\pm$ 0.06 ($\sigma$ = 0.22). In the bottom panel, the “NEW” meteoritic abundances (Lodders 2003) are correlated with our “NEW” photospheric ones (Table \[tab3\]). The data sources are denoted by different symbols in the figure: red open circles for photospheric abundances newly determined here and in Lawler (2009) for which Wisconsin-group lab data have been used; black filled circles for abundances reported in our previous papers; and blue open triangles for two elements with transition probability data adopted from other literature sources. Clearly the agreement is excellent: for 15 elements the formal mean difference is $<\log \epsilon_\odot - \log \epsilon_{met}>$ = 0.01 $\pm$ 0.01 ($\sigma$ = 0.05). No trends are discernible with the source of atomic data, or the abundance levels (as shown in the figure), or the number of lines that contribute to the photospheric abundances (Table \[tab3\]). With the possible exception of Hf (discussed in Lawler 2007 and in §\[discuss\]), and with repeated cautions about the photospheric abundances deduced from only one or two transitions, the two primary sources of primordial Solar-System abundances appear to be in complete accord.
The -Rich Low Metallicity Giant Stars\[rrichstars\]
---------------------------------------------------
Rare-earth abundances for the five -rich stars from this and our previous papers are collected in Tables \[tab4\] and \[tab5\]. For all stars the Pr, Dy, Tm, and Yb abundances are, of course, newly determined in this paper. We chose also to redo all the Ba abundances via new synthetic spectrum calculations, to ensure that these were determined in a consistent manner. We also performed new analyses for selected elements in individual stars (, Tb in ) when the original papers either did not report abundance values or did so with now-outdated atomic data.
Of particular interest is the very -enhanced star , which is a recent addition to our -rich star list. This star gained notoriety as the first -rich star with a convincing detection of U, a long-lived radioactive element of great interest to cosmochronology (Cayrel 2001). The first and most complete study of this star was published by Hill (2002). The mean difference between our RE abundances for this star and theirs is $<\log \epsilon_{Hill} - \log \epsilon_{us}>$ = $-$0.05 $\pm$ 0.03 ($\sigma$ = 0.10, 12 elements in common). We also compared our abundances with those of Honda (2004), with similar results: $<\log \epsilon_{Honda} - \log \epsilon_{us}>$ = +0.07 $\pm$ 0.03 ($\sigma$ = 0.09, 12 elements in common). The mean offsets are very small, and reflect minor differences in model atmospheres, observed spectra, analytical techniques, and atomic data choices. The element-to-element scatters are also reasonable, given the use of many more transitions in our study (a total of 342, Table \[tab4\]) compared to 95 in Hill and 49 in Honda Note that some portion of the $\sigma$’s in these comparisons arises because the Tb abundance differences are offset by $\simeq$0.2 dex from the mean differences (we derive larger values). Investigation of this one anomaly is beyond the scope of this work.
The abundance standard deviations of samples ($\sigma$) and of means that are given in Tables \[tab4\] and \[tab5\] refer to internal (measurement scatter) errors only. To investigate scale uncertainties, we determined the abundance sensitivities of eight RE elements to changes in model parameters (, , \[M/H\], ), to changes in the adopted model atmosphere grid, and to changes to line computations to better account for continuum scattering opacities. In Table \[tab9\] we summarize the results of these exercises. We began with a “baseline” model atmosphere from the Kurucz (1998) grid with parameters = 4750 K, = 1.5, \[M/H\] = –2.5, and = 2.0. Such a model is similar to the ones adopted for the -rich giants (Table \[tab2\]). We derived abundances with this model for 1-4 typical transitions each of the elements for the program star . Full account was taken of hyperfine and isotopic substructure for La, Pr, Eu, and Yb. We then repeated the abundance derivations for models with parameters varied as indicated in Table \[tab9\], including a trial using a model with baseline parameters taken from the new MARCS (Gustafsson 2008) grid.[^8] The inclusion of scattering in computations of continuum source functions, a new feature in our analysis code, is described in Sobeck (2008)
The Table \[tab9\] quantities are differences between abundances of the individual models and those of the baseline model. The uncertainties in stellar model parameters given in the original -star papers are typically $\pm$150 K in , $\pm$0.3 in , $\pm$0.2 in , and $\pm$0.2 in \[M/H\] metallicity. Application of these uncertainties to the model parameter dependences of Table \[tab9\] suggests that \[M/H\] and choices are not important abundance error factors. Temperature and gravity values obviously play larger roles. However, while the absolute abundances of individual elements change with different and choices, the relative abundances generally do not; in most cases, all RE abundances move in lock step. Assuming here that the atmosphere parameter uncertainties are uncorrelated, we estimate total abundance uncertainties for each RE element to be $\sim$0.15$-$0.20, but the abundance ratios have uncertainties of $\sim$0.01$-$0.05 (the exception is Yb, represented by only one very strong line in the UV spectral region; see §\[ybtext\]). More detailed computations that consider departures from LTE among RE first ions in the atmospheres of very metal-poor giant stars should be undertaken in the future. Some first steps in this direction have been undertaken for Ba and Eu by Mashonkina (2008), but such calculations will need to be repeated for many REs to understand the magnitude of corrections to the abundances reported here.
DISCUSSION\[discuss\]
=====================
We illustrate the RE abundances for , , , and in Figures \[f8\] and \[f9\]. For each star the abundances have been normalized at Eu, a predominantly $r$-process element. In Figure \[f8\] these relative abundances are shown in comparison to the Solar System -only predictions from Arlandini (1999) and Simmerer (2004). We note first the excellent star-to-star (relative abundance) agreement. Early RE abundance distributions of -rich metal-poor stars indicated large star-to-star scatter for a number of individual elements (, Luck & Bond 1985, Gilroy 1988). The combination of substantially better S/N and resolution of the stellar spectra and the experimental initiatives of this series of papers has dramatically reduced that scatter – all the RE elements are now in very good (relative) agreement for these five halo stars.
Figure \[f8\] also uses solid lines to illustrate the solar-system -only meteoritic abundances determined by Simmerer (2004) and Arlandini (1999). In both cases, these values were computed by subtracting the -only abundances from the total elemental abundances. The “classical” method (Simmerer ) matches smooth $\sigma$N$_s$ curves to those isotopes of elements whose production is essentially all due to the , and infers from those empirical curves the amounts of elements that can be produced by both the and . The solar system abundances are then just the residuals between total elemental and amounts. The “stellar” method (Arlandini ) uses theoretical models of nucleosynthesis instead of empirical abundance curves, and again infers the amounts by subtraction.
Our stellar abundances compare very well with the relative solar system distributions. In the past we and other investigators have found overall agreement, but on a more approximate scale. The new abundance determinations shown in Figure \[f8\] tighten the comparison, with deviations between the stellar and solar system r-process curves of typically less than 0.1 dex – probably the practical limit of what is currently possible. These abundance comparisons strongly support many other studies (see Sneden 2008, and references therein) arguing that essentially the same process was responsible for the formation of all of the contributions to these elements early in the history of the Galaxy in the element progenitor stars to the presently-observed -rich halo stars.
Despite this general level of elemental abundance consistency, there are some interesting deviations. In particular, the two solar system predictions differ by about 0.1 dex for the elements Ce and Nd (Table \[tab3\]). In both cases the stellar model predictions from Arlandini (1999) give a better fit to the stellar abundance data than do the standard model predictions from Simmerer (2004). This suggests that the Arlandini distribution might be superior for such abundance comparisons. This has been noted previously by others (, Roederer 2008) for isotopic studies.
There is also still some star-to-star scatter particularly at Ba, with several stellar elemental abundances appearing somewhat higher than the solar system r-process curves. This can be seen more clearly in Figure \[f9\], where we illustrate the difference between the relative (scaled to Eu) stellar RE and the scaled solar system r-process abundances (Arlandini 1999) in the five -rich stars. While most of the individual elemental abundance data lie close to the dotted line (indicating perfect agreement with the solar ), Ba and Yb have significant star-to-star scatter. But both elements have inherent observational problems, as they are represented by only a few very strong transitions that have multiple isotopic components whose relative abundances are sensitive to the relative $r$-/ dominance (recall the Yb discussion in \[ybtext\]). Abundance determinations for Yb and Ba are less reliable than those of most other RE elements, and should be treated with caution.
We also note that for the RE abundances relative to Eu appear to be somewhat higher their values in the other stars, particularly for the predominantly elements Ba and La. has a metallicity of \[Fe/H\] $\simeq$ –2.1 (Cowan 2002), so this star might be showing the signs of the onset of Galactic $s$-processing, which occurs at approximately that metallicity (Burris 2000). On the other hand with a similar metallicity (Ivans 2006) does not seem to show the same deviations for the elements, and thus the deviations for may be specific to that star.
We examine whether there is any correlation between the deviation of the stellar abundances from the solar system r-process values and the percentage of those elements in solar system material (from Simmerer 2004) in Figure \[f10\]. It is clear that there is little if any secular trend with the abundance differences with increasing solar-system abundance percentage. This lack of correlation was also found specifically for the element Ce by Lawler (2009).
To get a clearer sense of the overall abundance agreement with the solar-system abundances, we show in Figure \[f11\] the arithmetic averages of the elemental abundance offsets (from Figure \[f10\]) for the five stars, again as a function of percentage. Obviously these average offsets with respect to the solar-system values are very small. Including all elements the mean of the average offsets is $\log \epsilon$ = 0.05 ($\sigma$ = 0.05). Previously Lawler (2007) had found that the observed average stellar abundance ratio of Hf/Eu in a group of metal-poor halo stars is larger than previous estimates of the solar-system -only value, suggesting a somewhat larger contribution from the to the production of Hf. Our new analysis supports that finding, as the average Hf offset is larger than all of the other elemental abundances. If the solar system contribution was larger it would drive down the average offset illustrated in Figure \[f11\]. Ignoring the Hf results, the mean of the average offsets for all of the other RE elements is 0.04 ($\sigma$ = 0.03). This is essentially a perfect agreement within the limits of our observational and experimental uncertainties, as well as the uncertainties (observational and theoretical) associated with the solar system -only abundance values.
CONCLUSIONS\[conclude\]
=======================
We have determined new abundances of Pr, Dy, Tm, Yb, and Lu for the solar photosphere and for five very metal-poor, -rich giant stars. Combining these results with those of previous papers in this series (cited in §1), we have now derived very accurate solar/stellar abundances for the entire suite of stable RE elements.
With the single exception of Hf, the solar photospheric abundances agree with solar-system meteoritic values perfectly to within the uncertainty estimates of each. Our photospheric and stellar analyses have emphasized studying as many transitions of each species as possible (up to 46 lines in the Sun, up to 72 lines in ). The line-to-line abundance scatters are always small when the number of available transitions is large (typically $\sigma$ $<$ 0.07). This clearly demonstrates the reliability of the RE transition probabilities published in this series of papers. We argue that, with proper care in stellar analyses, trustworthy abundances of RE elements can be now be determined from spectra in which far fewer transitions are available.
Utilizing the new experimental atomic data we have determined far more precise stellar RE elemental abundances in five rich stars. These newly derived values show a dramatic decrease in star-to-star elemental abundance scatter – all the RE elements are now in very good (relative) agreement for these five halo stars. Furthermore, our newly derived values indicate an almost perfect agreement between the average stellar abundances and the solar system -only abundances for a wide range of elements in these five -rich stars. There is no evidence for significant contamination. The one exception appears to be a somewhat higher value of stellar Hf with respect to the solar system -only value for this element. This may indicate that further analysis of the solar $r$- and deconvolution for this element might be useful. These results for the five -rich halo stars confirm, and strongly support, early studies that indicate that the r-process was dominant for the elements early in the history of the Galaxy.
