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3 LIBRARY
:10»? Michigan State
University

This is to certify that the
dissertation entitled

THEORETICAL ESSAYS ON OPTIMAL SOURCING
STRATEGY UNDER PRICE UNCERTAINTY

presented by

SANTOSH KUMAR MAHAPATRA

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Business Administration

 

 

£44“ 74 Wfl/L’kx 4

 

Major Professor’s Signature
6'//2[06

Date

 

MSU is an Affinnative Action/Equal Opportunity Institution

—:-.---»--.—»--.-- -r-o

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/07 p:/ClRC/DateDue.indd-p.1

 

THEORETICAL ESSAYS ON OPTIMAL SOURCING STRATEGY
UNDER PRICE UNCERTAINTY

By

SANTOSH KUMAR MAHAPATRA

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Business Administration

2006

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ABSTRACT
THEORETICAL ESSAYS ON OPTIMAL SOURCING STRATEGY
UNDER PRICE UNCERTAINTY
By
SANTOSH KUMAR MAHAPATRA

This study examines the issue of sourcing strategy in the context of price
uncertainty. The extant literature has not addressed this issue adequately while
accounting for a risk averse buyer’s concern for magnitude and uncertainty of
cost. This dissertation investigates the issue analytically when the buying firm
adopts “contractual” sourcing arrangement with deterministic prices and “open
market” sourcing arrangement with stochastic prices to procure a product. The
investigation utilizes a disutility minimization perspective to analyze the problem.
The study utilizes the optimal control theory and proposes three alternative
models.

The first two models solve the problem in continuous time to find the
optimal fraction of expenses and the optimal fraction of units to procure from the
alternative arrangements when the buying firm does not incur switching cost for
shifting the units between the two alternative arrangements. The third essay solves
the problem in discrete time when the buying firm incurs switching cost for
shifting the units across the two alternative arrangements.

The study provides theoretical and managerial insights that have
implications for designing the sourcing mechanisms across different market states

that can be characterized by the varied price dynamics of the procured product.

Cepyright by

SANTOSH KUMAR MAHAPATRA
2006

 

 

Dedicated to my beloved daughter Meghna, my joy, inspiration and support.

iv

 

 

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Acknowledgements

It gives me immense pleasure to express my humble appreciation to my
advisor, Professor Ram Narasimhan. You were a constant source of inspiration,
support and guidance in completing this dissertation. I also express my earnest
gratitude for your care in preparing me for the future challenges. My sincere
thanks to my co-advisor Professor Shlomo Levental who helped me in learning
the stochastic calculus and checked my mathematical calculations meticulously. I
am deeply indebted to my other committee members, Professor Shawnee K.
Vickery, Professor Srinivas Talluri and Professor Kenneth K. Boyer for providing
insightful feedback and constructive comments to improve the quality of my
dissertation. My thanks to Mr. Santosh Kar who supported me by developing the
computer code for the discrete time numerical analyses.

I express my sincere appreciation to many enlightened Professors who
shared their wisdom with me during my doctoral study at Michigan State
University. I am especially thankful to Professor Steven A. Melnyk for inspiring
me to develop the habit of being simple yet incisive in addressing research
questions. I appreciate the guidance of Professor Roger J. Calantone for helping
me to recognize the importance of theory and methodology in conducting
meaningful research. I acknowledge the administrative and moral support
provided by the Department of Marketing and Supply Chain Management in
completing my doctoral study and in facing the roughest period of my life.

Finally, I express my deepest gratitude to my wife and daughter for their

constant support and patient endurance during my graduate studies.

 

'J)

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Table of Contents

LIST OF TABLES viii
LIST OF FIGURES x
1 Introduction .......................................................................... 1
2 Literature Review ................................................................... 7
2.1 Strategic sourcing arrangements under uncertainty ........................ 7
2.2 Price dynamics and sourcing strategy ...................................... 14
2.3 Sourcing strategy without price uncertainty considerations ............ 16
2.4 Sourcing strategy with price uncertainty considerations ................ 17
2.5 The impact of the emergence of electronic market ....................... 20

on sourcing strategy

Sourcing Strategy For Minimizing Expense Disutility ...................... 25
3.1 The problem context ......................................................... 25
3.2 The model ..................................................................... 30
3.3 Illustration of model application ............................................ 41
Sourcing Strategy For Minimizing Expense Disutility ....................... 56
While Procuring The Exact Number of Units

4.1 The problem context ......................................................... 56
4.2 The model ...................................................................... 58
4.3 Illustration of model application ............................................ 67
Optimal Sourcing Strategy With Proportional Switching Cost ............. 80
5.1 The problem context ............................................. . ............ 80
5.2 The model ..................................................................... 81
5.3 Illustration of model application ............................................ 92
Conclusion ........................................................................ 1 18
6.1 Theoretical contributions .................................................. 123
6.2 Managerial contributions .................................................. 138
6.3 Directions for future research ............................................. 145
Appendix A ....................................................................... 149
Appendix B ....................................................................... 151
Appendix C ....................................................................... 158
Appendix D ....................................................................... 162

vi

I;

Appendix E ....................................................................... 166

Bibliography ..................................................................... l 86

vii

 

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Table ;

Table 1

Table 5

Table 5

Table 3.1:

Table 3.2:

Table 3.3:

Table 4.1:

Table 4.2:

Table 4.3:

Table 4.4:

Table 5.1:

Table 5.2:

Table 5.3:

Table 5.4:

Table 5.5:

LIST OF TABLES

Illustrative data representing the price dynamics ......................... 42
across the PLC stages

The impact of price pattern and cost disutility ............................ 44
on optimal fraction (u ) spent on market sourcing

The impact of price pattern on units sourced .............................. 51

from alternative arrangements over time

Illustrative data representing the price dynamics ......................... 69
across the PLC stages

The impact of price pattern and cost disutility factor (n) ............... 71
on optimal fraction (u ) of units sourced from market

Optimal fractions of units (u‘) to source from market .................... 74
for gradually varying disutility function across the PLC stages

Optimal fraction of units (u‘) to source from market ..................... 75
for varied planning horizons in different stages of the PLC

Illustrative data representing the price dynamics ......................... 93
across the PLC stages

Optimal fraction of units to source from market .......................... 96
for beginning relative price of 1 and varied switching cost ratio
in the early market state '1'

Optimal fiaction of units to source fi'om market .......................... 97
for varied beginning relative prices and switching cost ratio of
10 in the early market state '1'

Optimal fraction of units to source from market .......................... 98
for beginning relative price of l and varied switching cost ratios
in the intermediate market state '7'

Optimal fraction of units to source from market ......................... 99

for varied beginning relative prices and switching cost ratio of
10 in the intermediate market state '7'

viii

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Table 5.6: Optimal fraction of units to source from market ........................ 100
for beginning relative price of 1 and varied switching cost ratios
in the late market state '13'

Table 5.7: Optimal fraction of units to source from market ........................ 101

for varied beginning relative prices and switching cost ratio
of 10 in the late market state ‘ 13’

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LIST OF FIGURES

Figure 1.1: A conceptual framework to describe the influence ........................ 5
of sourcing context on sourcing arrangement alternatives

Figure 3.1: Characteristic (hypothetical) price pattern across ........................ 43
different stages of the PLC

Figure 3.2: Optimal fraction of expense ‘u"on market sourcing .................... 45
for varying disutility factor 'n' and constant price parameters

Figure 3.3: Optimal fraction of expense ‘u"on market sourcing .................... 46
for varying market price standard deviation ‘ ,61 ’ parameter,
constant price growth rate differential and cost disutility factor (n=2)

Figure 3.4: Optimal fraction of expense ‘u"on market sourcing .................... 46
for varying price growth rate differential ‘ a ’,
constant standard deviation parameter ‘ fll'
and cost disutility factor (n = 2)

Figure 3.5: Optimal fraction of expense ‘u.’on market sourcing .................... 48
for characteristic price dynamics and cost disutility factors
across different stages of the PLC

Figure 3.6: Optimal fraction of expense ‘u” on market sourcing ................... 49
for characteristic price dynamics across different stages of the PLC
for constant disutility factor (n = 2)

Figure 3.7: Illustration of impact of price pattern on units sourced .................. 52
from alternative arrangements over time across different
stages of the PLC for constant disutility factor (n = 2)

Figure 4.1: Characteristic (hypothetical) price pattern across ........................ 70
different stages of the PLC

Figure 4.2: Patterns of optimal fraction of units to source from market for ........ 72
constant disutility factors (n = 2) and (n = 3) for a one year
planning horizon in different stages of the PLC

Figure 4.3: Pattern of optimal fractions of units to source from market ............ 75
for gradually varying disutility function over the PLC stages

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Figure 4.4: Patterns of optimal fraction of units to source ........................... 76
from market for varied planning horizons

Figure 5.1: Binomial Tree: The binomial representation of the ..................... 83
evolution of price process over four time intervals

Figure 5.2: Characteristic (hypothetical) price pattern across ........................ 93
different stages of the PLC

Figure 5.3: The binomial representation of the evolution of the ................... 102
relative price process with a beginning of 1 in the
early market state ‘ 1 ’of the PLC

Figure 5.4: Optimal fraction of units ‘u’ to source from market .................. 103
for the beginning relative price of 1 and switching cost ratio of l
in the early market state ‘1 ’ of the PLC

Figure 5.5: Optimal fraction of units ‘u’ to source from market ................... 104
for the beginning relative price of 1 and switching cost ratio of 50
in the early market state ‘1 ’ of the PLC

Figure 5.6: Optimal fraction of units ‘u’ to source from market ................... 105
the beginning relative price of 1 and switching cost ratio of 10
in the early market state ‘1 ’ of the PLC

Figure 5.7: Optimal fraction of units ‘u’ to source from market ................... 106
for the beginning relative price of 1 and switching cost ratio of 25
in the intermediate market state ‘7’of the PLC

Figure 5.8: Optimal fraction of units ‘u’ to source from market ................... 107
for the beginning relative price of 1 and switching cost ratio of 40
in the intermediate market state ‘7’of the PLC

Figure 5.9: Optimal fraction of units ‘u’ to source from market ................... 108
for the beginning relative price of 1 and switching cost ratio of 1 and 5
in the late market state ‘13’of the PLC

Figure 5.10: Optimal fi'action of units ‘u’ to source from market .................. 109
for the beginning relative price of 1 and switching cost ratio of 20
in the late market state ‘ 13’of the PLC

Figure 5.11: Optimal fraction of units ‘u’ to source from market .................. 110
for the beginning relative price of 1 and switching cost ratio of 20
in the market states ‘1’, ‘7’, and ‘13’of the PLC

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Figure 5.12: Optimal fraction of units ‘u’ to source from market .................. 111
for the beginning relative price process 1 and switching cost ratio of
1 in the market states ‘1’, ‘7’, and ‘13’ofthe PLC

Figure 6: 1: A descriptive comparison of the three analytical models ............. 120

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Chapter 1

Introduction

Purchased goods and services are one of the largest elements of cost for
many firms. Accordingly, purchasing has emerged as a strategic activity.
Bowersox et a1. (2002, pp. 135) noted:

‘In the average manufacturing firm in North America, purchased
goods and services account for approximately 55 cents of every sales
dollar. . .. it is clear that the potential savings from strategic management of
purchasing and sourcing can be considerable.’

A strategic activity involves establishment of operational objectives and
means to achieve competitiveness. Consequently, competitive priorities should
guide a firm’s procurement strategy (Narasimhan and Carter, 1998) and the
procurement strategy should be changed according to the changes in competitive
priorities with the evolution of product-market dynamics.

The manufacturing strategy literature suggests five competitive priorities:
cost, quality, delivery, flexibility, and innovation that a firm may focus on to
achieve competitive advantage (Wheelwright and Hayes, 1985). Among these,
cost is the common denominator against which benefits of other non-price value
elements (quality, delivery performance, flexibility etc.) are evaluated.
Consequently, cost is considered as the most important parameter in sourcing
decisions and supplier firms manipulate the price element to win against

competition and attempt to improve the non-price elements over a long-term since

these are difficult to adjust in the short-term.

 

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Cohen and Agrawal (1999) noted:

‘...managers have been slow to invest in long-term and
cooperative relationships, even though they claim to seek such
relationships. Furthermore, despite the extensive amounts of data collected
from suppliers on various technical and financial factors, most supplier
selection procedures are fairly subjective, with purchase price being the
most significant factor.’

The price of a product is influenced by its stage in the Product Life Cycle
(PLC), which influences the customer preferences, technological changes, and the
competitive dynamics at the market place (Reed, 2002, pp. 91; Thorelli and
Burnnet, 1981). These factors lead to varied patterns of price dynamics that make
formulation of long-term procurement strategy difficult. Consequently,
procurement arrangements that are developed with a long-term perspective are
likely to be sub-optimal in many circumstances (Cohen and Agrawal, 1999).

The impact of input price dynamics on a firm can be evaluated in terms of
the magnitude and uncertainty of cost of procurement and their relative
significance vis-a-vis other priorities. Firms adopt various procurement
arrangements to address the issues of procurement costs, changing priorities and
uncertainty in procurement due to the market dynamics of the procured product.
These arrangements may be broadly categorized into two types: long-term
contracts or partnership, and short-term contracts or open market arrangements
(Cohen and Agrawal, 1999). It is observed that the ‘open market’ arrangements

typically offer a lower average price than ‘contract’ arrangements but expose the

firm to a higher degree of price variability (Smith et al., 1999).

 

 

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Despite the lower average prices in the open market, the long—term
contractual arrangements have traditionally been preferred to the open market
arrangements. This is because, the open market arrangements involve multiple
arrangements over time leading to high transaction cost as a result of higher
search cost, monitoring cost and coordination cost than that of long-term
contractual arrangements. Moreover, the long-term contractual arrangements offer
better control mechanisms to address the supply side uncertainties relating to
input price, quality, quantity etc. However, it may be noted that the long-term
arrangements may lead to over-dependence on external supplier(s) that may result
in loss of control and exploitation by the supplier(s) in the long-term.
Furthermore, the long-term contracts prevent the buying firm to gainfully readjust
sourcing arrangements to realize the benefits of competition, which may result in
lower input-prices and superior input attributes. Therefore, buying firms should
have the flexibility to avoid the disadvantages of over dependence on a few
suppliers. Towards this end, firms may adopt a mix of supply arrangements,
which can lower purchasing costs and provide options to revise decisions as
conditions warrant (Narasimhan and Das, 1999).

In recent years, the emergence of electronic procurement mechanisms has
enabled buying firms to have access to a wider market and to benefit from market
efficiency (i.e., reduced cost of transaction and increased transparency).
Economic theory suggests that efficient markets maximize consumer surplus.
Thus it may be in the interest of the buying firm to procure from open market,

which is likely to be more efficient due to competitive influences. There is

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empirical evidence that provides support to the above theoretical premise. Several
firms in automotive, chemical, electronics, retail, energy, semiconductor, and
beverage sector have benefited from reduction in procurement cost, ordering cost,
and ordering time by successfully adopting electronic procurement mechanisms
that allow for sourcing efficiently both fiom contractual and open market
arrangements (Wu and Kleindorfer, 2005; Neef, 2001, pp. 84). In typical
electronic procurement settings a buying firm engages with a restricted supplier
base (e.g., one to five suppliers) for the long-term contract purchases, while using
the open market as a secondary source of supply and as a means to evaluate the
supplies received through long-term contractual arrangements. This provides an
opportunity to suitably integrate the long-term and short-term/open market
arrangements to safeguard against various supply side uncertainties while
achieving procurement efficiency.

The foregoing discussion on sourcing context and the need for integrated
sourcing arrangement may be summarized in the conceptual framework presented
in Figure 1.1. According to this framework, the PLC of the procured product
influences a buying firm’s purchasing priorities and uncertainty in the business
environment, which affect the suitability of ‘contract’ and ‘open market’ sourcing
arrangements for the firm. The recent emergence of electronic procurement
mechanisms presents opportunities for gainful integration of the ‘relatively
certain’ long-term contract arrangement with the ‘relatively uncertain’ open
market arrangements. In this context it is imperative for a buying firm to identify

the appropriate combination of sourcing arrangements (e. g., contractual and open

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arrangements.

Figure 1.1: A conceptual framework to describe the influence of sourcing context

on sourcing arrangement alternatives.

 
       
      
     
      
   

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Quality
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In the extant literature, while there are theories (i.e., transaction cost
theory, resource dependency theory etc.) that explain various sourcing
arrangements, these theories do not suggest suitability of alternative sourcing
arrangements in the context of varying importance of cost vis-a-vis other
priorities such as quality, delivery speed etc. across different stages of the PLC of
the procured product. This emphasizes the importance of examining two

important issues in sourcing that are the focus of the present research: a) deciding

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the optimal pattern of procurement across the two most commonly adopted
sourcing arrangements i.e., ‘long-term contractual purchase’ and ‘open/spot
market purchase’ in different stages of the procured product’s life cycle that may
be characterized in terms of the product’s price dynamics and the relative cost
priority of the buying firm over time, and b) identifying optimal time to switch
among sourcing arrangements. I examine these two issues in this dissertation.
Since, the issues pertain to optimal strategy, the research would use analytical
methodology.

The research is presented in six chapters. The following chapter discusses
the key literature on sourcing strategy under uncertainty that have implications on
the research issues addressed in this dissertation. The subsequent three chapters
present three analytical models and illustrate their applications in finding optimal
sourcing strategies in varied procurement contexts. The last chapter highlights the
key theoretical and managerial contributions of the research and suggests

directions for future research.

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Chapter 2

Literature Review

In this chapter the key literature on sourcing strategy under uncertainty is
reviewed with a special emphasis on the literature on strategy under price
uncertainty. There are several studies that deal with the issue of sourcing strategy
in general. However, there is hardly any study that has examined the issue of
sourcing strategy under uncertainty in the context of dynamically evolving

priorities of a risk averse buying firm.

The relevant literature may be studied along five broad themes: a)
sourcing strategies under uncertainty, b) price dynamics of the procured product
and sourcing strategy, c) optimal sourcing strategy without price uncertainty
considerations, (1) optimal sourcing strategy with price uncertainty considerations,
and e) impact of the emergence of electronic market mechanisms on sourcing

strategy.

2.1 Strategic sourcing arrangements under uncertainty

The literature on sourcing strategy under uncertainty may be examined in
terms of the appropriateness of various sourcing arrangements under varied types
of uncertainty that are experienced by the buying firm. The life cycle of a product
describes the composite effects of its product-market dynamics that influence the

nature of uncertainty, and intensity of competition. For example, in the early

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stage, uncertainty due to product characteristics is higher than that in the later
stages. Accordingly, the product life cycle (PLC) offers a useful framework to
examine the sourcing strategy of a firm. Purchasing being a boundary spanning
activity, the usefulness of such a framework is particularly high when the life
cycles of components and end products are strongly interlinked. The product life
cycles of electronic components, memory chips, hard drives, ZIP drives and of
their end products e.g., computers, electronic gadgets etc. are examples of such
strong linkages (Tibben-Lembke, 2002).

Based on empirical research, Rink and Fox (1999) developed a PLC
oriented procurement fi'amework which suggests distinct patterns of sourcing
strategy for inputs that are closely linked to sales of end product across the five
stages (i.e., pre-commercialization, introduction, growth, maturity, and decline) of
the end product’s PLC. According to the framework, in the pre-commercialization
stage, the component product standards are not yet developed; consequently costs
should be subordinated to the objective of prompt service on trial (small) orders
by vendors such that evaluation of new vendors and supplier development efforts
can be initiated. During the introduction stage, buying firms should employ
(short-term) contracts until a supplier’s capability with respect to the component
product’s varied attributes is demonstrated. This would encourage vendors to
develop appropriate capabilities and set the stage for long-term supply
arrangements by facilitating an orderly changeover from short-term contracts to
long-term supply arrangements. In the grth stage, buying firms should focus on

optimizing fixed costs by sourcing from in-house facilities or through long-term

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supply arrangements with one or a few key and large volume suppliers. In the
maturity stage, elimination of single sources of supply should be considered in
order to stimulate competition among vendors for achieving the lowest possible
cost. Their study implicitly suggests that across these stages, a buying firm’s
relative cost priority with respect to the components (importance of cost vis-a-vis
other non-price attributes) increases from introduction to maturity phase as the
end product becomes increasingly standard and that a firm’s sourcing strategy is
governed by efficiency maximization and uncertainty minimization objectives.

The findings of Rink and Fox (1999) converge with the normative
suggestions about the sourcing strategies made by the theoretical frameworks in
the literature. This convergence may be attributed to the common perspective of
efficiency maximization and uncertainty minimization in the theoretical
frameworks. Transaction cost theory, agency theory and resource dependency
theory are the three principal theories that explain the sourcing strategies of firms.
These theories have their origin in the field of institutional economics and classify
the sourcing strategies into two broad types: proactive strategy (long-term
arrangements) and reactive strategy (short-temr/open market arrangements)
(Noordewier et a1. 1990; Mabert and Venkatraman, 1998; Smeltzer and Sifred,
1998; Krause, 1999; McCutcheon and Stuart, 2000).

According to transaction cost theory, all transaction arrangements are
associated with costs and uncertainties due to the transaction specific investments,
the searching cost for the best supplier, the costs of establishing, monitoring and

enforcing the contracts, and the risk of opportunism (Williamson, 1985). The

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theory assumes the transacting parties to be risk neutral, limitedly rational and
opportunistic and the various transaction related costs to be subjective at the time
of setting up the arrangement (Chiles and McMackin, 1996). Agency theory
differs from transaction cost theory in terms of the assumptions held about the
transacting parties. It assumes the transacting parties to be risk averse and to
operate in a context of information asymmetry (Eisenhardt, 1989). Resource
dependency theory states that organizations should acquire and control resources
to avoid risky dependency on external organizations to minimize the chance of
exploitation (Ulrich and Barney, 1984). Application of these theories in a
sourcing context suggests efficiency and efficacy of transactions are realized by
matching the transaction governance structure (i.e., hierarchical, market oriented
or contractual governance structure with characteristics of both hierarchy and
market governance) with the contextual aspects of transaction (McCutcheon and
Stuart, 2000; Krause et al., 2000).

Proactive sourcing involves long-term arrangements that emphasize
potential for future benefits by sharing risks and current knowledge (e.g.,
collaborative technological innovations) with a few key suppliers. In contrast,
reactive sourcing involves short-terrn or open market arrangements that aim at
reduction of administrative (acquisition) and material expenditures by allowing
for opportunistic switch among suppliers in the rapidly changing business
contexts (e.g., discontinuous technological changes that may make technology
oriented alliances obsolete and ineffective). In a typical business context, a buying

firm may choose to adopt any one of the above two types of sourcing

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arrangements or some combination of these two types arrangements depending on
the fit among the sourcing arrangement, context and procurement priorities.

Proactive sourcing necessitates considerable investment of time, effort and
money to build the requisite trust and commitment for developing close
partnership type of arrangements. Thus these arrangements are associated with
higher risks of resource dependency, forward integration by supplier, and failure
of exchange mechanisms than that of reactive (open market sourcing)
arrangements (McCutcheon and Stuart, 2000). The risk of failure is particularly
high when the elements of uncertainty are out side the range of pre-specified tasks
of exchange. In such circumstances, the adjustment mechanisms of partnership
type of governance may not mitigate the risks, and the goals of exchange become
difficult to achieve (Noordewier et al., 1990). Consequently, true supplier
partnerships are somewhat uncommon and firms often engage in long-term
contractual relationships as a substitute to partnership type of arrangements.

It may thus be argued that proactive sourcing arrangements involve higher
development cost and entail a higher risk of failure if not designed and managed
properly. On the other hand, reactive (open market) sourcing is likely to be
beneficial when proactive sourcing is likely to be less cost-effective (i.e., in later
stages of the PLC with standard products and intense price-based competition) or
when managing the proactive sourcing arrangement is difficult (i.e., in early
stages of the PLC with unstable product characteristics and technology standards).
However, open market arrangements may be fraught with issues such as high

search and transaction costs and inconsistent quality and delivery standards due to

11

memui
ofsour

I
and cost

untena
actions
operafi.
compc:
Ihdl m,
uncen;

encour

long,
and “

lumen

imam

the multiplicity of supply arrangements. This calls for appropriate configuration
of sourcing arrangements while accounting for the varied types of uncertainties
and costs.

Sutcliffe and Zaheer (1998) categorized sourcing related uncertainties into
three broad types: primary uncertainty (e.g., uncertainty due to changes in
technology, customer preferences and other exogenous factors), competitive
uncertainty (e.g., price uncertainty and other product-attribute related competitive
actions), and supplier-related uncertainty (e.g., volume of supply, inconsistent
operations of suppliers etc.). They argued that the higher the primary and
competitive uncertainties, the lesser should be the emphasis on vertical integration
that may result in inappropriate investments, and the greater the supplier-related
uncertainty, the higher should be the propensity for vertical integration that may
encourage asset specific investments.

It may be conjectured that primary uncertainty is likely to be higher in the
early stages of the PLC, supplier uncertainty is likely to be higher in the
intermediate phases i.e., growth phases of PLC, and competitive uncertainty is
likely to be higher during the later stages of the PLC due to intense price-based
competition around a standard product. Moreover, long-term contracts that are
analogous to vertical integration would be involve high development costs (e. g.,
long-term supply arrangement cost and penalty costs for contract modification)
and would be difficult to design under high degrees of primary and competitive
uncertainties. Consequently, long-term contracts would be more useful during the

intermediate stages than in the earlier and later stages of the PLC.

12

35 “to

sun!)

