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DETERMINING THE OPTIMAL COLLECTION
PERIOD FOR RETURNED PRODUCTS IN A
STOCHASTIC ENVIRONMENT
- Shashank Kapadia
- Dr. Emanuel Melachrinoudis
2

• Introduction
– What is Supply Chain?
– Components of Reverse Supply Chain
– Importance and Impact
– Focus of this work
• Problem Definition
• Mathematical Formulation
– Generalized model
– Special Case: Poisson Distribution
– An Illustrative Example
• Conclusion and Future Work
Outline
3

• What is “Supply Chain”?
– The sequence of processes involved in the production and
distribution of a commodity
Introduction
Supply Chain
Forward Supply Chain Reverse Supply Chain
4

Introduction
Supply Chain
5
Components/
Raw Materials
Manufacturers
Wholesalers/
Distributors
Retailers
Customers

Introduction
6
Components/
Raw Materials
Manufacturers
Wholesalers/
Distributors
Retailers
Customers
Supply Chain

• Components of Reverse Supply Chain
Introduction
7
Reverse Supply Chain
Product
Acquisition
Inspection and
Disposition
Reverse Logistics Reconditioning
Distribution and
Sale

• Importance
– Environmental (regulations, consumer pressure etc.)
– Economic (value of used products, cost reduction etc.)
• Impact
– Macro level
• 20% of that is sold is returned
• According to Reverse Logistics Association, the volume of annual returns is
estimated between $150 billion and $200 billion at cost
• ~6% of the Census Bureau’s figure of $3.5 trillion total of US retail
• 21% increase in product returns cost in US electronics consumer and manufacturers
market since 2007, by Accenture in 2011
– Micro level
• Supply chain costs associated with reverse logistics average between 7% - 10% of
costs of goods
• Average manufacturer spends 9% - 15% of total revenue on returns
Importance and Impact
8

Focus of this Work
9
Supply Chain
Forward Supply
Chain
Reverse Supply
Chain
Product
Acquisition
Reverse Logistics
Distribution
Production
Planning
Inventory
Inspection and
Disposition
Reconditioning
Distribution and
resale
Specifically on the
collection of returned
products and the
economic driving force
that can bring direct
gains to the companies
in terms of cost
reduction

Problem Definition
10
Returned
Products
ICP3
ICP2
ICP1
CRC2
CRC1
Figure 1: Reverse logistics structure

Problem Definition
11
ICP CRC
• Objective
– To determine the finite collection time at an ICP before sending it to the CRC
• Prior work
– Although, the work has been done on reverse logistics in past, it is diverse and
heterogeneous. Recently, the dynamic interplay between shipping volume and the
collection period was examined
We propose a generalized model for stochastic product returns where the
rate of returns follows a discrete probability distribution
Figure 2: Sub-problem with one ICP and one CRC

Problem Definition
12
ICP
Inventory Cost Shipping Cost
$-
$5,000.00
$10,000.00
$15,000.00
$20,000.00
$25,000.00
1 2 3 4 5 6 7 8 9 10
Annualcost($)
Collection period (t days)
Annual Costs at ICP
Inventory Cost Shipping Cost
$18,000.00
$20,000.00
$22,000.00
$24,000.00
1 2 3 4 5 6 7 8 9 10
Total Annual Cost at ICP
Total Annual cost

Mathematical
Formulation
Indices
𝑖 Index for time periods in days
Decision Variables
𝑇 Length of the collection period in days
Model Parameters
𝑏
Daily inventory cost per unit, including the penalty of holding a
unit one more day
𝑤 Annual working days
𝑌𝑖
Discrete random variable representing the number of returned
products on the 𝑖 𝑡ℎ
day from all the customers; 𝑌𝑖 are assumed to
be independent and identically distributed random variables
according to a discrete mass function 𝑓 𝑦 = Pr 𝑌𝑖 = 𝑦
𝐹 Standard freight rate
𝛼𝑙
Freight discount rate depending on shipment volume from the
ICP to the CRC, 𝑙 = 1, … , 𝑚 and 𝛼0 = 1
𝑃𝑙
Preselected shipment volume breakpoints,𝑙 = 1, … , 𝑚 as shown
in Figure 3
The volume of accumulated returned
products over the period of 𝑡 days
as 𝑍(𝑡) = 𝑖=1
𝑡
𝑌𝑖.
The objective is to determine the
collection period 𝑇 for returned
products at ICP that minimizes the total
annual cost which is the sum of annual
inventory cost and annual shipping
cost.
Minimize: Total Annual Cost
(Annual Shipping Cost + Annual
Inventory Cost)
Assumptions:
1. Sufficient capacity
2. No transportation cost from
customers to ICP
3. Returned products are of same
kind
13

