Inventory Management in Response to an Unfolding
Epidemic
Siddhartha Paul, Jayendran Venkateswaran
Industrial Engineering and Operations Research,
Indian Institute of Technology Bombay,
Powai, Mumbai, MH 400076, India
siddhartha.paul@iitb.ac.in, jayendranQ@iitb.ac.in
Abstract
A generic production-i y (PI) t framework is developed
for a hospital to respond to an unfolding epidemic. The framework is modelled
as a closed feedback loop, where the future epidemic behaviour is governed by
dicine supply of current period which further influences the demand for
edicine in the future. A Susceptible-Infected-Recovered (SIR) disease diffu-
sion model is coupled with the PI model as a forecasting tool to anticipate the
dicine de ds of a novel epidemic during prod lead time, where the
forecasting model parameters in every decision cycle are estimated by calibrat-
ing forecasting disease model with past epidemic data. With an illustrative
example of a hypothetical outbreak, the performance of SIR model is compared
against a naive forecasting method (defined as next period’s forecast is cur-
rent period’s demand) and found that SIR model outperforms naive method in
terms of reducing epidemic impact and inventory leftover.
Keywords: production-inventory, integrated framework, SIR, naive model
1 Introduction
An epidemic could cause extreme damages by claiming many human lives
and incurring huge economic losses. The emergence of an epidemic requires
immediate attention to prevent it from going beyond controllable proportions.
Thus, for any treatable disease epidemic, an efficient management of medicine
supply chain in conjunction with epidemic outbreak information is required
to alleviate the suffering of infected people and reduce further spread of the
disease (Dasaklis et al., 2012). The medicines distributed in earlier periods’
will take effect in future periods and reduce demands for medicine (Paul &
Venkateswaran, 2015). This shows the importance of combined study of an
epidemic and its corresponding medicine supply chain.
There are not much literature until recent (Chick et al., 2008; Duint-
jer Tebbens et al., 2010; Liu et al., 2015; Paul & Venkateswaran, 2015; Liu
& Zhang, 2016) to study epidemics in light of medicine supply chain. Past
literature in planning and control of epidemics (Lee & Chen, 2007; Arinamin-
pathy & McLean, 2008; Ren et al., 2013; Yarmand et al., 2014) have primarily
focused on the allocation of critical resources such as medicine but overlooked
the supply chain aspect of these resources. But without ensuring the supply
of resources, resource allocation models are pointless (Dasaklis et al., 2012).
The stream of literature, combining epidemic diffusion and supply chain, are
discussed below. Chick et al. (2008) have proposed a variant of cost sharing
contract between government, healthcare sector, and medicine manufacturer
to improve the overall performance of vaccine supply chain using a game the-
oretic approach. Duintjer Tebbens et al. (2010) have integrated a polio epi-
demic model with its vaccine production model to minimize total public health
cost and vaccine production cost by solving an optimization problem to select
vaccine filling flows and production flows at each time period. Liu et al. (2015)
have proposed a time-space network model to study the effect of early peri-
ods’ medicine supply on the demands of later periods by defining a so-called
linear growth factor. This factor along with past supply information is used
to forecast demand for future periods. Later on, in another publication (Liu
& Zhang, 2016), the authors have pointed out following limitations of their
earlier work (Liu et al., 2015): the naive forecasting method served as a quick
corrector but not cross-validated against actual demand data, production lead
time and order sizes were not considered. Paul & Venkateswaran (2015) have
shown that the effect of inventory management on epidemic dynamics is signif-
icant by developing an integrated system dynamics (SD) model. They further
concluded that a communicable disease epidemic could occur due to either
severity of the disease or insufficient supply of medicine. Although they have
captured the feedback from medicine supply chain to disease diffusion process
and vice versa, they used exponential smoothing demand forecasting tech-
nique which is inadequate for estimating demand in disease diffusion process.
Liu & Zhang (2016) have addressed the limitations of their previous work and
solved a more realistic epidemic logistics planning problem with a large-scale
mixed integer programming formulation. In this model, the authors used a
disease diffusion model to forecast demands, leveraging past demand infor-
mation. However, they did not consider the effect of unmet demands (i.e.,
infected patients) on the spread of the epidemic in subsequent periods.