Proactiu
pannt’l':
fact!“
Suppilc
Coming
SOUICIn
and or
quillm
Of the

Xe"Err

Treleven and Schweikhart (1988) identified single and multiple sourcing
as two basic sourcing strategies that are useful in addressing various types of
supply related uncertainties and costs. Single sourcing is somewhat similar to
proactive sourcing and addresses supply uncertainty by establishing
partnership/long-term contract arrangements with one or two suppliers; this
facilitates a stable and high quality supply of key inputs at a fair price. It involves
supplier development efforts, paring of the supplier base, and works well with a
contingency plan to source from an alternative source. In contrast to single
sourcing, multiple sourcing may make use of spot market (reactive sourcing)
and/or multiple supply contracts simultaneously to achieve competitive price,
quality and delivery performance. Their paper did not discuss the relative efficacy
of the two strategies with reference to any specific pattern of uncertainty.
Nevertheless, it may be argued that single sourcing would be useful when the
uncertainty and cost concerns are limited and could be addressed by the supplier
in question, and multiple sourcing would be useful when the uncertainty and cost
concerns are wider and may not be within the capabilities of the supplier in
question (e.g., discontinuous technological changes that may nullify the
advantages of the existing supplier competent in an existing technology).

The foregoing discussion on sourcing arrangements suggests uncertainty
and cost considerations influence the efficacy of various sourcing arrangements.
Long-term/partnership (proactive) type sourcing (e.g., a single or very few supply
arrangements) is likely to be more effective during growth and early maturity

(intermediate) stages of the PLC that are characterized by moderate levels of risk

13

HCIOSS
term If

may

comps

aPPTOI
lenn 3
meta
IIE3C}
across
prOdUl
arrang

pUICh;

2.2 p

attn‘bL
into a
PfiCe
ofpfit
§\n10t
Ilmpac

across primary, competitive and supplier based uncertainties. Short-
term/transactional (reactive) type sourcing (e. g. spot market, multiple supply
arrangements etc.) is likely to be more effective in the introduction and maturity
stages of the PLC that are characterized by high levels of risk due to primary and
competitive uncertainties respectively.

However, such conceptual generalization is inadequate to suggest
appropriate sourcing arrangements (e.g., how much to buy from proactive/long-
term and/or reactive/short-term open market arrangements over time) to address
the cost and uncertainty concerns of the buying firm across different phases of the
life cycle of a product. This calls for studies to suggest optimal sourcing strategies
across the varied procurement contexts of the different phases of PLC. Since, the
product-market structure evolves dynamically, it is natural that the sourcing
arrangements are appropriately restructured over time according to the evolving

purchasing priorities of the buying firm.

2.2 Price dynamics and sourcing strategy

The price of a product is the key factor against which other product
attributes are evaluated. Consequently, the sourcing strategy of firms should take
into account the characteristic price pattern of the procured product over time.
Price dynamics of a product are often explained in terms of the economic theory
of price, suppliers’ pricing strategies, and PLC dynamics (Curry and Riesz, 1988).
Among these, the PLC represents the composite effects of several factors that
impact the product prices over time. Elasticity of demand, rate of demand

diffusion, rate of demand saturation, rate of innovation, experience curve

14

T-b

 

 

dmarr.

come: i

experts:
ei’ident
[hm SpC

perform

 

follou er

perform

dynamics, inflation, supplier’s planning horizon, technological changes, and
competition are the common factors that influence the price pattern over a PLC
(Hofer, 1975; Dolan and Jeuland, 1981; Clarke and Dolan, 1984; Rao and Bass,
1985; Krishnan et al., 1999, Smith et al., 1999). For example, the impact of
experience curve effect (i.e., reduction of price due to lower cost over time) is
evident from the empirical study by Larnbkin and Day (1989). This study found
that specialist pioneers with small scale operations are likely to be the best
performers in the early stage of market evolution; however, large-scale generalist
followers with established interest in related markets are likely to be best
performers in the developing and maturing stages of market.

Literature suggests that contractual prices are higher than competitive
market prices (Klein and Leffler, 1981; Horowitz, 1986; Rao and Bergen, 1992;
Rao and Monroe, 1996). Buying firms are likely to pay higher prices in contracts
for two principal reasons: one, they lack complete information about the true cost
of production of the supplier, and two, they give incentives to motivate suppliers
to reduce the supply uncertainty and meet the contractual terms with respect to
quality. Smith et al. (1999) examined the issue of price dynamics and the buying
firm’s willingness to pay price premiums in contractual arrangements empirically.
They observed that in high-tech, short PLC product markets, prices decline over
time as producers lower the prices of newer versions to reduce the switching costs
of the buyer. In contrast, in low-tech long PLC product markets, prices may
increase, remain constant or decline depending on the intensity of competition.

However, the standard deviation and rate of change of prices in both the product-

15

Tb

 

 

marker _
are ill;
monop II
and its“

to pa}

uncenr.

 

Specifi
Conan
offie:
ennui
more
The

the (
pl‘lci

market contexts generally decline over time. They also indicated that buying fums
are likely to pay a price premium in the presence of disadvantages due to supplier
monopoly, information asymmetry with respect to quality and cost of production,
and transaction specific investments. Nevertheless, the buying firm’s propensity
to pay a contractual premium is likely to decline with the decline of these
uncertainties.

It is thus apparent that prices offered by suppliers are dynamic, and
standard deviation of prices and contract premiums paid by the buyer decline as
the procured product becomes increasingly standard over time (e. g., competition
and customer’s awareness level about quality and cost of production increase).
The buying firm needs to consider these aspects while designing sourcing

arrangements and negotiating contracts with suppliers (Smith et al., 1999).

2.3 Sourcing strategy without price uncertainty considerations

The typical sourcing arrangements are contractual arrangements with
specific terms and conditions that define the supply arrangement. These terms and
conditions specify the pattern of prices over time, duration of contract, and degree
of flexibility with respect to conditions of supply. Thus contractual arrangements
eliminate or reduce supply related uncertainties significantly in exchange for a
more or less deterministic price pattern that the buying firm commits to abide by.
The existing analytical literature on supply contracts mainly focuses on deriving
the optimal price and payment structure (e.g., linear pricing, two part-non linear
pricing, options, discounts, risk sharing through quantity flexibility, purchase

commitments, returns etc.) as key coordination mechanisms between the buying

l6

TH

 

,o

Ill'm 3;".
19‘)"; .3.

‘1

Ct all '
in imp:
penod ”l
of Illt’r'u
examine
preclttit
context.
negotiati

Kleindm

 

are Predt
1994; M
Kulatjlak
adapt 501
Ca“ incr.
downsidi

Drug Te .

r‘

d.

firm and supplier under stochastic demand (Pastemack, 1985; Eppen and Iyer,
1997; Anupindi and Bassok, 1999; Lariviere, 1999; Corbett and Tang, 1999, Tsay
et al.1999). For example, Schuster et al. (2002) analyzed the benefits of options
in improving channel performance in the face of demand uncertainty in a two
period buyer-supplier system. The approaches for optimal contracts in this stream
of literature do not explicitly consider the issue of market price uncertainty, and
examine contractual arrangements with a single/limited number of suppliers that
preclude transactions in the open market. However, in a contractual purchasing
context, buyers consider the market price pattern as a benchmark for effective
negotiation or for deciding limits of contract price premiums (Smith et a1. 1999;
Kleindorfer and Wu, 2003). Consequently, analytical models on the design of
contractual sourcing arrangements should appropriately examine the relative cost

and benefits of contractual arrangements vis-a-vis open market arrangements.

2.4 Sourcing strategy with price uncertainty considerations

The studies on sourcing arrangements that account for price uncertainty
are predominantly influenced by the ‘real options’ literature (Dixit and Pindyck,
1994; McDonald and Siegel, 1986; Brennan and Schwartz, 1985; Kogut and
Kulatilaka, 1994). According to ‘real options’ perspective, a firm’s ability to
adapt sourcing strategies in response to altered economic or operating contexts
can increase its value by improving its upside potential while limiting its
downside losses. However, any change in strategy may involve additional
transaction costs, which is known as switching costs in the real options literature.

Thus real option valuation procedures are useful in deciding the timing, source,

17

T-h

 

and qt.-

H uchic

procure

 

relation
imestég
detemu
particul
piOposc
strateg}
Witchi-
Context
Source
Capacit
the pro
in 3 Sir
lattice)

Wiliie a

prOCure
rates 0_
Sllfi‘cifiC

imPIEm

and quantity of sourcing (Dixit, 1989; Huchzermeier and Cohen, 1996; Cohen and
Huchzermeier, 1999; Murthy et al., 2002; Kouvelis, 1999).

Murthy et a1. (2002) used the real option approach to analyze the dynamic
procurement and switching behavior of a buying firm that experiences different
relationship specific fixed costs with suppliers located in a foreign country. The
investigation assumed that the buying firm’s output price/revenue to be
deterministic and the firm can procure only from one of the suppliers in a
particular time-period. Kouvelis (1999) also used the real options approach to
propose a general modeling framework to examine multi-supplier sourcing
strategy with switching costs. The study derived the threshold price levels for
switching the preference from one supplier to another in a two-supplier sourcing
context, in a two-supplier sourcing context, and indicated that the buyer should
source simultaneously from both the suppliers only if the least cost supplier’s
capacity is inadequate. Li and Kouvelis (1999) developed models for minimizing
the procurement cost when the buyer uses risk-sharing (flexible) supply contracts
in a single or two suppliers sourcing context. They used a numerical (binomial
lattice) approach to identify optimal sourcing strategies for flexible contracts
while accounting for inventory costs and capacity constraints.

Two of the key assumptions in applying the real option approach to the
procurement context are the existence of security portfolios, which replicate the
rates of change of sourcing costs, and the existence of risk-less assets with a
specified risk fi'ee rate of return. Often, these assumptions are difficult to

implement, thereby limiting the applicability of these models. Furthermore, most

18

 

the per-
adopted

|
term cor
alzemari

dCVCIOp

the cor

funCliQ

moCiel

5011mm

of the existing analytical studies on sourcing in the context of price uncertainty
derive the optimal strategy for a risk-neutral buying firm. In contrast, as discussed
earlier, the theories on sourcing strategy justify the contractual arrangements on
the basis of the buying firm’s concern for uncertainty. Accordingly, optimal
strategies need to be derived fi'om the perspective of a risk averse buying firm.

Cohen and Agrawal (1999) examined the issue of optimal sourcing from
the perspective of a risk averse buying firm perhaps for the first time. They
adopted a utility maximization perspective to compare the advantages of short-
term contracts that have price uncertainty, but provide the flexibility to switch to
alternative suppliers, with that of long-term contracts that involve fixed cost of
developing the arrangement, but provide certainty of prices. Their results defied
the conventional notion that long-term contracts are superior to short-term
contracts and highlighted the fact that the advantage of a specific type of
arrangement is governed by transaction specific investments, length of planning
horizon, uncertainty of market prices, opportunity for cost reduction over time,
and decision makers’ risk preference. They employed the ‘mean-variance’
technique for modeling the buying firm’s utility function; this technique
associates only with quadratic utility function. In reality, a buying firm’s utility
function may be more varied than the ‘quadratic utility function’, which does not
model the theoretical relationships between risk aversion and magnitude of
expense appropriately (Eeckhoudt and Gollier, 1995, pp. 48).

The models in the extant studies solve for the case of adopting a single

sourcing arrangement at a time. In reality, a buying firm may have existing

19

TH

 

COIIII’JL

 

consec

signifi.
thatpr.

need It

perspet

 

 

 

the WWII

contract arrangements that may be difficult to cancel at a given time;
consequently it may be beneficial to suitably mix a number of sourcing
arrangements simultaneously (Billington, 2002). This is particularly of
significance due to the recent emergence of electronic procurement mechanisms
that provide for quick configuration of supply arrangements. Accordingly, models
need to be developed for optimally integrating multiple arrangements from the

perspective of a risk averse buying firm.

2.5 The impact of the emergence of electronic market on sourcing
strategy

The emergence of electronic market has affected the sourcing strategy of
firms significantly (Martinez-de-Albeniz and Simchi-Levi, 2005). To make a
proper assessment of its implications for sourcing strategy, its characteristics must
be assessed with reference to the sourcing strategy frameworks suggested in
literature. The various sourcing arrangements adopted by firms may be viewed to
lie on a continuum of long-term relational type to short-term transactional type
arrangements (Williamson, 1991; Hobbs, 1996; Melnyk and Swink, 2003).
Characteristically, the relational types are more suitable for non-standard/
strategic products with controllable but uncertain supply contexts, and the
transactional types are more suitable for standard/non-strategic products or for
non-standard/strategic products with less uncertain supply contexts (Kraljic 1983;
Martinez-de-Albeniz and Simchi-Levi, 2005). The standard products are usually

the products that have easy to specify attributes (Kleindorfer and Wu, 2003).

20

T-H

asasei

Clio cm.

COHITJCI

to suit.

compet:

 

de—Albe
Lem-er
mange?

er al., 2:

 

Relational arrangements ensure certainty of price and supply, and are
likely to result in low transaction cost in situations of high asset specificity, high
transaction frequency and uncertain supply contexts. However, true relational
arrangements are difficult to develop; consequently, long-term contracts are used
as a substitute to relational arrangements to achieve certainty of supply (Dyer,
Cho and Chu, 1998). From a buying firm’s perspective, the certainty through
contractual arrangements may involve two main costs: a) sacrificing the flexibility
to switch to an alternative source to benefit from favorable developments in
competitive market (McCutcheon and Stuart, 2000; Sanchez, 2003; and Martinez-
de-Albeniz and Simchi-Levi, 2005), b) payment of a price premium (Klein and
Leffler 1981; Horowitz, 1986; Smith et a1. 1999). Thus long-term contractual
arrangements may not always be optimal from a buying firm’s perspective (Peleg
et al., 2002; Cohen and Agrawal 1999). Rather, (open) market arrangements with
higher flexibility to switch to alternative sources are likely to be particularly
useful in situations of high relationship development cost, or when there are risks
of higher price premium and lock-ins in long-term contract commitments
(Sanchez, 2003). Since the upper and lower bounds of risks and relationship
development cost are context specific and difficult to quantify in clear terms,
complete dependence on short-term or open market arrangements to benefit from
the competitive market may expose a firm to an inordinate amount of price and
supply side uncertainty. In this context, integration of long-term contracts and
short-term contracts or open market arrangements can be effective in reducing the

magnitude and uncertainty of cost of procurement without affecting the non-

21

 

 

 

'
monetr.

tibeniz

SUCCESSf

PTOduCI

cost» a c,

monetary performance (Kleindorfer and Wu, 2003; Sanchez 2003; Martinez-de-
Albeniz and Simchi-Levi, 2005).

The emergence of variety electronic procurement mechanisms (e. g.,
private e-exchange, industry consortia, public e—market places, and ubiquitous
third party tools such as e-tendering, e-ordering, e-catalogues, e-reverse auctions
etc.) has facilitated such integration in several industries by increasing
transparency and transaction efficiency through reduced search cost and
coordination cost, and lower switching cost for new supply arrangements (Boer et
al., 2001; Billington, 2002, CAPS Research and McKinsey & Company, 2002,
Jap and Mohr, 2002; Kaplan and Sawhney, 2000; Johnson and Whang, 2002;
CAPS Research, 2003; Rabinovich et al., 2003). These mechanisms enable to
develop product specific sourcing arrangements for a wide range of product-
market contexts (Skjott-Larsen et al., 2003; Lancioni et al., 2003). For example,
industry consortia type electronic (B2B) exchanges such as Covisint have
successfully evolved to provide infrastructure to facilitate transaction of complex
product and service bundles (Wise and Morrison, 2000).

An analytical explanation of the benefits of integrated sourcing
arrangement may be found in the study of Peleg et al. (2002), who analyzed the
relative efficiency of the spot market purchase, the long-term contracts and the
combined strategy of spot market purchase and long-term contracts in a two
period stochastic demand context. They observed that while no particular strategy
has a clear dorrrinance over the others, in situations of negligible supplier search

cost, a combined strategy is superior to long-term contracts.

22

 

and lo:

COlllDll‘.

compo

d}1‘..‘i!7‘.2.
solutior‘
to be r.
literate:
Further:
element
Procuret
across :'
transac:
depend;
the 1m?
related ‘.

such co:

Strateg}.
across d;
ComraCls
gap in l

°Plima1

 

To summarize the literature: open market purchase or short-term contracts,
and long-term contracts are two principal alternatives for sourcing; these may be
combined to achieve optimality against the uncertainties over time. The
composite effects of market dynamics of a product are reflected in its price
dynamics. Most of the extant analytical investigations have derived optimal
solutions with a cost minimization perspective while considering the buying firm
to be risk neutral. In reality, as evident from the empirical and sourcing strategy
literature, firms are concerned about uncertainty and seek to reduce the risk.
Furthermore, buyers evaluate monetary cost of procurement against other value
elements or strategic considerations; consequently, their relative concern for
procurement cost vis-a-vis other (non-price) value elements changes over time
across the different phases of the PLC. For example, a buying firm may want to
transact with multiple suppliers, or to buy in the spot market to avoid over-
dependence on a particular supplier. Buying firms in above situations may view
the importance of procurement cost differently and may have different cost
related utility functions; thus the optimization perspective needs to accommodate
such considerations (Cohen and Agrawal, 1999).

In this context, it is imperative to investigate the issue of optimal sourcing
strategy taking into account the buying firm’s risk aversion and price uncertainty
across different phases of the PLC when the firm uses (open) market or short-term
contracts and long-term contracts for procurement. This dissertation addresses this
gap in the existing literature. It examines the following issues: a) what is the

optimal pattern of allocating supplies across the two most commonly adopted

23

 

arm-rig

arr «mg-c“
Varied

transac:
50min;

may SI."

 

sourcing arrangements i.e., ‘contractual purchase’ and ‘open/spot market
purchase’ while accounting for typical price patterns, transaction cost, and cost
disutility function of the buying firm in various stages of PLC, and b) what are the
implications of adopting optimal pattern of sourcing on developing the sourcing
arrangements.

The results of the investigation can provide insights on optimal sourcing
arrangements across different stages of the PLC that may be characterized by
varied patterns of input prices, buying firm’s cost disutility functions and
transaction cost or switching cost. These insights can be used for designing
sourcing strategies and generating hypotheses about sourcing arrangements that

may subsequently be verified through empirical investigations.

24

Hr

 

Chrl

 

Sourc

3.] Th

product
differs?
weenie.
Wichita

The proJ

C0mmOr

 

eValuare
Price M
element:
i“VESlm
for gUai
1999). 1
KleindC
COdifia

and Tel;

Chapter 3

Sourcing Strategy For Minimizing Expense Disutility

3.1 The problem context

The research examines the Optimal sourcing strategy for a risk averse
buying firm that uses long-term contractual arrangement and short-term market
arrangements in the following procurement context. The firm buys a standardized
product continuously for satisfying the specified levels of demand across the
different stages of the product’s life cycle. The attributes of the product can be
specified objectively in each procurement cycle, and there would be several
potential suppliers in the market who can supply according to the specifications.
The proposed models are based on the premise that the price of the product is the
common denominator against which other non-price value elements can be
evaluated over time (Melnyk and Swink, 2003). This premise enables to link the
price with the degree of product standardization which influences the various
elements of transaction specific costs and uncertainties e.g., cost of transaction,
investment in transaction specific assets, supply-demand instability, and premium
for guaranteed supply (Klein and Leffler, 1981; Horowitz, 1986; Smith et al.,
1999). This argument is somewhat similar to the conceptual framework offered by
Kleindorfer and Wu (2003), which suggests a direct relationship between cost of
codifiability (i.e., possibility to objectively specify the product/service attributes)

and relative cost of contract establishment and production.

25

T-H

 

 

integrr
oppon-
I
cost of
Martin.
these 5'.
context
Agra“;
Open at
3003; Ix

Both 01'

 

Context.

In the extant literature, there are studies that examine the issue of
integration of short-term and long-term arrangements, which provide
opportunities to trade-off certainty and higher cost against flexibility and lower
cost of transaction (Cohen and Agrawal, 1999; Kleindorfer and Wu, 2003;
Martinez de-Albeniz & Simchi-Levi, 2005; Wu and Kleindorfer, 2005). However,
these studies either consider an option theoretic framework in a discrete time
context in which only one type of arrangement is active at a time (Cohen and
Agrawal, 1999) or, consider the firm to be risk neutral in the context of stationary
open market price distribution during a specific operating horizon (Peleg et al.,
2002; Kleindorfer and Wu, 2003; Martinez de-Albeniz & Simchi-Levi, 2005).
Both of these considerations do not reflect the realities in the actual problem
context.

The recent emergence of BZB exchanges has motivated firms to adopt a
combined sourcing strategy, involving a dynamic mix of short-term contracts or,
open market purchase (with stochastic prices) and long-term contracts to buy the
product (CAPS Research and Mc Kinsey Consulting 2002). Such sourcing
arrangements help firms to manage risk and reduce cost in several industries such
as aluminium, auto-components, electricity, electronic components, chemical and
semiconductors (Kleindorfer and Wu 2003; Martinez de-Albeniz and Simchi-
Levi, 2005; Wu and Kleindorfer, 2005). For example, Hewlett Packard optimizes
the procurement cost of electricity or, memory products using such an integrated
approach (Billington, 2002). Past studies have recommend continuous time

modeling of such problems for better analytical insights (Cohen and Agrawal

26

1999; Kleindorfer and Wu 2003). The proposed model in this research addresses
these considerations.

The proposed modeling framework is distinct from existing studies in that
it explicitly considers the buying firm’s concern for uncertainty in terms of a
utility maximization perspective (Keeney and Raiffa, 1976, pp. 148). This
approach has been extensively used in the portfolio optimization problems in
finance (Korn and Korn, 2001, pp. 203). The efficacy of this approach lies in
estimating the subjective utility function appropriately. There are several
alternative approaches such as maximizing the value of ‘mean-variance’ or the
‘max-min’ function that can also address the issue of ‘concern for uncertainty’.
However, the utility maximization criterion scores over others due to the
following reasons. While, the ‘max-min’ criterion does not consider all possible
outcomes and thereby deforms the actual probabilities (Eeckhoudt and Gollier,
1995, chapter 4, pp. 39), the ‘mean-variance’ criterion is more justified when the
outcomes are normally distributed (Meyer, 1987). Consequently, optimization
based on maximization of suitable utility functions would be a more appropriate
approach to address the concerns of the magnitude and uncertainty of outcomes.

Utility functions being subjective are often difficult to assess across
multiple objectives; utility literature suggests substitution of easy-to-use simple
utility functions for such contexts (Hammond, 1974). Using the above rationale,
single attribute cost disutility functions that are polynomial in expense can be
derived to effectively describe the buying firm’s over all preference structure for a

product with specifiable attributes (Keeney and Raiffa, 1976, Chapter 5 and 6).

27

 

These d

l
procure:

i
5

Vendor.
pool of
certifies
manage:

tactical.

 

 

Opes of
OVer a It
the Con
“VET sl‘.
Operati.
market
quantii
(IEmar
firnq l
C Onse
Shale
PTOCL
pent:
Used

de- .-

.-
4

These disutility functions can be used to identify optimal sourcing strategies in
procurement contexts with specified price dynamics.

For the problem context in this study, it is assumed that a process of
vendor certification has already taken place, and thus the firm has available to it a
pool of contracts from which it must choose its sources of supply. The set of
certified vendors may be different in different stages of PLC. The supply
management decision made by the firm has two components i.e., strategic and
tactical. The strategic decision involves selection of contractual suppliers and
types of contract (e.g., fixed, flexible, or quantity buy-back etc.) that are reviewed
over a longer period of time may be once a year. Tactical decisions are made after
the contractual supplier is selected for actual ordering of the quantity of supply
over shorter time frames (e.g., may be weekly or even daily) to match firrn’s
operational requirements as per the existing demand and price conditions in the
market. Thus it allows for contract price renegotiation and/or procuring variable
’ quantity of supplies from alternative arrangements towards satisfying the exact
demand requirement over the tactical review period. It is further assumed that the
firm uses a single source of supply for each type of arrangement at a time.
Consequently, a long-term contractual supplier remains fixed for the entire
strategic review period, and open market supplier, contract prices and amount of
procurements from each of the sources may be adjusted across the tactical review
periods. These assumptions are consistent with the standard industry practices,
used in contractual sourcing arrangements (Cohen and Agrawal, 1999; Martinez

de-Albeniz and Simchi-Levi, 2005). Accordingly, for the above problem context

28

T-I'

 

the Opt:

 

arranue
.

patterns

 

justifie.
stable
premier
Lefiler.

COUiCi be

 

0f planrl
(low er 3

Diatllcz
(declini
Effect c
to the

marke‘

as$er

the optimal mix of the two extreme sourcing arrangements (e.g., open market
arrangements and long—term contracts or partnerships) is being determined.