Parts of this research were undertaken while CS was in residence at Osservatorio Astronomico di Padova; the Director and staff are thanked for their hospitality and financial support. We thank Anna Frebel, Katherina Lodders, Ian Roederer, and Jennifer Sobeck for helpful discussions. We appreciate the use of NASA’s Astrophysics Data System Bibliographic Services, and the privilege to observe on the revered summit of Mauna Kea. The solar abundance analyses of the present and previous papers of this series have greatly benefited from the availability of the photospheric spectrum in the BASS2000 Solar Survey Archive maintained by l’Observatoire de Paris. This work has been supported by the National Science Foundation through grants AST 05-06324 to JEL and EDH, AST 06-07708 to CS, and AST 07-07447 to JJC.
APPENDIX
There have been numerous experimental studies of hyperfine structure (hfs) in . We have reviewed the literature for measurements on the upper and lower levels of lines useful, or potentially useful, for elemental abundance studies. Six publications are relevant, as indicated in Table \[tab10\]. One sees generally good agreement among measured values of the hfs $A$ constants. Only a few, not very accurate, measurements of the hfs $B$ constants have been reported. Since the electric quadrupole interaction ($B$ constants) has a much smaller effect on the line component pattern than the magnetic dipole interaction ($A$ constant), it is often neglected and will be neglected here.
One of the best and fairly extensive set of measurements of hfs $A$ constants is that by Rivest (2002) using laser induced fluorescence. We adopted their measurements, if available, to compute the complete hfs line component patterns that are given in Table \[tab11\]. For levels which were not studied by Rivest , we used hfs $A$ constants from Ginibre (1989). Iv01 improved some energy levels using FTS data. The center-of-gravity wavenumbers in Table \[tab11\] are from the Iv01 energy levels in every case where an improved energy was reported for both the upper and lower level of the line. For other lines the center-of gravity wavenumbers are from the NIST energy levels (Martin 1978), because it is probably not a good idea to mix energy levels from two sources. Center-of-gravity air wavelengths were computed from wavenumbers using the standard index of air (Edlén 1953).
For we used the isotopic and hyperfine data of M[å]{}rtensson-Pendrill, Gough, & Hannaford (1994). We adopted the transition probabilities of Biémont (1998) renormalized to the lifetime results of Pinnington, Rieger, & Kernahan (1997): $_{3289}$ = +0.02, and $_{3694}$ = $-$0.30. These values are close to those derived from Biémont (2002), as given in the D.R.E.A.M. database[^9]: $_{3289}$ = $-$0.05, and $_{3694}$ = $-$0.32. Combining the transition probabilities, hyperfine and isotopic substructures, and the solar isotopic breakdown given in \[ybtext\] yields complete transition structures for these two lines; these are listed in Table \[tab12\].
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[^1]: We adopt the standard spectroscopic notation (Helfer, Wallerstein, & Greenstein 1959) that for elements A and B, \[A/B\] $\equiv$ log$_{\rm 10}$(N$_{\rm A}$/N$_{\rm B}$)$_{\star}$ $-$ log$_{\rm 10}$(N$_{\rm A}$/N$_{\rm B}$)$_{\odot}$. We use the definition $\equiv$ log$_{\rm 10}$(N$_{\rm A}$/N$_{\rm H}$) + 12.0, and equate metallicity with the stellar \[Fe/H\] value.
[^2]: IRAF is distributed by NOAO, which is operated by AURA, under cooperative agreement with the NSF.
[^3]: FIGARO is a part of the “Starlink Project”, which is is now maintained and being further developed by the Joint Astronomy Centre, Hawaii.
[^4]: The MIKE Pipeline is available from the Carnegie Observatories Software Repository at http://www.ociw.edu/Code/mike/
[^5]: Available at http://bass2000.obspm.fr/solar\_spect.php .
[^6]: Available at http://cfaku5.cfa.harvard.edu/
[^7]: Throughout this paper we will use the subscript symbol $\odot$ to indicate solar photospheric values, and the subscript $met$ to indicate solar system meteoritic values from Lodders (2003).
[^8]: Available at http://marcs.astro.uu.se/
[^9]: http://w3.umh.ac.be/$\sim$astro/dream.shtml
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Introduction
============
In this paper we introduce a general model of resource allocation with customer choice. In this model, there is a finite, continuous-time horizon. Over the horizon, there is a known set of resources, each finite in quantity and perishing at a known date. Each unit of a resource can be used to instantly make one of several products. Over the horizon, customers of various types arrive according to non-homogenous Poisson processes. The type of a customer is observable by the system. Upon a customer’s arrival, according to the time of arrival, the inventory of available resources, and the type of the customer, the system chooses an assortment of products to display. From among this assortment, the customer chooses a product according to some known, general model of choice. If the customer chooses a product, the system earns a customer-type-dependent and product-dependent reward, and the inventory of the corresponding resource is depleted by one. The system’s goal is to maximize the total expected reward that it earns over the horizon.
The problem above has application in a number of settings. In the revenue management of parallel flights, the resources are flight legs that share the same origin and destination. Each product is a ticket, which is defined by a flight leg, an associated fare, and a set of purchase restrictions or ancillary services for the corresponding fare class. A customer arriving at the system chooses dynamically among the assortment of tickets offered. The type of a customer can be based on information such as the customers’ past purchase history, their arrival time, the parameters of the customers’ search query, whether or not they belong to a loyalty program, their level within the loyalty program, their location, and any other information that can be observed by the system at the time of purchase.
In assortment-planning problems, the products and resources are one and the same. In this case, the type of a customer can be based on information similar to above, and whether they belong to a special program such as Amazon’s Prime program, Bloomingdale’s Loyallist rewards program, or Sephora’s Rewards Boutique program. Rewards are adapted to the customer type to capture a combination of personalized prices and the value of serving the customer type in the long term. For example, Target is known to send coupons to shoppers that it identifies as potential expecting mothers, because data shows that important life events, such as birth, can change people’s shopping habits. Target finds it beneficial in the long run to favor these shoppers in order to induce the habit to buy at Target [@duhigg2012companies].
In web- and mobile-based self-scheduling systems such as ZocDoc, the resources and products are the same and correspond to appointment slots that take place at different times, at various clinic locations, with various physicians. The patient type can be defined with reference to a patient’s status as new or existing patient, their insurance, the nature of concern, etc. The reward of assigning a slot to a patient can be chosen to capture a combination of the revenue expected for the visit and the patient’s priority.
The above class of problems are closest to choice-based revenue-management problems that have attracted intense interest over the past ten years. Researchers have focused on stochastic dynamic formulations of these problems. Gallego, Iyengar, Phillips, and Dubey (2004) are among the first to introduce a consumer choice model to the network revenue-management literature. The choice based linear program (CDLP) proposed by them has been widely used to approximate the stochastic dynamic optimal solution. They show that the optimal value of the stochastic dynamic problem is bounded above by the optimal solution to the CDLP, and further, that it approaches this upper bound asymptotically. There are various ways of using the CDLP solution to obtain practical policies, such as the bid price heuristic of Talluri and van Ryzin (2005) and the dynamic programming decomposition approach of Liu and van Ryzin (2008). Subsequently tighter approximation methods have been proposed to improve upon the CDLP upper bound, [@zhang2009af; @Meissner2012459; @KunnumkalT2010]. However, little is known about the theoretical performance of these methods outside of the asymptotic regime, and the tighter bounds usually come with significant computational cost.
In this paper, we introduce the first algorithms with theoretical performance guarantees for the above class of problems. Our approach offers both modeling flexibility, ease of implementation, and theoretical performance characterization. Several features of our model merit attention:
Personalization.
: We allow multiple customer types to be modeled. The customer types are based on information that can be observed by the system at the time of purchase. We offer assortments of products that are customized to the customer types. Personalization has been widely used by companies that collect data on customer characteristics. It has been used, for example, by Amazon to recommend products to customers based on their purchase history; by Groupon, Yelp, and Foursquare to offer discounts to customers based on their location; and by grocery stores to offer customers coupons based again on their purchase history (Golrezaei, Nazerzadeh and Rusmevichientong 2014). As we shall discuss, the literature on personalized revenue management is still limited.
Substitution across fare classes.
: We allow for simultaneous consideration of multiple products that might correspond to multiple fare classes, with no restriction on substitution behavior. Thus, we are able to capture the substitution of capacity across fare classes in revenue-management applications. Although the substitution effect has been studied, it has been limited to within fare classes (Zhang and Cooper 2005). This assumption turns out to be rather restrictive. Dai, Ding, Kleywegt, Wang and Zhang (2014) have found in recent analysis of empirical airline data that customers have much greater demand for the cheapest alternative than for the second cheapest alternative even when the price difference is small.
Substitution across time.
: We model multiple resources with expiry dates that can fall within the horizon. Thus, we are able to explicitly model inter-temporal demand substitution. Indeed, our model is among the first in the revenue-management literature to *explicitly* capture inter-temporal substitution. Previous models have usually assumed that the expiry dates fall beyond the horizon. Thus, they can only capture intertemporal substitution in an implicit manner, by changing the demand arrival process or the selection probabilities over time.
Non-stationarity.
: We allow demand arrivals to be non-stationary and stochastic over the horizon. Our performance guarantees are robust to dramatic changes in the demand rate. Past approximations, such as that of Liu and van Ryzin (2008), can be extended to the environment with time varying arrivals if the demand varies slowly over time, but perform poorly when the demand is volatile. Many of the remaining prior methods [@zhang2009af; @Meissner2012459; @KunnumkalT2010] have assumed stationary arrival.
Dynamic product substitution.
: We dispense with the commonly-used static-substitution assumption, which implies that a customer, who finds that his request is stocked out, will leave the system forever. We dynamically adjust our offered assortment according to current inventory levels, current demand type and future expected demands. We include only products with positive inventory levels in our assortments. For this reason, the customer behaviors that our model captures are much more realistic.
Our contributions are as followed.
- We propose the first column-generation algorithms to solve the CDLP in settings where this problem is NP-hard. Our algorithms can generate $\epsilon$-optimal solutions to the CDLP for any given $\epsilon>0$. It is applicable to a variety of choice models, including the Mixed Multinomial Logit (MMNL), which can approximate any random utility model to any precision level [@mcfadden2000]. Our algorithms build on existing polynomial-time approximation schemes (PTAS) or fully polynomial-time approximation schemes (FPTAS) that can approximately solve the underlying assortment-planning sub-problems.
- We derive theoretical performance characterization for our general class of choice-based resource-allocation problems. We prove that our algorithms are guaranteed to produce an expected reward no worse than $\frac{1}{2}(1-\epsilon)$ times that of $OFF$, where $\epsilon$ is the error in computing an optimal solution to the CDLP, and $OFF$ is an optimal offline algorithm that knows a priori the realization of all demand arrivals and makes optimal decisions given this information. To the best of our knowledge, this is the first constant relative performance characterization for this class of problems. Our algorithms are highly intuitive and simple to implement.
- We prove that $\frac{1}{2}$ is the best possible constant ratio that can be achieved between the expected reward of an online algorithm and the expected reward of $OFF$. Thus, our algorithms achieve within $(1-\epsilon)$ of the best possible constant relative performance. They achieve the upper bound of $\frac{1}{2}$ for a variety of choice models for which the CDLP can be solved exactly.
- We show that our algorithm has expected total reward at least as high as that of the deterministic algorithm proposed by Gallego, Iyengar, Phillips, and Dubey (2004) . We also prove that this classical algorithm has expected reward that is no worse than $1/e=0.368$ that of $OFF$. Thus, we prove that our algorithms are asymptotically optimal, therefore competitive with existing algorithms, according to the main theoretical performance characterization known prior to our work.
Literature Review
=================
We will summarize the streams of works that are most closely related to our paper, many of which are in revenue-management. We refer the reader to Talluri and van Ryzin (2005) for a comprehensive review of the larger revenue-management literature.
