Capacitated lot-sizing and scheduling with sequence-dependent, period-overlapping and non-triangular setups

TL;DR: This work presents the first linear mixed-integer programming extension for the capacitated lot-sizing and scheduling problem incorporating all the necessary features of sequence sub tours and setup crossovers.

Abstract: In production planning, sequence dependent setup times and costs are often incurred for switchovers from one product to another. When setup times and costs do not respect the triangular inequality, a situation may occur where the optimal solution includes more than one batch of the same product in a single period--in other words, at least one sub tour exists in the production sequence of that period. By allowing setup crossovers, flexibility is increased and better solutions can be found. In tight capacity conditions, or whenever setup times are significant, setup crossovers are needed to assure feasibility. We present the first linear mixed-integer programming extension for the capacitated lot-sizing and scheduling problem incorporating all the necessary features of sequence sub tours and setup crossovers. This formulation is more efficient than other well known lot-sizing and scheduling models.

Figures

Figure 8. U. panamensis microhabitat ordination biplot of latent variable posterior medians (Option 2 model – unconstrained ordination). Microhabitats are stated, numbers are positions of latent variables for individual snakes.

Figure 9. U. panamensis microhabitat residual correlations (from Option 2 model – unconstrained ordination). Only significant correlations are plotted (based on 95% credible intervals excluding zero). Correlations are represented by colours (red and blue for negative and positive correlations respectively), while the strength of correlations is represented by the size of the circles.

Figure 5. Boxplots of the morphometrics of female (n=5) and male (n=7) Ungaliophis panamensis.

Figure 6. Standardised Major Axis testing routine: SVL/Mass. Female U. panamensis (red), male (blue).

Figure 7. Dunn-Smyth residual fits of Bayesian ordination LVM (Option 2 – unconstrained ordination).

Figure 1. Location of sampled area, Barra del Colorado Wildlife Refuge, Limón, Costa Rica.

Figure 2. Location of sampled area, El Cope in Parque Nacional G. D. Omar Torríjos Herrera, in the Coclé and Panamá Oeste provinces of Panama.

Figure 4. Photographs depicting two individuals of Ungaliophis panamensis from El Cope in Parque Nacional G. D. Omar Torríjos Herrera, in the Coclé and Panamá Oeste provinces of Panama: individuals UPPA01 (A) and UPPA02 (B). Photographs taken by Julie M. Ray.

Table 1. Morphological and ecological data for 13 individuals of Ungaliophis panamensis from Costa Rica and Panama.

Figure 3. Photographs depicting two individuals of Ungaliophis panamensis from Barra del Colorado Wildlife Refuge in Limón, northeastern Costa Rica: individuals UPCR07 (A) and UPCR08 (B). Photographs taken by Alex Figueroa.

Full text

Ecology and morphology of the dwarf bromeliad
boa Ungaliophis panamensis (Squamata, Boidae,
Ungaliophiinae) in Costa Rica and Panama
Todd R. Lewis
1
, Rowland K. Grin
2
, Irune Maguregui Martin
3
,
Alex Figueroa
4
, Julie M. Ray
5
, Josh Feltham
6
, Paul B. C. Grant
7
1 UWE Bristol – Frenchay Campus, Coldharbour Ln, Bristol, BS16 1QY, UK
2 Indigo Expeditions, 37 Chamberlain St., Wells, Somerset, BA5 2PQ, UK
3 Estación Biológica Caño Palma, Tortuguero, Limón, Costa Rica
4 Department of Biological Sciences, University of New Orleans, 2000 Lakeshore Drive, New Orleans, LA
70148, USA
5 Department of Biology, University of Nevada-Reno, 1664 North Virginia Street, Reno, NV 89557, USA
6 Fleming College, 200 Albert St S., Lindsay, Ontario, K9V 5E6, Canada
7 4901 Cherry Tree Bend, Victoria, British Colombia, V8Y 1S1, Canada
Corresponding author: Todd R. Lewis (Todd.Lewis@uwe.ac.uk)
Academic editor: A.M. Leal-Zanchet|Received 21 August 2020|Accepted 1 May 2021|Published 18 May 2021
Citation: Lewis TR, Grin RK, Martin IM, Figueroa A, Ray JM, Feltham J, Grant PBC (2021) Ecology and
morphology of the dwarf bromeliad boa Ungaliophis panamensis (Squamata, Boidae, Ungaliophiinae) in Costa Rica
and Panama. Neotropical Biology and Conservation 16(2): 317–331. https://doi.org/10.3897/neotropical.16.e57872
Abstract
Ecological and morphological data on Ungaliophis panamensis is extremely limited as this species is
rarely encountered. ese knowledge gaps have been advanced in this study where data was analysed
from a small sample of snakes collected in two tropical forested environments in Costa Rica and Pana-
ma. Standardised major axis testing and a Bayesian latent variable ordination revealed that the species
is sexually dimorphic, closely associated with tree trunks in natural forested areas, and occasionally
discovered in rural buildings. Although further investigation into its natural history is warranted, this
study shows that even with just a few individuals it is possible to elucidate ecological information that
is relevant to the conservation of snake species.
Keywords
Bromeliad Boa, ecology, habitat, Latent Variable Ordination, snake, tropical forest, Ungaliophis
Neotropical Biology and Conservation
16(2): 317–331 (2021)
doi: 10.3897/neotropical.16.e57872
Copyright Todd R. Lewis et al. This is an open access article distributed under the terms of the
Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution,
and reproduction in any medium, provided the original author and source are credited.
RESEARCH ARTICLE