It is assumed that at the beginning of any review period, contract price
patterns are determined after accounting for the ‘contract premium’ that is
justified on the basis of supplier’s likely demand for such a premium for ensuring
stable supply according to contract terms, and buyer’s propensity to pay a
premium due to the lack of awareness about supplier’s true cost (Klein and
Leffler, 1981; Horowitz, 1986; Smith et a1. 1999). One of the ways to achieve this
could be, for the same contract and (expected) open market price at the beginning
of planning horizon, the contract price would increase (decrease) at a rate higher
(lower) than the expected grth (decline) rate of the open market price. From a
practical standpoint, this is reasonable as the expected market price growth
(decline) rate is likely to be lower than long-term contract due to the moderating
effect of competition. This approach to contract price determination is comparable
to the approach suggested by Cohen and Agrawal (1999). It is assumed that the
market is efficient with a competitive price, and the buying firm does not incur
any inventory holding cost i.e., the demand for the procured product may be
viewed as flow units per time or the product may be considered as non-storable
(i.e., electricity, services etc.). It is also assumed that there is no suppliers’
capacity and buyer’s monetary constraint that limit sourcing over time. Such
assumptions are typical of stochastic optimal control problems in extant literature
(e.g., Aytekin and Birge, 2004; Kouvelis, 1999). The buying firm can have

flexibility to procure from the contract arrangement a quantity that may vary to

29

some or
4 H
anC. i
market

compet;

3.2 Th

and P3
mean g:
price pr
time he:
Price a".
al-‘PFOpr
Such as

KOUVelj

aCIOSs I

 

 

procuw
firms 5
inexpen:
SOLIIcinE

Value (c

some extent around a specified average volume of procurement at the negotiated
price. While, the contract arrangement provides certainty of supply, the open
market sourcing arrangement enables the buying firm to benefit fi'om lower

competitive market price.
3.2 The model

Mathematically, the above problem can be described as follows. Let Pl
ansz be the open market and contract prices with al and a2 the respective

mean growth rates, and ,61 be the standard deviation parameter of the open market

price process that follows a geometric Brownian motion (GBM) in a specified
time horizon. GBM is a stochastic process, commonly used in modeling uncertain
price and exchange rate dynamics (Dixit and Pindyck, 1994), and is considered
appropriate for analytical examination of integrated supply arrangement problems
such as the present one (Cohen and Agrawal, 1999; Kouvelis, 1999; Li and
Kouvelis, 1999 etc.).

The firm searches for admissible and optimal sourcing arrangements
across the various procurement contexts in different stages of the PLC. The
procurement contexts may be characterized by varied price patterns and by the
firm’s subjective concern for uncertainty and magnitude of cost. The firm
inexpensively adjusts the percentage of expenses incurred on supplies from each
sourcing arrangement continuously over time in order to maximize (minimize) the

value (cost disutility) of the procurement strategy.

30

T-I>

deiiati

IWICSL'.’

 

parame:

 

 

[he DUN

 

Ekpecte

instzm
before
this p
the It
the ll
lincl
Im‘o
of

13m

13C]

The geometric Brownian motion with al and ,8] as drift and standard
deviation parameter and W, characterizing the Brownian process at time ‘t’ is

represented as:

9E]— = a1 dt+fl1dW,; andP1(t=O)=P10 (3-1)

P1
In the above price process, for a known initial price ‘Plo’ and price

parameters a1 and ,6] , prices remain uncertain during the operating horizon and

the buying firm pays the uncertain prices over time.

The deterministic long-term contract price process with a2 as the

expected price grth rate is represented as:

P
11: azdt ; and P2 (t = 0) = P20 (3.2)

P2

The above price processes are continuous in time. This implies if at any
instant one of the prices is greater than the other, then the prices need to be equal
before the originally greater price becomes smaller than the other price. During
this period of unequal prices, it is obviously optimal to buy all requirements from
the less expensive alternative until the prices become equal. It is to be noted that
the total procurement expense corresponding to any possible optimal arrangement
(including the singular arrangements that involve only one of the arrangements)
involving two continuous price processes would be continuous (i.e., the minimum
of two continuous functions is always continuous). While the optimal
procurement strategy is obvious when prices are likely to be unequal over the

tactical review period, in a competitive stochastic market price context it would

31

 

be diff.
heins
thong

-' 0'.
51Lrn1ir\

beconi;
loner p
than th.
btning

such so

 

comple-
the put.
subseq;
Thus ll"
Optimal

det'iatic

 

GECISIOI
Context,
Cheaper
tiEinsact

appeah;

be difficult to decide which type of arrangement would remain optimal right after
the instant when the prices are equal. Besides, an instantaneous complete switch
(‘bang-bang’ solution) to the cheaper alternative may be impractical due to the
significant amount of transaction or penalty costs

A feasible strategy at this instant may therefore be to wait until the prices
become unequal and to completely switch (‘bang—bang’ solution) in favor of the
lower prices once the uncertainty of prices are resolved. However, waiting longer
than the instant when the prices become unequal would not be optimal since the
buying firm is losing an opportunity to spend less due to such waiting. In case of
such sub~optimal waiting, the amount of money that could have been saved by a
completely optimal solution would be governed by the duration of time for which
the purchases are made at an unfavorable price. Moreover, for such waiting and
subsequent switching, the procurement expense pattern will be discontinuous.
Thus the strategy to wait to switch completely in favor of lower price is not
optimal, and would result in discontinuous expense pattern. The extent of
deviation from optimality is governed by the threshold values when switching
decisions are made. These decisions are tricky to make in a stochastic price
context. Besides, as discussed earlier an instantaneous complete switch to the
cheaper alternative may be impractical due to the significant amount of
transaction or, penalty costs. To summarize, implementing the intuitively
appealing ‘bang-bang’ policy is difficult to implement and sub-optimal.

Thus in situations of competitive stochastic market price context, it is

difficult to identify and instantaneously switch to the expense-minimizing

32

 

Sll'iilt? "

(1f

descrip

ditlicu.

stochas

 

confinu
pattern
discus:—
Condir
requir.

for (if:

SOurc
0“ p1
Spen
disu

adj \

f0”

alternative over the planning horizon. The above phenomenon of ‘non-
determinism’ due to irregular behavior of market price at an instant is recognized
as the ‘local time’ problem in stochastic literature (Oksendal, 1996, pp. 138). The
issue is structurally similar to the infeasibility of ‘Stop-Loss-Start-Gain’ trading
strategy in financial literature (Carr and Jarrow, 1990). A mathematical
description of the issue is presented in the Appendix A.

Since the price realizations that make the ‘bang-bang’ policy optimal is
difficult to identify a-priori at each instant of transaction in such continuous time
stochastic problems, it is justified to derive the optimal sourcing strategy for
continuous expense over time. Accordingly in the following model, the expense
pattern is considered to be continuous over the planning horizon (0,1). As
discussed earlier, for the same starting open market and contract prices, the

condition a2 > al is required to be satisfied for the contract price premium
requirement. Though this is not a modeling requirement, this needs to be satisfied
for drawing meaningful insights about the sourcing arrangements.

Let, X , >0 be the total procurement expenses of the firm across the two
sourcing arrangements at time t2 0; let ‘u’ be the fraction of expense to be spent
on purchase at open market price ( P1) and ‘I-u’ be the fiaction of expense to be
spent on purchase at long-term contract price (P 2) at time t to minimize expense

disutility. It is assumed that at time ‘t’ the buyer can select and inexpensively

adjust the fractions allocated to market sourcing and long-term contract sourcing

based on expenses at time X ,. This renders the procurement expense X t to

follow the Ito’s diffusion process with Markovian control ‘u’. Accordingly, as the

33

bunng

 

nmetE
This to
moan

phrat

nmeh

 

 

 

men.c

COnsi
lSaS
0Ver

{Tac-

buying firm allocates expenses across alternative arrangements optimally over
time, the expenses and units procured from alterative arrangements are random.
This random behavior is governed by the underlying price dynamics and disutility
function of the buying firm that influence the optimal fractions of expenses to

allocate across alternative arrangements.

The differential equation for expense X , can be written as follows. If ‘ u ’

is the fraction allocated to market, then:

Furthermore, if N l, and N 2, are units procured from market and contracts

then, change in expense at an instant rs: dX, = N lthlt + NlthZt

dP“ dPZt
= uX, —— + (1 — u)X, (3.4)
Plt PZt

 

Using equations (3.1) and (3.2), Equation (3.4) can be written as:
dX, =uXta1dt+uXtfllth +X,(1-u)a2dt (3.5)
It may be noted that at each instant of time in the planning horizon under
consideration, the change in expense over ‘dt’ is a function of change in prices. It
is assumed that resources required to accommodate the change in expense dX,
over ‘dt’ are immediately and continuously available as the firm adjusts the
fraction allocated to alternative sourcing arrangements to minimize the cost

disutility. After each ‘dt’ the firm rebalances the fraction spent on alternative

sources in order to minimize its cost disutility across the sourcing arrangements.

34

 

lebdirf'

tll'l'tlll‘;

process
equaltt
ofunus

llIlIlS In

 

(Wiener
Conditic
the suit
depend
eXPEHSc
Pfice ¢
CXpreSS
maths”
COnside

paIIEm

Thus with continuous expense, the equilibrium condition at the time of
rebalance is the following: expense before reallocation of fractions between
arrangements is same as expense after reallocation of fractions at each instant.
This implies marginal rate of substitution (MRS) at the time of rebalance is equal
to ratio of prices from alternative sourcing arrangements. This would result in
changing the number of units sourced from a particular source. Thus the exchange
process should be such that the overall number of units after exchange is at least
equal to the overall number of units procured at the initial state (if overall number
of units procured is of significance). The modeling of explicit consideration of the
‘units procured to be equal to demand’ constraint is hard and complex.

Consequently the problem is formulated first without such constraint
(referred as the base model in subsequent discussion) and subsequently the
conditions for verification of sufficiency of units are derived. The conditions for
the sufficiency of units to be satisfied at any instant in the operating period will
depend on the cost disutility minimization objective that influence the fraction of
expense spent on alternative sourcing arrangements, the initial prices, and the
price dynamics of alternative sourcing arrangements. The mathematical
expressions for the above conditions are presented in Appendices B and C. The
mathematical formulation of the base model (i.e., without the explicit
consideration of the units sufficiency condition) to derive the optimal expense

pattern is as follows. The buying firm wants to maximize (minimize) the value

(disutility) of procurement expenses over a time horizon [t, T]. Let F (t,x;‘ ) be

35

the cor

disutilz‘

fractior:
0f proc

Clan

Procur:
fractio:

Bellmaf

gii'en ir

and has

Markov

 

the continuous disutility rate function at time t, and S (xT) be the terminal cost

disutility function at time T.
Then the overall (expected) cost disutility functional can be written as:

T
J" (t,x) = Et’x[IF(r,x;‘ )dr+S(xT)] (3.6)
t

where, E(.) is the expectation operator and x? is the expense at ‘t’ when

fraction of expense allocated to market is ‘u’.

The buying firm wants to maximize (minimize) the value (cost disutility)
of procurement expenses over the entire time horizon under consideration. Let,
C (t,x) be the minimum of cost disutility functional at time ‘t’ due to
procurement expenses. It is assumed that the Markovian control parameter i.e.,
fraction purchased, u e U would lie between 0 and 1. The Hamilton-Jacobi-
Bellrnan (HJB) equation is used to solve the above stochastic control problem.

The HJB equation for the relevant (expected) cost disutility functional

given in equation (3.6) is:

infiLuC + F(t,x“)}= o for t < T and x > O;C(T,x) = S(x) (3.7)
u

In the above equation the differential operator L“ uses the Ito’s Lemma

and has the following form:

u _aC ac 1 2 2 zazc
(L Clint)—E+x(a1u+a2(1—u))a+§fl1u x gx-E— (3.8)

For obtaining the optimal solution, Equation (3.7) needs to be solved for

Markovian control u.

36

Tb

pttm. .' ‘

Ewe-iii:

lllClCLi>

 

 

 

 

Slit C

{merit

F0r th

and is

Proble;

prOpen

Let, F(t,x“)=x" , n >1 in (3.7), where ‘n’ represents the cost disutility
parameter i.e., the higher then, the higher the cost disutility of the buyer.

Equation (3.7) is solved to obtain u(t, x) :

(a1 " a2 )Cx
.312an

 

u(t,x) = — (3.9)

It is to be noted that the disutility functional C(t,x) should be convex and
increasing in x which makes both C x and C xx (first and second order derivatives
with respect to x) positive, thus rendering u(t, x) nonnegative and valid.

Substitution of the expression for optimizing u(t,x) from (3.9) in (3.7)

yields:

(“1 ‘02)2CJE

2 =0;fort<Tandx>O;C(T,x)=S(x) (3.10)
2:61 Cxx

 

x" +Ct+a2xCx —

Equation (3.10) is non-linear in C(t,x) and is difficult to solve for general
S(.). Consequently, it is solved for a special type of increasing convex function
given by:

S(x)=x",n >1 (3.11)
For this, let the trial solution be:

C(t,x) = f(t)x"; (3.12)

While, the above power functional form is a mathematical convenience

and is structurally similar to the utility function used in wealth maximization

problems in financial literature (Korn and Korn, 2001), it has also the desired

properties from a practical standpoint. The convex disutility function (x") is

37

monot.
chrrac:
greater
Ftnhef
hmepi
procur.
cont et

fora;

PUrchd

varICS .

 

monotonically increasing in ‘x’, represents the buying firms’ risk aversion
characteristics very well and ensures that higher the procurement expense the
greater is the disutility and risk aversion (Eeckhoudt and Gollier, 1995).
Furthermore, it facilitates incorporation of alternative values for it over distinct
time phases to allow for varied disutility patterns of the buyer over the PLC of the
procured product. Equation (3.10) is solved for the special case of the increasing

convex function given in (3.11), andC(t,x) given in (3.12) and the solution
for C(t,x) is obtained as:

T
C(t,x):[e‘9(T") +e‘9’ je9’dr](x") (3.13)
t

2
(at -a2) '1

2/31’ (n -1)

 

where, 6 = azn —

Substituting C (t,x) in (3.13) into (3.10) the optimal control expression is
obtained as:

a2 ‘al

131’ (n - 1)

u*(t,x) = where, 17 >1 and valid range for u‘is 0 _<_ u. 51 (3.14)

If for specified parameter values, u. lies outside the limits of 0 or 1, then

it would imply, the total expenses (X ,) would be completely spent on contract

purchase or market purchase respectively.

It may be observed that the optimal control function for expense disutility
minimization in the above model is constant and the fraction allocated to market
varies directly with difference in mean grth rates of prices, and varies inversely

with the variance of random market price and disutility parameter. The control

38

function influences the characteristics of expenses and units procured from
alternative arrangements. Consequently, for specified patterns of price dynamics
and buying firm’s cost disutility parametern , the optimal control firnction, pattern
of expenses, and the units procured from alternative arrangements can be
obtained.

The above base model does not consider the ‘units procured to be equal to
demand’ constraint to maintain mathematical tractability. However, for specified
parameters the conditions for sufficiency units over the planning horizon is
derived in Appendices B and C without and with the ‘unit sufliciency’ constraint

respectively. As per the results in Appendix B, for specified parameters, there is a

minimum operating horizon of length, ‘ ti ’, beyond which the units procured

would always be sufficient if the expenses are allocated as per optimal fractions.

It is observed that ‘ t' ’ equals zero (i.e., units procured as per optimal fractions
will always be sufficient) when the starting prices of both the sourcing

arrangements are equal. However, for unequal but comparable range of starting

prices at any instant before ‘ t. ’ there is a probability that total units procured as
per optimal fi'actions with the expense X , may be less than units procured at time

9

‘0’. It is observed for reasonable parameter values ‘1' and likely shortage are

very small to be of practical significance. If the possibility of having such

shortfall before ‘ t. ’ is unacceptable, then the buying firm may make provision for
additional expense to meet the shortfall, which may be allocated among
alternative arrangements as per the optimal fractions (e. g., u‘, I-u'). Alternatively,

as demonstrated in the Appendix C, the constraint for sufficiency of units can be

39

incorpv

 

may s

denim:

 

model.
proces:
contint

incum

 

 

 

lhf’ red
due to

leadin:

incorporated into the model. However, as the results indicate, incorporation of

sufficiency of units constraint leads ‘u” to depend on stochastic expense (X ,)
and market price (P1, ); this would require continuous monitoring and control of

the expense pattern across alternative arrangements.

It is to be noted that even if the model minimizes the expense disutility, it
may sometimes lead the buyer to accrue surplus units beyond the specified
demand. This is counterintuitive, and may be explained as follows. As per the
model, units sourced from alternative arrangements depend upon both the price
processes and the (optimal) expense fractions while the expense remains
continuous in time. The continuity of overall expense implies, the ‘expense
incurred before reallocation between arrangements is same as the expense after
the reallocation’ at each instant of adjustment. As a result, savings that may occur
due to rebalance at an instant is not withdrawn and is reutilized in purchasing
leading to surplus purchase.

For reasonable range of parameter values, and length of (tactical) review
period, the surplus units are observed to be of small order. However, to avoid
such a situation in practice, the buyer may start a fresh operating cycle at suitable
intervals (i.e., tactical review periods) with a revised initial expense and (surplus)
inventory-adjusted demand to procure the required units while maintaining the
optimal fractions (u‘, I-u‘) at the instant. This is feasible as per optimal results
presented in Appendix C that suggest the expressions for optimal fractions are
independent of planning horizon effect. Alternatively the surplus units may be

sold back in the open markets (Martinez de-Albeniz & Simchi-Levi, 2005). The

40

finale?
to bet...
degree
The 5:
may 1':
Martin
Parent.
across

(31).,

 

flexih.

3311i

pIOCUr
of mar
for ap
findinjl
PLC 0
Stages.

Paramy

 

emergence of electronic markets provides opportunities for such sell back
opportunities in many industrial situations (Mc Kinsey & Company, and CAPS
Research, 2000; Martinez de-Albeniz & Simchi-Levi, 2005).

It also needs to be recognized that procurement as per the optimal
fractions would make expenses and units procured from alternative arrangements
to behave randomly and such behavior is a function of the price dynamics, and the
degree of risk aversion that influence the disutility function of the buying firm.
The stochastic behavior of units to be procured from alternative arrangements
may thus necessitate the contracts to be of flexible types (Lariviere, 1999;
Martinez de-Albeniz & Simchi-Levi, 2005). In a given circumstance for specified
parameter values, the expected magnitude of procured units and their variance
across alternative arrangements can be computed as per the formulae (28), (29),
(31), and (32) in Appendix B. These results can be used for designing appropriate

flexible contractual arrangements.
3.3 Illustration of the model application

The model can be used for finding the optimal sourcing strategy in varied
procurement contexts that can be described in terms of specific price parameters
of market and contract price processes and the buying firm’s risk attitude. Thus
for appropriately chosen parameter values, the model may be employed for
finding the optimal strategy both in a specific stage or over multiple stages of the
PLC of the procured product. The application of the model is illustrated in three
stages. In the first stage, the impact of changes in price patterns and cost disutility

parameters of the buying firm on patterns of change in optimal fi'actions allocated

41

to ma"-
charm:
the Pit
degree

tactical

 

r‘ r .I_r_ll

-—‘

/-/,/./ ,/ / / / / ,/:.

r”, g/_ L

'/ /

 

to market (u‘) is illustrated separately. In the second stage, the implications of
characteristic price patterns and cost disutility factors across different stages of
the PLC on (optimal) sourcing pattern are demonstrated. Finally, for a specific
degree of cost disutility, the implications of price processes, and the duration
tactical review period on units sourced from alternative arrangements are
evaluated across early, intermediate and late stages of the procured product’s life
cycle.

The hypothetical data in Table 3.1 and Table 3.2 are used for the above
analyses. While, the data are arbitrary, these are stylized to represent the typical
contract and market price dynamics (Klein and Leffler 1981; Horowitz, 1986;
Smith et a1. 1999) and relative cost preference of firms (Reed, 2002, pp. 91;
Thorelli and Bumnet, 1981).

Table 3.1: Illustrative data representing the price dynamics across the PLC stages

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

State Expected Contract Difference Standard Expected Contract Standard
of the market price in contract deviation market price: deviation
market price annual and market parameter price: of market
across annual growth price of market initial initial price
the grth rates growth rates price price: price:
PLC rates ’ al' ‘ a2' ‘ a ’ process $100 $100
stages ' 151'
l -0.25 -0.2 0.05 0.75 77.88 81.87 52.70
2 -0.232 -O.185 0.047 0.67 61.75 64.73 36.86
3 -0.214 -0.17 0.044 0.6 49.86 52.10 26.50
4 -0.l96 -0.155 0.041 0.54 40.98 42.70 19.60
5 -O.178 -0.14 0.038 0.49 34.30 35.63 14.96
6 -0.16 -0.125 0.035 0.45 29.23 30.27 11.80
7 -0.142 -0.11 0.032 0.42 25.36 26.18 9.66
8 -0.124 -0.095 0.029 0.39 22.40 23.06 8.02
9 -0.106 -0.08 0.026 0.35 20.15 20.68 6.54
10 -0.088 -0.065 0.023 0.3 18.45 18.88 5.19
11 -0.07 -0.05 0.02 0.25 17.20 17.55 4.07
12 -0.052 -0.035 0.017 0.2 16.33 16.61 3.13
13 -0.034 -0.02 0.014 0.15 15.79 16.01 2.30

 

 

 

 

 

 

 

 

42

 

pnce .
and n;

are C'.

(I)

Coher

lhe eh

prlCe.

 

 

Thus in the hypothetical data in Table 3.1, a positive contract and market
price change rate differential is maintained to account for ‘contract premiums’,
and the ‘contract premiums’ and standard deviation parameter of market prices
are set to decline. The prices are constructed in line with approaches suggested in
Cohen and Agrawal (1999) and are graphically represented in Figure 3.1.

Figure 3.1: Characteristic (hypothetical) price pattern across different stages of
the PLC

 

  
   

 

‘ Price pattern ,1
I
l
1401X1j r
I I
120.00 J f
1oorxr— i
66% confidence i
j 9’: 30-00 “ interval :
. O |
A g 60.00 1. i
i “- l l
40.00 a l

201x1l "““‘--.._____‘_¥

1

 

 

12345678910111213

l State Of the market I +Expected market price
. across PLC stages

 

! P0=$100)
+ Pontract price

P0=$100)

|
i 0.00~I~—r—-r-—— .'“'—°*‘r . w r . r . .
l
—-— rketpricelimits_ _. i

 

 

To illustrate, using the formulae (24) and (26) for prices in Appendix B,

the expected market and contract prices are first constructed with an unit initial

price of $100, specified growth rates of -0.25 (a1) and -0.2 (02 ), and standard

43

 

 

 

Table
fractio-

:55;
the
marks
3Cross
the P1

Stages

arrrriiitr

-‘/
"J/
r

/

l

{3
/

0—4
DJ

deviation parameter of 0.75 (,81) for a planning horizon of one year. At the end of

one year, the contract price is equated to the expected market price (e.g.,
indicating the result of renegotiation); these negotiated prices are subsequently
used to compute the prices for the second year using the parameters

((11, ,6] and 0:2) for the second year. The process was repeated for each of the

subsequent years. The total duration of the PLC until the maturity stage in the
hypothetical dataset is considered to be of 13 years. Table 3.2 presents the cost
disutility factors (n) for two scenarios (cases) of buying firm’s preference over the
13 years of PLC, and the corresponding optimal solutions.

Table 3.2:. The impact of price pattern and cost disutility factors on optimal
fraction (u ) of expense spent on market sourcing

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

State of Difference Standard Cost Cost Fraction Fraction spent
the in contract deviation disutility disutility spent on on market
market and parameter factor ‘n’ factor ‘n’ market sourcing (u')
across market of market (case 1: (case 2: sourcing (constant
the PLC price price varying constant (u.) disutility
stages grth process disutility disutility (varying factor n = 2)
rates ‘ a ’ ' fll' factor) factor) disutility
factor)
1 0.05 0.75 1.10 2.00 0.89 0.09
2 0.047 0.67 1.20 2.00 0.52 0.10
3 0.044 0.6 1.30 2.00 0.41 0.12
4 0.041 0.54 1.40 2.00 0.35 0.14
5 0.038 0.49 1.50 2.00 0.32 0.16
6 0.035 0.45 1.60 2.00 0.29 0.17
7 0.032 0.42 1.70 2.00 0.26 0.18
8 0.029 0.39 1.75 2.00 0.25 0.19
9 0.026 0.35 1.80 2.00 0.27 0.21
10 0.023 0.3 1.85 2.00 0.30 0.26
11 0.02 0.25 1.90 2.00 0.36 0.32
12 0.017 0.2 1.95 2.00 0.45 0.43
13 0.014 0.15 2.00 2.00 0.62 0.62

 

 

 

 

 

 

 

44

 

3CCOUI‘.
are 561

pro-tit.

Optimal fractlon 'u' of expense

 

It may be observed that the cost disutility parameters are greater than 1 to
account for buying frrrn’s risk aversion; and, cost disutility parameters in case 1
are set to increase to represent the buying firm’s increasing concern for cost as the
product advances in the PLC.

For first stage of analysis, the base case considers a cost disutility factor

(n) of 2.00, spot (market) price standard deviation parameter ( ,61) of 0.75, and

expected price change rate difference (a2 —or1)of 0.05. Subsequently, the

patterns of optimal fiactions to source from market are obtained by separately
changing cost disutility factor, standard deviation parameter of market price, and
expected price growth rates differences. These results are plotted in Figure 3.2,
Figure 3.3 and Figure 3.4 respectively. The results are data dependent; however,
similar computation can be performed using relevant contextual data to gain
industry or product specific managerial insights.