Single-Leg Revenue Management
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Inspired by the news-vendor problem, Littlewood (1972) describes a simple technique for setting a booking limit for low-fare tickets when there are two fare classes and a single-leg flight. Later, Belobaba (1989) extends the model to the case with multiple booking classes and proposes an expected marginal seat revenue (EMSR) heuristic. Subsequently, a framework for determining booking limits for a single-leg flight with mutually independent demand classes that arrive in sequential blocks is developed. The earliest dynamic model is perhaps attributable to Mayer (1976), who introduces a dynamic-programming model and compares it with the static control derived from Littlewood’s rule. Dynamic model allows for a more precise formulation of the customer arrival process. For example, the low-to-high-fare arrival assumption can be relaxed. Lee and Hersh (1993) consider a general multiclass dynamic seat-allocation model with non-stationary demand. They show that the optimal policy is a monotone-threshold policy. That is, a ticket is open for purchase if and only if its fare is no smaller than the expected marginal revenue over the remaining booking horizon. Talluri and van Ryzin (2004) analyze a a single-leg revenue management problem with customer choice. They show that an optimal policy can be characterized by an ordered sequence of “efficient" offer sets, which can provide the most favorable trade-off between expected revenue and expected capacity consumption. They give conditions under which the efficient sets have a nested structure.
Within this literature, a number of papers address the design of policies for revenue management that are robust to the distribution of arrivals. @ball2009toward analyze online algorithms for the single-leg revenue-management problem. Their performance metric is the traditional competitive ratio that compares online algorithms with optimal offline algorithms under the worst instance of demand arrivals. They prove that the competitive ratio cannot be bounded by any constant when there are arbitrarily many customer types. In our work, we relax the definition of competitive ratio, and show that our algorithms achieve a constant competitive ratio (under our definition) for any number of customer types and for a more general multi-resource model. Qin, Zhang, Hua, and Shi (2015) study approximation algorithms for an admission control problem for a single resource when customer arrival processes can be correlated over time. They use as the performance metric the ratio between the expected cost of their algorithm and that of an optimal stochastic dynamic algorithm. Our performance metric is stronger than theirs as we compare our algorithms against an optimal offline algorithm, instead of an optimal stochastic dynamic algorithm. Qin, Zhang, Hua, and Shi (2015) prove a constant approximation ratio for the case of two customer types, and also for the case of multiple customer types with specific restrictions. In addition, they allow only one type of resource to be allocated. In our model, we assume arrivals are independent over time, but we allow for multiple customer types and multiple resources without additional assumptions.
Online Matching
---------------
Our work is closely related to works on online bipartite matching problems. In these problems, the set of available resources is known and corresponds to one set of nodes. Demand requests arrive one by one, and correspond to a second set of nodes. As each demand node arises, its adjacency to the resource nodes is revealed. Each edge has an associated weight. The system must match each demand node irrevocably to an adjacent resource node. The goal is to maximize the total weighted or unweighted size of the matching. There is no choice process that is modeled.
The online unweighted bipartite matching problem is originally shown by Karp, Vazirani and Vazirani (1990) to have a best competitive ratio of $0.5$ for deterministic algorithms and $1-1/e$ for randomized algorithms. Our work generalizes the online weighted bipartite matching problem. When demands are chosen by an adversary, the worst-case competitive ratio of this problem cannot be bounded by any constant [@mehta2012online]. Many subsequent works have tried to design algorithms with bounded performance ratios for this problem for more regulated demand processes.
Specifically, three types of demand processes have been studied. The first type of demand processes studied is one in which each demand node is independently and identically chosen with replacement from a *known* set of nodes. Under this assumption, @jaillet2013online [@manshadi2012online; @bahmani2010improved; @feldman2009online] propose online algorithms with competitive ratios higher than $1-1/e$ for the unweighted problem. Haeupler, Mirrokni, Vahab and Zadimoghaddam (2011) study online algorithms with competitive ratios higher than $1-1/e$ for the weighted bipartite matching problem.
The second type of demand processes studied is one in which the demand nodes are drawn randomly without replacement from an unknown set of nodes. This assumption has been used in the secretary problem (Kleinberg 2005, Babaioff, Immorlica, Kempe, and Kleinberg 2008), ad-words problem [@goel2008online] and bipartite matching problem (Mahdian and Yan 2011, Karande, Mehta, and Tripathi 2011).
The third type of demand processes studied is one in which each demand node requests a very small amount of resource. This assumption, called the *small bid* assumption, together with the assumption of randomly drawn demands, lead to polynomial-time approximation schemes (PTAS) for problems such as ad-words [@Devanur09theadwords], stochastic packing (Feldman, Henzinger, Korula, Mirrokni, and Stein 2010), online linear programming (Agrawal, Wang, Zizhuo and Ye 2009), and packing problems [@molinaro2013geometry]. Typically, the PTAS proposed in these works use dual prices to make allocation decisions. Devanur, Jain, Sivan, and Wilkens (2011) study a resource-allocation problem in which the distribution of nodes is allowed to change over time, but still needs to follow a requirement that the distribution at any moment induce a small enough offline objective value. They then study the asymptotic performance of their algorithm. In our model, the amount capacity requested by each customer is not necessary small relative to the total amount of capacity available. Therefore, the analysis in these previous works does not apply to our problem.
Our work builds upon recent results by Wang, Truong and Bank (2015), who propose online algorithms with competitive ratio of $0.5$ for the bipartite matching problem with heterogeneous demands. They assume that demands arrive according to non-homogenous Poisson processes. For this class of problems, they show that the bound of $0.5$ achieved by their algorithm is tight. Our model extends theirs to allow for endogenous customer behavior, and decisions that control assortments of products to offer, rather than those that match customers directly to resources.
Choice-based network revenue management
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Our work extends the literature on choice-based network revenue management (CNRM). These models assume that there is some fixed, finite amount of resources, for example, flight legs. Products are sold, which are built from one or more resources. The models focus on dynamically adjusting the set of offered products as a function of remaining capacity of the resources and remaining time in the selling horizon. They assume that consumer demands are dependent on the set of offered products.
Our algorithms build on the use the CDLP that was first introduced by Gallego, Iyengar, Phillips, and Dubey (2004). They propose an efficient column generation algorithm to solve the CDLP and suggest a deterministic control policy that uses the primal optimal solution as the proportion of time to offer each assortment. More recently, Gallego, Ratliff and Shebalov (2014) reformulate the CDLP as a polynomial-size sales-based LP under a general class of attraction models. Liu and van Ryzin (2008) give a dynamic-programming formulation for CNRM. They come up with a heuristic algorithm by decomposing over flight legs, with the opportunity cost of each flight leg being generated from the dual values of the capacity constraints in the LP. Kunnumkal and Topaloglu (2010) propose another dynamic-programming-decomposition algorithm for CNRM; they allocate revenue associated with itinerary among the different flight legs and solve a single-leg revenue-management problem for each leg. Zhang and Cooper (2005) dynamically control the inventory of parallel flights for a common itinerary while assuming demands are substitutable across different schedules, but not across different fare classes. Zhang and Adelman (2009) present an affine approximation to the value function. Jasin and Kumar (2012) study the performance of the heuristic of resolving the deterministic LP periodically while setting all random variables at their expected future values. They provide both upper bound and lower bound for the expected revenue loss compared to the optimal policy. All of the above papers assume a single market segment, or multiple market segments with disjoint consideration sets. They also assume that the customer type is not observable to the seller, so the seller must decide on a common assortment for all segments.
As Bront, Mendez-Diaz, and Vulcano (2009) point out, when customers belong to overlapping segments, even solving for the CDLP is NP-hard. They propose a heuristic column-generation algorithm for solving the CDLP. Our model is different in that we can observe the customer type and can provide personalized assortments.
Jaillet and Lu (2012) design near-optimal learning-based online algorithms for dynamic resource-allocation problem. They do not model consumer choice. Further they assume that demand arrivals are stationary, and that the amount of resource used by each demand unit is very small relative to the capacity. In contrast, we will model consumer choice, allow non-stationary demands, and allow individual demand requirements to be large relative to capacity.
Revenue management of parallel flights
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Zhang and Cooper (2005) consider the seat-inventory control of multiple parallel single-leg flights in the presence of dynamic customer-choice behavior. Each customer’s choice is modeled by his preference mapping, and they assume the preferences only shift among different flights, not across fare classes. They use lower and upper bounds to approximate the expected revenue from the optimal stochastic programming, and suggest control policies based on the approximated marginal value. Dai, Ding, Kleywegt, Wang and Zhang (2014) describe a revenue-management problem of a major airline that operates in a very competitive market involving two hubs and having more than 30 parallel daily flights. They observe demand discontinuities from the industrial date, i.e, demand spikes for the cheapest available fare classes and for fully refundable fare classes. They incorporate the discontinuity into a logit model and focus on a deterministic formulation of the problem. In a departure from previous literature, they take the competitor’s response into consideration by modeling the attractiveness of the no-purchase option with a random coefficient. They also show that under some conditions, the CDLP can be solved efficiently by column generation.
Assortment planning
-------------------
Assortment planning problems are special class of revenue-management problems in which the products sold are the resources themselves. That is, the products are not built from one or more simpler resources. In assortment planning, a retailer needs to decide the set of products to offer at various times over a selling horizon. Usually this decision is jointly considered with inventory decisions.
Earlier works considered the static assortment problem. In these problems, customers have no knowledge about the status of inventory. They make purchase decision only based on the offered product set. If their selection is stocked out, a second choice will not be made. Van Ryzin and Mahajan (1999) show that an optimal assortment is composed of the most popular products when all the products have the identical price and cost and customers’ demands are governed by the Multinomial Logit model of choice. Subsequent literature has considered various choice models and prices and costs structures. Cachon, Terwiesch and Xu (2005) use a more general choice model in which they incorporate search costs and shows that ignoring customer search will lead to less assortment variety since in equilibrium the seller needs a bigger sized assortment to attract more customers. Topaloglu (2013) works on a similar problem as van Ryzin and Mahajan (1999), but instead of deciding a single offering set, he allows multiple assortments and decides the duration of time that each is offered.
Our paper features a dynamic, or stockout-based model of consumer choice. In these models, customers base their choice from the products that are in stock upon their arrival. Mahajan and van Ryzin (2001) choose initial inventory levels to maximize the expected profit under dynamic substitution. However, they show the objective function is not even quasiconcave. Therefore, they suggest a stochastic-gradient algorithm for the problem. Kok and Fisher (2007) propose an estimation method to obtain both the original demand rate and the substitution rate. They also present an iterative heuristic to solve the problem of joint assortment planning and inventory optimization. Honhon, Gaur and Seshadri (2010) consider multiple customer types. They derive some structural properties for the optimal assortment.
Within the literature on dynamic consumer choice, our paper is closest to the literature on personalized dynamic assortment planning. These assortments are dynamically optimized depending on inventory levels, and tailored to the customer segment, provided that the system can observe the customer’s segment and knows the preferences of each segment. Bernstein, Kok and Xie (2010) are the first to propose the idea of assortment customization in presence of heterogeneous customer segments. To obtain some structural properties of the optimal policy, they assume all products are not functionally differentiated and have the same price; and they propose a heuristic by implementing a newsvendor-type approximation to the marginal revenue of each product. [@chan2009stochastic] relax the restrictive uniform price constraint and show that a myopic policy achieves at least 0.5 the expected revenue of the stochastic optimal policy in non-stationary settings. Golrezaei, Nazerzadeh and Rusmevichientong (2014) extend this result by showing that under adversarially chosen demand there is an online algorithm that achieves at least 0.5 the cost of an optimal offline policy in the worst case.