On the other hand, in supply chain modelling literature, Forrester (1961)
and Sterman (2000) have provided a generic production-distribution
which serves as a basic framework for many decision-making analyses
stock management control, raw material ordering etc. Most of the analyses
in this domain (Disney & Towill, 2003; Venkateswaran & Son, 2007; Bijulal
et al., 2011) were conducted for stationary demand setting while epidemic
demand pattern follows a bell-shaped curve over time.
In this paper, we attempt to fulfil the research gaps as pointed out in
literature survey. An integrated framework, similar to the one discussed in
(Paul & Venkateswaran, 2015), is proposed for managing inventory and order-
ing decisions in healthcare institutions in response to an unfolding epidemic.
We describe the problem under study as follows: although in post-epidemic
response planning process, the disease true behaviour of a novel epidemic
remains unknown, but it is known that the disease can be alleviated by pro-
viding medicines to infected patients on time otherwise the infected people
would keep spreading the disease into other susceptible population. This situ-
ation demands a good inventory management which depends on the ordering
policy and hence the forecasting method. Due to lack of information about
the disease true behaviour, we try to anticipate its dynamics through other
disease diffusion models. The novelty of this work lies in combining both
epidemic diffusion and inventory management models with a dynamic disease
forecasting model. The proposed framework is simulated for three different
settings of actual disease diffusion model and the overall performance is com-
pared against a naive forecasting method.
The rest of the paper is organised as follows: Section 2 describes the
conceptual framework of the proposed model, then a detailed disease out-
break model and an inventory management model are outlined. In section 3,
simulation results of the proposed framework with an illustrative example are
demonstrated and performance is compared with the naive method. Section 4
discusses observations and future work.
2 Model Description
In this section, we will first describe the conceptual framework of our proposed
integrated model and then provide a detailed description of each module in
the subsequent sections. The symbols, notations and then abbreviations as
used in this section are listed in Table 1.
2.1 Conceptual Model
The basic framework of integrated model, presented by Paul & Venkateswaran
(2015), is adopted here to capture the interaction of disease outbreak and its
corresponding medicine supply chain.
In this paper, we have modified the major components viz. inventory
management module, disease diffusion module (which depicts patients flow
in the hospital), and demand forecasting module of (Paul & Venkat
2015) in order to capture the reality better and build a decision support syste
in response to an unfolding epidemic.
The mental model of our proposed system is displayed in Fig. 1. In this
model, with the inception of a novel epidemic, infected patients start com-
ing to the hospital, denoted as compartment “I”, and generate demands for
Table 1: Notations Used
Symbol Description Units
¥ Infection probability (p) x Contact rate (A) (rate) 1/day
N Total population person
P Incubation period day
TRM Time to recovery with medicine day
TR Time to recovery without medicine day
TW Waning time day
Sa Susceptible population at time d person
Ey Exposed population at time d person
Ta Infected patients secking treatment at time d person
T, Infected patients under treatment at time d person
Ra Recovered population at time d person
Ra Infection rate at time d person/day
ERa Exposure rate at time d person /day
ARy Hospital admission rate at time d person/day
RRMa Recovery rate with medicine at time d person/day
RRy Recovery rate without medicine at time d person /day
WRa Waning rate at time d person /day
a Fractional rate of adjustment of medicine on order discrepancy (rate) 1/week
8 Fractional rate of adjustment of medicine in stock discrepancy (rate) 1/week
¢ Fractional rate of adjustment of medicine backlog discrepancy (rate) 1/week
n Medicine unit require per person units/person
TL Production lead time week
TS Time to ship medicine week
TO Time to fill order week
DC Desired stock coverage week
ss Safety stock coverage week
Diy Desired inventory of medicine at time w units
DOw Desired order of medicine at time w units
FDy Demand forecast for time period w units/week
MBy Medicine order backlog at time w units
MIy Available inventory or stock at time w units
MO» Medicine quantity on order at time w units
Alw Adjustments for inventory discrepancy at time w units/week
AO. Adjustments for medicine on order discrepancy at time w units/week
ABy Adjustments for medicine backlog discrepancy at time w units/week
DRa Medicine demand rate at time d units/day
ORw Medicine purchase order rate at time w units/week
PRy Order receiving rate at time w units/week
SRa Shipment rate of medicine at time d units/day
MSa Maximum Shipment rate of medicine at time d units/day
Inventory (wveekly basis) Epidemic Outbreak (daily basis)
Figure 1: Schematic of proposed framework.