Figure 3.2: Optimal fraction of expense ‘u"on market sourcing for varying
disutility factor 'n' and constant price parameters (a2 — a1 = 0.05, ,6] = 0.75)

 

 

 

 

1.00 |
0.80 « .
. t
0.40 t
0.20 ~

on market sourcing

A A
V

A
v A
v v v

0'00 T l I fl T I T r r— r v r v
1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.75 1.80 1.85 1.90 1.95 2.00

Cost disutility factor 'n'

 

 

 

Optimal fraction 'u' of expense

 

 

l
l

 

45

1 : ()1

b
.

figure 3.3: C
market price 5

Ian-Cl

.iJ r are EU no «u rt I i
tommmo Dom/(r mmmmzo
a 0 4. u .U 0. 0. 0. 0. M nnuu.
c.0530. easel»: CO .3. 3 m mu. 0 95
OICOQIO $0 C0309: .lE-uno We 6. m 0 L p: 0|
0: .C a 8 IE :0 .3. C0209... .IE‘uQO
.l n N
F p p

Figure 3.3: Optimal fraction of expense ‘u”on market sourcing for varying
market price standard deviation ‘ ,61’, and constant price grth rate differential

(a2 — a1 = 0.05) and cost disutility factor (n=2)

 

 

 

l
-l

 

_s

O

O
f

. .0 _o .
or on
o o

'u' on market sourcing
O
.A
O

 

 

 

 

Optimal fraction of expense

0.00 . ' . . . . . . . l
0.20 0.25 0.3 0.35 0.39 0.42 0.45 0.49 0.54 0.6 0.67 0.75 l
l

i Market price standard deviation

 

Figure 3.4: Optimal fraction ‘u" of expense on market sourcing with varying
price growth rate differential ‘ a ’, and constant standard deviation
parameter (,61 = 0.75) and cost disutility factor (n = 2)

 

 

l
I
l

 

 

 

 

 

2 0.10 .
g .
c 0.08 4 I
P l
.2 3 0.06 4 .
2

o
3 3 0.04 - l
E "’ ‘
- l
'3 0.02 ~ i
E
g. 0.00 . . . . . . . l

I 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.05 0.05

L Difference In contract price and market price growth rates

Lam __ l

 

46

D?!
l

31

bl

Cl

The plots for the illustrative dataset indicate the following general

patterns:

a) the higher the cost disutility parameter, the lesser the fraction of
expense spent on market for a given set of price parameters; this
implies buyers, concerned about uncertainty would procure less from
market,

b) the greater the spot (market) price uncertainty ( ,Bl ), the lesser the
fiaction spent on market when the price growth rate differential and
cost disutility parameter remain unchanged; and

c) the higher the expected price grth rate differential, the larger the

fraction sourced from open market when the market price uncertainty

(,61 ) and cost disutility parameter remain unchanged.

These patterns suggest the need for appropriate sourcing strategies for
different procurement contexts as characterized by cost disutility parameters and
price regimes. For example, it may be noted from Figure 3.2 that buyers with
lower cost disutility parameter have higher risk of adopting sub-optimal strategy;
this is because, the optimal fi'action curve has a higher slope in the region of lower
disutility factor representing higher sensitivity to inaccurate specification of
disutility parameter.

The implications of characteristic price patterns and buyer’s cost disutility
on optimal fractions to source from alternative arrangements across various stages
of the PLC is examined with reference to the illustrative data set in Table 3.1 and

Table 3.2. As noted earlier, the data are stylized to reflect the typical pattern of

47

price of a product and the likely increase in cost disutility as a product shifts from
early stages of PLC to the mature phases of PLC. The optimal fractions of
expense to allocate to market are plotted in Figure 3.5 and Figure 3.6 for cases of

increasing cost disutility and constant cost disutility respectively.

Figure 3.5: Optimal fraction of expense ‘u" on market sourcing with
characteristic price dynamics and cost disutility factors (i.e., n varies from 1.1 to
2.0) across different stages of the PLC

 

 

1.00
: E» 0.90«
l E 0.80«
l I: 0.70~
O O
‘550m~
: g g 0.50 -
‘6 3 0.40 —
E 0.30 -
g 0.20«
i 8 0.104
L 0.00

 

 

‘ 12345678910111213
Stateofindustry

 

 

It is observed that for the case of increasing disutility factor, the optimal
fractions for market expense have a spoon shape indicating phases of initial
decline (i.e., from u. = 0.89 at the early stage), a low level of (i.e., u. = 0.25 at the
intermediate stage) followed by an increase towards the later stages (i.e., u. = 0.62
at the late stage). However, as evident from Figure 3.6, the optimal fractions for a

firm having constant disutility factor across various stages of the PLC have a ‘J’

48

‘23
ch I
v Q
n A

5111;
‘

pattern indicating a gradual increase over time (i.e., from u' = 0.09 at the early
stage to 0.62 at the late stage for disutility factor n = 2).
Figure 3.6: Optimal fraction of expense ‘u.’ on market sourcing for characteristic

price dynamics across different stages of the PLC and for constant disutility factor
(n = 2)

 

 

 

0.70 -. — ,_,,-______ - 3 ,
0.60' l

0.50 1

 

0.40. '

 

0.30 t,
0.20 t

0.10

 

Optimal fraction 'u' on market sourcing

 

 

 

0.00i—_——. 4 .- . . .
12345678910111213

i State of Industry J

 

 

 

 

Comparison of Figure 3.5 and Figure 3.6 indicates for specified price
dynamics, optimal fractions to spend on alternative arrangements are dependent
on buying firrn’s degree of risk aversion that is represented by the cost disutility
factor. For given price dynamics in a planning period, a firm with higher cost
disutility would spend proportionately less on market arrangement.

The above patterns have a variety of implications. These indicate the
advantage of sourcing from open market in the early stages of the PLC
(introduction to early grth stages) that are characterized by high contract and

market price grth rate differential, high standard deviation of market prices and

49

lot

501

if

lower cost disutility. The plots highlight the advantage of contract-price based
sourcing during the intermediate (moderate price growth rate differential) stage
when long-term suppliers are likely to charge a reasonable price premium over the
moderately uncertain spot (market) price due to competition. It also suggests the
increased importance of spot (market) price based sourcing towards the later stage
of PLC when the product becomes standardized and functional as competition
intensifies, and companies become more cost conscious.

The impact of contract and market price dynamics and duration of tactical
review periods on units sourced may be evaluated by computing the mean and
standard deviation of number of units sourced from alternative arrangements at
different times using the formulae 28, 29, 31, 32 specified in Appendix B.

Illustrative results of such computation for the price grth rates (a1 and a2)
and standard deviation parameters ( A) across early (state 1), intermediate (state

7) and late (state 13) stages of market state specified in Table 3.1 are given in
Table 3.3 and are plotted in Figure 3.7. These results are obtained for demand of
100 units, and starting prices of $100.00 at the beginning of planning horizon
across each of the stages for a risk averse firm with n = 2.

The results indicate that the average total units sourced from both the
arrangements in each of the stages are sufficient to meet the demand, but increase
with the duration of tactical review period (i.e., for the early stage, the average
units sourced increased from 100.23 for t = 0.05 to 106.24 for t = 1.0; for the late
stage, the average units sourced increased from 100.03 for t = 0.05 to 100.54 fort

= 1.0). This suggests that the longer the review period the higher the expected

50

surplus, and the magnitude of expected surplus is more in the earlier stages of the

PLC as compared to the later stages of the PLC due to higher volatility of market

prices in the earlier stages that presents opportunity to buy at lower prices.

Table 3.3: Impact of the price pattern over time on units sourced from alternative
arrangements over time (Demand = 100 units, n=2, P10: 100.00, P20= $100.00,

X0 =$10000.00)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Units sourced from alternative arrangements at various times
State of time in year
Industry =0 t=0.05 t=0.25 t=0.5 t=1
Market Cont- Market Cont- Market Cont- Market Cont- Market Cont-
-ract -ract -ract -ract -ract
Eam Stagflfl
Mean 8.89 91.11 9.14 91.09 10.22 91.0 11.75 90.91 15.53 90.71
1
Standard 0.00 0.00 1.44 1.36 4.14 3.03 7.97 4.28 20.94 6.03
deviation
Intermediate Stage (7)
Mean 18.14 81.86 18.30 81.84 18.93 81.7 19.76 81.62 21.51 81.39
4
Standard 0.00 0.00 1.42 1.39 3.42 3.11 5.31 4.39 9.04 6.17
deviation
Late Stage (13)
Mean 62.22 37.78 62.26 37.76 62.44 37.7 62.65 37.62 63.08 37.45
0
Standard 0.00 0.00 0.79 0.79 1.78 1.76 2.53 2.47 3.63 3.47
deviation

 

 

 

 

 

 

 

 

 

 

 

 

Accordingly, even when the buying firm minimizes the expense disutility

there is a probability of surplus due to higher sourcing from market arrangements.

These surpluses may be traded back in the open market for additional indirect

savings or, may be carried over to the subsequent period and the buying firm may

procure units after adjusting for any such surplus from the previous period. The

results show that the variability of units sourced from alternative arrangements

declines towards later stages in the PLC and increases with the duration of tactical

review period, and the variability is more pronounced for open market purchases.

51

Funh:
marginally d;
90.71 fort =
market Milli.
results highl;
renegotiation
firm and the :
buying firm r
contracts for
The formulas
units sourced
design such c

Figure 3.7:

arrangements
factor n : 2

 

Units Sourccd

Furthermore, the average units sourced from contractual arrangement
marginally declines with longer review periods (e.g., from 91.09 for t = 0.0 to
90.71 for t = 1.0 in the early stage); this is due to the higher sourcing from open
market arrangement, which is expected to provide a price advantage. These
results highlight the significance of quantity flexible contracts and periodic
renegotiation of contracts at suitable intervals such that the expectations of buying
firm and the supplier firm are appropriately matched over time. For example, the
buying firm may renegotiate a long-term contract as a combination of short-term
contracts for reducing a variance in purchases made from alternate arrangements.
The formulae presented in Appendix B for expected magnitude and variance of
units sourced from alternative arrangements would enable the buying firm to
design such contracts for varied tactical review periods.

Figure 3.7: Impact of the price pattern on units sourced fi'om alternative

arrangements over time across different states of the industry for cost disutility
factor n = 2

 

 

 

 

 

 

 

 

 

 

 

1 Units sourced from alternative arrangements
100.00 — — ——- ~_ __-___._._
i n
I 5 I—————I——I——-——l——I
l . - >e -’ x >< r: I —-—~-n.
3 80"” —-—Market (early)
2 H——x—*———x i
5;, 60°00 “ +Contract (early) l
. l
1% 40'00 7 "'—’—’—’——' —-—Market (intermediate)i_
D ’____. I
20°00 ‘ ;__.___-____./l —-—x———Contract r
1 intermediate) ’
0 0.05 0.25 0.5 1 1
Time + Contract (late) '5

l
I ,t
I

.L

 

 

52

The increase in emphasis on market sourcing as evident fiom the optimal
results, seem to conform with the PLC based framework of Rink and Fox (1999)
that recommends a shift in emphasis of purchasing approaches for key
components that have a close correspondence with the PLC of the end product:
gradual paring of supply base (in favor of reduced number of suppliers) as the
products progresses towards growth stages, emphasis on contractual arrangements
in intermediate/ growth stages for assured supply, and increased sourcing from
spot market to benefit from competitive market in later stages of PLC.
Accordingly, the changing pattern of optimal fractions for varied disutility
patterns as suggested in Figure 3.5 may be explained in terms of contextual
characteristics and buying firm’s likely risk attitude in the three major phases of
PLC i.e., introduction-early growth, late growth, and maturity phases.

In the early stages, technological and customer preference uncertainty
would be high; there may not be adequate number of capable suppliers for
contractual arrangements to meet buyer’s (unsteady) requirements. While, this
may lead to higher price differential between contract price and spot market price,
the buying firm may be relatively less risk averse and more willing to try with
multiple suppliers due to lesser degree of standardization of the product.
Therefore, some purchase from open market at this stage would facilitate supplier
assessment, and would foster competition among potential suppliers. This would
drive down the price differential and standard deviation of market price.
Moreover, it may encourage suppliers to develop appropriate capabilities to match

buyer’s expectations thus setting the stage for long-term contract arrangements

53

towards the intermediate (growth) phases that are typically characterized by
product availability uncertainty. The decline in fractions on contract purchase in
favor of open market purchase as the product advances from intermediate phase to
mature phase may be justified on the basis of higher cost sensitivity and price-
based competition towards later phases that reduces price growth rate differential
and spot (market) price uncertainty.

In parctice such reallocation is plausible only if the market price becomes
truly competitive with highly capable suppliers, reduced supplier search cost and
switching cost, and stagnation in innovations such that the product becomes more
or less a standard product. Consequently, the pattern of sourcing will be a
function of the duration of various stages of the PLC that may be governed by
technological changes, market competition, and emergence of fi'ictionless
efficient markets.

The degree of smoothness of the optimal fraction curve would indicate the
severity of the impact of transitions across different phases of the PLC that has
implications on designing appropriate arrangements and organizational
capabilities for optimal sourcing. Thus to increase allocation of share to market
arrangement with minimal severity may necessitate the buying firm to proactively
indicate the contract suppliers about the emphasis on open market purchase
towards later phases, to stimulate timely competition among suppliers such that
the market becomes efficient with reduced cost without any compromise on non-
price attributes, and to train procurement managers with capabilities for tracking

and forecasting of open (spot) market prices, designing of flexible contracts, and

54

 

Valuation of ‘l
increasing st;
howex‘er be a‘
to match the
Anheuser Bu
cans in the b
optimal solut

and market

arrangements

 

valuation of a portfolio of sourcing arrangements. It is to be recognized that the
increasing shifl towards open market arrangement towards the later stages may
however be avoided if the buying firm can induce the existing contractual supplier
to match the prices in the competitive open market periodically. To illustrate,
Anheuser Busch adopts such an approach in sourcing aluminium for producing
cans in the beverages industry (Wu and Kleindorfer, 2005). In this scenario, the
optimal solution in the model, which reflects the integrated valuation of contract
and market offers, can be used as a benchmark to negotiate contractual

arrangements.

55

Chapt

Sourcing
While Pr

4.] The pr
The r
of expenses
arrangement
procured are
Specified int
equal 10 a 5
meet the re
gleatet thar
reqmrell‘tent
accrual PIC:
back in the
if there is z
m°d€ling t
YESuh in E
Comm of
Chaplet

As

Chapter 4

Sourcing Strategy For Minimizing Expense Disutility
While Procuring The Exact Number of Units

4.1 The problem context

The model described in the previous chapter identifies the optimal fraction
of expenses to be allocated between contract and open market sourcing
arrangements to minimize the expense disutility while ensuring that the units
procured are at least equal to the requirement. In the model, it was found that for
specified initial market and contract prices, the planning horizon should be at least
equal to a specified duration to ensure that the units procured are sufficient to
meet the requirement. Furthermore, it was observed that for planning horizons
greater than this minimum, the total units procured might exceed the original
requirement due to the stochastic nature of market price process. While such
accrual presents an opportunity for additional indirect savings by way of trading
back in the spot market, a buying firm may want to avoid such accrual of surplus
if there is a net cost in handling such materials. Consequently, there is a need for
modeling the optimal sourcing problem such that the total units procured do not
result in any accrual of surplus at any time. The corresponding models in a
context of continuous and cost-less readjustment process are developed in this
chapter.

As discussed in the previous chapter, the absolutely optimal procurement

strategy would have continuous expense and would be of ‘bang-bang’ type (i.e.,

56

to buy all rec
equal) if the
times of trar
decide “hie
equal. Since
difficult to
time $10cha:
continuous
expense pat
context as t]
A r
Strategy int
Open mark
PTOduct m
minimiziné
modding
expllCltly (
maximizaI
(billet, 1(
from a Set
The so“
‘laclical‘ '

Agrawal

to buy all requirements fiom the less expensive alternative until the prices become
equal) if the arrangement with favorable prices can be known in advance at all
times of transaction. However, in a stochastic market price context it is difficult to
decide which type of arrangement would remain favorable when the prices are
equal. Since the price realizations that make the ‘bang-bang’ policy optimal is
difficult to identify ‘a-prz'orz" at each instant of transaction in such continuous
time stochastic problems, it is justified to derive the optimal sourcing strategy for
continuous expense over time. Accordingly, the current model also considers the
expense pattern to be continuous. The model is developed for the same problem
context as that in the previous chapter and is stated below for brevity.

A risk averse buying firm considers adopting a dual (combined) sourcing
strategy involving a mix of long-term contract with ‘deterministic prices’ and
open market arrangement with ‘stochastic prices’ to procure a standardized
product over time to satisfy specified levels of demand over time while
minimizing the expense disutility. As noted in the previous chapter the proposed
modeling framework is distinct from existing analytical research in that it
explicitly considers the buying firm’s concern for uncertainty in terms of a utility
maximization perspective (Keeney and Raiffa 1976, pp. 148; Eeckhoudt and
Gollier, 1995, chapter 4, pp. 39). The firm has available to it a pool of suppliers
from a set of certified vendors from which it must choose its sources of supply.
The sourcing arrangements are implemented through a series of ‘strategic’ and
‘tactical’ decisions in lines similar to standard industry practices (Cohen and

Agrawal, 1999; Martinez de-Albeniz & Simchi-Levi, 2005). It is assumed that at

57

  
  
  
  
  
  
  
  
  
  
  

the beginning
while accoun
basis of supp‘.
contract term
the premium
able to avoid
Smith et al. 1
contract pric-
increase (dec
of the Open
EXPCCIEd ma
Contract due
context the
arm“gt-them

Supplier arra

4'2 The m

Mat}
Process p1 is
and Stand ar
Thus the g0
dPI ,1 Pt
The determ

EXPCCted pr

the beginning of any ‘tactical’ review period, contract price patterns are chosen
while accounting for the issue of ‘contract premium’, which is justified on the
basis of supplier’s likely demand for such a premium for stability of supply as per
contract terms while absorbing various risks, and the buyer’s propensity to pay
the premium due to the lack of awareness about supplier’s true cost, and for being
able to avoid market price uncertainty (Klein and Leffler, 1981; Horowitz, 1986;
Smith et al. 1999). One of the ways to achieve this would be: for the same starting
contract price and (expected) open market price, the contract price is negotiated to
increase (decrease) at a rate higher (lower) than the expected growth (decline) rate
of the open market price. From a practical standpoint, this is reasonable as the
expected market price grth (decline) rate is likely to be lower than long-term
contract due to the moderating effect of competition. Accordingly, in this problem
context the optimal mix of two extreme sourcing arrangements (e. g., open market
arrangements and long-term contracts or partnerships) in the continuum of buyer

supplier arrangements is being determined for minimizing the expense disutility.
4.2 The model

Mathematically, the model can be described as follows. The market price

process P1 is described as a geometric Brownian motion with a, and A as the drift
and standard deviation parameters and W, characterizing the Brownian process.
Thus the governing equation is:

dirt/Pi = ardt+adWa andPi(0)=P10 (4.1)
The deterministic long-term contract price process P2 is described with a!2 as the

expected price growth rate. Hence, the corresponding equation is:

58

dip: P2 :
As discussed
condition a;
requirement.
for draxting

procurement

Let .

sourcing Sir;
oPen market
term contrac
in the Plann;
Change in 6‘
Prices. It is

expense OVC

the fractionj
each idl‘ til
(without Wit.
over the p13
is: expense

after realloc

COITeSDOnd

X |

I

 

sz /P2 = azdt; and P2 (0) = P20 (4.2)
As discussed earlier, for the same starting open market and contract prices, the
condition a2 > al is required to be satisfied for the contract price premium
requirement. Though this is not a modeling requirement, this needs to be satisfied
for drawing meaningful insights about the sourcing arrangements in typical
procurement contexts.

Let X , >0 be the total procurement expenses of the firm across two
sourcing strategies at time t2 0; and ‘ ul (t) ’ be the fraction of units to purchase at
open market price (P1) and ‘ u2(t) ’ be the fraction of units to purchase at long-
term contract price (P2) at time t to minimize the expense disutility. Furthermore,
in the planning horizon under consideration, for a given ‘ u1(t) ’ and ‘ uz (t) ’, the
change in expense ‘ dX, ’over infinitesimal time ‘ dt ’ is a function of change in

prices. It is assumed that resources required to accommodate the change in
expense over time are immediately and continuously available as the firm adjusts
the fractions allocated to alternate strategies to minimize the cost disutility. After
each ‘ dt ’ the firm can adjust the fractions sourced from alternative arrangements
(without withdrawing money fi'om the system) to minimize the expense disutility
over the planning horizon. Thus the equilibrium condition at the time of rebalance
is: expense before reallocation of fractions among sources is same as expense
after reallocation of fractions’ at each discrete time point.

Correspondingly, the following equations hold good at any instant ‘t’.

X: =“1(t)P1(t)+u2(t)P2(t) (4.3)

59

The
altemative st
planning ho:
horizon. the
lumps acros:
any addition
continuous ;
implications
Appendix E

continuous

 

Constant uh
Standard Ito
‘0 the folio“
Claim; FOr 1
[101 Change (
Thus if Cont
cortengndi
Proof;

The market

It is to be in

dX(t)=u1(t)dP1(t)+u2(t)dP2(t) (4.4)
1 = “l (t) + u2 (t) (4.5)
The above equations define the expenses, the units procured from
alternative sources at any instant ‘t’, and how the expenses change over the
planning horizon. It may be noted that at any instant of time over the planning
horizon, the buying firm can readjust the fractions procured continuously or in
jumps across the market and long-term contract arrangements without incuning
any additional expense. In the following discussion the analysis is limited to
continuous adjustment of the fractions procured from the arrangements. The
implications of adjustment in discrete jumps at an instant are presented in
Appendix D. It may be observed from the following analysis that for such
continuous adjustment, units procured fiom alternative arrangements remain
constant when equations (4.3), (4.4) and (4.5) hold good. The analysis involves
standard Ito Calculus (Oksendal, 1996) and is presented as a mathematical proof
to the following claim.
Claim: For fixed total purchase, units purchased from alternative arrangements do
not change over time in a planning horizon (0 S s S t) .
Thus if conditions (4.3), (4.4) and (4.5) hold good then u1(t) = u1(s) = u1(0) and
correspondingly uz (t) = uz (s) = uz (0).
Proof:
The market and contract price processes are defined as in (4.1) and (4.2).

It is to be proved that ul (t) = u1(0) <=> dul = 0.

60

Let the argument and the subscript ‘t’ of the functions be dropped in the following
discussion for notational convenience.

Equations (4.3), (4.4) and (4.5) can be rewritten as:

X = ulPl +u2P2, dX = uldPl +u2dP2 and 1: ul + u2 respectively.

(1X = uldPl + 142sz =u1dP1 + uszazdt (4.6)
Let Y = X/PZ, 171 = 101 /P2 ; Hence, die; : 1—’-1[(a1 -a2)dt + flldW)]

Then, at)? = d(X-P2'1) = Xd(P2'1) + Pz'ldX

=Xd(P2") + P2'1[u1dP1 +u2dP2]

=- 612de + 102—lump, + uzazdt

= u1(-a2171dt + PfldPl) , (since, )7 = ulljl + u2)

= uldFl (4.7)
Since, the market price is stochastic, for optimal sourcing, the (required) fractions
of units procured from arrangements at an instant should also behave

stochastically. Let the continuous change of units sourced from market

arrangement be represented as:
dul = dulc = A(t)dt + B(t)dW, (4.8)
Here, A(t) and B(t) are the drift and standard deviation parameters, and W,

represents the standard Weiner process.

X=u1}—’1 +112

:> d}? = uldP; +031 —1)du1c + duf ~dI_’1 (4.9)

61

 

Equating (4
Ozl-p} -1-jti.
=lpl -lld'u:

L'sing (4.8) ‘.

 

(Hi—’1 -1);
It may be oh
b0th.l(t)an\

restriction u

(4.3). (4.4).

“1“): mt»
This implie
intending 10

Would 50ch

 

The Optima}

 

the firm
cumbersom
disutility TU
aversion 0n

Lei F u.

I, “’hen th

Equating (4.7) and (4.9) one obtains:
0 = (7’1 —1)duf + duf ~dl_’1

=(Fl —1)duf + duf 471m1 - a2)dt + flldW,)] (4.10)
Using (4.8) in (4.10), and ignoring the higher order terms of dt it can be written:
0 = (131 -1)[A(t)dt + 8(0th ] + B(t)F1,61dt (4.11)

It may be observed that, since W, is random RHS of (4.1 1) can be zero only when

both A(t) and B(t) are zero. Thus dulc is zero, and correspondingly, for the

restriction ul (t) + u 2 (t) = 1 to hold good, one must have, duz = 0.

Hence, it is observed that units procured from an arrangement under conditions
(4.3), (4.4), and (4.5) remain unchanged over time, i.e.,

“1(1) = “1(0) and “20) = “2(0)-

This implies for price processes defined by (4.1) and (4.2), a buying firm
intending to minimize disutility of expenses while satisfying (4.3), (4.4) and (4.5)
would source a fixed number of units from a particular arrangement.

The optimal fraction of units to be procured from alternative arrangements when
the firm minimizes the cost disutility is derived next. The derivation is
cumbersome for general disutility function; consequently, results for a specific
disutility function are derived to illustrate the implications of degree of risk

aversion on optimal sourcing.
Let F (t,x;‘ 1 ) be the instantaneous procurement disutility of a buying firm at time

I, when the buying firm incurs a total expense of X t and chooses to procure a

62

 

  
   
  
   
  
  
   
 
  

fraction of n
lhe buflng
over the plar‘

disutility fur:-

Cu: (1.x) =

t
Where. E ( . )is

Let r" be th'
the cost dis
the buyer. .
is a mathen
used in the

2001)- Th0

fraction of u] from the open market arrangement over the planning horizon [t, T].

The buying firm intends to minimize the overall procurement cost disutility of
over the planning horizon. Accordingly, the minimum of overall (expected) cost

disutility fianctional can be written as

0<u1 <1

T
C“1 (t, x) = min Et’xl: IF(s, x?‘ )ds] (4. 12)
t

where, E(.)is the expectation operator.

Letxnbe the disutility rate function F(.) in (4.12), where n(> 1) and represents

the cost disutility parameter i.e., the higher then , the higher the cost disutility of
the buyer. As discussed in the previous chapter the above power fianctional form
is a mathematical convenience and is structurally similar to the utility function
used in the wealth maximization problems in financial literature (Korn and Korn,
2001). Though this functional form is a mathematical convenience, it has also the
desired properties from a practical and theoretical standpoint. The monotonically
increasing convex function characterizes the buying firms’ risk aversion and
ensures that the higher the procurement expense the greater is the disutility.
Furthermore it facilitates incorporation of alternative values for n over distinct
time phases to address the issue of varied disutility function of buyers over the

PLC. Equation (4.12) requires to be solved to obtain to the optimal u1(t,x). Since

obtaining the optimal u1(t,x) for general x” is cumbersome, it is derived for the

disutility function with n = 2 below to illustrate the approach and only the results

for the disutility fimction with n = 3 are presented. From a practical standpoint,

63

 

the power dr‘
the buting fr
Earlier in (It
dfl = Rita.
Let (a1 - u

dTOPPEd {mt

notational c.