Similar to Golrezaei, Nazerzadeh and Rusmevichientong (2014), we propose simple and robust online algorithms for dynamically determining the set of offered products, based on the inventory of products available and the time that remains in the horizon. However, our model captures rewards that depend on the customer type and not just on the products sold. In revenue-management applications, these rewards capture differentiated fares for the same seat capacity. In assortment planning applications, our rewards capture personalized prices or discounts, and differentiated rewards for serving different customer groups, such as Amazon Prime customers versus regular customers. In self-scheduling systems, our rewards capture differentiated priorities among different customer groups such as urgent versus non-urgent patients, new versus existing patients, and regular versus follow-up visits, etc. In the context of online matching, the models of Golrezaei, Nazerzadeh and Rusmevichientong (2014), [@chan2009stochastic], and others in the assortment-planning literature extend the *vertex-weighted bipartite matching problem*, where the revenue earned is a function of only the resource nodes that are used. This problem is a special case of the *edge-weighted bipartite matching problem* that we extend. In these more general problem, the revenue earned is a function of both the resource nodes that are used, and the demand nodes that they are matched with.
Appointment Scheduling with Choice
----------------------------------
Our work is related to the literature on appointment scheduling [@guerriero2011operational; @may2010surgical; @cardoen2010operating; @gupta2007surgical]. Patient preferences are an important consideration in many real scheduling systems. In the literature considering patient preferences, @gupta2008revenue consider a single-day scheduling model where each arriving patient picks a single slot with a particular physician. The clinic accepts or rejects the request. Our model generalize their framework to a multi-period setting. We also characterize the theoretical performance of algorithms in an online setting, whereas they use stochastic dynamic programming as the modeling framework and develop heuristics.
A multi-day, single-patient-type, stationary model has been studied by Feldman, Liu, Topaloglu and Ziya (2014). They assume that patients have preferences for slots that can be captured by the multinomial logit model. This model is essentially a dynamic assortment-planning model. They derive structural results for the optimal policy, and exhibit a heuristic that is asymptotically optimal. In contrast, we characterize the theoretical performance of our algorithms in both asymptotic and non-asymptotic regimes. We also model heterogenous patients and non-stationary demand, both of which features are especially important in healthcare settings, where patients frequently have differing priorities and demands are usually non-stationary (Wang, Truong and Bank 2015). Finally, we allow a very general model of patient choice to be used.
Our model closely follows the previously discussed model of Wang, Truong and Bank (2015) for multi-day, multi-patient-type settings. Their model is useful in applications in which the system can observe *revealed* patient preferences before assigning appointment slots to them. Requiring patients to explicitly specify their preferences can be a cumbersome exercise. Therefore, some systems such as ZocDoc make scheduling more user-friendly by offering sets of open slots to patients and allowing them to choose. Our model captures the latter setting. Thus, our contribution is to add the control of assortments of slots to offer to the model of Wang, Truong and Bank (2015).
Model
=====
We consider a continuous horizon $[0,1]$. There are $L$ different resources, $N$ different products, and $K$ customer types. Each resource $l$, $l=1,\ldots,L$, has a capacity $C_l$ and an expiration time $t_l$. Each product $n$, $n=1,\ldots,N$, consists of a single resource $l_n$. A product might be a resource offered with a specific price and a certain set of restrictions. For expositional simplicity, each product $n$ earns a reward of $r_n$ regardless of the customer type served. However, the reward can be made to depend on the customer type served without changing any of the results that follow. In the more general case, the reward would be indexed by both the product and the customer type.
Customers of type $k$ arrive according to a non-homogenous Poisson process with [*known*]{} rate $\lambda_k(t)$. We assume that the customer type is observable by the system. When a customer of type $k$ is presented with an assortment $S$, the customer will choose a product from the assortment following a [*general*]{} choice model. Let $P^k(n,S)$ be the probability that a customer of type $k$ chooses product $n$ from assortment $S$. Note that *each customer type can be modeled with a different choice model.* We assume that the no-purchase option, or product $0$, is included in each assortment, with $r_0=0$.
When a customer of type $k$ arrives, the system must decide which assortment to offer to the customer. If the customer purchases product $n$ from the offered assortment, the reward $r_n$ is earned and the inventory of resource $l_n$ is depleted by $1$. The objective of the system is to maximize the expected total reward.
Let $c = (c_1,c_2,...,c_L)$ denote the vector of remaining inventory, where $c_l \in \{0,1,...,C_l\}$ represents the remaining inventory of resource $l$. Under a policy $\Pi$, let $V^\Pi(c,t)$ denote the expected future reward and $A_k^\Pi(c,t)$ denote the assortment offered to customers of type $k$ at time $t$ with inventory $c$. In general, $A_k^\Pi(c,t)$ can be random.
Let $N_l \equiv \{n \in \{1,2,...,N\}: l_n = l\}$ be the set of products that consist of resource $l$. Let $e_l$ be the unit vector with the $l$-th position being $1$. The dynamics of $V^\Pi(c,t)$ is governed by $$\label{eq:Dynamics}
\frac{\partial V^\Pi(c,t)}{\partial t} = - \sum_{k=1}^K \sum_{l=1}^L R_t^{kl}(A_k^\Pi(c,t), \Delta_l V^\Pi(c,t)),$$ where $$\label{eq:DeltaV}
\Delta_l V^\Pi(c,t) \equiv \left\{ \begin{array}{ll} V^\Pi(c,t) - V^\Pi(c-e_l,t) &\text{ if } c_l>0\\ \infty & \text{ if } c_l =0\end{array} \right.$$ is the marginal value of expected future reward with respect to resource $l$, and $$\label{eq:RewardRate}
R_t^{kl}(S,z) \equiv \lambda_k(t) \sum_{n \in N_l} \mathbf{E}[\mathbf{1}(n \in S) P^k(n,S)] (r_n - z)$$ is the rate at which the expected future reward of resource $l$ changes due to customers of type $k$. Here the expectation in (\[eq:RewardRate\]) is taken over $S$. $V^\Pi(c,t)$ must satisfy the boundary conditions $V^\Pi(0,t) = 0$ and $V^\Pi(\cdot,1) = 0$.
The total expected reward of policy $\Pi$ is $V^\Pi(C,0)$, where $C =(C_1,C_2,...,C_L)$ is the vector of initial inventory.
Action space
------------
Existing models differ in whether an assortment is allowed to contain products with zero inventory. A model assumes *static substitution* if an assortment can contain any product. A customer who chooses a product with zero inventory leaves the system without affecting the total reward. A model assumes *dynamic substitution* if assortments must not contain products with zero inventory.
Dynamic substitution is much more realistic than static substitution, although dynamic substitution requires more complex analysis. Recent literature has mostly focused on dynamic substitution.
Our paper assumes dynamic substitution. The formal definition is as follows.
For any given state $(c,t)$, if $c_l =0$, we must have $N_l \cap A_k^\Pi(c,t)=\emptyset$.
We will propose a policy that achieves the best performance guarantee under this assumption. To motivate the idea of the policy, however, we will first analyze some intermediate policies that generate performance bounds under the static-substitution assumption. We will show that our final policy that works under dynamic substitution dominates these intermediate policies.
More on time-dependent effects
------------------------------
Note that in our model, the resources might perish over the horizon. That is, it is possible for $t_l < 1$ for certain resources $l$. We will define customer types to ensure that any customer $k$ arriving after time $t_l$ has selection probability $P^k(n,S)=0$ for any product $n$ that is based on $l$, and any assortment $S$ that includes $n$. As time moves forward, the types of customers who are arriving with positive probabilities will change in our model in order to capture any time-dependent changes in demand behavior.
The horizon is also finite in our model. As the end of horizon approaches, there might be products that expire after the end of horizon entering into customers’ consideration sets. In the language of revenue management, as the end of horizon approaches, customers might be increasingly choosing to purchase flights that depart after the end of the horizon. Existing models can address this effect by changing the attractiveness of the no-purchase option as the end of horizon nears. A similar strategy can be applied here. However, our model can be used to capture inter-temporal substitution more explicitly near the end of the horizon as follows. For concreteness, we explain the strategy in the language of revenue management.
- Include consideration sets with flights that depart up to time $T + \Delta$, where $\Delta$ might be a week.
- Allocate capacity $c_l$ to flights $l$ with departure time $t_l > T$ for sale during $[0,T]$. In this case $c_l$ may be less than the actual capacity for flight $l$. The quantity $c_l$ may be set by management or may be the solution to a higher-level optimization problem. As an example, if the capacity of a flight that departs between $T$ and $T+ \Delta$ is 100, we might allocate 85 seats to the optimization problem over $[0,T]$ if we estimate that we can sell 15 seats at a higher price during the interval $[T, T+\Delta]$.
By including post-horizon flights and some of their capacity, we will be able to incorporate most of the inter-temporal substitution behavior explicitely into our model.
Competitive ratio
-----------------
It is practically impossible to compute the optimal policy for our problem because of the “curse of dimensionality” in both the state and decision spaces. Instead, our goal is to give an online algorithm with expected total reward that is bounded by a constant factor of an optimal offline policy $OFF$.
A policy $\Pi$ is online if the decision $A_k^\Pi(c,t)$ is adapted to the information up to time $t$, including the future arrival rates $\lambda_k(t)$, for $k=1,2,...,K$ and $t \in [0,1]$, that are known a priori. On the other hand, the optimal offline decision depends on the information of all realizations of future arrivals over the horizon, but the randomness in customer choices is still exogenous to *OFF*. We will also require that $OFF$ follows the dynamic-substitution assumption in the sense that it never offers a product with zero inventory.
Let $V^{OFF}$ denote the expected reward of $OFF$. Note that $V^{OFF}$ does not need to satisfy the dynamic equation (\[eq:Dynamics\]) for online algorithms. The following is our definition of competitive ratio.
An algorithm $\Pi$ is *$\alpha$-competitive* if its total expected reward $V^\Pi(C,0)$ satisfies $$V^\Pi(C,0) \geq \alpha \cdot V^{OFF}$$ under any values of $r_{n}, \nu^k_n, \lambda_k(t), C_l$, for $n=1,2,...,N;\ k = 1,2,...,K;\ l = 1,2,...,L;\ t\in [0,1]$.
Choice-Based Deterministic Linear Program
=========================================
Before introducing our algorithms, we first characterize an upper bound on $V^{OFF}$ using a widely-used choice based deterministic linear programming (CDLP) formulation proposed by @gallego2004managing. @LiuV2008 have shown that the CDLP is an upper bound on the expected reward of an optimal stochastic policy. In this paper, we prove a stronger result that the CDLP is an upper bound on the expected reward of an optimal offline policy *OFF*. Our algorithms and bounds will build on this CDLP.
Let $\cal{S}$ denote the set of all assortments. Let $\Lambda_k \equiv \int_0^1 \lambda_k(s) ds$ denote the average total number of arrivals of type $k$ customers over the horizon. In the following choice-based CDLP, the decision $x_k(S)$ represents the probability of showing assortment $S \in \cal{S}$ to a type-$k$ customer upon his arrival. Note that this quantity is independent of the time of that arrival.
$$\begin{aligned}
\begin{split} \label{eq:ChoiceBasedLP}
V^{CDLP}=& \max \sum_{k=1}^K \sum_{S\in \cal{S}} \Lambda_k x_k(S) \sum_{n\in S} P^k(n,S) r_n \\
\text{s.t. } & \sum_{k =1}^K \Lambda_k \sum_{n \in N_j} \sum_{S \in \cal{S}: n \in S} x_k(S) P^k(n,S) \leq C_j , \,\,\, \forall j = 1,2,...,L,\\
& \sum_{S \in \cal{S}} x_k(S) \leq 1, \,\,\, \forall k =1,2,...,K;\\
& x_k(S) \geq 0, \,\,\, \forall k =1,2,...,K,\ \forall S \in \cal{S}.
