Zahra Azadi, Harsha Gangammnavar, Sandra Eksioglu
Department of Industrial Engineering
Clemson University
Introduction
Solution Approach
•
•
•
•
• Method: Stochastic Benders Algorithm.
• Dataset: The data used is obtained from a clinic in Iran.
• Platform: The single-cut version of the L-shaped method was implemented in C programming language on a 64 bit Intel Core i5 [email protected]
GHz with 8 GB RAM.
• Setup: The results present optimal solutions (optimality gap < 0.001) to the problem.
Vaccines are distributed in single- and multi-dose vials.
Doses of opened vials should be utilized within their safe use time.
Usage time of a dose is short if not refrigerated.
Single dose vials have zero wastage.
Research Motivations
• The Global Alliance for Vaccines and Immunizations (GAVI) requests
that countries take measures to decrease vaccine wastage rates.
• Replenishment decisions in remote locations where no refrigeration
exists are challenging.
• Patient arrival is unpredictable. Thus, using multi-dose vials may
lead to wastage during low demand periods.
• Highest vaccine wastage occurs at service delivery level (27% for
DPT and 61% for BCG at outreach session site)
• Purchasing and inventory costs for single-dose vials are high.
Results
Mean arrival rate
Optimization
Obj. Value
192.69
443.04
544.94
766.17
796.44
1
5
10
15
20
Table1: Results with varying
mean arrival rate.
Evaluation
Mean Std. dev.
15
0.21
13
0.47
573.37
0.61
779.46
0.71
840.57
00.82
Instance Optimization
P1
P2
P3
P4
P5
P6
P7
P8
CI
[193.07, 193.2]
[452.51, 454.37]
[572.16, 574.58]
[778.06, 780.86]
[838.96, 842.18]
Obj. Value
195.52
224.44
344.4
355.2
359.49
362.17
365.23
370.16
Evaluation
Mean Std. dev.
CI
195.84
0.22
[226.21, 227.2]
226.71
0.25
[195.39, 196.29]
346.61
0.26
[346.008, 347.14]
357.87
0.37
[357.15, 358.6]
362.05
0.4
[361.27, 362.84]
361.07
0.46
[360.22, 362.28]
363.17
0.48
[364.29, 366.18]
372.22
0.49
[371.288,373.17]
Table2: Results for instances
with varying purchase cost.
the.
(b) Wastage
(a) Unserved demand
Research Objectives
• Identify optimal replenishment schedules for vaccine vials in remote
clinics using a stochastic optimization framework.
• The model captures:
 The timing and quantity of orders for different sized vials
 The timing of vial-opening decisions
 Trade-offs between wastage and inventory holding costs
Figure3: Vial portfolio at varying arrival rate.
• Provide analytical support for immunization programs.
Figure4: Effect of varying purchase cost on dose utilization.
Master Problem
Vaccine Vial Replenishment Model
Fixed ordering costs
min
s Nt
s n
 V
Expected value of Wastage and Penalty costs
Conclusions
 (ft z t   c x t )    d s  E {h(u,ˆ )}
tT

Inventory costs
Purchase costs
 V
 V nN
 s (Nt 1)  x t 1  u Nt
 s n 1  u n
x t
n

t
N
 M t zt
 T
,
\ {N, 2 N, ..., TN},
t
 T
.
• Experiments with varying purchase cost illustrate the
achievable immunization level for a given budget (Figure 4).
Sub Problem
Wastage costs
h(u,  )  min
Figure 1: Inventory dynamics in vaccine vial replenishment model.
Demand loss costs
• The computational analysis indicate that the model proposed
is an effective tool to design economic immunization
programs.
 (gy
n (n  )  prn )
nN
n 

 q u
ynm 


n
m n
 V
n

ymn  rn  D n (  )
m  n  1
n

N
n

N.
Non-anticipative Decision
zt
x t
s n
u n
Figure 2: Dose utilization for n = 2 with t = 6.
• Experiments using different demand arrival rates indicate that
replenishment decisions at outreach centers are based on
regional demographics (Figure 3).
,
• Pandemic outbreak evaluation and demand prediction
models is part of our future works.
Contact:
Binary variable equal to 1 when a replenishment is made at time t; 0 otherwise.
Number of vaccine vial size υ ordered at time period t.
Zahra Azadi
Clemson university
Inventory level of vials size υ at time period n.
Number of vaccine vial size υ opened at time period n.
662-6174292
Anticipative Decision
Number of doses obtained from vials opened in period n, and used in
ynm
period m.
rn
Number of unserved patient arrivals at time period n.
Email: [email protected]