Integrating '

 

 

the power disutility functions with n greater than 3 are unlikely to be specified by

the buying firm.

Earlier in (4.7) it was seen that d )7 = uldFl , where
4351 = Pittal — and: + flide.

Let (a1 — a2 ) =a , in the following discussion. Also, let the subscript 1 in Fl be

dropped and the limits of planning horizon be changed from [t, T] to [0, t] for

notational convenience.

Integrating both sides of d X = u1d1_’ , one obtains:

=> E: Jo +u11’Ii—P_o>=Z='fB—u1P_o+u1E
=u1("1%—1)+1—u1P_o+u1E
=(1_u1)+uIFt

For a disutility function with n = 2, the objective functional will be:

t_ t t _ t
minE( jxfds) = minE[ [(1 —u1)2ds + (a? Pfids + j2u1(1—ul)Psds]
“1 0 “1 0 0 0

t t _ t
=min[j(1—u1)2ds +u12 IEU’SZMS + 2a1 (1 —u1)IE(PS)ds]
“1 0 0 o

t t_ 2 t
=min[ [(1— u1)2ds + ul2 IP02e(2a+fl‘ ”are + 2al (1 — ul) fiewds]
“l 0 0 0

64

 

=miniil " it
"1

Taking the

-2(1 “u; ill,

:9 2111!“-

0
3U1:_

I-

The expres

can be Ct

cOTTESpondT
“Ellen as:

0
U1:

§\
3

Po‘[/

The
market Vat
mi’firsely “

To

 

The approf

 

2 l— 7’— 2 2 2
=min[(1-u1)zt+ u1( “1)0(em—1)+__”1_’b_2-(e(20’+fl1)’—1)] (4.13)
“l a (2a+,81)

 

Taking the first order condition for optimization:

 

 

 

21-2 _ 2u P2 2
—2(l-—u1)t+ ( “1)1’0 (ear_1)+ 1 0 (8(2a+fl1)t_1)=0
a (2a+,61)
4 P— 2u F5 2 —
:> zap—Mam —1)+—1°—2(e(2"+fl1 )’ —1)= 2t—fl(e“’ —1)
a (2a+,61) a
.. t—Eaa’ —1)
:>u1= _ a .7 (4.14)
at
t— 2P0(e —1)+ Po (8(2a+,612)t _1)
a (20+1312)

The expression for optimal fraction of units sourced from market at each instant
can be computed by construing the problem as zero planning horizon;
correspondinglyuf can be determined by using L’Hospital’s rule and can be
written as:

P—0(a2 —a1) (4.15)

 

“1 = — _
Po’rflf - 2(a2 —a1)1+ 2Pota2 — an

The above expression indicates that instantaneously optimal allocation to
market varies directly with difference in mean price growth rates, and varies
inversely with the variance of the stochastic price.

To verify that first order condition indeed gives the minimum, it is to be
ensured that the second derivative is positive.

The appropriate condition for second derivative of (4.13) may be written as:

65

[’0' |

 

D I
(20+;

The abOt'e
specific p;
between 1r.
With €Xp0i‘.
Thus {34.10

 

<0
“1‘: E (0

>1
Tnlines SllT
With n :3‘
“1.10::

— 2;; 2
4P0 (eat — 1) + ——0—(e(2a'+’61 )t — 1) > 0

21——
a (20: + ,612)

 

— 7 — 7
2 t+ 2P0 + P0 e<2a+fllzy > EEO—em + i—
a (zone?) a (2mm

—2 _
P 2 2 P
=> -——9—2—(e(2“+/’1 )’ — 1) +1 > —‘l(ea’ — 1) (4.16)
(2a + ,6] )t at
The above expression indicates that existence of minimum will depend upon the
specific parameter values. Since, as per market condition a (i.e., difference

between market price and contract price) is always negative, RHS and the part

with exponential term in LHS are always positive, convex and increasing in t.
Thus (4.16) is likely to be satisfied when ,612 > (a2 —a1). Correspondingly, the
likely range of values for u| is:

<0 lfaz <a1
uf' = e(0,l) if [312 >a2 —a1

>1 1f,612<a2—a1
In lines similar to above, the expression for optimal fraction for disutility function

with n = 3 (i.e., x3) can be derived as:

—Y+\/Y2—4XZ (417)

u; (t) = ;

 

2X

where,

66

 

 

It is to be p
Values, (h;
Contract p;
(4-14) and
EOVEmed t
the initial 1

It 11.
Units SOUr

pTOCUI'e fit

a SpeClfied

 

 

TX = —3z + 3.41003 — 9191002 + 9cP—O

t Y = 6t + 631002 426%

 

 

 

 

Z = 3cP_O—3t
i
e3(a+fl12)t _1
A = 2
3(a + '31 )
(2a+,612)t _
and, < B = e 2 1
(20: + ,61 )
at
C ___ e 1
0!
i

 

It is to be noted that, if u; lies out side the limits of 0 or 1 for specified parameter

values, then it would imply, all the procurements would be from either from
contract purchase or market purchase respectively. It is evident from equation
(4.14) and (4.17) that the optimal fraction of units to procure from a source is
governed by the price parameters, duration of planning horizon and the prices at
the initial instant.

It is important to note that as per the model, for continuous adjustment of
units sourced from alternative arrangements, the optimal fraction of units to
procure from a source depends on the duration of planning horizon; however, for

a specified planning horizon it remains constant during the planning horizon.
4.3 Illustration of model application

As discussed in the previous chapter, the model can be used for finding the
optimal sourcing strategy in varied procurement contexts that can be described in

terms of specific price parameters of market and contract price processes and the

67

buting fin:
model ma}:|
oi the pro;
application
of Optimal
over time .1
disutility {T
of the cha:
across tan.

T0)
market an;
it" With 11

studied, 1

 

 

index of
dlSUtility n
data set Us

Presented j

 

buying firm’s risk attitude. Thus for appropriately chosen parameter values, the
model may be employed for finding the optimal strategy both in a specific stage
of the procured product’s life cycle and over multiple stages of the PLC. The
application of the model is illustrated in two stages. In the first stage, the patterns

of optimal fractions (ul) sourced from open market arrangement are examined

over time across different stages of the PLC, when the risk averse buying firm has

disutility functions x" with n = 2 and n = 3. In the second stage, the implications
of the changing concern for cost (i.e., varied disutility) on (optimal) sourcing
across various stages of the PLC of the procured product is investigated.

Towards this end the pattern of change in optimal fractions to source from

market arrangement when the buying firm’s cost disutility fimction changes from

x" with n = 2 in earlier stages to x" with n = 3 towards the later stages of PLC is
studied. It is to be noted that since a disutility function with n = 3 has a higher
index of power, it would represent a higher degree of risk aversion than a
disutility function with n = 2 (Keeney and Raiffa, 1976). The hypothetical price
data set used in the previous chapter is also used for the above analyses and is

presented in Table 4.1 for reference.

68

Table 4.1: Illustrative data representing the price dynamics across the PLC stages

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

State Expected Contract Difference Standard Expected Contract Standard
of the market price in contract deviation market price: deviation
market price annual and market parameter price: of market
across annual growth price of market initial initial price
the growth rates grth rates price price: price:
PLC rates ’ a,’ ‘ az' ‘ a ’ process $100 $100
stages ' ,61'
1 —0.25 -0.2 0.05 0.75 77.88 81.87 52.70
2 -0.232 -0.185 0.047 0.67 61.75 64.73 36.86
3 -0.214 -0.17 0.044 0.6 49.86 52.10 26.50
4 -0.196 -0.155 0.041 0.54 40.98 42.70 19.60
5 -0.178 -0. 14 0.038 0.49 34.30 35.63 14.96
6 -0. 16 -0. 125 0.035 0.45 29.23 30.27 11.80
7 -0.142 -0.11 0.032 0.42 25 .36 26.18 9.66
8 -0.124 -0.095 0.029 0.39 22.40 23.06 8.02
9 -0.106 -0.08 0.026 0.35 20.15 20.68 6.54
10 -0.088 -0.065 0.023 0.3 18.45 18.88 5.19
11 -0.07 -0.05 0.02 0.25 17.20 17.55 4.07
12 -0.052 -0.035 0.017 0.2 16.33 16.61 3.13
13 -0.034 -0.02 0.014 0.15 15.79 16.01 2.30

 

 

 

 

 

 

 

 

The dataset has been stylized to reflect the effect of PLC on the price

pattern. The price pattern is graphically presented in Figure 4.1. As evident from

the Figure 4.1, the average market prices is lower than contract prices, and both

the market and contract prices, and the variance of the market prices decline over

time in line with the theoretical and empirical literature (Klein and Leffler, 1981;

Horowitz, 1986; Smith et al, 1999).

69

 

Figure 4
mePLC

buy fit
COrttrac
4.2 Dre
flunk.
fTactic,

hon'Zor

Figure 4.1: Characteristic (hypothetical) price pattern across different stages of
the PLC

_ i —— 1“ —5 Price pattern T

 

140.00 1 , l

120.00 1

  

100.00 «
66% confidence

  

 

 

5 30-00 ‘ interval
l 3:3 60.00~
1;
l 40.00~
l
I 20.00 .
cool .-...—. . .4

 

 

12345678910111213
State of the market _._ Expected market

across PLC stages +8" cetr(l::(t)=$100)
on a pnoe

 

 

 

. +thgil=<gfggcelimits l

 

For the above price pattern, a risk neutral buying firm should consistently
buy from the market as the expected market price is always lower than the
contract price. However, a risk averse buying firm would act differently. Table
4.2 presents the cost disutility factors (n) for two scenarios (cases) of buying
firm’s risk preference over the 13 years of PLC, and the corresponding optimal
fractions of units to source from market arrangement for a one year planning

horizon. The optimal fractions are computed using formulae (4.14) and (4.17).

70

Table-4.2: The impact of price pattern and cost disutility factor (n) on optimal

fraction (u; ) sourced fiom market

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

State of Difference Standard Cost Cost Fraction Fraction
the in contract deviation disutility disutility spent on spent on
market and parameter factor factor market market
across market of market ‘n’= 2 ‘n’= 3 sourcing sourcing
33.2.22“C ‘2222. £22... (“I )f‘” (“1)
, , ‘n’= 2 for ‘n’= 3
rates a 'fl,‘
1 0.05 0.75 2.00 3.00 0.08 0.04
2 0.047 0.67 2.00 3.00 0.09 0.05
3 0.044 0.6 2.00 3.00 0.11 0.06
4 0.041 0.54 2.00 3.00 0.13 0.06
5 0.038 0.49 2.00 3.00 0.15 0.07
6 0.035 0.45 2.00 3.00 0.17 0.08
7 0.032 0.42 2.00 3.00 0.18 0.09
8 0.029 0.39 2.00 3.00 0.19 0.09
9 0.026 0.35 2.00 3.00 0.21 0.10
10 0.023 0.3 2.00 3.00 0.25 0.13
11 0.02 0.25 2.00 3.00 0.32 0.16
12 0.017 0.2 2.00 3.00 0.42 0.21
13 0.014 0.15 2.00 3.00 0.62 0.31

 

 

 

 

 

 

 

 

The pattern of optimal fractions to source from the market for disutility

functions (x") with n = 2 and n = 3 are obtained using formulae (4.14) and
(4.17). The results are presented in Table 4.2 and are plotted in Figure 4.2. The
results are data dependent; however, similar computations can be carried out
using relevant contextual data to gain firm, industry or product specific

(managerial) insights.

71

 

Figure 4.2: Patterns of optimal fraction to source for disutility function with n = 2
and disutility function with n = 3 for a one year planning horizon in different
stages of the PLC

 

Optimal sourcing pattern overtime *
(Planning horizon = 1 year) i

0.80 l
0.60 . 1
0.40 -

020‘1

0.00~ . .......
12345678910111213

State 0fthe market across PLC [:o—disutility function with n= 2
stages

 

 

 

 

 

 

 

Optimal fraction of units from
market

l +disuti|ity function with n= 3 (

L Li!. ____________,.

 

The plots for the illustrative dataset indicate the following general
patterns:

a) ceteris paribus, the higher the risk aversion (i.e., higher the power of

disutility function), the lesser the fraction procured fi'om open market; this

implies buying firms with higher cost concern would procure less from the

open market,

b) ceteris paribus, the greater the ratio of market price uncertainty (,81 ) to

the difference in contract and market prices (growth rates), the lesser the
fraction procured from open market; consequently, for a specific degree of
risk aversion, contract arrangement is more important in the earlier stages
of the PLC and market arrangement gradually becomes more important in

the later stages of PLC,

72

c) the general characteristics of optimal trajectories are similar. Hence for

disutility functions with index ‘n’ between 2 and 3, the optimal fractions

may be obtained by appropriate interpolation. The results of such

interpolation are presented in Table 4.3 and Figure 4.3.

The above results suggest that the optimal sourcing strategies should be dynamic,
and need to be adjusted in accordance with the degree of risk aversion and price
pattern across different procurement contexts.

The implications of varied planning horizons for the characteristic price
patterns and buyer’s cost disutility function on optimal sourcing strategy in
different stages of the PLC can be examined in terms of the optimal results
obtained using formulae (4.14) and (4.17). These results are presented in Table

4.4 and Figure 4.4.

73

Table 4.3: Optimal fractions of units to source fi'om market for gradually varying
disutility function over the PLC stages (the fractions during transition phase are
computed by interpolation and hence approximate)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

State of Optimal Optimal Disutility function Optimal fi'actions
the market fragtion fraction (11?) from market
across the ( “1 ) for ( “1 ) for
PLC . . . . . .
sta es disutility drsutrlrty
g function function
n = 2 with n = 3
1 0.08 0.04 Disutility fimction with 0.08
n = 2
2 0.09 0.05 Disutility fimction with 0.09
n = 2
3 0.1 1 0.06 Disutility function with 0.11
n = 2
4 0.13 0.06 Disutility function with 0.13
n = 2
5 0.15 0.07 Intermediate between
disutility function with 0.15 + 0.07 _ 0 11
n = 2 and disutility _ '
function with n = 3
6 0.17 0.08 Intermediate between
disutility function with 0.17 + 0.08 _ 0 12
n = 2 and disutility — ‘
function with n = 3
7 0.18 0.09 Intermediate between
disutility function with 0.18 + 0.09 _ 0 13
n = 2 and disutility — '
L function with n = 3
8 0.19 0.09 Intermediate between 0.19 + 0.09 _ 0 14
disutility function with " '
n = 2 and disutility
‘ function with n = 3
9 0.21 0.10 Intermediate between 0.21+ 0.10 _ 0 16
disutility function with — '
n = 2 and disutility
\ function with n = 3
10 0.25 0.13 Disutility function with 0.13
K n = 3
11 0.32 0.16 Disutility function with 0.16
\ n = 3
12 0.42 0.21 Disutility function with 0.21
X n = 3
13 0.62 0.31 Disutility function with 0.31
L n = 3
74

 

Figure 4.3: Pattern of optimal firactions of units to source fi'om market for
gradually varying disutility function across the PLC stages

 

Optimal fraction of units to source

from market

 

Optimal sourcing pattern for one year planning horizon for

0.35

varied disutility over the PLC

 

(130 .
025 .
1120 —
015 .
040 . ,
1105 -

 

 

 

0.00
0

5

10

State 0fthe market across PLC stages

 

Table 4.4: Optimal fraction of units to source from market for varied planning
horizons in different stages of the PLC

 

Planning
horizon

Early state of
industry (1)

Late state of industry
(13)

 

Optimal
fraction
(u; ) for
disutility
fimction
with

n = 2

Optimal
fraction
(u; ) for
disutility
function
with

n = 3

Optimal
fraction
(u; ) for
disutility
function
with

n = 2

Optimal
fraction
(u; ) for
disutility
fimction
with

n = 3

 

0.08

0.04

0.62

0.31

 

0.05

0.02

0.62

0.31

 

0.04

0.01

0.62

0.31

 

0.02

0.00

0.63

0.31

 

 

\Oflmwr—t

 

0.01

 

0.00

 

0.63

 

0.31

 

 

75

Figure 4.4: Patterns of optimal fraction (u; ) of units to source from market for
varied planning horizons (the values represent results in Table 4.3)

 

I Optimal sourcing pattern for varied planning horizons

 

 

 

 

 

 

 

 

 

 

 

g 0.70
g E 0.60 « l—-—*——“——*—‘—‘* + disutility function w ith n l
'8 g 0.50 1 = 2 (early stage) 1 .
g g 0.40 . + disutility function w ith n
° 3 0.30 . I—I—H—I = 3 (em 5996)
E '8 0.20 . +disutility function w ith n l
g g 0.10 , =2(iate stage) i
. 8' 0 0.00 M.— —.—cubic disutility function
1 2 3 4 5 (late stage) '
Duration of planning horizon
l

 

 

It is evident from the results that the optimal fractions to source from
market arrangement diminishes fast with the duration of planning horizon in the
earlier stage of PLC as compared to the later stage of PLC. This suggests that in
the earlier stages, the risk averse buying firm can minimize the expected cost
disutility either by sourcing higher fractions form market with shorter planning
horizons or by sourcing lesser fractions from market with a longer planning
horizon. Any of these approaches would enable to maintain a balance between the
exposure to cost uncertainty and overpayment to the contractual supplier. It is
also observed that for similar durations of planning horizons the rate of decline is
faster for lesser degree of risk aversion (e.g., disutility function with n = 2) as
Compared to that for higher degree of risk aversion (e. g., disutility function with n
= 3). This implies the duration of planning horizon moderates the impact of the

degree of risk aversion. Consequently, the propensity to source from market and

76

to readjust the optimal fractions by firms with higher and lesser degree of risk
aversion would reduce with increase in planning horizon.

The above patterns have a variety of strategic implications. First, these
highlight the significance of mixed sourcing strategy to balance uncertainty of
cost with the opportunity to reduce cost according to the risk aversion
characteristics of the buying firm. The results illustrate the advantages of some
sourcing from open market even in the earlier stages of the PLC that are
characterized by high price growth rate differential, and high market price
variance. Such dual strategy enables the buying firm to present a competitive
tension to the contractual suppliers; this would discourage them to unreasonably
overcharge the buying firm. Second, the patterns highlight the relative importance
of contract based sourcing during the early/intermediate (with moderate price
growth rate differential) stage to ensure certainty of supply even when long-term
contractual suppliers are likely to charge a price premium over the expected
(competitive) market prices. Third, the patterns suggest the increased importance
of market price based sourcing towards the later stages of PLC (when the product
becomes standardized and functional) as competition intensifies, and firms
become more cost conscious. The decline in fraction of contract purchase in favor
of open market purchase (as illustrated in Figure 4.2) as the product advances
fiom intermediate phase to mature phase may be justified on the basis of higher
cost sensitivity and price-based competition towards later phases that reduces
price growth rate differential and market price uncertainty. However, in reality

such reallocation is plausible only if the market becomes truly competitive with

77

highly capable suppliers, and supplier search cost and switching cost are
insignificant. Thus the pattern of optimal sourcing is a function of price patterns
across various stages of the PLC that may be governed by technological changes,
market competition, and emergence of efficient transaction mechanisms. The
above results seem to conform to the framework of Rink and Fox (1999) that
suggests varied emphasis on alternative sourcing approaches along the PLC of
end product that often has a close correspondence with the PLC of the
components. The presence of close correspondence between the life cycles
influences the relative cost priorities of components over time and results in
paring of supply base as the products progress fi'om early stage to growth stage,
higher preference for long-term contractual arrangements in intermediate stages to
reduce fixed cost by enabling the suppliers to derive economies of scale, and
higher purchase fiom market to stimulate competition among suppliers in later
stages of the life cycles.

While the proposed model helps to identify optimal fi'action of units to
source from alternative arrangements across different stages of the PLC, the
extent of smoothness of the trajectory would indicate the nature of transition
between the stages and the need for planned changes in sourcing arrangements.
Thus reallocation of share to market sourcing arrangement over time may
necessitate the buying firm to proactively indicate or signal the suppliers about the
emphasis on price based open market purchase towards later phases. This would
stimulate timely competition among them such that the market becomes truly

efficient (i.e., reduced price uncertainty and growth rate differentials) with

78

 

reduced transaction cost and without the buying firm having to compromise with
other non-price attributes. As noted in the previous chapter, the increasing shift
towards open market arrangement towards the later stages may however be
avoided if the buying firm can induce the existing contractual supplier to match
the prices in the competitive open market periodically. In this scenario, the
optimal solution in the model would reflect the integrated valuation of contract
and market price processes and may be used as a mechanism for benchmarking

the expense patterns for minimizing the expense disutility over time.

79

 

Chapter 5

Optimal Sourcing Strategy With Proportional Switching
Cost

”5.1 The problem context

The models discussed in the previous two chapters for a typical (single
product) procurement context of a firm that adopts a dual (combined) sourcing
strategy involving a mix of long-term contract and open market arrangements
over continuous time consider the procurement expense function to be continuous
and ignore the presence of switching cost to readjust the units sourced between
arrangements. These models are appr0priate for a procurement context in which
the costs for switching allocations between alternative arrangements are
negligible and the firm procures continuously over time.

However, in practice there are procurement contexts in which a firm may
incur a noticeable cost of switching units procured between arrangements due to
contract penalty clauses or structural and infi'astructural change requirements.
Moreover firms may procure materials at a significant interval of time such that
the patterns of expenses are discontinuous in time. In extant literature, these issues
have been addressed to an extent primarily fi'om the perspective of a risk neutral
buying firm (Kouvelis, 1999; Li and Kouvelis (1999). There seems to be only one
study (Cohen and Agrawal, 1999) that examines the relative advantages of long
and short-term contracts from a risk averse buying firm’s perspective when the

buying firm chooses one of the alternative arrangement at a time. Thus this model

80

does not consider the possibility of simultaneous consideration of both
alternatives. Moreover their model uses the ‘mean-variance approach’ to evaluate
the cost; this approach is less general in a sense that the formulation implies the
variance is the only measure of risk and the utility function of that characterizes
the risk aversion is of quadratic nature only that does not satisfy the theoretical
relationships between risk aversion and magnitude of expense appropriately
(Eeckhoudt and Gollier, 1995).

In this chapter, the procurement problem with switching costs and
transactions in discrete time is modeled for minimizing procurement expense
disutility over a specified time horizon. In contrast to the ‘mean-variance
approach’, the model formulated here considers a power disutility function that
can address more general cases of risk aversion. In this model, it is assumed that
the costs for switching allocations between arrangements are proportional to the
magnitude of switch. Such a situation is plausible when the supplier charges a
premium for changing the volume sourced in the previous period or the buying
firm can translate the impact of switching cost as a percentage of the unit price of

the product.

5.2 The model

As discussed in earlier models, the market price process (P1) is continuous
and may be described as a geometric Brownian motion with al and ,61 as drift
and standard deviation parameters and W, characterizing the Brownian process.

Thus the governing equation is:

81

‘\

dP
—‘—=a1dz+,61dw,; and 101(0)=1D10 (5.1)

P1
The long-term contract price process (P 2 ) is a continuous process with 0:2 as the

expected price growth rate. Hence, the corresponding equation is:

fig: azdt . and P2 (0) = P20 (5.2)

P2
As discussed in earlier chapters, the contract prices would be higher than

the average competitive market prices over time (Klein and Leffler, 1981; Smith

et al., 1999). Correspondingly, it may be assumed thataz > a1 will hold good for

the same beginning prices. It is to be noted that the condition ‘ a2 > a1 ’ is not a

requirement for the development of the model here. However, it is a plausible
outcome of competitive market economics and may be used for drawing

meaningful theoretical managerial insights.

For the above context, the relative price process P=If/I§ may be described as:

dP = P [(cr1 — a2 )dt + flldW)] (5.3)
The above relative price process may be considered to evolve as an equivalent
binomial process with probability of moving up and down ( p, l — p) (Hull, 2003,

p. 407). Thus the relative price process for a time interval of At between two

successive nodes may be described as:

P: Poe(a1 —a2)Atifllx/E; (5.4)

where, p=1/2 and 1-p=1/2 and P0 is the ratio of prices at the beginning of

operating/planning horizon i.e., at t = 0. Thus a binomial tree with five time stages

82

 

will represent transactions over four time intervals in a planning horizon of one
year (Figure 5.1). For example, for a planning horizon of one year, these four

intervals may be considered as four quarters in a year.

Figure 5.1: Binomial Tree: The binomial representation of the evolution of
relative price process p = pl / p2 over four time intervals. The integers represent

the nodes corresponding to a specific price state with a specific probability of

 

 

 

occurrence.