\end{split}\end{aligned}$$
\[thm:upperbound\] $V^{CDLP}$ is an upper bound on $V^{OFF}$.
Our algorithms rely on an optimal solution of the CDLP. Since there are $L+K$ constraints, we know that at optimality, the solution involves at most $L+K$ different assortments with positive displaying probabilities. However, with $K2^N$ variables, this CDLP could be very difficult to solve in practice. As suggested by Gallego, Iyengar, Phillips, and Dubey (2004), column generation techniques can be used to circumvent these difficulties.
Solving the CDLP by Column Generation
-------------------------------------
Our algorithm requires a solution to the CDLP. We propose solving the CDLP by column generation. Column generation works by expanding a set $\cal{H}^d$ of *active* assortments, i.e., those with positive displaying probabilities, in each iteration $d$. We start with a limited number of columns, namely $\cal{H}^{1}$. We then solve a reduced LP that involves only these columns. Conditional on the dual values from the reduced LP in the current iteration, we then calculate the reduced cost of potential assortments that have not yet been considered. If the reduced cost is strictly positive, we incorporate the corresponding column into a new active set and re-optimize. If there is no potential assortment with positive reduced cost, the current solution is optimal. A detailed illustration is as follows:
1. Initially, let $\cal{H}^1$ be any collection of assortments. Set $d\leftarrow 1$.
2. Solve the following reduced linear program, in which $x_k(S)$ is the decision variable, for all $k = 1,2,...,K$ and for all $S \in \cal{H}^{d}$. $$\begin{aligned}
\begin{split}
V^d \equiv \max &\,\,\,\,\,\,\, \sum_{k=1}^K \sum_{S \in \cal{H}^{d}} \Lambda_k x_k(S) \sum_{n \in S} r_n P^k(n,S)\\
\text{ s.t. } & \sum_{k=1}^K \Lambda_k \sum_{n \in N_j} \sum_{S\in \cal{H}^{d}: n \in S} x_k(S) P^k(n,S) \leq C_j, \,\,\forall j = 1,2,...,L\\
& \sum_{S \in \cal{H}^{d}} x_k(S) \leq 1, \,\,\, \forall k = 1,2,...,K\\
& x_k(S) \geq 0, \,\,\, \forall k = 1,2,...,K,\ \forall S \in \cal{H}^{d}.
\end{split}\end{aligned}$$ Let $\pi(1), \pi(2), ..., \pi(L), \sigma(1), \sigma(2),...,\sigma(K)$ be the optimal dual variables for the above reduced LP. The reduced cost corresponding to $x_k(S)$, which is the offering probability of assortment $S \in \cal{S}$ to consumer type $k$, is $$\Lambda_k \sum_{n \in S, n>0} [r_n - \pi(l_n)] P^k(n,S) - \sigma(k).$$ We must check whether there is any assortment $S \notin \cal{H}^{d}$ that has a strictly positive reduced cost. This can be done by solving the following optimization problem for each $k$ $$\label{eq:CGSubproblem}
L_k\equiv \max_S \{ \Lambda_k \sum_{n \in S, n>0} [r_n - \pi(l_n)] P^k(n,S) \}$$ and compare the value with $\sigma(k)$. Note the above is just an assortment optimization problem with $r_n - \pi(l_n)$ being the profit of product $n$.
3. Let $k^* \in {argmax}_k L_k-\sigma(k)$, if $L_{k^*} -\sigma(k^*)\leq 0$, the solution of the current reduced LP is optimal for the primal CDLP. Stop.
4. Else, set $\cal{H}^{d+1} \leftarrow \cal{H}^{d} \cup \{S^d_{k^*}\}$, where $S^d_{k^*} \in {argmax}_S \Lambda_{k^*} \sum_{n \in S, n>0} [r_n - \pi(l_n)] P^{k^*}(n,S)$ and set $d \leftarrow d + 1$
Column generation is successful only if can be solved efficiently for all values of $(r,\pi)$. Clearly, this is impossible for arbitrary assignments of the choice probabilities $P(n,S), S \in \cal{S}$.
Solving the Column-Generation Subproblem
----------------------------------------
Now let us analyze the complexity of each column-generation step. Gallego, Iyengar, Phillips, and Dubey (2004) show that if the customer choices follow the structure $$P^k(n,S) = \frac{\mu_{kn} + \nu_{kn}}{\sum_{i=1}^N \mu_{ki} + \sum_{i\in S} \nu_{ki} + 1}, \,\,\,\,\,\forall n \in S,$$ then the column-generation subproblem can be solved exactly by a simple sort. The above class of models cover a wide range of choice models, for example, the Independent-Demands, Multinomial Logit (MNL), and General Attraction models (GAM). Recently, Gallego, Ratliff and Shebalov (2014) show that under the above class of choice models, the CDLP can be reformulated as a more compact sales-based linear program with only a polynomial number of variables.
There are other choice models under which the optimal assortment can be found in polynomial time, for example, the $d-$Level Nested Logit (Li, Rusmevichientong and Topaloglu 2015) and Markov-Chain models (Blanchet, Gallego and Goyal 2013). For those choice models, column generation can efficiently return the optimal value. However, under more complex choice models, the assortment problem might be NP-hard. For example, Bront, Mendez-Diaz, and Vulcano (2009) show that the column generation sub-problem under the Mixed Multinomial Logit (MMNL) model is NP-hard. Moreover, when an optimal solution to the CDLP cannot be obtained, there is currently no algorithm that can generate an approximate solution to the CDLP that is guaranteed to be $\epsilon$-optimal for any given $\epsilon>0$, where $\epsilon$ is the error of the approximate solution defined as follows.
Let $x_k(S)$, for $k=1,2,...,K$ and $S \in \cal{S}$, be a feasible solution to (\[eq:ChoiceBasedLP\]). We say that $x_k(S)$ is $\epsilon$-optimal if $$\sum_{k=1}^K \sum_{S \in \cal{S}} x_k(S) \sum_{n \in S} r_n P^k(n,S)\geq (1-\epsilon)V^{CDLP}.$$
We propose the first column-generation algorithms specifically designed for the case when the sub-problem is NP-hard. Our algorithm can generate $\epsilon$-optimal solutions to the CDLP, for any given $\epsilon>0$, for a variety of choice models, including the MMNL, which can approximate any random utility model to any precision level [@mcfadden2000]. Our algorithm builds on existing polynomial-time approximation schemes (PTAS) or fully polynomial-time approximation schemes (FPTAS) that can approximately solve the assortment-planning problem. Compared with the column-generation algorithm defined before, we make a slight change in each iteration step. Instead of searching for the optimal assortment $S_k^*(\pi) \in \cal{S}$ for all $k$, which is NP-hard, we aim to find an assortment $S_k^d$ that satisfies $$\sum_{n \in S_k^d, n>0} P^k(n,S_k^d) [r_n - \pi(l_n)] \geq (1 - \frac{\epsilon}{1 + \epsilon}) \sum_{n \in S_k^*(\pi), n>0} P^k(n,S_k^*(\pi)) [r_n - \pi(l_n)] .$$ This modified problem can be solved in polynomial time using a PTAS or FPTAS that applies to a broad class of choice models, including the MMNL [@doi:10.1287/opre.1120.1093]. The algorithm terminates when at the $\epsilon$ precision level, we cannot find any assortment $S_k^d$ such that $$\Lambda_k \sum_{n \in S_k^d, n>0} [r_n - \pi(l_n)] P^k(n,S_k^d) - \sigma(k) >0 ,$$
\[thm:nearOptimalCG\] For any given $\epsilon > 0$, the above column-generation algorithm generates an $\epsilon$-optimal solution.
Upper Bound on the Competitive Ratio
====================================
In this section, we show that $0.5$ is the best competitive ratio that any online algorithm can achieve for our model. This result implies later that our algorithms achieve the best performance guarantee.
For the dynamic assortment-planning problem, no algorithm can achieve a competitive ratio higher than $0.5$.
Consider the following special case of our model. There is only one product and one associated resource. Then the problem becomes a single-resource reward management problem. Wang, Truong and Bank (2015) have shown that the best competitive ratio for such single-resource problem is $0.5$ when the arrival rates are non-homogeneous. Therefore, for the general multi-product, multi-resource problem, the best competitive ratio is also $0.5$.
The First-Come-First-Served Algorithm
=====================================
To motivate our main ideas, we first give an analysis of a simpler first-come-first-served ($FCFS$) algorithm. While $FCFS$ has a small proven performance guarantee under relaxed conditions, it leads to the full intuition behind our main algorithm.
For ease of exposition, we introduce the following somewhat impractical assumptions that relax the action space of online algorithms for the current section and Section \[sec:PrimalRouting\]. Later in section \[sec:OPR\], we will show that these assumptions can be easily removed without loss of generality:
1. An assortment can contain products whose corresponding resource has zero inventory. Customers’ choices are unaffected by the inventory status of products. If a customer chooses a product with zero inventory, this customer is treated as being rejected. Recall that this assumption is also called the *static substitution* assumption.
2. Even after a customer has chosen a product with positive inventory, an algorithm can still reject the customer, preventing the chosen product from being purchased. That is, inventory will be unchanged after the interaction. This assumption is used only in Section \[sec:PrimalRouting\].
The $FCFS$ algorithm is a naive implementation of an optimal solution to (\[eq:ChoiceBasedLP\]):
1. [(Pre-processing Step) Solve the CDLP (\[eq:ChoiceBasedLP\]) to obtain an $\epsilon$-optimal solution $x^*_k(S)$ for $k = 1,2,...,K$ and $S \in \cal{S}$. Note that $\epsilon$ can be $0$ if it is possible to compute an exact optimal solution. For customers of type $k$, let $\cal{O}_k \equiv \{S \in \cal{S} : x_k^*(S) > 0\}$ be the set of assortments with positive displaying probabilities.]{}
2. [(Random Offering Step) Upon an arrival of a type-$k$ customer at time $t$, offer an assortment $A_k^{Static}$ that is randomly picked from $\cal{O}_k$ such that $P(A_k^{Static} = S) = x_k^*(S)$ for all $S \in \cal{O}_k$. Note that $A_k^{Static}$ is independent of the state of inventory and time $t$.]{}
3. ($FCFS$ Step) If the customer chooses a product $n \in A_k^{Static}$ and $n>0$, let the customer purchase it if the corresponding resource $l_n$ has positive remaining inventory. Otherwise, reject the customer.
$FCFS$ is essentially a generalization of the deterministic algorithm of Gallego, Iyengar, Phillips, and Dubey (2004) to a non-stationary, multi-customer-type environment.
Since $FCFS$ assumes static substitution, its expected future reward $V^{FCFS}(c,t)$ does not satisfy (\[eq:Dynamics\]). Instead, the dynamic equation that $FCFS$ follows is $$\label{eq:DynamicsFCFS}
\frac{\partial V^{FCFS}(c,t)}{\partial t} = - \sum_{k=1}^K \sum_{l=1}^L \mathbf{1}(c_l >0) R_t^{kl}(A_k^{Static}, \Delta_l V^{FCFS}(c,t)).$$ Note that the only difference between this equation and (\[eq:Dynamics\]) is the additional term $\mathbf{1}(c_l >0)$, which implies that customers who choose products with zero inventory have no impact on the system.
Using an interesting analysis based on properties of Poisson processes, we can show that $FCFS$ already gives a constant performance guarantee under static substitution, as stated in the following theorem.
\[thm:FCFS\] If $x^*$ is $\epsilon$-optimal then $V^{FCFS}(C,0) \geq \frac{1}{e}(1-\epsilon) V^{OFF}$.