medicine which in turn are fulfilled from the medicine stock of hospital, “MI”,
subject to its availability. The hospital records its daily demand of medicine
from incoming epidemic patients and leverages it for predicting future demand
for medicine. It is assumed that the hospital sends medicine order to the man-
ufacturer after every “h” days, defined as the length of planning horizons or
decision cycles (= T;/h, where Ty is the final simulation runtime) i.e., the
ordering takes place only at the beginning of each planning horizon. This as-
sumption is justified by frequent changes in order rate incurs a high cost and
causes an increase in dynamics as the order flows to upstream of the supply
chain (Lee et al., 1997). The hospital sends the order to the manufacturer
based upon the forecasted demand for the next planning horizon and inventory
profile (measured as units/h days) and receives shipment after an average pro-
duction lead time. Thus, it is noted that the actual epidemic outbreak and the
forecasting process run on a daily basis, but the production-inventory model
runs on the time-length of planning horizon basis (i.e. variables are updated
after every “h” days). The simulation process of this model is shown in Fig. 2.
2.2 Integrated Inventory Management and Disease
Diffusion Framework
In this section, each component of integrated framework is demonstrated.
2.2.1 Hospital Disease Diffusion Component
To depict the real unknown epidemic at hospital, a variant of popular compart-
mental treatment model (Brauer, 2008) viz. Susceptible-Exposed-Infected-
Treatment-Recovered-Susceptible (SEITRS) is taken here (see Fig. 3). In
a manner similar to the conventional SD epidemic models (Sterman, 2000;
Brauer, 2008), this model also divides the total population into five com-
partments namely, Susceptible (S), Exposed (E), Infected patients seeking
treatment (I), infected patients under Treatment (T), and Recovered (R).
Fix number of
decision cycles Tt/h,
initialize integrated
model & simulate
for t=0
Calibrate past period
estimate parameters
Using forecasting
model predict next
planning horizon
cumulative demand
(Simulate integrated supply
chain & disease model;
adjust inventory & other
variables after shipping
medicine to hospital
Stop & output
the results
Figure 2: Flow chart diagram of proposed framework.
teste!
This model is an extension of the model proposed by Paul & Venkateswaran
(2015), where the earlier Infected (I) compartment is split into two compart-
ments viz. Infected (I) and Treatment (T) to capture the medicine supply
delay explicitly. The assumptions of our model are as follows:
1. The hospital was in stable condition before the introduction of the epi-
demic.
2. Each patient requires “n” units of drugs to recover.
3. The total population remains fixed during the course of the epidemic.
4. The mixture of people within each compartment is homogeneous.
wing Rate
faning Time Shipment Ra ae
™ ing SR) eso oe Re
la _@ = L
Pr} DY, asa WY Teatne ee
[is hrm Rate Lt | eposire Rate mission Rate to L__(1)__J Recovery Rate with L_(R)
IR) @ (ER) ee any Medicine (RRM)
ft y Ar) 4
{ SF cubation Time to Recover Time to Recover with
\ Period (P without Medicine Medicine (TRM
Infetvty 1) ee R) =
il Infectivity of
0 reatment Class Recovery Rate
Popa Ge. “TentaeetCiew without Medicine
RR)
Figure 3: Stock flow diagram of disease diffusion component.
A susceptible (S') individual may get exposed to the disease if he comes in
contact with an infected (J) or treatment (I) class individual (see Eq. (1)).