15; p = 0.0625
7 16; p = 0.0625
17; p = 0.0625

3
8 18; p = 0.0625
19; p = 0.0625

1 9

20; p = 0.0625
21; p = 0.0625

1- p = 0.5 4
10 22;p = 0.0625

0
23; p = 0.0625
p = 0.5

11 24; p = 0.0625
25; p = 0.0625

5
l2 26;p = 0.0625
2 27; p = 0.0625

13
28; p = 0.0625
6 29; p = 0.0625
11
30; p = 0.0625
4 4 x At >

83

Let ‘u’ and ‘1 -u’ refer to the optimal fi‘action of units to be sourced from
the market and contract arrangements respectively at an instant. At any
transaction time stage ‘j’, the risk averse buying firm may change the pattern of

sourcing from (u j_1, 1—u j_1) to (u j, l— u j) . Correspondingly, the minimum of

expense disutility C (.) at any instant (j) of transaction may be described as:

C(uj—I’j)
= min{ [uj -1>j +1—uj 4.01110j +az)|uj—uj_1|]" +Ej-[C(uj,j+1)]} (55)

Here, E j[C (u, j +1)] refers to the expected minimum disutility at instant ‘j’ due

to future sourcing pattern and is zero for terminal nodes. Let 2.1 and 212 be the

coefficients of switching cost (expressed in percentage of unit cost) for a unit
change in allocation pattern across market and contractual arrangements at an
instant. ‘n’ is the cost disutility index, which can be of any value greater than 1 for
the risk averse buying firm. In the current model, the formulations are developed
for n = 2. Formulations for higher values of n can be carried out in similar ways.
It is to be noted that at the beginning instant (t=0), the buying firm does not incur
any switching cost because it starts the procurement arrangements fresh. Let ‘N’
be the number of time stages in the operational planning horizon when the
transaction occurs. So the optimal cost disutility equations for the entire

procurement process over the operational planning horizon becomes:

CE min u -P +1—u 7-
OSuo,...uNSl{[ 0 0 O]
(5.6)

N
41.13..j .12j +1—uj +(41Pj +2.2)“:j -uj_1 112)}
j=1

84

d_\t
the

C017

fon
tim.
dep-
pret
beyc
the c
COml

Opiin

 

Thus at t=0,

CE min {[110 .PO +1—u0]2 +E0{C(UO,1)} (5.7)

O<ll0 <1

For ISjSN—l wehave:

C(uj-laj)

-min{[u-.P +1—u+(2P +2)| -— - |]2+E-[C( '+1)]} (5'8)
_. J j lj 2 u] “1-1 ,1 u’J

OSuSI
For j = N at terminal time instant,
C(uN_1,N)=min{[uj-PN+1—uj+(21PN+22)|uj-u,(,_1|]2} (5.9)

It is to be noted that the above equations use the Bellrnan’s principle of
dynamic optimization in a recursive manner (Dixit and Pindyck, 1994, p.100). For

,

the above cases optimal ‘u - values at every instant of transaction are to be

J

computed for obtaining C (.). However, it may be observed that deriving closed
form expressions for optimal ‘ u j ’ values becomes difficult beyond the last two-

,

time stages of the planning horizon because the optimal ‘u j values are path

’

dependent. As a result, ‘u - values at subsequent stages are influenced by the

J
previous values; consequently, the expressions become unwieldy as one proceeds
beyond two stages fiom the terminal stage. Hence, as an alternative to applying
the dynamic optimization principle recursively for the entire operating horizon, a
combination of computational and analytical procedures is adopted for obtaining

optimal ‘ u j ’ values over the operating horizon.

85

 

acuon
sage
penul
tenni
anec1
pretn
begnt
expre
proce
cone

the p

the it

as tin

First, in line with the principles of dynamic optimization, the optimal
actions (i.e., transaction pattern ‘u N ’ and ‘l-u N ’) at the nodes in the terminal

9

stage are obtained for arbitrarily specified patterns of transaction ‘u N4 in the

penultimate stage. Subsequently, using the optimal expressions for ‘u N ’ at the

terminal state, optimal expressions ‘ uN_1’ at the nodes in the penultimate stages

are computed with respect to the arbitrarily specified pattern of transaction in the
previous stage ‘u N—2,- This process could have been continued until the
beginning instant ‘t = 0’. However, as discussed in the previous paragraph the
expressions become unwieldy after this stage. Consequently a numerical
procedure is adopted after this stage. As per this procedure, the disutility

corresponding to alternative actions (i.e., discrete values of 'uj 6 [0,1]') at each of
the previous stages (i.e., uo, u], u2, u j, ...u N_2) are evaluated to obtainC at

the initial instant I = 0. These results represent an optimal pattern of transactions
as time progresses.

,

It is to be noted that the computational approach used to obtain ‘ u J

values in stages previous to terminal two stages is robust but naive. As a result,
although in principle a purely numerical approach as described above can be used
for obtaining optimal values at all nodes from start to end, the exponential
increase in numerical operations with higher number of discrete values of

'u j 6 [0,1]' prohibits the use of the computational approach beyond three time

stages (i.e., 0-6 nodes). Thus in the absence of a more efficient alternative

numerical approach, the naive approach is used here for a limited range of

86

discrete

approac
implen:
present
pattem
princip
Conch:
The co

descrir

allocai

 

discrete values u j 6 [0,1]' for the first three time stages and an analytical

approach is used for the terminal two stages. The computational process is
implemented using a computer software code written in C-language. The code is
presented in Appendix E. The algebraic expressions for optimal transaction
patterns for terminal two stages are derived using the dynamic optimization
principle and are presented below.

Conditions for optimality at terminal nodes (i=N)

The condition for minimizing the expense disutility function at the terminal stage

described in expression (5.9) gives the optimal 0 S u N S1 as:

“N = “N—1+SN;

 

 

Where,
1+ P —1 . . . . . .
SN = ( N )uN '1 represents a posrtrve actron1.e., 1ncrease1n market
(Haul—’12 —P}
1 1+11 N

allocation with respect to u,.,_1 , and is valid when 0 S s S 1 — “iv—1 ;

 

 

or,
1+ P -1 . . . .
sN = ( N MIN-1 represents a negative actlon i.e., decrease 1n market
(1—4 Hub —P}
1 1-41 N

allocation, and is valid when — uN_1 S s < 0.

The above two conditions can be simplified as:

87

ut' ‘—'

Thus i

C(llt‘ I

Candi

The c
given
it“
Pm'io
follou

Let A

Ther

For (h

 

 

 

 

 

 

 

 

 

 

To if PN>1+12
l—xil
. T—ITQ lei-12
u =<u _ 1f <P S 5.10
N N 1 1+Ill N l—Zl ( )
1 if PNsl’fz
L 1+1]
Thus for any uN_1, u N can be computed. Correspondingly,
T
[14(2le+212)u,v_1]2 if PN>1+22
" 1
1—2 1+1
Cu_,N=<u_P +1—u_2 if 2<PS 2 5.11
(N1)(N1N N1) 1+1] [VF/11 ()
2 . 1—12
[PN+(21PN+22)(1—uN—t)l 1f PNS
T 1+1]

 

Conditions for optimalin at penultimate nodes (i=N—1)
The condition for minimizing the expense disutility at the penultimate stage as

given in expression (8) can be used to obtain the optimal 0 S uN_1 S 1 such that:
u N-l = u N—2 + sN_1; where, sN_1 is the change in allocation with respect to
previous pattern of transaction ‘ u N_2 ’. The expressions for sN_1 can be written as

follows.

Let Aj =(11Pj't'12), Bj =Pj—l, and C]: =Pj+Aj
The relative price states at terminal nodes may be represented as the following:

Pill (up state)

PN4<:::::::
Pg,” (down state)

For the above scenario, there can be six plausible scenarios.

88

(D
\e
to

m:

if actio

if actic

S\_1

1‘
-1

-
-—‘

 

Easel

1+12
1—21

 

Pilv, PIT/+l>

if action (3N4) is positive, then

SN—l
_12tBN .uN 2+1)(AN 1+BN 0+(AN+AN’")+uN 2t<ANi +(AN*‘)"-}1
214N1+BN0 +{(AN) HAN”) }

 

if action (SN_1) is negative, then

SN-l
_12(BN luN 2+1)(BN 1-AN 0+1AN+AN*‘)+uN 2{(AN)2+(A'N"1) )1
2(BN 1—AN 1) +{tANi 22MAN“) 1

If the results are inconsistent (i.e., sN_1des not have the expected sign) then no
change in allocation pattern is required. Thus uN_1 = u N—2-
Case 2

Pi, >1+29and Liz—<30“ «IL/ll
1-11 1+1] 1-11

 

if action (s,(,_l )is positive, then

SN— 1
_12(BN iuN 2+1)(AN 1+BN 1)+(AN+BN*‘)+uN 2{(Aiv) +(Biv+1)2 11
2(AN 1+BN 1) +{tANi +(BN“)2 1

 

if action (sN_,) is negative, then

SN—l
_12(BN iuN 2+1)(BN 1—AN- 11+(AN+BN*‘)+uN 2{(Aiv) 22MEN“) )1
2(BN 1—AN 1) +1(AN) HEN”) }

 

89

lithe rt

change

Cae3

 

l-Zw

 

 

1+2]

ifacu

ifactit

55"-]

[2

~‘N

lithe

If the results are inconsistent (i.e., sN_1des not have the expected sign) then no
change in allocation pattern is required. Thus u N—l = u N-2-

Case 3

_—___1;(Q<PIIV Pllv't'l< 1+12
1+1] l—lll

 

if action (3,“) is positive, then

SN—i
_[2(BN—1uN—2+1)(AN—1+BN—1)+(BN +13l+ll+uN2{(19N)2+(19iv+1)2 }]
214N1+BN0 +{(BN) HEN“) }

 

if action (sN_,) is negative, then

SN—l
_12(BN..1uN 2+1)(BN i—AN 1)+(BN+BN*‘)+uN 2{(Biv) MEN”) )1
2(BN-1—AN 1) +{tBNi +(BN’1)2}

 

If the results are inconsistent (i.e., sN_1des not have the expected sign) then no

change in allocation pattern is required. Thus uN_1 = u N—2-

 

 

 

Case4
1—12<P};,<1+/12andPi+1<1_’12
lie/Tl 1-11 1+2]

if action (3 N4 ) is positive, then

SN—t
12(BN-1uN-2+1)<AN1+BN-1)+(BN —AN+‘CN+‘)+uN zttBN ) MAN”) 11
2(AN-1+BN1) +ttBN ) 22HAN“) }

 

90

if action (SAM) is negative, then

SN—l
_121BN1uN 2+1)119N1—4N_.)+1BN—AN+1 CN*‘)+uN 2{(Biv) ’+1AN+‘)’ 11
21BN 1—AN_1)’+11BN)’+1AN+‘)’}

 

If the results are inconsistent (i.e., sN_1des not have the expected sign) then no

change in allocation pattern is required. Thus uN_1 = u N—2-

Case 5

1—22

Pl ,Pl+l <
N N 1+1]

 

if action (sN_1) is positive, then

SN—l
=—[2(BN luN— 2+1)(AN-1+BN—1)- (CNAN +441“ CivHHuN— 2{(Aiv) 2i~(/1iv+1)2 }]
214N1+BN1)’ +114N) ’+1AN*‘)’ }

 

if action (3N4) is negative, then

SN—l
_1218N-1uN 2+1)1BN1-AN_1>-1CNZAN+AN“CN+‘)+uN 2{(«‘1iv)2+(/1iv+1)2 11
213N-1-AN-1)’+11AN)’+1AN"‘)’}

 

If the results are inconsistent (i.e., s,,,_l des not have the expected sign) then no
change in allocation pattern is required. Thus u N_1 = u N_2 .

Case 6

1+ andPl+1< _112

Pi >
N 1—,11’1+,11

 

if action (sN_1) is positive, then

91

51-1

1:.

 

iiacu.

 

lithe
chang
[sun
solut
ofth:

acho

5.3 l

descr
prese
This
stage
101 b

lOtt'e

 

sN-l
= _L2(BN—luN—2 +1)(AN—1 +BN—1)+(Aiv —Ai)71CitTl)+uN—2{(AN)2 +(Ait71)2}]
21AN-1 +BN-1)’ +11AN)’ +1AN‘)’}

 

if action (sN_1) is negative, then

SN—t
= _121BN-1uN-2 +1)(BN—l -AN-1)+1AN -AN‘CN‘)+uN-2 11AN)’ +1AN‘)’}1
21BN-1 —AN-1)’ +11AN)’ +1AN‘)’1

 

If the results are inconsistent (i.e., sN_1des not have the expected sign) then no
change in allocation pattern is required. Thus u,(,_l = u N4 .

Using the above expressions for any u N_2, uN_l can be computed. Thus the
solution procedure evaluates optimal uN_1 and u N for alternative actions at each
of the previous stages (i.e., uo , u], uz , u j, ...uN_2 ), and for the combination of

actions that obtains C are the optimal set of actions.
5.3 Illustration of the model application

The application of the above numerical model for 4 time stages as
described in Figure 5.1 is illustrated with respect to the hypothetical data set
presented in Table 5 .1. This is the same dataset that is used in previous chapters.
This data represent the typical price pattern of a product over time across different
stages of the PLC. The price pattern is graphically presented again in Figure 5 .2
for brevity. It may be noted that the competitive market price is on an average

lower than the contract prices.

92

 

Table 5.1: Illustrative data representing the price dynamics across the PLC stages

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
 

 

 

 

 

 

State Expected Contract Difference Standard Expected Contract Standard
of the market price in contract deviation market price: deviation
market price annual and market parameter price: of market
across annual growth price of market initial initial price
the grth rates growth rates price price: price:
PLC rates ’ al' ‘ az' ‘ a ’ process $100 $100
stages ' [31'
l -0.25 -0.2 0.05 0.75 77.88 81.87 52.70
2 -0.232 -0.l85 0.047 0.67 61.75 64.73 36.86
3 —0.214 -0.17 0.044 0.6 49.86 52.10 26.50
4 -0.196 —0.155 0.041 0.54 40.98 42.70 19.60
5 -0.178 -0.14 0.038 0.49 34.30 35.63 14.96
6 -0. 16 -0. 125 0.035 0.45 29.23 30.27 11.80
7 -0. 142 -0.11 0.032 0.42 25.36 26.18 9.66
8 -0.124 -0.095 0.029 0.39 22.40 23.06 8.02
9 -0.106 -0.08 0.026 0.35 20.15 20.68 6.54
10 -0.088 -0.065 0.023 0.3 18.45 18.88 5.19
11 -0.07 -0.05 0.02 0.25 17.20 17.55 4.07
12 -0.052 -0.035 0.017 0.2 16.33 16.61 3.13
13 -—0.034 -0.02 0.014 0.15 15.79 16.01 2.30
Figure 5.2: Characteristic (hypothetical) price pattern across different stages of
the PLC
Tm _; _ _____ Price pattern
‘ 140.00
120.00 ~
100.00 «
66% confidence
.. 390° . interval
. 8
] 1?; 60.00 ~
0.
1 40.00 «
! 20.00 «
0.00 S

 

 

 

1

W2345678910111213

State of the market
across PLC stages

 

 

H“- b; *‘L‘T'

93

+ Expected market

grice (P0= $100)

gontract price
+hlb =$100)
rket price limits

i

 

T

is apg‘
relatix
V311 EC
suite?
at an
base
switc

anal}

Comr

PLC

in 161
Cch

Plam

alter

 

The computational illustration is carried out in two stages. First, the model

15 applied to a base case of sw1tch1ng cost ratio (SCR) (e.g., ’1—2) and initlal
1

relative price (Po). Subsequently, switching cost ratios and initial relative price are

varied to conduct sensitivity analysis to examine the impact of changes in

switching cost and initial prices. In all cases of analysis, the model considers 'uj

at an interval of 0.1 (i.e.,‘uj e [0, O.l,....1.0]') for the first six nodes. Thus the
base case considers beginning relative price of P0 =1 (i.e., 191(0)=P2(0)),
switching costs of 2,1 = 0.01, and 12 = 0.1 and At = 0.25 . Subsequently, sensitivity

analysis is carried out for a range of relative price and switching cost ratios (e. g.,

3};

P0 = [O.95,0.96,0.97,0.98,0.99] and A = [1,5,....50] when/l1 = 0.01 ) to examine

1
how the optimal sourcing patterns change in different stages of PLC.

Since, the supply market is likely to be more competitive in later stages,
contract readjustment costs are likely to be smaller towards the later stages of
PLC. Hence, the upper limit of switching cost ratios in later stages of the PLC is
kept smaller than in earlier stages. Thus the application of the model is illustrated

in terms of finding the optimal sourcing pattern for a risk averse buying firm with

expense disutility function (x") with n = 2, when the firm procures four times in a

planning horizon while incurring switching costs to reallocate units sourced from.

alternative arrangements.

94

cmum
mocm
plannii
allow
is be

pnce.
time

the m

nmge

In the previous two chapters, it was seen that continuous time models with
continuous expense that do not consider switching costs yield optimal fractions of
procurement across arrangements to be somewhat constant for a specified
planning horizon. In contrast, the current discrete time model with switching costs
allows the optimal pattern of procurement to vary over the planning horizon. This
is because the procurement pattern depends upon past history i.e., the state of
prices and previous decision patterns influence the optimal pattern in the current
time period. The characteristic pattern of optimal fractions of units sourced from
the market over time across early, intermediate and late stages of PLC for a varied
range of beginning relative price and switching cost ratios are presented in Tables
5.2-5.7. Graphical representations of the price and sourcing patterns are presented

in Figures 5.3-5.12.

95

 

 

 

 

 

Table 5 .2: Optimal fraction of units to source from market for beginning relative

price P(O) =1 and varied switching cost ratio (SCR) % with A, = 0.01at the

early market state '1'.

 

Relative price P = P1 /P2

overtime and switching cost ratio ‘SCR’

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nodes Relative Price SCR: 1 SCR: 20 SCR: 30 SCR: 35 SCR: 40 SCR: 50
0 1 0 0 0 0 0 0
1 1.44 0 0 0 0 0 0
2 0.68 1 1 1 0.9 0.6 0.3
3 2.06 0 0 0 0 0 0
4 0.98 0 0 0 0 0 0
5 0.98 1 1 1 0.9 0.6 0.3
6 0.46 1 1 1 1 1 1
7 2.97 0 0 0 0 0 0
8 1.40 0 0 0 0 0 0
9 1.40 0 0 0 0 0 0

10 0.66 1 1 1 1 1 0.6
1 1 1.40 0 0 0 0 0 0.0
12 0.66 1 1 1 1 1 0.8
13 0.66 1 1 1 1 1 1.0
14 0.31 1 1 1 1 1 1.0
15 4.26 0 0 0 0 0 0.0
16 2.01 0 0 0 0 0 0.0
17 2.01 0 0 0 0 0 0.0
18 0.95 1 0 0 0 0 0.0
19 2.01 0 0 0 0 0 0.0
20 0.95 1 0 0 0 0 0.0
21 0.95 1 1 1 1 1 0.6
22 0.45 1 1 1 1 1 1.0
23 2.01 0 0 0 0 0 0.0
24 0.95 1 0 0 0 0 0.0
25 0.95 1 1 1 1 1 0.8
26 0.45 1 1 1 1 1 1.0
27 0.95 1 1 1 1 1 1.0
28 0.45 1 1 1 1 1 1.0
29 0.45 1 1 1 1 1 1.0
30 0.21 1 1 1 1 1 1.0

 

 

 

 

 

 

 

 

96

 

Table 5.3: Optimal fraction of units to source from market for switching cost ratio

:11 =10 with A, = 0.01 and varied beginning relative price P(O) at the early

market state '1 '.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Price P = P1 /P2 at time t=0

Nodes 0.95 0.96 0.97 0.98 0.99 1
0 1 1 1 0.3 0 0

1 0 0 0 0 O 0
2 1 1 1 1 1 1
3 0 0 0 O 0 0
4 0 0 0 0 0 0
5 1 1 1 1 1 1
6 1 1 1 1 1 1
7 0 0 0 0 0 0
8 0 0 0 O 0 0
9 0 0 0 0 0 0
1O 1 1 1 1 1 1
1 1 0 0 0 0 O 0
12 1 1 1 1 1 1
13 1 1 1 1 1 1
14 1 1 1 1 1 1
15 0 0 0 0 0 0
16 0 0 0 0 0 0
17 0 0 0 0 0 0
18 0 0 0 0 0 0
19 0 0 0 0 0 0
20 0 0 0 0 0 0
21 1 1 1 1 1 1
22 1 1 1 1 1 1
23 0 0 0 0 0 0
24 0 0 0 0 0 0
25 1 1 1 1 1 1
26 1 1 1 1 1 1
27 1 1 1 1 1 1
28 1 1 1 1 1 1
29 1 1 1 1 1 1
30 1 1 1 1 1 1

 

 

 

 

 

 

 

97

Tabk

pnce

fluent

lNode

ll

 

 

 

 

 

f/ / / / / / / NWTH/WH 1711 i

 

Table 5.4: Optimal fraction of units to source from market for beginning relative

2:12- with 21:0.01 at the

price P(0) =1 and varied switching cost ratio

intermediate market state '7'.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nodes -
Relative price P = P1 /P2 overtime and switching cost ratio ‘SCR’
Relative

Price SCR: 1 SCR: 20 SCR: 25 SCR: 30 SCR: 35 SCR: 40

0 1 0 0 0.1 0.1 0 0
1 1.22 0 0 0 0 0 0
2 0.80 1 1 0.9 0.5 0.2 0.1
3 1.50 0 0 0 0 0 0
4 0.98 0 0 0 0 0 0
5 0.98 1 1 0.9 0.5 0.2 0.1
6 0.65 1 1 1 1 1 1
7 1.83 0 0 0 0 0 0
8 1.20 0 0 0 0 0 0
9 1.20 0 0 0 0 0 0
10 0.79 1 1 1 1 0.50 0
1 1 1.20 0 0 0 0 0.03 0.1
12 0.79 1 1 1 1 0.63 0.1
13 0.79 1 1 1 1 1 1
14 0.52 1 1 1 1 1 1
15 2.24 0 0 0 0 0 0
16 1.47 0 0 0 0 0 0
17 1 .47 0 0 0 0 0 0
18 0.97 1 0 0 0 0 0
19 1 .47 0 0 0 0 0 0
20 0.97 1 0 0 0 0 0
21 0.97 1 1 1 1 0.50 0
22 0.64 1 1 1 1 1 0
23 1.47 0 0 0 0 0 0
24 0.97 1 0 0 0 0.03 0.1
25 0.97 1 1 1 1 0.63 0.1
26 0.64 1 1 1 1 1 0.1
27 0.97 1 1 1 1 1 1
28 0.64 1 1 1 1 1 1
29 0.64 1 1 1 1 1 1
30 0.42 1 1 1 1 1 1

 

 

 

 

 

 

 

 

98

 

Table 5.5: Optimal fraction of units to source from market for switching cost ratio

iii—=10 with 2, =0.01 and varied beginning relative price P(0) at the

intermediate market state '7'.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Price P = P, /P2 at time t=0

Nodes 0.95 0.96 0.97 0.98 0.99 1
0 1 1 1 1 0.2 0

1 0 0 0 0 0 0
2 1 1 1 1 1 1
3 0 0 0 0 0 0
4 0 0 0 0 0 0
5 1 1 1 1 1 1
6 1 1 1 1 1 1
7 0 0 0 0 0 0
8 0 0 0 0 0 0
9 0 0 0 0 0 0
10 1 1 1 1 1 1
1 1 0 0 0 0 0 0
12 1 1 1 1 1 1
13 1 1 1 1 1 1
14 1 1 1 1 1 1
15 0 0 0 0 0 0
16 0 0 0 0 0 0
17 0 0 0 0 0 0
18 0 0 0 0 0 0
19 0 0 0 0 0 0
20 0 0 0 0 0 0
21 1 1 1 1 1 1
22 1 1 1 1 1 1
23 0 0 0 0 0 0
24 0 0 0 0 0 0
25 1 1 1 1 1 1
26 1 1 1 1 1 1
27 1 1 1 1 1 1
28 1 1 1 1 1 1
29 1 1 1 1 1 1
30 1 1 1 1 1 1

 

 

 

 

 

 

99

Table 5.6: Optimal fraction of units to source from market for beginning relative

42

price P(O) =land varied switching cost ratio I with A, = 0.01 at the late

market state '13'.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative price P = P, /P2 over time and switching cost ratio ‘SCR’

Nodes Relative Price SCR: 1 SCR: 5 SCR: 10 SCR: 15 SCR: 20
0 1 0 0 0.100 0.4 0.4
1 1.07 0 0 0.000 0.4 0.4
2 0.92 1 1 1.000 0.5 0.4
3 1.15 0 0 0.000 0 0
4 , 0.99 0 0 0.000 0.4 0.4
5 0.99 1 1 1.000 0.5 0.4
6 0.85 1 1 1 .000 1 1
7 1.24 0 0 0.000 0 0
8 1.07 0 0 0.000 0 0
9 1.07 0 0 0.000 0.4 0.4
10 0.92 1 1 1.000 0.59 0.4
11 1.07 0 0 0.017 0.5 0.4
12 0.92 1 1 1.000 0.66 0.4
13 0.92 1 1 1 .000 1 1
14 0.79 1 1 1.000 1 1
15 1.33 0 0 0.000 0 0
16 1.15 0 0 0.000 0 0
17 1.15 0 0 0.000 0 0
18 0.99 0 0 0.000 0 0
19 1.15 0 0 0.000 0.4 0.4
20 0.99 0 0 0.000 0.4 0.4
21 0.99 1 1 1.000 0.59 0.4
22 0.85 1 1 1.000 0.59 0.4
23 1.15 0 0 0.000 0.5 0.4
24 0.99 0 0 0.017 0.5 0.4
25 0.99 1 1 1.000 0.66 0.4
26 0.85 1 1 1.000 0.66 0.4
27 0.99 1 1 1.000 1 1
28 0.85 1 1 1 .000 1 1
29 0.85 1 1 1 .000 1 1
30 0.73 1 1 1 .000 1 1

 

 

 

 

 

 

 

100

Table 5.7: Optimal fraction of units to source from market for switching cost ratio

3:11: 10 with xi, =0.0l and varied beginning relative price P(O) at the late

market state ‘13’.