The Primal Routing Algorithm {#sec:PrimalRouting}
============================
The FCFS algorithm gives the following intuition. If we view the Random Offering Step as exogenous, the problem separates into $L$ single-resource revenue-management problems. The demands arriving at each resource can be considered as [*independent demands*]{}. It is well-known that $FCFS$ is not optimal for the single-resource revenue-management problem, as it might not be optimal to always accept customers who arrive first. The following Primal Routing Algorithm (PR) optimally solves the single-resource revenue-management problem for each resource, by rejecting customers who want to purchase products at prices that are too low. In this sense, $PR$ improves upon the performance of $FCFS$. However, $PR$ has many disadvantages even compared to $FCFS$. In particular, our analysis of $PR$ still assumes the *static substitution* assumption that we made for $FCFS$. In addition, $PR$ might offer a product as part of an assortment, then upon a customer’s choosing the product, reject the customer. This might happen even if the product has positive inventory.
Nevertheless, the following performance analysis of $PR$ will help us to build intuition for the analysis of a more practical, but more intricate, algorithm that we will develop in Section \[sec:OPR\].
The difference between $PR$ and $FCFS$ lies in the Primal Routing Step that replaces the FCFS Step:
1. (Pre-Processing Step) Same as in FCFS.
2. (Random Offering Step) Same as in FCFS.
3. [(Primal Routing Step) Let $c = (c_1,c_2,...,c_L)$ be the vector of inventory at time $t$. Suppose that a customer chooses a product $n \in A_k^{Static}$. Suppose that $n>0$ and $l = l_n$. Let the customer purchase the product if and only if $c_l>0$ and $$r_n \geq \Delta_l V^{PR}(c,t) ,$$ where $V^{PR}(c,t)$ stands for the expected future reward of $PR$ given state $(c,t)$, and the definition of $\Delta_l V^{PR}(c,t)$ follows (\[eq:DeltaV\]). The dynamic equation that $V^{PR}(c,t)$ must satisfy is $$\label{eq:DynamicsPR}
\frac{\partial V^{PR}(c,t)}{\partial t} = - \sum_{k=1}^K \sum_{l=1}^L \cal R_t^{kl}(A_k^{Static}, \Delta_l V^{PR}(c,t)),$$ where $$\cal R_t^{kl}(S,z) \equiv \lambda_k(t) \sum_{n \in N_l} \mathbf{E}[\mathbf{1}(n \in S) P^k(n,S)] [r_n - z]^+.$$ Note that $\cal R_t^{kl}(S,z)$ is different from $R_t^{kl}(S,z)$ defined in (\[eq:RewardRate\]) as $\cal R_t^{kl}(S,z)$ applies the $[\cdot]^+$ operator to the difference $r_n-z$ between the reward of product $n$ and the cost value $z$.]{}
The dynamic equation (\[eq:DynamicsPR\]) can be decomposed by resources as follows. For each resource $l$, consider the following single-resource reward function $V_l^{PR}(c,t)$ defined as $$\label{eq:dp}
\frac{\partial V_l^{PR}(c,t)}{\partial t} = - \sum_{k=1}^K \cal R_t^{kl}(A_k^{Static}, \Delta_l V_l^{PR}(c,t))$$ with boundary conditions $V_l^{PR}(0,t) = 0$ and $V_l^{PR}(c,1) = 0$.
It is easy to see that (\[eq:dp\]) is just the Hamilton-Jacobi-Bellman equation that computes the optimal expected future reward for resource $l$ under arrivals over the horizon of $|N_l|$ demand classes, with demand class $n \in N_l$ bringing unit reward $r_n$ and arriving at the rate $$\sum_{k=1}^K \lambda_k(t) \mathbf{E}[\mathbf{1}(n \in A_k^{Static}) P^k(n,A_k^{Static})].$$
This immediately leads to the following result
The expected total reward of $PR$ is $$V^{PR}(C,0) = \sum_{l=1}^L V_l^{PR}(C,0).$$ In particular, the expected future reward that $PR$ earns from resource $l$, starting with state $(c,t)$, is $V_l^{PR}(c,t)$.
This theorem is a direct result of the properties of the Hamilton-Jacobi-Bellman equation (\[eq:dp\]).
Core to our competitive analysis of PR is the following property of the reward function.
\[thm:SingleLegCR\] For each resource $j=1,\ldots,L$, $$V_j^{PR}(C,0) \geq 0.5 \sum_{k=1}^K\Lambda_k \sum_{n \in N_j} \sum_{S \in \cal{O}_k : n \in S} x_k^*(S) P^k(n,S) r_n.$$
The proof proceeds by proving that the expected revenue generated by each resource $j$ under $PR$ is at least half of what $OFF$ obtains. Wang, Truong and Bank (2015) have proved a special case of this theorem when $C_j = 1$. The proof in the Appendix extends their result to the case $C_j > 1$.
\[thm:PRBound\] If $x^*$ is $\epsilon$-optimal then $V^{PR}(C,0) \geq 0.5(1-\epsilon) V^{CDLP}$.
From Theorem \[thm:SingleLegCR\] we know that $$\sum_{j=1}^L V_j^{PR}(C,0) \geq 0.5 \sum_{j=1}^L \sum_{k=1}^K\Lambda_k \sum_{n \in N_j} \sum_{S \in \cal{O}_k : n \in S} x_k^*(S) P^k(n,S) r_n = \sum_{k=1}^K \Lambda_k \sum_{S \in \cal{S}} \sum_{n \in S} x_k^*(S) P^k(n,S) r_n.$$
Since $\sum_{j=1}^L V_j^{PR}(C,0)$ is the expected total reward of $PR$, and the right hand side is at least $(1-\epsilon)V^{CDLP}$, the expected reward of $PR$ is at least $0.5(1-\epsilon) V^{CDLP}$.
Since $PR$ optimally manages the capacity allocation for each resource $l$, the expected future reward that $PR$ earns from each resource must dominate that of $FCFS$, which is stated in the following theorem.
\[thm:PRDominance\] $V^{PR}(c,t) \geq V^{FCFS}(c,t).$
It is easy to check that $\cal{R}_t^{kl}(S,z) \geq \mathbf{1}(c_l >0) R_t^{kl}(S,z)$. The result follows after we combine this condition with (\[eq:DynamicsFCFS\]) and (\[eq:DynamicsPR\]).
The Optimized Primal Routing Algorithm {#sec:OPR}
======================================
In this section, we propose a new algorithm called the Optimized Primal Routing ($OPR$) algorithm. The advantage of $OPR$ is that it is much more practical than $PR$. It does not perform random routing of customers. Therefore, it exhibits much more stable behavior. It also never offers an assortment that contains a product with $0$ inventory. At the same time, it retains the performance guarantee of $PR$.
For $OPR$, we only make one reasonable assumption on choice models, namely, that if we remove all products with non-positive rewards from an assortment, the expected reward of the assortment does not decrease.
\[OPRAssumption\] For any assortment $S$ and any reward values $r_n$, $n=1,2,...,N$, let $H = \{ n \in S : r_n > 0\}$ be the set of products in $S$ with positive reward values. We must have for any customer type $k$, $$\sum_{n \in H} r_n P^k(n,H \cup \{0\}) \geq \sum_{n \in S} r_n P^k(n,S).$$
Note that this assumption holds for all random-utility models.
$OPR$ performs an additional optimization step compared to $PR$. Specifically, for each arriving customer, the algorithm locates an assortment that is at least as good as the assortment given by $PR$, according to the marginal values $\Delta_l V^{PR}$, $l=1,\ldots,L$, that are calculated in the same way as in $PR$:
1. (Pre-Processing Step) Same as in FCFS.
2. [(Marginal Allocation Step) Let $c = (c_1,c_2,...,c_L)$ be the vector of inventory at time $t$. Upon an arrival of a type-$k$ customer at time $t$, offer an assortment $A_k^{OPR}(c,t)$ that aims at maximizing the marginal reward $$\label{eq:OPRStepTwo}
A_k^{OPR}(c,t) \in \argmax_{S\in \cal{S}} \{ \sum_{l=1}^L R_t^{kl}(S, \Delta_l V^{PR}(c,t))\}.$$ When $S$ is a deterministic assortment the marginal reward becomes $$\sum_{l=1}^L R_t^{kl}(S, \Delta_l V^{PR}(c,t)) = \lambda_k(t) \sum_{n \in S} P^k(n,S) \cdot (r_n - \Delta_l V^{PR}(c,t)).$$ Thus, is equivalent to solving an assortment-optimization problem for a single customer with $r_n - \Delta_l V^{PR}(c,t)$ being the price for product $n$. In this step, $A_k^{OPR}(c,t)$ can be found by any approximation algorithms or heuristics. We only require that the expected marginal reward of $A_k^{OPR}(c,t)$ be at least the marginal reward that $PR$ earns from this customer. Mathematically, we require that $$\label{eq:OPRRequirement}
\sum_{l=1}^L R_t^{kl}(A_k^{OPR}(c,t), \Delta_l V^{PR}(c,t)) \geq \sum_{l=1}^L \cal R_t^{kl}(A_k^{Static}, \Delta_l V^{PR}(c,t)).$$ Note that it is trivial to satisfy this requirement because according to Assumption \[OPRAssumption\], we could just obtain $A_k^{OPR}(c,t)$ by taking the assortment in $\cal{O}_k$ that has the largest marginal reward, and then removing products with non-positive price $r_n - \Delta_l V^{PR}(c,t)$. The idea of is to enhance the empirical performance $OPR$ even further by conducting a broader search. ]{}
The requirement (\[eq:OPRRequirement\]) also implies that $OPR$ satisfies the dynamic-substitution assumption. Therefore, the expected future reward $V^{OPR}(c,t)$ of $OPR$ satisfies the dynamic equation (\[eq:Dynamics\]).
\[thm:OPR\] The expected total reward of $OPR$ dominates that of $PR$. That is, $V^{OPR}(C,0) \geq V^{PR}(C,0)$.
The proof defines a series of algorithms $\Pi^{(i)}$ that are intermediate to $PR$ and $OPR$. Algorithm $\Pi^{(i)}$ is defined as follows. For the first $i$ customers, apply *OPR*. Afterward, for the $(i+1)$-th, $(i+2)$-th,..., customers, apply *PR*. Thus, $\Pi^{(0)}$ resembles $OR$ and $\Pi^{(\infty)}$ resembles $OPR$. The proof in the Appendix shows that the expected revenue of each algorithm $\Pi^{(i)}$ improves upon that of $\Pi^{(i-1)}$.
Using Theorems \[thm:PRDominance\] and \[thm:OPR\] and Corollary \[thm:PRBound\], we can obtain our main result for $OPR$.
If $x^*$ is $\epsilon$-optimal then $OPR$ is $0.5(1-\epsilon)$-competitive. Moreover, $OPR$ is asymptotically optimal as we scale up the total demand and capacity simultaneously.
The competitive ratio is a direct result of Theorem \[thm:OPR\] and Corollary \[thm:PRBound\]. It is well known that $FCFS$ is asymptotically optimal. Since $PR$ dominates $FCFS$ and $OPR$ dominates $PR$, $OPR$ must also be asymptotically optimal.
Appendix
========
[**Proof of Theorem \[thm:upperbound\].**]{} Let $\delta_k$ be the actual number of arrivals of type-$k$ customers during the entire horizon. Let $I_k(S)$ be the number of times that a customer of type $k$ is shown assortment $S$ under *OFF*. Let $J_{kn}$ be the number of times that a customer of type $k$ purchases product $n$ under *OFF*.