It is assumed that for the under treatment class (T'), the infectivity of an
individual is k%. The infection rate of S is governed by the
rameter 7, which is defined as the product of contact rate (A, the number of
people who interact per time period) and infection probability (p, the chance
of getting the infection from contacting an infected person). Thus y = x p.
The exposed individuals remain asymptomatic for an average incubation time
period P (see Eq. (2)). As soon as the patient develops a symptom or falls
sick, he is shifted from the “E” to the “J” compartment (see Eq. (3)). The
admission rate (AR) to treatment class (“ZT”) is governed by the shipment
rate of medicine from the inventory management model and number of pa-
tients waiting for treatment, ie, ARa = min(SRy,2Ra+Ia—RRa)/n. It
is assumed that when medicines are not available, then the patients recover
from the disease after an average time of natural recovery (T'R) (ie., natural
burnout). Patients under treatment (T’) are assumed to recover after an aver-
age recovery time (IRM) (see Eq. (4)). To make the treatment beneficial, it
umed that TR > TRM. After recovering from the disease, an individual
loses immunity (this loss could be due to the mutation of the disease virus)
and again becomes susceptible to the disease after an average time period
ctivity pa-
is
of “TW” (see Eq. (5)). It is noted that hospital component variables are
updated every day.
Ra _ yx Ua+k x Ta) x Sa
Sa = Sata W (1)
x Uda +k x Ta) x §,
Bay, = By t ECAH RX TO * Sa _ yp @)
ew.
Eka
lua = lat Be _ min(SRa,BRa+la- RRa) _ Iy/TR (3)
n —“s—
RRa
ae min(SRy,ERa+la—RRa) Ta
Tai = Tat rn cn 717i (4)
_ Ta Ta Ra
Ron = Rat pea} rR” Tw (5)
2.2.2 Inventory Management Component
A generic production-distribution system, similar to those discussed in (For-
rester, 1961; Sterman, 2000; Venkateswaran & Son, 2007) is taken here for
depicting the inventory management of hospital (see Fig. 4). Although the
inventory management component, in contrary to SEITRS disease diffusion
component, runs with a time step of “h” days (ie., inventory management
model variables are updated after every planning horizon) but there are few
variables such as, medicine inventory (MJ), backlog (MZB) etc. which in-
teracts with actual disease model, are updated on both daily and “h” day
(planning horizon) basis.
Ordering policy and demand forecasting
The ordering strategy is the key decision variable to control on hand in-
ventory. In the case of demand uncertainty, ordering strategy becomes highly
dependent on the accuracy of forecasting method. Production-inventory mod-
els studied in past literature (Sterman, 2000; Venkateswaran & Son, 2007),
have considered conventional exponential smoothing method for forecasting
demand which is designed for not to respond immediately to a sudden change
of demands. However, in the case of an epidemic, the inventory management
model is expected to react proactively to a small change in demands to reduce
the further mismatch between demand and supply. Fig. 5, demonstrates the
effect of demand fulfillment of only one planning horizon (in e.g. 2Ӣ planning
horizon is taken considering h=7 days) on the dynamics of the entire epidemic
span.
Since the actual disease outbreak information is unknown, we chose the
popular Susceptible-Infected-Recovered (SIR) disease diffusion model (Ster-
man, 2000) arbitrarily from many other compartmental models as the demand
forecasting tool and compared its performances with naive forecasting model,
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1 BS Quarter
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Time (Day)
Figure 5: Epidemic dynamics when medicine supply between 7“"-14"" day
is made full, half and quarter of the actual demand respectively.
defined as the next period’s forecasted demand is current period’s demand,
for an unfolding epidemic. We have used superscript “F’” to distinguish the
disease forecasting model variables from the actual outbreak model (see Sec-
tion 2.2.1) variables. We elucidate the SIR forecasting process below.