Todd R. Lewis et al.318
Introduction
e natural history of tropical snakes is oen little understood. is is due in part
to their cryptic behaviour and remarkable camouage, which results in infrequent
detection and is a frequent issue for snake biologists (Durso and Seigel 2015; Shel-
ton et al. 2018). e genus Ungaliophis consists of two species that are restricted to
tropical Mesoamerica; they are considered highly cryptic and very little data exists
regarding their natural history (Bogert 1968; Savage 2002; Köhler 2008). e north-
ern species, U. continentalis, is found from southern Mexico to northern Nicaragua,
whereas the southern species, U. panamensis, occurs from southern Nicaragua to
northwestern Colombia (Villa and Wilson 1990). e genus was rst described in
1880 by Müller who designated an individual from Guatemala as Ungaliophis con-
tinentalis (Savage 2002). It was not until 1933 that Schmidt described the species
U.panamensis. Furthermore, U. panamensis was not conrmed from Costa Rica
until 1974 (Corn 1974).
Since then, very little has been reported about the ecology and natural his-
tory of either species. e little data that exists regarding the natural history
of either species of Ungaliophis strongly suggests that they are arboreal species
found from lowland rainforests to high elevation cloud forests (Köhler 2008).
Corn (1974) found several individuals of U. panamensis in banana plants and
bromeliads. As such, many authorities believe that Ungaliophis naturally occurs
in dense epiphytic growth in the tree canopy. Both species of Ungaliophis are
small snakes capable of reaching between 470 mm and 482 mm in total length
(Köhler 2008). Historically, the diet of wild Ungaliophis was unknown; however,
several recent publications regarding the diet of U. panamensis in Costa Rica
have included observations of bats (Solórzano and Carillo 2017), birds (Dwyer
2017), and geckos (Espinoza and Barrio-Amorós 2018) being taken by the spe-
cies. Ungaliophis continentalis have been recorded to give birth to between 5 to 6
neonates per clutch in captivity (Ross and Marzec 1990; Burger and Ford 2007).
While there is no information regarding reproduction of U. panamensis, it is
likely to be similar to its sister species.
Identifying the habitat preferences of a given species is critical to understand-
ing the extent of functional habitat available to that species in the wider landscape
(Ocampo-Peñuela et al. 2020). Eective conservation strategies can be built once the
available habitat for a species, or assemblage, has been quantied and understood
(Riva and Nielsen 2020). Herein, we describe the microhabitat preferences and
morphometry of 13 individuals of U. panamensis, ten of which were encountered in
Barra del Colorado Wildlife Refuge in northeastern Costa Rica, and three from the
Coclé and Panamá Oeste provinces of Panama. We hypothesise that U.panamensis
has an association with natural forest habitats and that its morphology does not
inuence its habitat preference.