Mathematical
Formulation
Inventory Cost
The cost associated with storing the
returned products at the ICP.
The expected annual inventory
cost 𝔼 𝐼𝐶𝑌 𝑡 can be derived as
𝐼𝐶 1 = 𝑏𝑌1
𝐼𝐶 2 = 𝑏𝑌1 + 𝑏 𝑌1 + 𝑌2
= 𝑏 2𝑌1 + 𝑌2
𝐼𝐶 𝑡 = 𝑏 𝑡𝑌1 + 𝑡 − 1 𝑌2 + ⋯ + 𝑌𝑡
𝔼 𝐼𝐶 𝑡 =
𝑏𝑡 𝑡 + 1
2 𝑦
𝑦𝑓(𝑦)
Therefore, accounting for
𝑤
𝑡
cycles in a
year, we have
𝔼 𝑰𝑪𝒀 𝒕 =
𝒃𝒘 𝒕 + 𝟏
𝟐 𝒚
𝒚𝒇(𝒚)
14
$-
$2,000.00
$4,000.00
$6,000.00
$8,000.00
$10,000.00
$12,000.00
1 2 3 4 5 6 7 8 9 10
Annualcost($)
Expected Annual Inventory Cost at ICP
Inventory Cost

Mathematical
Formulation
Shipping Cost
The shipping cost is a function of accumulated
returned products over the collection period of 𝑡
days,𝑍(𝑡), and the freight discount rate.
The shipping cost can be expressed as:
𝑆𝐶 𝑡
=
𝐹𝛼𝑙 𝑍(𝑡)
𝐹𝛼 𝑚 𝑍(𝑡)
𝑓𝑜𝑟 𝑃𝑙−1 ≤ 𝑍 𝑡 < 𝑃𝑙, 𝑙 = 1, … , 𝑚
𝑓𝑜𝑟 𝑃𝑚 ≤ 𝑍 𝑡
By defining another breakpoint at infinity,
i.e.𝑃 𝑚+1 = ∞, we can express above equation as:
𝑆𝐶 𝑡 = 𝐹𝛼𝑙 𝑍 𝑡 , 𝑓𝑜𝑟 𝑃𝑙−1 ≤ 𝑍 𝑡 < 𝑃𝑙, 𝑙 =
1, … , 𝑚 + 1, and its expected value can be
expressed as
𝔼 𝑆𝐶 𝑡 = 𝐹
𝑙=1
𝑚+1
𝛼𝑙−1
𝑘=𝑃 𝑙−1
𝑃 𝑙−1
𝑘𝑓𝑍 𝑡
𝑘
Therefore, accounting for
𝑤
𝑡
cycles in a year, we
have
𝔼 𝑺𝑪𝒀 𝒕
=
𝑭𝒘
𝒕
𝒍=𝟏
𝒎+𝟏
𝜶𝒍−𝟏
𝒌=𝑷𝒍−𝟏
𝑷𝒍−𝟏
𝒌𝒇 𝒁 𝒕
𝒌
15
$9,000.00
$11,000.00
$13,000.00
$15,000.00
$17,000.00
$19,000.00
$21,000.00
1 2 3 4 5 6 7 8 9 10
Annualcost($)
Expected Annual Shipping Cost at ICP
Shipping Cost
𝐹𝛼 𝑚𝐹 𝐹𝛼1 𝐹𝛼2 𝐹𝛼3 …
𝑃0 = 0 𝑃1 𝑃2 𝑃3 𝑃𝑚
Figure 3: Preselected shipment volume breakpoints

Mathematical
Formulation
𝔼 𝑆𝐶𝑌 𝑡
=
𝐹𝑤
𝑡
𝑙=1
𝑚+1
𝛼𝑙−1
𝑘=𝑃 𝑙−1
𝑃 𝑙−1
𝑘𝑓𝑍 𝑡
𝑘
Above equation can be simplified
using approximation as:
𝔼 𝑺𝑪 𝒕 ≅ 𝑭𝜶∗
𝒕 𝔼 𝒁 𝒕
where,
𝜶∗
𝒕 =
𝒍=𝟏
𝒎+𝟏
𝜶𝒍−𝟏
𝒌=𝑷𝒍−𝟏
𝑷𝒍−𝟏
𝒇 𝒁 𝒕
𝒌
𝜶∗
𝒕 can be considered as the
effective discount rate.
The approximation was extensively
tested and was found to be quite good.
16
$-
$5,000.00
$10,000.00
$15,000.00
$20,000.00
$25,000.00
1 2 3 4 5 6 7 8 9 10
Annualcost($)
Shipping Cost Approximation
Comparison
Shipping Cost Theoretical(Approx) Shipping Cost Theoretical (Exact)