The governing equations of the SIR model in discrete time are
Shu = Si—a" x Sp x IT /NP (6)
ge
IRE
F iF F EF
PF] op YX SG XT Ij
linn = La + —\F > FRMF (7)
RE, = rey (8)
aed TRMF
The demand of medicine depends on the patient’s inflow rate to infected
compartment (i.e, ERy in hospital model; see Eq. (2)) and medicine vials
required per person (n), as shown in Eq. (9). It is noted that this equa-
tion serves as a linking constraint from hospital disease model to inventory
management /supply chain model.
DRa=ERaxn (9)
In order to estimate the disease forecasting (SIR) model parameters, the
demand rate, DRY (= IR} xn, where IR} is given by Eq. (6)), is calibrated
10
with previous planning horizon actual demand data (DRa, i.e. the demand
signal sent from the hosp ee Eq. (9)) using the least square regression
method. That is, in i!" decision cycle Vi = 1,2,..., Ty/h, the objective is to
minimize the root mean squared error (RMSE) between the forecasted model
demand data and the original demand signal data of (i—1)"" planning horizon,
sent from the hospital as shown in Eq. (10), where, 9, and L are the set of
all parameters to be optimized, and input parameter ranges respectively.
yy (DRE — DR)?
RMSE; = min
EL h (10)
After estimating the parameters from Eq. (10) in i” planning horizon, fore-
casting model (i.e., SIR model, see Eq. (6) & (7)) is run with these new set
of parameters to forecast the demand for i” planning horizon, denoted by
F'D;, as shown in Eq. (11). In Eq. (11), we accumulate forecasted demand
rate because our inventory management model runs on planning horizon basis
while forecasting model runs on a daily basis.
(é+1)h-1 .
FD:= > DRE (11)
d=ih
Model description
The functionality of inventory management model is described below. In
the following discussion, it should be noted that the variables which updated
every day, are denoted with subscript “d” and those updated every “h” days,
are denoted with subscript “w”.
In inventory management model, cumulative forecasted demand is calcu-
lated for next planning horizon as per Eq. (11). The hospital maintains a daily
order backlog for medicine (7 Ba), as shown in Eq. (12), where the inflow
and outflow are modelled as a coflow of infected class (I). Here, the coflows
keep track of the amount of medicine in backlog based upon the number of
people waiting in compartment J. Since, the inventory model runs with a time
step of “h” days, so the hospital also keeps a record of the medicine backlog
of every “h” days, denoted as MB,,, calculated as the sum of M By’s of last
planning horizon (see Eq. (13)). The order quantity, modelled as a general
replenishment-rule depends on the forecasted demand F'D,,, the discrepancy
between desired and actual orders in pipeline (AO,,), the discrepancy between
desired and actual stock or inventory on hand (AJ,,) and the discrepancy be-
tween actual and normal medicine order backlog (AB,,) (see Eq. (14)). The
discrepancy in orders in the pipeline, in stock, and in backlog are adjusted
using the tuning parameters a (= 1/T\yo, the reciprocal of time to adjust
MOw), 8 (=1/Tr, the reciprocal of time to adjust MI), and ¢ (= 1/Tup,
the reciprocal of time to adjust MW B,,), respectively. The desired orders in the
pipeline, as per Little’s Law (Little, 1961), is computed as a product of FD,
1
with the production lead time TL (see Eq. (15)). The desired stock is also
governed by the F'D,, and the desired coverage level DC and safety stock S'S
(see Eq. (16)). Equations (18), (20), and (21)) are the balance equations for
orders in the pipeline (1/O,,), daily medicine inventory (MJ,) and “h” days
medicine inventory (MJ,,) respectively. The production order receiving rate
(PRw») is modelled as a first order material delay to capture the mixing of
products and variabilities in production times. The delivery or sales rate SRa
of medicines, Equations (22-24), govern the actual supply of medicines from
the stock to the hospital, subject to actual availability.