Ecology and morphology of the dwarf bromeliad boa
319
Methodology
Study area
Individuals of U. panamensis were recorded across multiple locations within the
Barra del Colorado Wildlife Refuge (BCWR) in the NE region of Costa Rica (Fig.1),
and from Coclé and Panamá Oeste provinces, Panama (Fig. 2). e BCWR region
is dominated by lowland tropical wet forest (Holdridge 1967) and Manicaria (Are-
caceae) swamp forests (Myers 1990; Lewis et al. 2010). e regional habitat in both
Coclé and Panamá Oeste, Panama, is mid-elevation mature secondary cloud forest
and primary premontane cloud forest (Ray et al. 2012; Ray 2015).
Field methods
Data collected for this study involved a pool of surveys that were performed from
multiple longer term studies spanning 1997–2012 (Ray 2009; Lewis et al. 2011;
Lewis et al. 2013; Zipkin et al. 2020). Most individuals were detected using visual
encounter surveys in forested environments (Heyer et al. 1994; Lovich et al. 2012).
However, some were observed by chance encounter in ceiling raers of buildings
and others reported to us by local people. Surveys at both sites were conducted from
November 2005 to September 2008, between 20:00 and 02:00 in a variety of habitats:
canal edge areas, deep forest and riparian gallery forest. Surveys averaged three sur-
veyors and were conducted four nights a week for up to six hours per survey.
Biometric data comprised sex determined by careful probing and examination
of the anal spurs, snout-vent length (SVL, mm), tail length (TL, mm) and mass (g).
Microhabitat data comprised observations where the snake was located. If the in-
dividual was in Forest, specic substrate classes were recorded (Tree, Palm, Shrub),
along with relevant structural features (Trunk, Leaf, Branch, Twig). Although infre-
quent, when snakes were found in rural environments, oen buildings, they were
assigned their own identity (Buildings).
Data analysis
Encounter rates of U. panamensis at both sites were calculated by dividing the num-
ber of individuals by the number of surveys and multiplying by 100, a technique
adapted from Kiszka et al. (2007). For the purposes of exploring patterns of associa-
tion between the morphology of U. panamensis and its microhabitat preferences we
considered individuals to be the sampling unit.
e morphometric relations of U. panamensis were analysed using Standard-
ised Major Axis (SMA) estimation created by the function SMA within the package
smatr within the program R (Warton et al. 2012a; R Core Team 2020). An advantage

Todd R. Lewis et al.320
Figure 1. Location of sampled area, Barra del Colorado Wildlife Refuge, Limón, Costa Rica.
Figure 2. Location of sampled area, El Cope in Parque Nacional G. D. Omar Torríjos Herrera, in the
Coclé and Panamá Oeste provinces of Panama.
of using SMA to test for common slope allometric relations is its inclusion of tting
factor groups for y against x (Warton and Weber 2002). We also utilised Huber’s M
estimation in place of a least squares method using the robust function in smatr in
order to make the t more inclusive for marginal outliers (Taskinen and Warton
2011). Likelihood ratio tests evaluated the slope t.
We chose to use a multivariate Bayesian, instead of univariate or distance based,
approach for ecological analysis. is was because transforming data to meet as-
sumptions for univariate and distance based approaches is problematic for smaller
data sets (Warton et al. 2012b; Warton 2017). Microhabitat data were therefore ana-
lysed using a Bayesian unconstrained ordination based on a latent variable model
(LVM) (Hui et al. 2015) using the package Boral (Hui 2016). Bayesian LVM’s are

Ecology and morphology of the dwarf bromeliad boa
321
useful at explaining multivariate composition while accounting for residual correla-
tion. ey are superior to non-metric multidimensional scaling (NMDS) because
they make provision for possible mean-variance relationships in data without con-
founding location with dispersion (Warton et al. 2012b; Hui et al. 2015). Boral ts
three types of model; 1) Covariates with no latent variables (tting independent re-
sponse GLMs where columns of y are assumed independent; 2) With no covariates
(a pure LVM that constructs unconstrained ordination); and 3) Combined covari-
ates with latent variables (tting correlated response GLMs, with latent variables).
We explored U. panamensis microhabitat preference using modelling option 2. e
model comprised the following;
Option 2 (pure LVM),
log(µ
ij
) = α + Ɵ
0j
+ z
i1
× Ɵ
j1
+ z
i2
× Ɵ
j2
= α + Ɵ
0j
+ ȥ
i
T
Ɵ
j
,
where µ
ij
is considered the mean response at microhabitat level i for an individual
snake j, Ɵ
0j
is the individual-specic intercept, ȥ
i
= (ȥ
il
, ȥ
i2
)
T
is a vector of two latent
variables, and Ɵ
j
= (Ɵ
j1
, Ɵ
j2
)
T
are the corresponding individual-specic coecients.
is modelling approach enabled biplots to visualise the data in a similar way to a
two-dimensional NMDS. From the model we extracted posterior median values of
latent variables and used these as coordinates on ordination axes to plot microhabi-
tat association (Hui et al. 2015).
For Boral models, estimation is performed using Bayesian Markov Chain Mon-
te Carlo (MCMC) methods via soware JAGS (Plummer 2003). Our model com-
prised two latent variables, used a negative binomial family, ran 40000 iterations
with 10000 discarded for burn-in, and was thinned by 30. Priors were set using
Boral’s modest automated uniform normal distribution detected and set through
JAGS (priors = ~ dnorm (0,0.1)). Convergence was assessed using MCMC trace
plots retrieved from package Coda and inspection of Geweke convergence diag-
nostic (Geweke 1992), a test which is similar to the Gelman statistic potential scale
reduction factor (PSRF) (Gelman et al. 2013), but applicable given Boral operates
with only a single MCMC chain (Hui 2016). Model assumptions of mean-variance
and log-linearity were examined using Dunn-Smyth residuals vs. t plots and nor-
mal quantile plots (Dunn and Smyth 1996).
Correlation and residual correlation were checked by plotting a residual covari-
ance matrix of latent variable regression coecients using function get.residual.cor
in Boral and package Corrplot. A strong residual covariance/correlation between
factor variables can be interpreted as evidence of autocorrelation in a model; how-
ever, acceptable levels have been recognised as indicative of an interaction/asso-
ciation (Pollock et al. 2014). e residual precision matrix in Boral can be used to
directly identify association between factors (in our case microhabitats and indi-
vidual snakes) (Ovaskainen et al. 2016). Two factors exhibiting a zero result in such
plots may remain correlated, indicating they do not directly interact, but also can
remain correlated through other factors. Residual precision matrix results should

