NEDSI-2015_Final

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 ReconditioningDistribution 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

InventoryInspection 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

Annual

cost

($)

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

FormulationIndices

𝑖 Index for time periods in days

Decision Variables

𝑇 Length of the collection period in days

Model Parameters

𝑏Daily 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 𝑖𝑡ℎ 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

𝛼𝑙Freight discount rate depending on shipment volume from the

ICP to the CRC, 𝑙 = 1,… ,𝑚 and 𝛼0 = 1

𝑃𝑙Preselected 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=1𝑡 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

FormulationInventory 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 = 𝑏𝑌1𝐼𝐶 2 = 𝑏𝑌1 + 𝑏 𝑌1 + 𝑌2= 𝑏 2𝑌1 + 𝑌2

𝐼𝐶 𝑡 = 𝑏 𝑡𝑌1 + 𝑡 − 1 𝑌2 +⋯+ 𝑌𝑡

𝔼 𝐼𝐶 𝑡 =𝑏𝑡 𝑡 + 1

2

𝑦𝑦𝑓(𝑦)

Therefore, accounting for𝑤

𝑡cycles in a

year, we have

𝔼 𝑰𝑪𝒀 𝒕 =𝒃𝒘 𝒕 + 𝟏

𝟐

𝒚𝒚𝒇(𝒚)

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$-

$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

Annual

cost

($)


Expected Annual Inventory Cost at ICP

Inventory Cost

Mathematical

FormulationShipping Cost

The shipping cost is a function of accumulated

returned products over the collection period of 𝑡days,𝑍(𝑡), and the freight discount rate.

The shipping cost can be expressed as:

𝑆𝐶 𝑡

= 𝐹𝛼𝑙𝑍(𝑡)𝐹𝛼𝑚𝑍(𝑡)

𝑓𝑜𝑟 𝑃𝑙−1 ≤ 𝑍 𝑡 < 𝑃𝑙 , 𝑙 = 1,… ,𝑚

𝑓𝑜𝑟 𝑃𝑚 ≤ 𝑍 𝑡

By defining another breakpoint at infinity,

i.e.𝑃𝑚+1 = ∞, we can express above equation as:

𝑆𝐶 𝑡 = 𝐹𝛼𝑙𝑍 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=𝑃𝑙−1

𝑃𝑙−1

𝑘𝑓𝑍 𝑡𝑘

Therefore, accounting for𝑤

𝑡cycles in a year, we

have

𝔼 𝑺𝑪𝒀 𝒕

=𝑭𝒘

𝒕

𝒍=𝟏

𝒎+𝟏

𝜶𝒍−𝟏

𝒌=𝑷𝒍−𝟏

𝑷𝒍−𝟏

𝒌𝒇𝒁 𝒕𝒌

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 10A

nnual

cost

($)


Expected Annual Shipping Cost at ICP

Shipping Cost

𝐹𝛼𝑚𝐹 𝐹𝛼1 𝐹𝛼2 𝐹𝛼3 …

𝑃0 = 0 𝑃1 𝑃2 𝑃3 𝑃𝑚

Figure 3: Preselected shipment volume breakpoints

Mathematical

Formulation𝔼 𝑆𝐶𝑌 𝑡

=𝐹𝑤

𝑡

𝑙=1

𝑚+1

𝛼𝑙−1

𝑘=𝑃𝑙−1

𝑃𝑙−1

𝑘𝑓𝑍 𝑡𝑘

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


𝑷𝒍−𝟏

𝒇𝒁 𝒕𝒌

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

Annual

cost

($)


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


𝑷𝒍−𝟏

𝒇𝒁 𝒕𝒌

• 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.

∎

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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

ExampleWe 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 = 𝟏and 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 𝑷𝟏= 𝟐𝟎𝟎, 𝑷𝟐 = 𝟒𝟓𝟎and 𝑷𝟑= 𝟔𝟓𝟎, beyond which the

freight discount rate decreases

to 𝜶𝟏= 𝟎. 𝟖, 𝜶𝟐 = 𝟎. 𝟔 and 𝜶𝟑=𝟎. 𝟓, respectively.

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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

Annual

cost

($)


Expected Annual Total Cost

Table 2: Expected annual costs

Conclusion and

Future WorkOne 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 PeriodTheoretical Total

Cost

Simulated Total

CostError (%)

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%

THANK YOU

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NEDSI-2015_Final

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