MBay: = MBg+DRy—(ARa+ RR) xn (12)
ef -
OAa OFy
wh-1
MBy = >> MBa (13)
d=(w—1)h
OR, = max {0,FD,+a(DO, —MO,)
AOw
+ B(DIw — MIy) + C(MBw — NBw) (14)
hase as a
Alw ABy
DOw = TLxFDy (15)
DI, = (SS+DC)x FDy (16)
NBy = FD,xTO (17)
MOwi1 = MO,+OR,—PRy (18)
MOw
PRy = 1
Ry = ae (19)
Mla = MIg—SRa (20)
Ml, = MIg+PRy (21)
SRy = min{MSy,DSu} (22)
MBa
DSi = FG (23)
ve. Mla
MSi = Fg (24)
2.3. Model Verification
Before going into the simulation experimentation, verification of our proposed
integrated model is examined in two ways. Firstly, the dimensional consis-
tency is verified and secondly, model robustness at extreme conditions are
tested (Barlas, 1996). For verific: ‘8S, initially exposed and all
infected population (ie. I & T) of disease model was set to zero and con-
12
firmed that no epidemic occurs and all stock values remain unchanged while
rate variables become zero. Further, the stability of inventory management
module is confirmed against constant and step input of demand.
3 Experiments and Analyses
In this section, we study and compare the performances of naive and SIR
forecasting models in terms of three performance measures namely, infected
peak (Ipeak; see Eq. (25)), final epidemic size (Roo; see Eq. (26)), and
inventory leftover (MIjes,).
IPeak = omy, +O) (25)
ty
Ro = )\ (RRM) +RRi) (26)
i=0
The proposed framework is simulated for three different settings of epi-
demic severity, measured in terms of infectivity parameter y.
3.1 Simulation Settings and Optimization
Simulation experiment of our difference equation based integrated model is
carried out on Microsoft Excel (2010). Euler integration method with a time
step of 1 was used for discrete time simulation and approximating the model
variable trajectories. The total simulation runtime for the integrated model
was set to 70 days (Ty). For stock management component, we chose L = 1
week (ic. L = h = 7 days), TO = 1 week, TS = 1 week, ¢ = 1/week,
a = 1/week, 38 = 1/week, SS = 0.1 week (chosen arbitrarily but lesser system
inventory preferred). For the SEITRS (epidemic) component of integrated
model, we have taken k = 0.5 (chosen arbitrarily), n = 1 unit/person, TW =
365 days (Paul & Venkateswaran, 2015), P = 2 days (CDC, 2015), TR =
10 days, TRM = 3.8 days, Sg = 2211 persons, Ey = 15 per:
person, 7y = 78 persons, and Ro = 0 person. We chose three
infectivity parameter (y) as 0.572 (low infectivity), 0.596 (base infectivity),
and 0.62 (high infectivity). For MOw, MIw, Mla, MBu, MBa, and FDy
initial values are chosen as DO,, DI, DI, F Dw, Ip, and 100 units/week
respectively. Initial FD,, is kept lesser than first planning horizon actual
demand so that the system does not possess excess inventory and the problem
becomes interestingly dependent upon the forecasting technique.
Calibration of the models using the least square regression (see Section 2.2.2)
was carried out using the Generalized Reduced Gradient (GRG) (Fylstra et al.,
1998), a gradient-based method, in Solver® optimisation toolbox of Microsoft
Excel (2010). This simulation-based optimisation works as follows. The op-
timisation solver (i.e. GRG) chooses a set of parameter values and passes it
13
on to the simulation (SD) model. The model runs the simulation, evaluates
the objective function (ie. RMSE), and returns the same to the optimisa-
tion solver. The solver, then intelligently chooses the next set of parameters
for evaluation by the simulation model. This iterative scheme continues until
a predefined stopping criterion is met or the current solution meets a “slow
progress” test (Fylstra et al., 1998), and the best result obtained is reported.
3.2 Comparison of Disease Forecasting Models
The input parameter ranges for calibration of SIR forecasting model with
actual disease outbreak demand data are taken as $f € [2000, 10000], 7” €
(0, 1], TRM* € [2.5, 10] and J¢’ is always set to 79 persons (ie. Io + Th of
actual outbreak model).