 

Relative Price P = P, /P2 at time t=0
Nodes 0.95 0.96 0.97 0.98 0.99

 

 

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.1
01

AAA-sa-n-s-s-s-a—s—soooo-sA—x—s-x—soo-s-s-so-s—x—s
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AdAAA-sao-s—s-soooooaa—x—s—s-soo-s-n-so-x-s-s
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—s—x-s—-xdoc-s-soooooo-sa-so—sooo-s—soo-soo-s

 

 

 

 

 

 

 

03
O

 

101

Figure 5.3: The binomial representation of the evolution of relative price process

over one year for P(O) = P, /P2 = 1.0 in the early market state ‘ 1 ’of PLC.

4.26; p = 0.0625

.97 2.01;p = 0.0625
2.01;p = 0.0625

.0
1.40 0.95; p = 0.0625
2.01;p = 0.0625

.44 1.40

0.95; p = 0.0625
0.95; p = 0.0625

1- p = 0.5 0.98
0.66 0.45; p = 0.0625

2.01;p = 0.0625
p = 0.5 1.41
0.95; p = 0.0625
0.95; p = 0.0625
0.98

0.66 0.45; p = 0.0625
0.68 0.95; p = 0.0625

0.66

0.45; p = 0.0625
0.46 0.45; p = 0.0625

0.31
0.21;p = 0.0625

V

 

A

4xAt

102

Figure 5.4: Optimal fraction of units ‘u’ to source from market over one year
for P(0) = P, / P2 =1.0 and switching cost ratio i=1 with A, = 0.01 in the early

market state ‘ l ’ of PLC.

0.00; p = 0.0625
0.00 0.00; p = 0.0625
0.00; p = 0.0625
10.0

0.0.1.00; p = 0.0625
0.00; p = 0.0625

0.00 0.01
1.00; p = 0.0625
1.00; p = 0.0625

1- p = 0.5 0.00
1.00 1.00; p = 0.0625
0.0
' 0.00; p = 0.0625
p = 0.5 0.01
1.00; p = 0.0625
1.00; p = 0.0625
.00

1.00 1.00; p = 0.0625
1.00 1.00; p = 0.0625

1.00
1.00; p = 0.0625
1.00 1.00; p = 0.0625

1.00
1.00; p = 0.0625

4 4 x At 7‘

 

 

103

Figure 5.5: Optimal fraction of units ‘u’ to source from market over one year

forP(0) = P, ll"2 =1.0 and switching cost ratio 123—=50 with ,1, =0.01 in the
1
early market state ‘ 1’ of PLC.

0.00; p = 0.0625

.00 0.00; p = 0.0625
0.00; p = 0.0625

00.0

0.00 0.00; p = 0.0625
0.00; p = 0.0625

0.00 0.01
0.00; p = 0.0625
0.60; p = 0.0625

1- p = 0.5 0.00
0.60 1.00; p = 0.0625
0.0
0.00; p = 0.0625
p = 0.5 0.00
0.00; p = 0.0625
0.80; p = 0.0625
0.30

0.80 1.00; p = 0.0625
0.30 1.00; p = 0.0625

1.00
1.00; p = 0.0625
1.00 1.00; p = 0.0625

1.00
1.00; p = 0.0625

< 4 x At P

 

 

104

Figure 5.6: Optimal fraction of units ‘u’ to source from market over one year
forP(O)=P, /P2 =1.0 and switching cost ratio %=10 with [1,, =0.01 in the

early market state ‘1 ’ of PLC.

0.00; p = 0.0625
0.00 0.00; p = 0.0625
0.00; p = 0.0625
00.0

0.01 0.00; p = 0.0625
, 0.00; p = 0.0625

0.00 0.01
0.00; p = 0.0625
1.00; p = 0.0625

1- p = 0.5 0.00
1.00 1.00; p = 0.0625
1.0
0.00; p = 0.0625
p = 0.5 0.00
0.00; p = 0.0625
1.00; p = 0.0625
.00

1.00 1.00; p = 0.0625
1.00 1.00; p = 0.0625

1.00
1.00; p = 0.0625
1.00 l.00;p = 0.0625

1.00
l.00;p = 0.0625

< 4 x At ’

 

 

105

Figure 5.7: Optimal fraction of units ‘u’ to source from market over one year
forP(O) = P, /P2 =l.0 and switching cost ratio %=25 with 2., =0.01 in the

intermediate market state ‘7’of PLC.

0.00; p = 0.0625
0.00 0.00; p = 0.0625
0.00; p = 0.0625
00.0

0.00 0.00;p = 0.0625
0.00; p = 0.0625

0.00 0.01
0.00; p = 0.0625
1.00; p = 0.0625

1- p = 0.5 0.00
1.00 1.00; p = 0.0625
0.1
0.00; p = 0.0625
p = 0.5 0.01
0.00; p = 0.0625
1.00; p = 0.0625
0.90

1.00 1.00; p = 0.0625
0.90 1.00; p = 0.0625

1.00
1.00; p = 0.0625
1.00 1.00; p = 0.0625

1.00
1.00; p = 0.0625

4 4 x At >

 

 

106

Figure 5.8: Optimal fraction of units ‘u’ to source from market over one year

for P(0) = P, /P2 =1.0 and switching cost ratio £3: 40 with ,1, = 0.01 in the
1
intermediate market state ‘7’of PLC.

0.00; p = 0.0625

I.OO 0.00; p = 0.0625
0.00; p = 0.0625

II.0
0.01 0.00; p = 0.0625
0.00; p = 0.0625

0.00 0.0I

0.00; p = 0.0625
0.00; p = 0.0625

1- p = 0.5 0.00
0.00 0.00; p = 0.0625

0.00; p = 0.0625
p = 0.5 0.“
0.10; p = 0.0625
0.10; p = 0.0625
I.10

0.10 0.10;p = 0.0625
0.10 1.00; p = 0.0625

1.00

1.00; p = 0.0625
1.00 1.00; p = 0.0625

1.00
1.00; p = 0.0625

V

 

4xAt

A

107

Fig
for

the

 

 

 

Figure 5.9: Optimal fraction of units ‘u’ to source from market over one year

for P(0) = P, /P2 = 1.0 and switching cost ratio 1%: l and 5 with A, = 0.01 in
1
the late market state ‘ l 3’of PLC.

0.00; p = 0.0625

I.OO 0.00; p = 0.0625
0.00; p = 0.0625

 

 

11.0
0.0I 0.00; p = 0.0625
0.00; p = 0.0625
0.00 0.0I
0.00; p = 0.0625
1.00; p = 0.0625
1- p = 0.5 0.00
1.00 1.00; p = 0.0625
0.0
0.00; p = 0.0625
p = 0.5 0.0I
0.00; p = 0.0625
1.00; p = 0.0625
1.00
1.00 1.00; p = 0.0625
1.00 1.00; p = 0.0625
1.00
1.00; p = 0.0625
1.00 1.00; p = 0.0625
1.00
1.00; p = 0.0625
4 4 x At >

108

 

 

 

Fig
for

lati

 

Figure 5.10: Optimal fraction of units ‘u’ to source from market over one year

forP(O) = P, /P2 =1.0 and switching cost ratio 123: 20 with ,1, = 0.01 in the
1
late market state ‘ l 3’of PLC.

 

 

0.00;p = 0.0625
I.OO 0.00; p = 0.0625
0.00; p = 0.0625
00.0
0.0I 0.00; p = 0.0625
0.40; p = 0.0625
0.40 0.4I
0.40; p = 0.0625
0.40; p = 0.0625
1- p = 0.5 0.40
0.40 0.40; p = 0.0625
0.4
0.40; p = 0.0625
p = 0.5 0.40
0.40; p = 0.0625
0.40; p = 0.0625
0.40
0.40 0.40; p = 0.0625
0.40 1.00; p = 0.0625
1.00
1.00; p = 0.0625
1.00 1.00; p = 0.0625
1.00
1.00; p = 0.0625
4 4 x At P

109

Figure 5.11: Optimal fraction of units ‘u’ to source from market over one year
forP(O) = P, /P2 =1.0 and switching cost ratio %= 20 with 2., = 0.01 in the

alternative market states ‘ 1’, ‘7’, and ‘13’ofPLC. The numbers in the parentheses
represent the optimal fractions in states 1, 7, and 13 respectively.

(0,0,0); p = 0.0625

II,0, (0,0,0); p = 0.0625
(0,0,0); p = 0.0625

 

 

(0,0,0

(0,0,0 (0,0,0); p = 0.0625

(0,0,0.4); p = 0.0625
(0,0,0.4) (0,0,0.4
(0,0,0.4); p = 0.0625
(1,1,0.4); p = 0.0625
1- p = 0.5 (0,0,0.4)
(l,l,0.4) (1,1,0.4); p = 0.0625
(0,0,0.4)
(0,0,0.4); p = 0.0625
p=05 mfifiA
(0,0,0.4); p = 0.0625
(1,1,0.4); p = 0.0625
(1,1,0. ‘

,(l,1,0.4) (1,1,0.4); p = 0.0625

(l,l,0.4) (1,1,1);p=0.0625

(1,1,1
(1,1,1);p = 0.0625
(1,1,1) (1,1,1);p=0.0625
(1,1,1)
(1,1,1);p = 0.0625
4 4 x At ‘7‘

110

Figure 5.12: Optimal fraction of units ‘u’ to source from market over one year
forP(O) =P, /P2 =1.0 and switching cost ratio —:13= 1 with 2., =0.01 in the

alternative market states ‘ 1’, ‘7’, and ‘13’ofPLC. The numbers in the parentheses
represent the optimal fractions in states 1, 7, and 13 respectively.

(0,0,0); p = 0.0625

11,0, (0,0,0); p = 0.0625

 

 

(0,0,0); p = 0.0625

(0,0,0
(0,0,0 (1,1,0);p = 0.0625
(0,0,0); p = 0.0625
(0,0,0) (0,0,0
(1,1,0);p = 0.0625
(1,1,1);p= 0.0625
1- p = 0.5 (0,0,0)
(1,1,1) (1,1,1);p= 0.0625
(0,0,0)
(0,0,0); p = 0.0625
p=05 wflfl
(1,1,0);p = 0.0625
(1,1,1);p= 0.0625
(1,1,1)

(1,1,1) (1,1,1);p= 0.0625
(1,1,1) (1,1,1);p= 0.0625

(1,1,1
(1,1,1);p= 0.0625
(1,1,1) (1,1,1);p= 0.0625

(1,1,1)
(1,1,1);p = 0.0625

4 4 x At 4

111

The results are data dependent; however, similar computations can be can‘ied out

using relevant contextual data to gain firm, industry or product specific
(managerial) insights. The results indicate the following general patterns.

a) The optimal sourcing pattern is not always of the ‘bang-bang’ variety

i.e., it is optimal to procure all units from the cheaper alternative at a

given time. This result is in contrast to the bang-bang type of outcome

that occurs when the buying firm is assumed to be risk neutral

(Kouvelis, 1999; Li and Kouvelis 1999). It is observed that for

specified price parameters, switching costs, planning horizon and

initial prices, there are threshold price levels or price bands at which

the optimal fractions to procure from an arrangement need to be

readjusted over time. For example, for a typical price pattern and a set

of switching cost ratio, it is observed from the sourcing pattern in the

early stage of the PLC as presented in Table 5.2 and described in

Figures 5.4-5.6, that though the optimal sourcing arrangement is

purely contract type at the beginning, it is optimal to switch a specified

fraction (i.e., 1 or 0.3) of procurement to market arrangement when the

relative price falls to 0.68, and to continue with the specified level of

market arrangement until the relative price goes up to 1.40 or drops

further. Thus when the relative price lies between 0.68 and 1.40, it is

optimal to maintain the status quo until the last transaction at the

terminal time. These patterns are somewhat similar to the concept of

target optimal portfolio boundaries described in financial literature

112

(Shreve and Soner, 1994). As per the aforesaid financial literature, the
portfolio (e.g., ratio of risky shares to risk less bonds) space can be
divided into three disjoint regions, which can be specified as the buy
region, the sell region, and the no trading or status quo region. The
implications are: the initial portfolio structure need not be changed
until prices of equity shares hit specified limits; when prices hit these
limits, portfolio can be rebalanced for certain target structure by buy or
sell actions. It is also observed that the optimal pattern of sourcing
over time depends upon the duration of (remaining) operating horizon,
and the switching cost ratios. The longer the (remaining) time horizon,
the wider is the price band in which the status quo may be maintained;
and the greater is the switching cost ratios, the lesser the magnitude of
readjustment for a specified price band. These characteristics are
similar to the concept of hysteresis that describes the inertia against
(quick) readjustment in the presence of transaction cost and price
uncertainty (Dixit, 1989). It may be noted that the optimal pattern of
sourcing also depends upon the initial prices. Typically, the lesser the
market to contract price ratio the greater is the initial allocation to
market (Table 5.3). As evident from Tables 5.4-5.7 and Figures 5.7-
5 .10, the above general patterns of sourcing are observed across
intermediate (market state ‘7’) and late stages (market state ‘13’) of

PLC as well.

113

b) While, the general patterns of optimal fractions to source from
alternative arrangements as described above across different market
states are similar, the sensitivity of fractions to switching costs and
initial prices vary across different market states. For example,
comparing the results in Tables 5.3, 5.5 and 5.7 for varied relative
prices at a specified switching cost ratio of 10 across early,
intermediate and late stages of the PLC, it may be concluded that the
propensity to procure from market arrangement is higher towards the
later stages of the PLC for the same relative price at the beginning of
planning horizon. This may be attributed to relatively greater (cost)
benefit compared to the uncertainty by sourcing from the market
towards the later stages. Similarly, a comparison of fractions sourced
from alternative arrangements for switching cost ratios of 20 and 1
from Tables 5.2, 5.4, and 5.6 indicates that the buying firm should
source more from the market arrangement in the later stages of PLC
than in the earlier stages in a situation of relatively higher switching
cost of contract arrangement. This implies relatively higher switching
costs or penalties for readjusting units sourced from contractual
arrangements are likely to be counterproductive for the contractual
supplier in later stages of PLC. Consequently, it is in the interest of the
contractual supplier to provide more contract flexibility, and provide

matching market price in later stages.

114

c) The greater the ratio of market price uncertainty (,61) to the difference

in contract and market prices (growth rates), the lesser the probability
of procurement from open market; accordingly, contract sourcing is
more important in the early stage of the PLC; however, towards the
later stages of the PLC, market sourcing gradually becomes more
important. In the illustrative dataset presented in Table 5.1, the ratio of

market price uncertainty ( ,61) to the difference in contract and market

prices (grth rates) declines from 15 in market state ‘1’ through 13 in
market state ‘7’ to 10.7 in market state ‘13’. The optimal patterns of
sourcing across early, intermediate and late stages of the PLC for
specific switching cost ratios (i.e., 20 and 1) are graphically presented
in Figures 5.11 and 5.12. It is apparent that, when the contract
switching costs are relatively higher, as per the optimal strategy, the
buying firm will have a higher propensity to buy from market
throughout the operational plamring horizon, and the sourcing
arrangements would be more stable (e.g., number of switches are less
and magnitudes of switch are smaller) in later stages of PLC than in
earlier stages of PLC. However, it is to be noted that this stability is
achieved with a price both for the buying and the contractual supplier
firm. The supplying firm gets lesser allocation and the buying firm has
to procure more fi‘om market thus being subjected to higher

uncertainty.

115

The above characteristics of sourcing patterns suggest that the optimal
strategy for a risk averse buying firm is not of ‘bang—bang’ type. Furthermore, the
sourcing strategies need to be redesigned in accordance with the attitudes towards
risk, relative switching cost and state of prices over time across different
procurement contexts. The results suggest that in the earlier stages it is generally
optimal to buy from contractual arrangements and it is only advantageous to
switch to market arrangements under very favorable market price conditions when
the relative switching cost ratios are smaller. However, in later stages of PLC, it is
optimal to buy more from market when the contract readjusting costs are
relatively higher and market prices are more favorable. This implies the
contractual supplier may be able to charge a higher penalty for contract
readjustment only during the earlier stages and not towards the later stages.
Correspondingly, the contractual supplier needs to be more responsive to market
price patterns in the later stages than in the earlier stages. Cohen and Agrawal
(1999) observed a somewhat similar pattern for cases of high fixed cost and stable
market conditions in that it is optimal to buy from market arrangement (e.g., short
term contract) at the beginning of planning horizon and not to switch to long-term
arrangement in the planning horizon. The increased importance of market price
based sourcing towards the later stage of PLC (when the product becomes
standardized and functional) as competition intensifies and firms become more
cost conscious conforms to the industry practice (Rink and Fox, 1999; Cohen and

Agrawal 1999).

116

The model discussed here may be improved along the following lines.
First, the proposed model helps to identify optimal fraction of units to source from
alternative arrangements in the presence of proportional switching cost. The
numerical results indicate there are many situations when it is optimal to
completely procure from either contract or market states during the intermediate
period of the operating horizon. This implies complete abandonment of a sourcing
arrangement in those situations. It is to be noted that in reality a complete
abandonment of a sourcing arrangement under very favorable prices of alternative
arrangement though rational may, be difficult to implement considering the
implications on buyer-supplier relationships or the enormity of cost to restart the
arrangement again in the future. Thus a simultaneous consideration of fixed and
proportional switching cost would be more appropriate and may be considered as
an area for future research. Second, the model illustrated here considers a power
disutility function of cost. In reality, a buying firm’s utility function may be more
complex. Thus studies may be carried out for obtaining solutions for more general
utility functions. The current numerical solution approach is limited by the
exponential increase in numerical operations due to increase in solution space
with finer discretization in time and fractions allocated. Research should be
undertaken to develop efficient solution techniques that can substantially reduce
the solution space thereby providing more precise estimation of optimal

outcomes.

117

Chapter 6

Conclusion

This research makes a contribution to the sourcing literature by proposing
a set of analytical models for formulating integrated contract and open market
sourcing strategy. The modeling framework resembles the portfolio optimization
in the finance literature. It is different from the existing approaches .in the
sourcing literature in that it considers the buying firm to be risk averse and allows
for the simultaneous use of both the sourcing mechanisms. In contrast, the related
sourcing literature predominantly considers the buying firm to be risk neutral and
assumes the use of one sourcing alternative at a time unless there is a capacity
constraint with the least cost supplier that justifies multiple arrangements
simultaneously (Li and Kouvelis, 1999; Peleg and Lee, 2002; Kleindorfer and
Wu, 2003; Martinez-de-Albeniz and Simchi-Levi, 2005).

The study by Cohen and Agrawal (1999) is probably the only one that
considers the buying firm to be risk averse. Their model examines the trade-off
between a high fixed cost long-term contract with allowance for cost reduction
due to continuous learning (an adjustment mechanism), and price-uncertain short-
term contract arrangement without any cost reduction opportunity, when these are
adopted singularly. In practice, a buying firm is more likely to use a portfolio of
long-term and the short-term contract arrangements simultaneously (Billington,
2002; Kleindorfer and Wu, 2003; Kleindorfer and Wu, 2005). This is because the

market conditions are dynamic and uncertain, and single type of sourcing

118

arrangement would entail either the risk of excessive supplier opportunism or
unstable supply conditions. From this perspective, the proposed models with dual
sourcing arrangement have more practical relevance.

In the present study three analytical models were developed. Two of these
were continuous time models that do not consider the (indirect)
transaction/switching cost and the other is a discrete time model that explicitly
considers the (indirect) transaction/switching cost. These models solve for optimal
fractions to be sourced from open market arrangements with ‘uncertain prices’
and contractual arrangements with ‘deterministic prices’. A comparative
discussion of the key characteristics of models, their relative advantages and
limitations is presented in Figure 6.1. The models provide insights on two key
areas: a) how do the tradeoffs between procurement cost and uncertainty
influence the sourcing strategy of a risk averse buying firm, and b) what are the
likely challenges in implementing the sourcing strategy, and how to address these
challenges over time. The models are applicable to procurement contexts where
product and service attributes can be specified objectively in a verifiable manner.

The proposed models make several theoretical contributions and provide
significant managerial insights regarding the optimal sourcing strategy. These

insights and directions for firture research are discussed next.

119

Figure 6.1: A descriptive comparison of the three analytical models.

 

 

 

Model Key findings Advantages Limitations
description
Model-1 The 1. For equal contract and 1. The model can 1. Accrual of
continuous market prices at the obtain optimal surplus may be
time model beginning of planning fractions of problematic
developed in horizon, the optimal expense for when inventory
Chapter 3 fractions are constant and power disutility cost is high.
that derives depend on price function xn with .
optimal parameters, and buying ">1. 2. The umts
fiactrons of firm’s risk attitude. Consequently, it procured from
expense to can provide alternative
allocate 2. For unequal contract solutions for a arrangements are
across and market prices at the wide range of stochastic; this
contractual beginning of planning power disutility calls for quantity
and .market horizon, the optimal functions of the flexrble
sourcrng fractions are stochastic n contractual
arrangements and depend on price type x - arrangements.
to minimize parameters, buying .
the firm’s risk attitude, 2- The. solutron 3. The approach
procurement instantaneous expense, results in 360111.31 involves solving
expense and units of demand. 0f surplus umts of the HJB
disutility, over time even equation, which
when the 3. The optimal fractions when the calls for trial
expense is do not depend on the expense . functions that are
continuous duration of planning ‘1‘-51111119’ {5 difficult to guess
over time. horizon. mrmrnrzed. T1115 for general
provrdes . disutility
4. The average units WROMW .for firnctions that are
procured from indirect savrngs different from

 

contractual arrangement
decline over time; the
average units procured

from market arrangement

increase over time.
However, the variance of
units procured from both
arrangements increase
over time. The rates of
these decline and
increase are smaller for
the contractual
arrangement than for the
market arrangement.

 

by selling back
the surplus in
secondary
markets.

 

power disutility
function.

 

120

 

F igure6.1 (cont’d)

 

 

 

Model Key findings Advantages Limitations
description

Model-2 The 1. The optimal fractions 1. The approach 1. The model is
continuous depend on price can be extended not amenable to
time model parameters, buying to obtain optimal power disutility
in Chapter 4 firm’s risk attitude, and fractions for function xn with
that derives duration of planning disutility non-integer
optimal horizon. function that are values for n >1.
fractions of more general in

units to be 2. The fractions procured fimctional form

sourced fi'om market than the power

across arrangements decline for disutility

contractual longer planning horizons. function xn.

and market

sourcing 3. The fractions may be 2. The solution

arrangements adjusted in jumps when

to minimize
the
procurement
expense
disutility,
when the
expense is
continuous
over time.

 

the prices are equal
during a planning
horizon.

 

exactly meets the
demand and the
units procured
from the
arrangements are
deterministic.

 

 

121

 

Figure 6.1 (cont’d)

 

 

 

 

 

Model Key findings Advantages Limitations
description

Model-3 1. The optimal fractions 1. Accounts for 1. The solution
The discrete depend on price the effect of approach is
time model parameters, buying proportional numerical and
time model firm’s risk attitude, switching cost in requires finer
in Chapter 5 switching cost, and shifting across discretization for
that derives remaining duration of the sourcing precise results.
optimal planning horizon. arrangements. However,
fractions of exponential
units sourced 2. The optimal fractions 2. Provides for increase in
across are path dependent and savings due to computational
contractual can be adjusted in jumps readjustment of effort with
and market when the prices are units across the increase in
sourcing favorable during the arrangements in discretization
arrangements planning horizon. discrete time. limits the
to minimize precision.

the 3. The higher the

procurement switching cost, and the 2. Does not
expense longer the remaining address the issue
disutility, planning horizon, the of fixed cost of
when there is greater the inertia against switching across
proportional readjustment of fraction arrangements.
switching of units sourced across

cost due to arrangements.

the shift

between

arrangements

 

 

 

 

 

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6.1 Theoretical contributions

The proposed models build on the theoretical frameworks on sourcing
strategy and provide insights on the combined impact of price dynamics, risk
aversion, and switching cost on a firrn’s sourcing strategy. The results suggest that
a dual sourcing arrangement that provides strategic flexibility is preferable to
single sourcing in uncertain procurement contexts.

The models indicate the optimal sourcing strategy described in terms of
fractions of units to source from alternative arrangements depends upon the
degree of risk aversion, price parameters, the switching cost, and the duration of
the planning horizon. However, the optimal strategy described in terms of
fractions to spend on alternative arrangements additionally depends upon units of
demand and instantaneous expense. As a result, the units sourced from alternative
arrangements are stochastic when the firm controls the fraction of expense. It is
found from the models in Chapters 3 and 4 that ceteris paribus, the propensity to
source from market is directly proportional to rate of contract price premium and
inversely pr0portional to market price uncertainty and the buying firm’s degree of
risk aversion. Moreover, the propensity to source from market is convex in market
price uncertainty and degree of risk aversion i.e., the rate of decline in market
sourcing increases with increase in price uncertainty and degree of risk aversion.
It is to be noted that the rate of contract price premium is governed by the relative
bargaining strength and negotiation between buying firm and contractual supplier;

this can act as a managerial lever to influence the sourcing strategy.

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Consequently, the optimal sourcing strategy should be an outcome of uncertain
product-market environment, risk attitude and managerial capability.

The application of the models on the illustrative data indicates that the
sourcing strategy is dynamic and a buying firm needs to adjust the amount
sourced from c0ntractual and open market arrangement over time as the procured
product advances through its product life cycle. In general, for a firm with
constant risk attitude the amount sourced from open market should increase over
time. This may be explained as the increasing attractiveness of open market
procurement when the product becomes increasingly standard and the market
becomes intensely competitive and liquid towards the later stages of the product
life cycle (Mendeleson and Tunca, 2003). The results also conform to Kleindorfer
and Wu (2003)’s conceptual framework that justifies greater suitability of market
arrangement when the product attributes are easy to codify and cost of developing
contractual arrangement is relatively high.