We must have $$\label{eq:LPProof1}
\sum_{S \in \cal{S}} I_k(S) = \delta_k,\,\,\, \forall k = 1,2,3...,K,$$ $$\label{eq:LPProof2}
\sum_{n=1}^N \sum_{k=1}^K J_{kn} \cdot \mathbf{1}(l_n = j) \leq C_j, \,\,\,\forall j = 1,2,...,M,$$ $$\label{eq:LPProof3}
\mathbf{E}[J_{kn}] = \mathbf{E}[\sum_{S \in \cal{S} : n \in S} I_k(S) \cdot P^k(n,S)].$$
Note that equation (\[eq:LPProof3\]) is a result of the dynamic-substitution assumption.
Taking expectation on both sides of (\[eq:LPProof1\]), we get $$\label{eq:LPProof4}
\sum_{S \in \cal{S}} \mathbf{E}[I_k(S)] = \Lambda_k,\,\,\, \forall k = 1,2,3...,K.$$
Taking expectation on both sides of (\[eq:LPProof2\]), we get $$\sum_{n=1}^N \sum_{k=1}^K \mathbf{E}[J_{kn}] \cdot \mathbf{1}(l_n = j) \leq C_j$$ $$\Longrightarrow \sum_{n=1}^N \sum_{k=1}^K \mathbf{E}[\sum_{S \in \cal{S} : n \in S} I_k(S) \cdot P^k(n,S)] \cdot \mathbf{1}(l_n = j) \leq C_j$$ $$\label{eq:LPProof5}
\Longrightarrow \sum_{S \in \cal{S}} \sum_{n \in S} \sum_{k=1}^K \mathbf{E}[ I_k(S) ] \cdot P^k(n,S)\mathbf{1}(l_n = j) \leq C_j .$$
From (\[eq:LPProof4\]) and (\[eq:LPProof5\]) we know that $\mathbf{E}[ I_k(S) ]/ \Lambda_k$ is a feasible solution to LP (\[eq:ChoiceBasedLP\]). Thus $$\mathbf{E}[R^{OFF}] =\sum_{k=1}^K \sum_{S\in \cal{S}} \Lambda_k \mathbf{E}[ I_k(S) ] \sum_{n\in S} P^k(n,S) r_n$$ is at most the optimal objective value of (\[eq:ChoiceBasedLP\]).
[**Proof of Theorem \[thm:nearOptimalCG\].**]{} When the column-generation algorithm terminates, we must have, $\forall k = 1,2,...,K$, $$\begin{aligned}
\begin{split}
&\Lambda_k \sum_{n \in S_k^d, n>0} [r_n - \pi(l_n)] P^k(n,S_k^d) - \sigma(k) \leq 0 ,\\
\Longrightarrow &\Lambda_k \sum_{n \in S_k^*(\pi), n>0} [r_n - \pi(l_n)] P^k(n,S_k^*(\pi)) \cdot \frac{1}{1 + \epsilon} - \sigma(k) \leq0\\
\Longrightarrow &\Lambda_k \sum_{n \in S_k^*(\pi), n>0} [r_n - \pi(l_n)] P^k(n,S_k^*(\pi)) - \sigma(k) \leq \epsilon \sigma(k)\\
\Longrightarrow &\Lambda_k \sum_{n \in S, n>0} [r_n - \pi(l_n)] P^k(n,S) - \sigma(k) \leq \epsilon \sigma(k), \,\,\,\forall k = 1,2,...,K, \ \forall S \in \cal{S}.
\end{split}\end{aligned}$$
Now let us look into the dual formulation of the CDLP given by (\[eq:ChoiceBasedLP\]): $$\begin{aligned}
\begin{split} \label{eq:dualofChoiceBasedLP}
V^{CDLP}\equiv \min &\,\,\,\,\,\,\, \sum_j C_j\pi(j)+\sum_k \sigma(k) \\
\text{s.t. } & \Lambda_k \sum_{n \in S, n>0} [r_n - \pi(l_n)] P^k(n,S) - \sigma(k) \leq 0 , \,\,\,\forall k = 1,2,...,K, \ \forall S \in \cal{S},\\
&\pi(j) \geq 0, \,\,\, \forall j=1,2,...,L\\
& \sigma(k) \geq 0, \,\,\, \forall k =1,2,...,K.
\end{split}\end{aligned}$$ Let $\pi(j), j=1,...,L$ and $\sigma(k), k=1,...,K$, be optimal dual values for the reduced CDLP at termination of the column-generation algorithm. Clearly, these variables will satisfy the dual constraint if we relax the first constraint in the linear program (\[eq:dualofChoiceBasedLP\]) by $$\Lambda_k \sum_{n \in S, n>0} [r_n - \pi(l_n)] P^k(n,S) - \sigma(k) \leq \epsilon \sigma(k), \,\,\,\forall k = 1,2,...,K, \forall S \in \cal{S}.$$ That is, the optimal dual variables $\pi(j), j=1,...,L$ and $\sigma(k), k=1,...,K$ are feasible for the relaxed LP defined by $$\begin{aligned}
\begin{split} \label{eq:reldualofChoiceBasedLP}
V^R\equiv \min &\,\,\,\,\,\,\, \sum_j C_j\pi(j)+\sum_k \sigma(k) \\
\text{s.t. } & \Lambda_k \sum_{n \in S, n>0} [r_n - \pi(l_n)] P^k(n,S) - \sigma(k) \leq \epsilon \sigma(k), \,\,\,\forall k = 1,2,...,K, \ \forall S \in \cal{S},\\
&\pi(j) \geq 0, \,\,\, \forall j=1,2,...,L\\
& \sigma(k) \geq 0, \,\,\, \forall k =1,2,...,K.
\end{split}\end{aligned}$$ Let us write down the primal of the relaxed dual program (\[eq:reldualofChoiceBasedLP\]): $$\begin{aligned}
\begin{split} \label{eq:dualrelaxChoiceBasedLP}
V^R\equiv \max & \,\,\,\,\,\,\, \sum_{k=1}^K \sum_{S\in \cal{S}} \Lambda_k x_k(S) \sum_{n\in S} P^k(n,S) r_n \\
\text{s.t. } & \sum_{k =1}^K \Lambda_k \sum_{n \in N_j} \sum_{S \in \cal{S}: n \in S} x_k(S) P^k(n,S) \leq C_j , \,\,\, \forall j = 1,2,...,L,\\
&(1+\epsilon) \sum_{S \in \cal{S}} x_k(S) \leq 1, \,\,\, \forall k =1,2,...,K;\\
& x_k(S) \geq 0, \,\,\, \forall k =1,2,...,K,\ \forall S \in \cal{S}.
\end{split}\end{aligned}$$ By strong duality, we know the optimal value returned by (\[eq:reldualofChoiceBasedLP\]) and (\[eq:dualrelaxChoiceBasedLP\]) should be the same and we denote it by $V^R$. Since (\[eq:reldualofChoiceBasedLP\]) is less restrictive than (\[eq:dualofChoiceBasedLP\]), then $V^R\leq V^{CDLP}$. Note that compared with the CDLP given in (\[eq:ChoiceBasedLP\]), (\[eq:dualrelaxChoiceBasedLP\]) has nothing changed except that the left hand side of the second constraint is multiplied by $1+\epsilon$ and thus the entire problem becomes more restrictive. Now let us multiply the left hand side of the first constraint by $1+\epsilon$ $$\begin{aligned}
\begin{split} \label{eq:morerelaxedChoiceBasedLP}
V^{R'}\equiv \max &\,\,\,\,\,\,\, \sum_{k=1}^K \sum_{S\in \cal{S}} \Lambda_k x_k(S) \sum_{n\in S} P^k(n,S) r_n \\
\text{s.t. } & (1+\epsilon)\sum_{k =1}^K \Lambda_k \sum_{n \in N_j} \sum_{S \in \cal{S}: n \in S} x_k(S) P^k(n,S) \leq C_j , \,\,\, \forall j = 1,2,...,L,\\
&(1+\epsilon) \sum_{S \in \cal{S}} x_k(S) \leq 1, \,\,\, \forall k =1,2,...,K;\\
& x_k(S) \geq 0, \,\,\, \forall k =1,2,...,K,\ \forall S \in \cal{S}.
\end{split}\end{aligned}$$ Obviously, program (\[eq:morerelaxedChoiceBasedLP\]) should have the same offering sets as those of (\[eq:ChoiceBasedLP\]) at optimality, and the optimal value satisfies $V^{R'}=\frac{1}{1+\epsilon}V^{CDLP}$. Moreover, since (\[eq:morerelaxedChoiceBasedLP\]) is more restrictive than (\[eq:dualrelaxChoiceBasedLP\]), we know $V^R \geq V^{R'}$. Given that the values of current $\pi(j), j=1,...,L$ and $\sigma(k), k=1,...,K$ are feasible to (\[eq:reldualofChoiceBasedLP\]), we have $\sum_j C_j\pi(j)+\sum_k \sigma(k) \geq V^R$. Combining all the information together, we know $\sum_j C_j\pi(j)+\sum_k \sigma(k)>\frac{1}{1+\epsilon}V^{CDLP}>(1-\epsilon)V^{CDLP}$. In other words, the value obtained from the column- generation algorithm $\sum_j C_j\pi(j)+\sum_k \sigma(k)$ is at least $1-\epsilon$ times $V^{CDLP}$.
[**Proof of Theorem \[thm:FCFS\].**]{}
Let $$s_{kn}^* \equiv \sum_{S \in \cal{O}_k : n \in S} x_k^*(S) P^k(n,S)$$be the probability that a customer of type $k$ chooses product $n$ from $A_k^{Static}$.
Under $FCFS$, the total number of customers who will choose resource $j$, including those who are rejected due to a lack of inventory, is a Poisson random variable $D_j$ with mean $$\mathbf{E}[D_j] = \sum_{k=1}^K\Lambda_k \sum_{n\in N_j} s_{kn}^*,$$ which is at most $C_j$ according to the capacity constraint of (\[eq:ChoiceBasedLP\]).
Conditioned on the event that a customer of type $k$ successfully purchases a product associated with resource $j$, the probability that the purchased product is $i$ is $$\frac{ \mathbf{1}(l_i = j) s_{ki}^*}{ \sum_{n\in N_j} s_{kn}^*},$$ which is independent of the choice of other customers.