Anticipated medicine demands from the naive and SIR forecasting models
are presented in Table 2. Fig. 6 and Fig. 8 shows the comparison of actual
demand data and forecasted demand data in right axis while corresponding
order rate data are plotted on the left axis, for naive and SIR forecasting
models respectively. The dynamics of the corresponding supply chain of naive
and SIR models are also shown in Fig. 7 and Fig. 9 respectively, in terms of
medicine backlog (MBq), inventory (MJ) and shipment rate (SRy) on a
daily basis. Finally, the overall comparison of demand forecasting models is
summarised in Table 3.
Table 2: Predicted demands of forecasting models
Low Infectivity (7 = 0.572) Base Infectivity (1 = 0.596) High Infectivity (7 = 0.62)
Time (Week) _ Naive SIR Naive SIR Naive SIR
1 100 100 100 100 100
2 124.897 218.795 131.051 238.700 259.877
3 201.448 268.333 2: 129 328.504 394.290
4 335.240 514.030 419.983 672.289 512.190 844.009
5 596.551 375.369 700.309 390.138 764.730 359.713,
6 528.830 196.567 468.568 127.053 381.071 82.599
7 201.118 65.926 134.276 41.215 86.403 20.660
8 38.500 19.004 23.001 10.356 14.496 5.530
9 9.134 6.407 5.134 3.250 3.189 1.664
10 2.384 2.257 1.265 1.067 0.768 0.526
Fig. 6 shows that for naive demand forecasting model, estimated epidemic
peak demand rate is exactly delayed by one week (i.e. length of planning hori-
zon) of actual epidemic peak demand rate which further worsen the epidemic.
In Fig. 8, it is seen that the forecasting SIR model has overestimated the epi-
demic peak demand rate by 4.38%, 14.086%, and 28.12% for low, base, and
high infectivity (y) parameters respectively. Moreover, the forecasted peak
demand occurred at exactly the same time of actual epidemic peak demand
which is good from epidemic planners’ perspective. However, the realization
of the epidemic peak is just one planning horizon before actual which results
14
OrderRate (Unis/W eek)
in high order rates and eventually increases the medicine inventory even after
the eradication of epidemic. This situation is reflected in Fig. 9, where a sharp
increase in inventory is seen after day 35 or week 5, i.e. a week after the epi-
demic peak demand (see Fig. 8; peak occurs at week 4) where the maximum
order is placed and transformed into inventory after a week later (since the
lead time TL=1 week). For the same reason, we see sharp growth in inventory
of Fig. 7.
‘on
0
zoe
ste Ms ean
a >
wo 8 Pw seo
see. g Fi ee
nt F :
z § im 2x0
20 5
seo 3 rE 220
a os
100 ts
Popa be fas Paid ed ts
Tie Week) The Weck)
(a) Low Infectivity (7 = 0.572) (b) Base Infectivity (y = 0.596)
OrderRate Unit/W eck)
8
Tine eek)
(c) High Infectivity (y = 0.62)
Figure 6: Comparison of weekly forecasted demand rate (£'D,,), actual
demand rate (DR,,), and order rate (OR,,) for naive forecasting method.
From Table 3, we see that SIR model outperforms naive method in all
three settings of y, in terms of Ipeax, Roo, and MIjef, reduction. For e.g.,
15
(spun) aey pue weg
ventory Units)
gs 8 8
8
(se a/stun) ae yque wtys'(srun) Boy2e9
venry Unis)
Tine Day) Tine Day)
(a) Low Infe
ivity (y = 0.572)
Infectivity (7 = 0.596)
&
1
4000. [roe enenenencnnne aaa
&
g
we
2010 :
4 00
5 a
Bom “4
a a &
1080 e
e
g
a 2 oo @
Tine Day)
(c) High Infectivity (y = 0.62)
Figure 7: Comparison of daily backlog (1/7 Bz), shipment rate (S'Ra), and
inventory (M/Jq) dynamics of naive forecasting method.
16
(Kea/seun) aexnua wdtys‘(srun) BayDe8
OrderRate UniisAl eek)
seo vw.
