Applied Stochastic Processes

1
A Supplier Selection-Order Allocation Problem
with Stochastic Demands
Xiaobo Zhao, Lei Zhao and Jianhua
Jiang Department of Industrial Engineering Tsinghu
a University China
2
SCOR Model
Supply-Chain Council in 1997 developed its first
version of a business process reference model
3
Order Supplier Activity
SUPPLIER

CONSUMER
SUPPLIER
4
Outline

• Introduction
• Model formulation
• A solution approach
• Computational experiment
• Conclusions

5
Introduction
Purchased parts, components, and supplies can
cost up to 40 60 of an end products sales
value (Ballou 2004).
Relatively small cost reduction in the
acquisition activities can have a significant
impact on profits.
Two issues 1. Supplier selection among a set of
candidates 2. Order allocation to the selected
supplier(s)
6
Introduction

• Supplier selection
• Analytic hierarchy process (AHP)
• Multiple criteria of many levels are determined
  with different
• weights assigned to the criteria. Select the
  supplier with the
• highest score. (e.g. Akarte et al. 2001, Liu and
  Hai 2005)
• Total cost of ownership (TCO)
• Costs associated with service, quality,
  delivery, administration, communication,
  failure, maintenance.
• (e.g. Degraeve et al. 2000, 2004, 2005,
  Ghodsypour and Obrien 2001, Ellram and Siferd
  1993)

7
Introduction

• Order allocation
• Ganeshan et al. (1999) two suppliers, one
  serves as a regular
• supplier with normal price, the other one serves
  as s candidate
• with discount price.
• Kawtummachai and van Hop (2005) multiple
  suppliers for a
• single period deterministic model.
• Menon and Schrage (2002) Paper industry for
  single period
• deterministic model as an integer programming.

8
Introduction

• Supplier disruptions
• In recent years, several supply disruptions
  caused by force majeure such as natural disasters
  result in serious impact on the business
  operations of some companies.
• Fire at Royal Philips Electronics plant in
  Albuquerque, New Mexico, USA, in 2000, affected
  Ericsson and Nokia.
• Fire at Toyotas brake suppliers plant in Japan
  in 1997.
• Earthquake in Taiwan, China, in 1999 affected
  Dell, HP, Apple, etc.
• Supplier diversification
• 1) Reduce or avoid losses in case of supply
  disruptions.
• 2) Occupy the dominant position for purchasing
  prices in
• negotiation.

9
Introduction

• Multiple suppliers
• A rule of thumb
• Each supplier is allocated with an order not
  exceeding a specified percentage (?) of the total
  replenishing quantity.
• (Ghodsypour and Obrien 2001, Tempelmeier 2002,
  and Leenders et al. 2006)
• This paper
• A system with multiple suppliers for supplier
  selection-order allocation in a dynamic
  multi-period setting with stochastic demands.
• Decisions by retailer in each period (based on
  inventory level)
• the replenishing quantity
• supplier selection
• order allocation

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Model Formulation
Notation T Total number of periods Dt Demand
(stochastic) in period t Bt Upper bound on Dt,
Bt lt8 Ft() Probability distribution function of
Dt N Total number of suppliers, N 1, ,
N Order allocated to supplier n in period t
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Model Formulation
It Inventory level at the beginning of period t
For It i, Xt is bounded by
The state space of It
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Model Formulation
It Inventory level at the beginning of period t
For It i, Xt is bounded by
The allocation space of
13
Model Formulation
Cost Fixed purchasing cost from supplier n in
period t Unit purchasing cost from supplier n
in period t ht Unit holding cost in
period t pt Unit shortage penalty cost
in period t
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Model Formulation
Ct(i) The minimum expected system cost from
period t to T, given It i.
Problem 1
for
and t 1, , T ,
The value space of Xt for It i
15
Model Formulation
Given Xt, the allocation space
Problem 2
Proposition 2.1 Problem 1 is equivalent to
Problem 2.
16
A Solution Approach
Inner level optimization
17
A Solution Approach
Inner level optimization
Proposition 3.1 implies that an optimal solution
to Sub-problem 1 can be formed by setting
and zeros to all other suppliers.
18
A Solution Approach
Inner level optimization
A polynomial algorithm
The algorithm (when ? 1)
2
3
1
Cost, Kt ct xt
order quantity, xt
19
A Solution Approach
Inner level optimization
A polynomial algorithm
The polynomial algorithm (when ? lt 1)
2
1
Cost, Kt ct xt
3
4
xt?
xt?
order quantity, xt
20
A Solution Approach
Inner level optimization
A polynomial algorithm
The polynomial algorithm (when ? lt 1)
2
1
Cost, Kt ct xt
3
4
xt?
xt?
xt?
order quantity, xt
21
A Solution Approach
Inner level optimization
A polynomial algorithm
The polynomial algorithm (when ? lt 1)
2
1
Cost, Kt ct xt
3
4
xt?
xt?
order quantity, xt
22
A Solution Approach
Inner level optimization
A polynomial algorithm
The polynomial algorithm (when ? lt 1)
2
1
Cost, Kt ct xt
3
4
xt?
xt?
order quantity, xt
23
A Solution Approach
Inner level optimization
A polynomial algorithm
The polynomial algorithm (when ? lt 1)
2
1
Cost, Kt ct xt
3
4

xt?
xt?
order quantity, xt
24
A Solution Approach
Outer level optimization
A Standard Dynamic Programming approach
25
Computational Experiment
T 10, N 10

• Allocation rules
• Uniform constant ?
• Piecewise constant ?(Xt)

26
Computational Experiment
Uniform constant ?
27
Computational Experiment
Piecewise constant ?(Xt)
Scenario 1
?(Xt) ? 1
Scenario 2
?(Xt) ? 1
?(Xt) ? 2
Scenario 3
?(Xt) ? 1
?(Xt) ? 3
?(Xt) ? 2
. . .
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Computational Experiment
Piecewise constant ?(Xt)
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Conclusions

• Consider supplier selection order
  allocation simultaneously
• Easy to implement order allocation rules
• Polynomial algorithm for inner level
  optimization
• Uniform constant ? vs. piecewise constant
  ?(Xt)
• Applicable in deterministic setting
• Applicable in lost-sales setting

30
Questions?