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Ecology and morphology of the dwarf bromeliad boa ungaliophis panamensis (squamata, boidae, ungaliophiinae) in costa rica and panama" ?

These knowledge gaps have been advanced in this study where data was analysed from a small sample of snakes collected in two tropical forested environments in Costa Rica and Panama. Although further investigation into its natural history is warranted, this study shows that even with just a few individuals it is possible to elucidate ecological information that is relevant to the conservation of snake species. 

Q2. What are the future works mentioned in the paper "Ecology and morphology of the dwarf bromeliad boa ungaliophis panamensis (squamata, boidae, ungaliophiinae) in costa rica and panama" ?

With such low detection rates, the authors expect future research and observations of U. panamensis to further the knowledge of the ecology of the genus substantially. 

Q3. What is the advantage of using SMA to test for common slope allometric relations?

An advantageof using SMA to test for common slope allometric relations is its inclusion of fitting factor groups for y against x (Warton and Weber 2002). 

Q4. What is the way to assess the association between factors?

The residual precision matrix in Boral can be used to directly identify association between factors (in their case microhabitats and individual snakes) (Ovaskainen et al. 2016). 

Q5. What was the p-value for the two models?

All Geweke Z-score p-values for both models were >0.05 (results of <0.05 is approximately equivalent to a PSRF of 1.96 which would exceed MCMC Geweke upper values and indicate poor convergence). 

Q6. What is the importance of identifying the habitat preferences of a given species?

Identifying the habitat preferences of a given species is critical to understanding the extent of functional habitat available to that species in the wider landscape (Ocampo-Peñuela et al. 2020). 

Q7. What is the purpose of the Open Database License?

The Open Database License (ODbL) is a license agreement intended to allow users to freely share, modify, and use this Dataset while maintaining this same freedom for others, provided that the original source and author(s) are credited. 

Q8. What is the regional habitat of U. panamensis in Panama?

The regional habitat in both Coclé and Panamá Oeste, Panama, is mid-elevation mature secondary cloud forest and primary premontane cloud forest (Ray et al. 2012; Ray 2015). 

Q9. What were the results of the data collected?

Data collected for this study involved a pool of surveys that were performed from multiple longer term studies spanning 1997–2012 (Ray 2009; Lewis et al. 

Q10. What were the surveys conducted at both sites?

Surveys at both sites were conducted from November 2005 to September 2008, between 20:00 and 02:00 in a variety of habitats: canal edge areas, deep forest and riparian gallery forest. 

Q11. Why did the authors choose to use a Bayesian approach to analyse microhabitat data?

This was because transforming data to meet assumptions for univariate and distance based approaches is problematic for smaller data sets (Warton et al. 2012b; Warton 2017). 

Q12. What is the way to assess the correlation between the two factors?

Two factors exhibiting a zero result in such plots may remain correlated, indicating they do not directly interact, but also can remain correlated through other factors. 

Q13. What was the purpose of the study?

For the purposes of exploring patterns of association between the morphology of U. panamensis and its microhabitat preferences the authors considered individuals to be the sampling unit.