• Let us now assume that 𝑌𝑖, 𝑖 = 1, … , 𝑡 are independent and identically distributed random
variables following the Poisson distribution with mean 𝜆 = 𝑟. Then 𝑍 𝑡 ~𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝑟𝑡 .
• The expected annual inventory cost:
𝔼 𝑰𝑪𝒀 𝒕 =
𝒃𝒘𝒓 𝒕 + 𝟏
𝟐
• The expected annual shipping cost:
𝔼 𝑺𝑪𝒀 𝒕 ≅ 𝑭𝒘𝒓𝜶∗
𝒕 ≅ 𝑭𝒘𝒓
𝒍=𝟏
𝒎+𝟏
𝜶𝒍−𝟏
𝒌=𝑷 𝒍−𝟏
𝑷 𝒍−𝟏
𝒇 𝒁 𝒕
𝒌
• The total cost which is the sum of inventory cost and the shipping cost can be expressed as
𝑬 𝑻𝑪𝒀 𝒕 ≅
𝒃𝒘𝒓 𝒕 + 𝟏
𝟐
+ 𝑭𝒘𝒓𝜶∗
𝒕
Special Case: Poisson Distribution
17

Special Case:
Poisson
Distribution
Proposition 1
Based on the approximation and
extensive simulation, there is drop
in expected annual total cost
wherever,
𝜶∗
𝒕 − 𝟏 − 𝜶∗
𝒕 >
𝒃
𝟐𝑭
Reduction in the number of
possibilities for optimal collection
period.
∎
18
Collection
Period t (days)
Total Cost α*(t) α*(t-1)-α*(t)
1 $ 22,000.00 1.000
2 $ 22,993.54 1.000 0.000
3 $ 20,011.88 0.801 0.199
4 $ 21,000.00 0.800 0.001
5 $ 21,965.98 0.799 0.001
6 $ 19,296.21 0.618 0.181
7 $ 19,893.86 0.596 0.022
8 $ 19,098.54 0.506 0.090
9 $ 20,000.00 0.500 0.006
10 $ 21,000.00 0.500 0.000
Table 1: Results that illustrate Proposition

An Illustrative
Example
We consider a cluster of
customers where the total daily
returns volume follows a Poisson
distribution with 𝒓 = 𝟖𝟎, the
daily inventory cost 𝒃 = 𝟎. 𝟏, the
unit standard freight rate is 𝑭 = 𝟏
and the annual working days
are 𝒘 = 𝟐𝟓𝟎. All the customers
return products to a single
designated ICP and subsequently
the products are collected during a
period 𝑇 before they are shipped
to a single designated CRC. The
shipment volume breakpoints
are 𝑷 𝟏= 𝟐𝟎𝟎, 𝑷 𝟐 = 𝟒𝟓𝟎
and 𝑷 𝟑= 𝟔𝟓𝟎, beyond which the
freight discount rate decreases
to 𝜶 𝟏= 𝟎. 𝟖, 𝜶 𝟐 = 𝟎. 𝟔 and 𝜶 𝟑=
𝟎. 𝟓, respectively.
19
Time Period Inventory Cost Shipping Cost Total Cost
1 $ 2,000.00 $ 20,000.00 $ 22,000.00
2 $ 3,000.00 $ 19,993.54 $ 22,993.54
3 $ 4,000.00 $ 16,011.88 $ 20,011.88
4 $ 5,000.00 $ 16,000.00 $ 21,000.00
5 $ 6,000.00 $ 15,965.98 $ 21,965.98
6 $ 7,000.00 $ 12,296.21 $ 19,296.21
7 $ 8,000.00 $ 11,893.86 $ 19,893.86
8 $ 9,000.00 $ 10,098.54 $ 19,098.54
9 $ 10,000.00 $ 10,000.00 $ 20,000.00
10 $ 11,000.00 $ 10,000.00 $ 21,000.00
$18,000.00
$19,000.00
$20,000.00
$21,000.00
$22,000.00
$23,000.00
$24,000.00
1 2 3 4 5 6 7 8 9 10
Annualcost($)
Expected Annual Total Cost
Table 2: Expected annual costs

Conclusion and
Future Work
One of the first paper to tackle the reverse
logistics network problem involving random
returned products
The proposed model can aid in coping up
with a new challenge of uncertainty in
product returns
From the theoretical standpoint, the proposed
model was proven to be efficient in
determining a functional relationship between
the expected total inventory cost and the
shipping cost, and with the returns collection
period.
Future work
• The model can be extended to reflect the
continuous time collection period and
freight rates fluctuations
• Future research should be able to tackle
different types of product returns with
multiple ICPs and CRCs
• Model could be validated for large-sized
real-world problems with actual data
20
Time Period
Theoretical Total
Cost
Simulated Total
Cost
Error (%)
1 $ 22,000.00 $ 21,967.99 0.146%
2 $ 22,993.54 $ 23,040.60 -0.204%
3 $ 20,011.88 $ 20,022.07 -0.051%
4 $ 21,000.00 $ 21,007.55 -0.036%
5 $ 21,965.98 $ 22,009.25 -0.197%
6 $ 19,296.21 $ 19,295.07 0.006%
7 $ 19,893.86 $ 19,939.87 -0.231%
8 $ 19,098.54 $ 19,139.38 -0.213%
9 $ 20,000.00 $ 20,052.77 -0.263%
10 $ 21,000.00 $ 21,058.91 -0.280%

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