409. ad
an 2 >
Fl E uw. soa
a =
300 2 ee. 400
e 3
oo
= : 0
mE 2 a
: 3 20
A 5 we
we $
- uo
ar ay ea
Tine Week) Tine Week)
(a) Low Infe
ivity (y = 0.572)
Infectivity (7 = 0.596)
g
8
&
(428 m/stu) ey pue we g
OnerRate (UnitsM eck)
¥
Ea
Tine Week)
(c) High Infectivity (y = 0.62)
Figure 8: Comparison of weekly forecasted demand rate (F'Dw), actual
demand rate (DR,,), and order rate (OR,,) for SIR forecasting method.
17
(499 W/STURH ey pue e.g
ventory Units)
(ke aysun) aeyaua weys (stun) Bay2eg
venbry Unis)
8
=
16007 [-— —snpnentnae]
he
3
Tine Day)
(a) Low Infectivity (+ = 0.572)
ventory Units)
» ®
Tine Day)
o @
Infectivity (7 = 0.596)
&
bi
¥
&
Tine Day)
(c) High Infectivity (y = 0.62)
Figure 9: Comparison of daily backlog (1/7 Bz), shipment rate (S'Ra), and
(Keq/atup) aeynua utys‘(stu Sax2°9
inventory (MJq) dynamics of SIR forecasting method.
18
Dem and Rate Unis/W eek)
in case of low 7, 32.02%, 8.67%, and 63.18% reductions in Ipeax, Roo, and
Mie 4 are possible with SIR model than naive forecasting model.
Table 3: Comparison of forecasting techniques
Low Infectivity (7) Base Infectivity (y) High Infectivity (7)
Naive
SIR Naive SIR Naive SIR
Trcak 503.593
Ry 2117.224
Mj: 2786.161
383.139 656.316 467.870 712.648 525.034
1933.683 2186.260 2023.300 2226.859 — 2079.768
1025.977 3206.021 1489.188 3960.500 1916.775
Den and Rate Unis/ eek)
&
Tints Supply
[ SIR
yi
Time (Week)
(a) Low Infectivity (7 = 0.572)
Demand Rate (Unis/H eck)
Tafinte SuppIy
IR
Time (Week)
(c) High Infectivity (y = 0.62)
é
Time (Week)
(b) Base Infectivity (y = 0.596)
Figure 10: Comparison of weekly epidemic demand rate (DR,,) for fore-
casting policies under consideration and infinite supply.
In Fig. 10, actual demand rates of two forecasting models under consider-
ation are plotted against the demand rate when infinite/unlimited medicine
19
inventory are available (i.e. no stock out) to reflect the scope of further im-
provement. In Fig. 10, the demand rate forecasted by SIR model can be
reduced by 48.508 %, 51.797%, and 52.467% for low, base, and high infec-
tivity respectively while the same can be done by 57.493%, 59.439%, and
59.054% respectively for naive forecasting method.
4 Conclusions and Future Work
A framework for inventory management in response to the post-epidemic dec-
laration is developed, where the demand forec:
diffusion model and updated dynamically based on past epidemic behaviour.
An SEITRS epidemic model is integrated with the inventory management
model for simulation purpose of a real epidemic diffusion. The performance
of two different forecasting models (SIR and naive) are compared in anticipa-
tion of the unknown epidemic. It is found that SIR model outperforms naive
method in terms of reducing epidemic impact (i.e., infected peaks and final
epidemic sizes) and is also proven to be relatively economic in terms of inven-
tory leftover. However, there is significant scope for improving the proposed
model as shown in Fig. 10.
Further, analyses are currently being carried out in the following direc-
tions: comparing and finding the appropriateness of other variants of disease
diffusion models as a forecasting method; quantifying the effect of planning
horizon length (h), and inventory management parameters (for e.g., a, 3) on
disease dynamics and system co:
one parameter (h) for both reviewing demand and the number of days to accu-
mulate forecasted demand. Future research can explore what if there are two
different parameters say, a fixed demand review period (r) and another for
the number of periods, to accumulate forecasted demands (h) which will help
to cope with managing the excess demand during the peak of an epidemic.
. In our current model, we have considered
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