Then, conditioned on the event that a customer of type $k$ successfully purchases a product associated with resource $j$, the expected reward that the customer brings is $$\label{eq:proofFCFS1} \frac{\sum_{n \in N_j} s_{kn}^* r_n}{ \sum_{n \in N_j} s_{kn}^*}.$$
According to the properties of Poisson processes, conditioned on the event $D_j = d$, each of the $d$ customers can be seen as randomly and independently picked from the entire horizon. The probability that each of the $d$ customers is of type $k$ is $$\label{eq:proofFCFS2} \frac{\Lambda_k \sum_{n \in N_j} s_{kn}^*}{ \sum_{i=1}^K \Lambda_i \sum_{n \in N_j} s_{in}^*} = \frac{\Lambda_k \sum_{n \in N_j} s_{kn}^*}{\mathbf{E}[D_j]}.$$
Under FCFS, for any integer $d \leq C_j$, if $D_j = d$, all of these $d$ customer demands will be satisfied. Combining (\[eq:proofFCFS1\]) and (\[eq:proofFCFS2\]), we know that for any $d \leq C_j$, $$\begin{aligned}
& \mathbf{E}[\text{Total reward obtained from resource $j$} | D_j = d ] \\
= & d \cdot \mathbf{E}[\text{Total reward obtained from resource $j$} | D_j = 1]\\
= & d \cdot \sum_{k=1}^K \frac{\sum_{n \in N_j} s_{kn}^* r_n}{ \sum_{n \in N_j} s_{kn}^*} \cdot \frac{\Lambda_k \sum_{n \in N_j} s_{kn}^*}{\mathbf{E}[D_j]}\\
= & d \cdot \frac{\sum_{k=1}^K \Lambda_k \sum_{n \in N_j} s_{kn}^* r_n}{\mathbf{E}[D_j]}.\end{aligned}$$
Then, $$\begin{aligned}
&\mathbf{E}[\text{Total reward obtained from resource $j$}]\\
= &\sum_{d=0}^{\infty} P(D_j = d) \times \mathbf{E}[\text{Total reward obtained from resource }j |D_j = d]\\
\geq &\sum_{d=0}^{C_j} P(D_j = d) \times \mathbf{E}[\text{Total reward obtained from resource }j |D_j = d]\\
= & \sum_{d=0}^{C_j} \frac{\mathbf{E}[D_j]^d}{d!}e^{-\mathbf{E}[D_j]} \times d \cdot \frac{ \sum_{k=1}^K\Lambda_k \sum_{n \in N_j} s_{kn}^* r_n}{ \mathbf{E}[D_j]}\\
\geq & \frac{1}{e} \cdot \sum_{k=1}^K\Lambda_k \sum_{n \in N_j} s_{kn}^* r_n.\end{aligned}$$ The last step follows from the fact that for any non-negative value $x$, it always holds that $$\sum_{i=0}^{\lceil x \rceil} \frac{x^i}{i!} e^{-x}\cdot \frac{i}{x} \geq \frac{1}{e}.$$
Thus, the expected total reward that FCFS earns from all resources is at least $$\sum_{j=1}^L \frac{1}{e} \cdot \sum_{k=1}^K\Lambda_k \sum_{n \in N_j} s_{kn}^* r_n = \frac{1}{e} \cdot \sum_{k=1}^K\Lambda_k \sum_{n=1}^N s_{kn}^* r_n,$$ which is $1/e$ times the objective value of (\[eq:ChoiceBasedLP\]) corresponding to solution $x_k^*(S)$. Since $x_k^*(S)$ is $\epsilon$-optimal, and $V^{CDLP}$ is an upper bound on $V^{OFF}$, the total expected reward that FCFS earns is at least $\frac{1}{e}(1-\epsilon)\cdot V^{OFF}$.
[**Proof of Theorem \[thm:SingleLegCR\].**]{} Wang, Truong and Bank (2015) have proved a special case of this theorem when $C_j = 1$. In this proof we extend their results to the case $C_j > 1$.
Let $$s_{kn}^* \equiv \sum_{S \in \cal{O}_k : n \in S} x_k^*(S) P^k(n,S)$$be the probability that a customer of type $k$ chooses product $n$ from $A_k^{Static}$.
The total expected demand $\mathbf{E}[D_j]$ that is allocated to resource $j$ is defined by the optimal solution $x^*$ of (\[eq:ChoiceBasedLP\]) $$\mathbf{E}[D_j] \equiv \sum_{k=1}^K \Lambda_k \sum_{n \in N_j} \sum_{S \in \cal{O}_k : n \in S} x_k^*(S) P^k(n,S) = \sum_{k=1}^K\Lambda_k \sum_{n \in N_j} s_{kn}^*,$$ which is at most $C_j$ according to the constraints of (\[eq:ChoiceBasedLP\]).
Consider the following sub-optimal policy applied to the single-resource revenue-management problem for resource $j$. Divide the horizon $[0,1]$ into $C_j$ intervals $[t(0),t(1)]$, $[t(1), t(2)]$,...,$[t(C_j-1),t(C_j)]$, such that $t(0) = 0$, $t(C_j) = 1$ and $$\label{eq:PRProofa}
\int_{t(i-1)}^{t(i)} \sum_{k=1}^K \lambda_k(u) \sum_{n\in N_j} s_{kn}^* du = \frac{\mathbf{E}[D_j]}{C_j} \leq 1, \,\, \forall i = 1,2,...,C_j.$$ In other words, the average number of arrivals at resource $j$ in each interval has the same value $\mathbf{E}[D_j] / C_j$. Then, we view the $C_j$ units of resource $j$ as $C_j$ different resources with unit inventory. The $i$-th resource only accepts customers, if any, arriving during the $i$-th interval $[t(i-1),t(i)]$, for $i=1,2,...,C_j$. We optimally solve the admission-control problem for each resource with unit inventory. Let $g_i(t)$ be the expected future reward obtained from the $i$-th resource. It satisfies $$\label{eq:PRProofb}
\frac{dg_i(t)}{dt} = - \sum_{k=1}^K \cal R_t^{kj}(A_k^{Static},g_i(t)), \,\,\, \forall t \in (t(i-1),t(i))$$ with boundary condition $$\label{eq:PRProofc}
g_i(t(i)) = 0.$$
Wang, Truong and Bank (2015) have shown that, once conditions (\[eq:PRProofa\]), (\[eq:PRProofb\]) and (\[eq:PRProofc\]) hold at the same time, the total expected reward obtained from resource $i$ is at least $$g_i(t(i-1)) \geq 0.5 \int_{t(i-1)}^{t(i)} \sum_{k=1}^K \lambda_k(u) \sum_{n \in N_j} s_{kn}^* r_n du.$$
Thus, under this sub-optimal policy, the expected total reward obtained from all $C_j$ resources is at least $$\sum_{i=1}^{C_j} g_i(t(i-1)) \geq 0.5 \sum_{i=1}^{C_j} \int_{t(i-1)}^{t(i)} \sum_{k=1}^K \lambda_k(u) \sum_{n \in N_j} s_{kn}^* r_n du = 0.5 \sum_{k=1}^K \Lambda_k \sum_{n \in N_j} s_{kn}^* r_n .$$ Since $V_j^{PR}(C,0)$ is the optimal expected reward when the decisions are made for the entire resource $j$, $V_j^{PR}(C,0)$ must be at least the expected total reward of the sub-optimal policy. In other words. $$V_j^{PR}(C,0) \geq 0.5 \sum_{k=1}^K \Lambda_k \sum_{n \in N_j} s_{kn}^* r_n.$$
[**Proof of Theorem \[thm:OPR\].**]{}
We want to show that $$V^{OPR}(c,t) \geq V^{PR}(c,t)$$ for every given state $(c,t)$.
Define an algorithm $\Pi^{(i)}$ as follows. For the first $i$ customers, apply *OPR*. Afterward, for the $(i+1)$-th, $(i+2)$-th,..., customers, apply *PR*. Let $h^{(i)}(c,t)$ be the expected future reward when policy $\Pi^{(i)}$ is applied starting at time $t$ with remaining inventory $c(t)$, and *assuming that no customers have arrived prior to time $t$*. We must have $$h^{(0)}(c,t) = V^{PR}(c,t),$$ $$\lim_{i\to \infty} h^{(i)}(c,t) = V^{OPR}(c,t) .$$
The dynamic programming equation for algorithm $\Pi^{(1)}$ is $$\begin{aligned}
\label{eq:OptimizedAlgProof2}
\begin{split}
\frac{\partial h^{(1)}(c,t)}{\partial t} &= - \sum_{k=1}^K \left[ \sum_{l=1}^L R_t^{kl}(A_k^{OPR}(c,t), h^{(1)}(c,t) - h^{(0)}(c-e_l,t)) - \lambda_k(t) \mathbf{E}[P^k(0,A_k^{OPR}(c,t))] \Delta^{(1)}(c,t) \right] \\
& = - \sum_{k=1}^K \left[ \sum_{l=1}^L R_t^{kl}(A_k^{OPR}(c,t), \Delta_l V^{PR}(c,t)) - \lambda_k(t) \Delta^{(1)}(c,t) \right],
\end{split}\end{aligned}$$ where $$\Delta^{(1)}(c,t) \equiv h^{(1)}(c,t) - V^{PR}(c,t)$$ is the difference in the expected future reward between $\Pi^{(1)}$ and *PR* (*PR* $\equiv \Pi^{(0)}$).
Combining (\[eq:OPRRequirement\]), (\[eq:OptimizedAlgProof2\]) and the dynamic equation (\[eq:DynamicsPR\]) for $PR$, we can obtain $$\frac{\partial h^{(1)}(c,t)}{\partial t} \leq \frac{\partial V^{PR}(c,t)}{\partial t} + \sum_{k=1}^K \lambda_k(t) \Delta^{(1)}(c,t).$$ This equation implies that, if at some time $t_0$ we have $\Delta^{(1)}(c,t_0)< 0$ or equivalently $$\label{eq:OptimizedAlgProofb}
h^{(1)}(c,t_0) - V^{PR}(c,t_0) < 0,$$ then we must have $$\frac{\partial h^{(1)}(c,t)}{\partial t} < \frac{\partial V^{PR}(c,t)}{\partial t}, \,\,\, \forall t \in (t_0,1]$$ and $$\label{eq:OptimizedAlgProofa}
h^{(1)}(c,t) < V^{PR}(c,t), \,\,\, \forall t \in (t_0,1].$$ However, since we know that $h^{(1)}(c,1) = V^{PR}(c,1) = 0$, (\[eq:OptimizedAlgProofa\]) cannot be true, and thus (\[eq:OptimizedAlgProofb\]) cannot be true. Therefore, we have proved
$$\label{eq:OptimizedAlgProof4}
h^{(1)}(c,t) \geq V^{PR}(c,t), \,\,\, \forall t \in [0,1].$$
Next, we show that $$\label{eq:OptimizedAlgProof5}
h^{(i)}(c,t) \geq h^{(i-1)}(c,t), \,\,\, \forall t\in [0,1]$$ by induction on $i$.
Equation (\[eq:OptimizedAlgProof4\]) already proves the base case $i=1$. Suppose for some $\bar i > 1$, (\[eq:OptimizedAlgProof5\]) holds for all $i < \bar i$. Now we show that it also holds for $i = \bar i$. By definition, for any $\bar i > 1$, algorithms $\Pi^{(\bar i)}$ and $\Pi^{(\bar i-1)}$ must offer the same assortment to the first customer, for they both apply *OPR* to the first customer. Thus, $\Pi^{(\bar i)}$ and $\Pi^{(\bar i-1)}$ earn the same reward from the first customer, and then transit into the same state. After that first customer, $\Pi^{(\bar i)}$ continues to apply $\Pi^{(\bar i-1)}$ pretending that no customer has ever arrived, while $\Pi^{(\bar i-1)}$ continues to apply $\Pi^{(\bar i-2)}$. By induction, the expected future reward of $\Pi^{(\bar i-1)}$ is at least that of $\Pi^{(\bar i-2)}$. Therefore, the expected future reward of $\Pi^{(\bar i)}$ is at least that of $\Pi^{(\bar i-1)}$.
Thus, we have proved (\[eq:OptimizedAlgProof5\]). It immediately follows that $$h^{(\infty)}(c,t) \geq h^{(0)}(c,t) \Longrightarrow V^{OPR}(c,t) \geq V^{PR}(c,t).$$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
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author:
- 'Michael Shell, John Doe, and Jane Doe, [^1][^2][^3]'
title: 'Bare Demo of IEEEtran.cls for TCOM'
---
[Submitted paper]{}
IEEEtran, journal, LaTeX, paper, template.
Introduction
============
demo file is intended to serve as a “starter file” for IEEE journal papers produced under LaTeX using IEEEtran.cls version 1.7 and later. I wish you the best of success.
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Subsection Heading Here
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### Subsubsection Heading Here
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Conclusion
==========
The conclusion goes here.
Proof of the First Zonklar Equation
===================================
Appendix one text goes here.
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Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank...
[1]{}
H. Kopka and P. W. Daly, *A Guide to LaTeX*, 3rd ed.1em plus 0.5em minus 0.4emHarlow, England: Addison-Wesley, 1999.
[^1]: M. Shell is with the Department of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332 USA e-mail: (see http://www.michaelshell.org/contact.html).
[^2]: J. Doe and J. Doe are with Anonymous University.
[^3]: TCOM version based on Michael Shell’s barejrnl.tex version 1.3.
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