The dual arrangement with some sourcing from market arrangement in
introduction stage, dominance of contractual arrangement in growth stage
followed by the dominance of open market arrangement in the maturity stage of
the procured product’s life cycle indicates the importance of managing suppliers
through a varying mix of relational and competitive arrangements across different
stages of the PLC. Examination of the above results in light of transaction cost
theory, agency theory, and resource based theory justifies the significance and
scope for implementing such integrated sourcing strategy in terms of contextual

relevance and functional appropriateness across different stages of the PLC.

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To illustrate, the introduction stage of the PLC may be characterized by
the simultaneous existence of technological, supplier capability, product design,
and price uncertainties. In the face of these varied types of uncertainties, risk
averse buying firms may significantly differ in terms of their risk attitudes
towards the magnitudes and uncertainty of cost. While a relatively more
aggressive buying firm with less concern for cost uncertainty may experiment
with several suppliers to assess the future potential of building a long-term supply
arrangement with a specific set of suppliers, a relatively less aggressive buying
firm with more concern for cost uncertainty may limit the interactions to a few
well-known suppliers. Correspondingly, the former type may emphasize trial
purchases using short-term arrangements, which are comparable to market
oriented sourcing arrangements. In contrast, the latter type may emphasize
relatively more long-term contractual arrangements. Interestingly, the application
of the models on the illustrative dataset of PLC oriented price patterns suggests
such alternative sourcing strategies in the early stage the PLC.

These strategies can also be justified on the following theoretical grounds.
Agency theory assumes that the actions of contracting parties are guided by
opportunism, bounded rationality and risk aversion. It explains the principal-agent
relationship in a contractual arrangement in terms of organization of information
sharing for verification of actual behavior and for sharing of risk. Accordingly,
there are two types of contracts: outcome based contracts e. g., market governance,
and behavior based contracts e. g., hierarchical governance (Eisenhardt, 1989). In

a sourcing context, it may be reasoned that short-term open market arrangement

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and long-term contract arrangement would correspond with Eisenhardt’s
outcome-based contract and behavior-based contract respectively.

Eisenhardt posits that there will be a preference for outcome-based
contracts in the presence of goal incongruence between transacting firms;
however, in the presence of difficulty in specifying tasks, there will be a
preference for behavior-based contracts in the presence lower outcome
(performance) measurability and higher outcome (performance) certainty. In the
early stage of the procured product’s life cycle, a buying firm may not settle on
the product requirements and may therefore need to redefine the task definition
frequently; furthermore, the supplying firm’s goals may not clearly converge on
that of the buying firm, when the product characteristics are incompletely
described. Consequently, agency theory would suggest that a buying firm that
recognizes the merit of changing specifications and measurement system for task
performance in the early stage, and the capability of implementing such change in
specifications and measurement system will emphasize a market arrangement. In
contrast, a buying firm not capable of implementing such change in specifications
and measurement system will emphasize a contract arrangement. Thus, it may be
contended that the optimal strategy identified by the model in the introduction
stage is in alignment with agency theory.

Transaction cost theory and resource-dependency theory make a case for
alternative sourcing strategies (e.g., relatively higher and lower emphasis on
contract arrangement) on the basis of a buying firm’s relative attitude for reducing

transaction costs and uncertainties (Noordewier et al., 1990). While more

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emphasis on trial purchases in the early stage of the PLC from multiple suppliers
enables the buying firm to avoid the cost of risky dependence and idiosyncratic
investments in (uncertain) product/technology required of contractual
arrangements, more emphasis on contractual arrangements reduces uncertainty of
transaction cost due to multiplicity of arrangements and facilitates development of
relational arrangement, which if successful would be of higher value in future.
Consequently, a buying firm may choose either of the arrangements, depending
on the perceived value of the trade-off between transaction cost and high
uncertainty. In the early stage, contractual arrangement may increase the
transaction cost due to lesser transaction frequency, but this increased cost may be
worthwhile considering the higher degree of uncertainty from open market
sourcing arrangement.

The resource-based theory posits that firms need to acquire rare, tacit,
difficult to imitate, causally ambiguous, and heterogeneously distributed resources
to achieve sustainable competitive advantage (Barney, 1991). Since, in the early
stage, it is not apparent which supplier would provide access to such critical
resources, the resource-based perspective would suggest the buying firm should
engage in exploring strengths and weakness of alternative suppliers through direct
experience in early stages of the PLC. Accordingly, it may be conjectured that
firms focusing on building resource based strategic capabilities are more likely to
have lower cost related risk aversion and may gain from trial supply arrangements

with multiple suppliers in the early stage of the PLC.

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As the product advances to the growth stage, the sourcing context changes
to one of greater stability regarding technological changes, increasingly
standardized product design, higher competitive intensity, and higher uncertainty
regarding availability of adequate amount of supply. Consequently, buying firms
tend to be increasingly concerned about magnitude and uncertainty of input cost
and volume of supply, and are likely to look for opportunities for cost reduction
and increased certainty of supply and input cost (Rink and Fox, 1999). The
applications of model on the data corresponding to this stage for a risk averse
buying firm suggests a sourcing strategy with relatively higher emphasis on
contractual arrangement. These results may also be justified from transaction cost,
resource based and agency theory perspectives.

While, transaction theory suggests that transaction arrangements should
reduce the direct and indirect costs of transaction and uncertainty, the resource-
based theory recommends that a firm should have access to difficult to imitate
critical resources. The buying firm’s transactions costs increase with supplier
specific (idiosyncratic) investments and uncontrollable transaction uncertainty,
and decrease with frequency of transaction with a specific supplier. In the
intermediate (growth) stage, the uncertainty regarding supplier capabilities,
product features and firnctionalities decrease over time. There are opportunities
for greater goal congruence between the buying and the supplier firms, lesser
difficulty in task specification and performance measure and lesser information
asymmetry. Also, there exists opportunities for higher volume of transaction over

a sustained period of time with specific suppliers who have proven capabilities to

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satisfy the requirements of the product in the introduction stage. According to
agency theory under such circumstance, the buying firm should choose long-term
contractual arrangements. Such arrangement can: reduce the impact of asset
specific investments over time; present opportunities for mutual learning by
information sharing and sustained cost reduction through collaborative planning
and control initiatives; offer learning curve benefits, and reduce uncertainties
regarding supplies. These sequential outcomes can reduce transaction costs, and
provide for the critical resources to the advantage of the buying firm. However,
long-term arrangements can entail the risk of supplier opportunism, and lock-ins.
Agency theory suggests that in case of such dysfunctional relationship with a
supplier it may be in the interest of the buying firm to rely on outcome (market)
based contracts to address the negative outcomes in a contract (Eisenhardt, 1989).
This would enable the buying firm to maintain a competitive mechanism for the
contractual supplier by making purchases through short-term or open market
arrangements. For a buying firm this may however mean practices such as
alternative sourcing arrangement, inducing short-term suppliers to make
idiosyncratic investments, flexible/reversible supplier specific investments to
reduce excessive dependence on a supplier, and redesign of incentives through
periodic renegotiation of contracts etc. The optimal sourcing strategy in the
intermediate stages suggested by the models is consistent with these theoretical
suggestions. The models’ results suggest integration of competitive open market

and contractual sourcing arrangements that can lead to long-term competence

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building and process innovation while preventing dysfunctional contractual
outcomes.

In the maturity phase of the PLC, the product becomes highly standardized
with increase in competition among more or less homogeneous suppliers. The
uncertainties regarding various non-price attributes, asset specificity, and
availability of the product are negligible during this stage. Furthermore, the
buying firm becomes aware of the true cost structure of suppliers and does not
experience uncertainty regarding the availability of the product, leading to
emphasis on cost-based procurement. Consequently, suppliers tend to compete on
prices. Rink and Fox (1999) noted that buying firms should increase the market
orientation during this stage. The application of models on the illustrative data
corresponding to this stage also suggests a sourcing strategy with relatively higher
emphasis on market-oriented arrangement, which may also be explained from
transaction cost, resource based and agency theoretic perspectives.

The sourcing context in the maturity phase of the PLC is characterized by
highest order of information symmetry, performance verifiability and certainty of
outcomes. According to agency theory, the outcome (market) oriented contracts
would be appropriate in such circumstances (Eisenhardt, 1989). It can also be
argued that, the information symmetry, performance verifiability and certainty of
outcomes in this stage can facilitate structured specification of tasks and goal
congruence between the buying firm and the supplier, which can justify
verifiable, long-term (behavior based) contractual arrangement (Eisenhardt,

1989). Thus agency theory does not clearly explain the suitability of a sourcing

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arrangement; however, the transaction costs theory perspective provides a better
answer for the appropriate sourcing strategy in this context.

According to transaction cost theory, for a limitedly rational buying firm,
competitive mechanism is likely to reduce transaction cost and uncertainty in a
situation of lower risk of asset specificity and higher risk of supplier opportunism
(Williamson, 1985). This implies in the maturity phase, since a buying firm is
mainly exposed to the risk of supplier opportunism, market competition led
contractual arrangements are likely to be most efficacious in ensuring certainty
and minimal cost of transaction. Consequently the appropriate mechanism should
be a combination of open market and market driven contractual arrangements. A
buying firm may achieve this integration either by simultaneously procuring
through both open market and contractual arrangements or by frequently
modifying the terms of contractual arrangements according to competitive market
conditions. The requirement for relatively higher emphasis on market
arrangement would necessitate a buying firm to have unique capabilities to
quickly understand the developments in the product market and readjust sourcing
arrangements with supplier(s). Accordingly, from the resource-based perspective,
the critical resource to acquire in this stage would be the capability to quickly
reconfigure the supply arrangements and/or to modify the contractual terms with
the existing suppliers. A firm having sourcing arrangements with highly capable
supply base is likely to have the greatest opportunity to integrate the contractual
and market arrangements in the maturity phase of the PLC. For example, firms

such as Anheuser Busch and Hewlett Packard use such approach while procuring

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aluminium, electricity, electronic components, and semiconductors (Billington,
2002; Wu and Kleindorfer, 2005).

The foregoing theoretical interpretation of the results supports the
argument that applicability of the dominantly held view that ‘supplier alliance or
partnerships is crucial for effective supply management’ is not universal and is
rather context specific (McCutcheon and Stuart, 2000). The findings are in line
with the perspective that characteristics of inter-organizational alliances evolve
dynamically in accordance with the nature of competition and criticality of
external resources over time (Gulati, 1998). In the early stages of the procured
product’s life cycle there are likely to be a small number of capable suppliers who
can meet the buying firm’s requirements. Consequently, the key factors that
would affect the success of a sourcing arrangement during this stage are:
commitment of suppliers for a sustained relationship, existence of inter-
organizational communication systems, and existence of a mechanism to balance
the costs and risks in the arrangement. Adoption of long-term contractual
arrangements with scope for periodic renegotiation can provide mechanisms for
continuity, trust, and commitment in relationship while facilitating risk and
reward sharing (Krause, 1999; McCutcheon and Stuart, 2000; Carr and Smeltzer,
2002). Consequently, the relational arrangements in managing suppliers would be
of high utility during this stage (Rink and Fox, 1999; Dyer, 1997, Saeed et al.,
2005)

While the long-term contractual arrangements can enable the development

of relational mechanisms for supplier management, the use of short-term open

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market arrangements can enable the buying firm to maintain a competitive and
constructive tension with the contractual supplier that may be useful for market
development, improvement of the contractual supplier, and achieving strategic
flexibility. The empirical sourcing literature also highlights the efficacy of such an
approach in buying firm’s efforts to improve supplier performance (Krause, et al.,
2000). This line of research suggests that supplier development activities can be
divided into two groups: extemalized activities and internalized activities. While
extemalized activities entail use of a combination of competitive pressure,
supplier assessment, and supplier incentives in the form of purchase volumes to
motivate suppliers to improve, internalized activities involve direct involvement
of the buying firm in supplier’s development through various types of
investments. Their study observes that systematic use of extemalized mechanisms
can be effective in supplier development. Thus by maintaining an alternative
market sourcing arrangement, the buying firm can employ such extemalized
mechanisms to induce the contract suppliers to continuously improve in line with
the development in competitive market place.

In view of above discussion and on the basis of results of the numerical
analysis of the illustrative dataset with the PLC perspective, it can be
hypothesized that while relational arrangements are likely to be more relevant in
earlier stages of the PLC, competitive arrangements would be more relevant
towards the later stages of the PLC. The execution of such a souring strategy with
a dynamic perspective to match with the internal and external contingencies

would however require appropriate implementation of systems that facilitate

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boundary-spanning activities (e.g., information exchange, supplier selection,
assessment and monitoring, purchasing etc.) that manage the buyer-supplier
relationships. The related issue is how should the buying firm go about adopting
and implementing the requisite (electronic) procurement mechanisms that are to
support a mix of relational and competitive buyer-supplier relationships across
different phases of the procured product’s life cycle.

Studies on the impact of information technology and inter-organizational
systems on process efficiency and sourcing leverage (e.g., close buyer-supplier
relationship and benefits from competition among suppliers) indicate, while
extensive information sharing mechanisms facilitate integration of buyer-supplier
systems for higher efficiency, systems for higher number of interconnections with
suppliers lead to higher sourcing leverage (Stump and Srirarn, 1997; Saeed et al.,
2005). Furthermore, buying firms who drive the information integration processes
with many suppliers can develop increased awareness of suppliers’ capabilities.
This knowledge can subsequently be used for achieving sourcing
leverage/competitive efficiency among suppliers. Consequently, firms with long-
terrn contractual arrangements will have the opportunity to revise the transaction
terms according to the market conditions. This may however require the buying
firm to dynamically structure and sequence the various functionalities of the
electronic systems for advantageous execution of relational and competitive
mechanisms over time. This is crucial because many firms have tended to focus
on collaboration oriented electronic procurement mechanisms that support the

pre-existing long-term relationships which has been the traditional mode of

134

exchange process; often these firms are not inclined to adopt the alternative of
efficient open market sourcing mechanisms even when the supporting electronic
mechanisms are available for adoption (Wang and Archer, 2004). Accordingly,
there is a need for examining the correspondence between the alternative types of
sourcing strategy and electronic procurement firnctionalities with a contingency
perspective.

In addition to above strategic implications, the models suggest several
insights that have tactical and operational implications. The continuous time
model in Chapter 3 indicates that units procured from alternative arrangements
are variable and the variability increases with the duration of planning horizon.
Furthermore, the average units sourced from contractual arrangement declines
marginally with longer review periods due to the higher sourcing from the open
market arrangement, which is expected to provide a price advantage over
contractual arrangement. These results highlight the significance of quantity
flexible contracts with higher flexibility for longer-terrn contracts. In case a
supplier firm is not willing for such contracts, the buying firm may consider
renegotiating the long-term contract as a sequence of short-term contracts for
reducing the variance in purchases made from alternative arrangements.

The continuous time model in Chapter 4 exactly matches the units
constraint, suggests that while the market arrangement is generally attractive
towards later stages of PLC, a risk averse firm with longer planning horizon
would buy lesser fraction of units from market arrangement than that with shorter

planning horizon. This is because the arrangements with longer planning horizon

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incorporates more market uncertainty than the one with shorter planning horizon.
Since, the earlier stages of the PLC are likely to have higher price uncertainty, this
implies a buying firm may choose to employ short-term tactical planning horizons
in the earlier stages of the PLC as compared to the later stages of PLC to benefit
from competitive market price. This would allow the buying firm to minimize the
expected cost disutility by limiting the exposure to cost uncertainty. The results
thus suggest that long-term contracts may have to be renegotiated more frequently
during the earlier stages than in the later stages.

The optimal sourcing strategy with switching cost as per the discrete time
model in Chapter 5 is somewhat different. This is because the firm readjusts
fractions between arrangements infrequently and the decision to readjust depends
upon both the price dynamics, and the cost and benefits of switching over the
remaining time period. It is observed that a firm’s switching frequency (i.e., the
number of time the fraction of units sourced from alternative arrangements are
adjusted) falls (rises) when the switching cost from/to the contract arrangement is
high (small). However, in case of a switch, the magnitude of switch is high
(small) if the price differences between arrangements are high (small). Thus,
higher switching cost and lower price differential present higher inertia against
switching and provide a case for greater stability of the sourcing arrangement.
This implies relatively high switching costs/penalties for readjusting units sourced
from contractual arrangements are likely to be counterproductive for the
contractual supplier in later stages of PLC when the product is very standardized

to command a price premium.

136

The research extends the contingency perspective in sourcing strategy by
explicitly including the buyer’s risk attitude, transaction/switching cost and
competitive price dynamics in the analysis. The results have normative
implications on how contracts should be designed and renegotiated, how buyer-
supplier relationships are to be managed, how electronic procurement
mechanisms are to be implemented, and how much importance is to be given to
market arrangements over time. This conceptualization aids in understanding the
importance and operationalization of dynamic readjustment of sourcing strategy
over time and contributes to future theory-building activities in the broad area of
managing buyer-supplier relationships and adopting electronic procurement
mechanisms. Some of the insights developed from the study are presented below
as hypotheses regarding the sourcing strategy for a standardized product.

1. The buying firm should have a higher emphasis on long-term
contractual sourcing mechanisms with scope for renegotiation in
the earlier stages of the procured product’s life cycle.

2. The buying firm should have a higher emphasis on open market
sourcing mechanisms in later stages of the procured product’s life
cycle.

3. The long-term contractual arrangements in later stages of the
procured product’s life cycle do not necessitate frequent
renegotiation.

4. The contractual arrangements are likely to have lesser

penalty/switching cost in later stages of the PLC.

137

i.

5. The buying firm is likely to emphasize open market oriented
frmctionalities offered in public electronic procurement
mechanisms in later stages of the procured product’s life cycle.

6. The buying firm is likely to emphasize relation building
functionalities offered in private or consortia based electronic
procurement mechanisms in earlier/intermediate stages of the

procured product’s life cycle.

6.2 Managerial contributions

The models formulated in this dissertation can be applied to many
product-market contexts (e. g., steel, chemicals, electricity, semi—conductors etc.)
in which the product attributes are easy to standardize and codify. The
applications to context specific data can enable managers to formulate context
specific sourcing strategies. For example, in light of the results of models applied
to the illustrative data set, it may be observed that for a buying firm with
negligible switching cost and transaction cost, it is optimal to have dual sourcing,
and the fraction of sourcing through the open market arrangement should increase
over time due to the changes in procurement contexts across different stages in
the PLC.

The procurement contexts that affect the optimal fractions can be
described in terms of contractual and open market price dynamics and risk
aversion. The optimal sourcing patterns have practical relevance in the context of
the recent emergence of electronic procurement arrangements. These

arrangements have enabled firms to achieve information transparency and to

138

reduce the costs and time of identifying new suppliers, switching and conducting
transactions (Boer et al., 2001; CAPS Research, 2003; Flynn, 2004). A joint study
by McKinsey and CAPS Research mentions ‘B2B e-market places can provide
real value to buyers and the buyer should move quickly in finding ways to capture
that value. The best approach is to view B2B e-market places from a portfolio
perspective, blending traditional purchasing best practices with the specific online
benefits that each type of B2B market places can provide’ (McKinsey and CAPS
Research, 2000). A subsequent study further indicates that e-procurement can
accommodate a total cost of ownership perspective (TCO) perspective when the
buying firm’s requirements across price and non-price factors are clearly specified
and could be implemented for direct, indirect, capital goods and services that
represent the commodities, leverage (e. g., high value and low supply complexity)
and bottleneck (e.g., low value and high supply complexity) products in the
purchasing strategy matrix suggested in Kraljic’s 2003 study (CAPS Research,
2003)

Accordingly, the key issue is how does a firm implement the suggested
optimal pattern of sourcing over time without negative outcomes. This highlights
the significance of developing appropriate strategic (e.g., conducting spend
analysis, identifying new opportunities through analysis of information, managing
suppliers etc.), tactical (e.g., setting specifications, selecting suppliers, negotiating
and contracting etc.), and operational (e.g., ordering, tendering, expediting etc.)
approaches to actualize the identified ‘optimal sourcing strategy’ effectively. It

implies that the buying firm needs to develop appropriate organizational

139

arrangements (e.g., structure, process, people, and technology) for contractual and
market sourcing over time. Successful implementation of the optimal strategy
may thus call for appropriate adoption and execution of e-procurement
alternatives over time such that there exists alignment among the strategy and
organizational structure, culture, operational processes, and portfolio of e-supply
tools over time (CAPS Research study, 2002; Flynn, 2004). These studies
highlight the role of sourcing strategy in achieving a rationalized supply base,
appropriate buyer-supplier relationships, and economies of scale and scope, and
recommend that sourcing strategy should drive the e-sourcing strategy. Such a
perspective is critical from the buying firm’s perspective as several studies raise
concerns about the negative implications of (purely cost oriented) e-procurement
mechanisms on buyer-supplier relationships and long-term relation specific
investments by suppliers (J ap, 2003; Williams et al., 2002; Emiliani, 2000).
Research indicates there are several possible alternative e-sourcing
arrangements (e.g., private e-sourcing exchanges, industry consortia, public e-
market places, and ubiquitous third party non-collaborative tools such as e-
tendering, e-ordering, e-catalogues, e-reverse auctions etc.), which can facilitate
product specific sourcing arrangements spanning long-term contractual sourcing
and spot market sourcing in a varied range of product-market contexts (Skjott-
Larsen et al., 2003; Lancioni et al., 2003; CAPS Research and Mc Kinsey &
Company, 2002). These arrangements help identify a possible supply base,
establish terms of purchase while accounting for price transparency over time,

aggregate demand across firms, facilitate transactions and execute purchases with

140

 

the key objective of optimizing the direct and indirect costs and benefits from
both contract and open market sourcing (Mc Kinsey & Company and CAPS
Research, 2000). There are studies that suggest alternative frameworks to assess
the impact of various e-procurement tools/techniques (Boer et a1. 2001; CAPS
Research and Mc Kinsey & Company, 2002; Skjott-Larsen et al., 2003, Hamik,
2005). These frameworks may be used to identify most befitting procurement
arrangements for adopting both contractual and market arrangements over time.
To illustrate, it is apparent that the buying firm needs to focus on
developing internal capabilities (e.g., ability and skills for effective negotiation of
flexible contracts, IT infrastructure for effective B2B transactions etc.) to
implement both contractual and market arrangements over time. Towards this
end, buying firms may emphasize customized e-procurement mechanisms (e. g.
ubiquitous e-procurement tools for a specific function, or private exchanges with
reverse auctions, e-catalogs, and data-mining tools) for handling varied non-
standard information requirements in earlier stages of the procured product’s
PLC, industry consortia for industry-wide collaboration to manage (increasingly)
standardized information and large volume of procurement in the intermediate
stages, and public market exchanges for commodity products in the later stages
(CAPS Research and Mc Kinsey & Company, 2002). Such a strategy would
enable buying firms to cost-effectively deal with products in different stages of
the PLC. For example, Hewlett Packard used the private auction mechanism

‘TradingHubs’ and private exchange called ‘GetSupply’ when dealing with non-

141

standardized or proprietary products and the public exchange Converge when
dealing with relatively more standardized products.

The pattern of optimal sourcing indicating decline in fraction of contract
purchase in favor of open market purchase as the product advances in the PLC
highlights the importance of building proactive buyer-supplier relationship over
time. This is because it is plausible to increasingly source from market only if the
market price becomes truly competitive with highly capable suppliers that make
the product more or less a standard product over time. This may necessitate the
buying firm to proactively indicate to the contract suppliers about the likely
emphasis on open market purchase over time unless they outperform the
alternatives in the competitive market on a sustained basis. It may thus require the
buying firm to stimulate appropriate type of competition among suppliers in a
manner such that they gain insights about their strengths and weaknesses in the
efficient market, which may be characterized by reduced price, insignificant
transaction cost, and superior non-price attributes (CAPS Research, 2003).

A firm needs to take several measures towards achieving the above
objectives. While undertaking such measures the firm may explore opportunities
for achieving economies of scale and scope across multiple products and across
firms (e. g., auto-parts exchange by the auto manufacturers). Some of the required
measures are:

a) Integrating finance and operations to implement the dual

sourcing arrangement. This implies the focus of sourcing

functions should expand beyond cost, quality, dependability

142

 

 

b)

d)

issues to include financial and operational risk management
activities such as tracking and forecasting of open (spot) market
prices, design of flexible contracts, and valuation of a portfolio
of sourcing arrangements.

Designing contracts with apprOpriate clauses for revising and/or
terminating contracts, and timely re-bidding and re-negotiation
of contracts to adjust with changes in the competitive market
place.

Cost-effective ad0ption and modification of e-procurement
mechanisms over time, which is challenging given the fact that
firms find it hard to change the existing practices to leverage the
technology (CAPS Research and Mc Kinsey & Company,
2002). This means development of standardized, robust
procurement processes both for market and contract sourcing,
and integration of requisite e-procurement tools to
organizational structure, culture, people and information
systems of the buyer and participating suppliers (Flynn, 2004;
Harnik, 2005).

Rationalizing the pre-qualified supplier base such that
constructive competition is stimulated among the suppliers in
the earlier stages (CAPS Research study, 2003; Flynn, 2004).

This would involve inducing adequate number of capable

143

 

 

suppliers to participate in e-procurement process in a
responsible and economically sustainable manner.

Developing clear specifications for products and practicing
clear communication. The greater the codifiability of a
product’s attributes and the clearer the communication the
easier its adoption to e-procurement mechanisms (Kleindorfer
and Wu 2003; CAPS Research, 2003; Huber et al., 2005).
Handling internal and inter-organizational challenges. These
challenges have to be met while implementing e—procurement
strategies (CAPS Research and Mc Kinsey & Company, 2002;
Booz-Allen and Hamilton, 2000; and Huber et al., 2005). The
challenges may be due to significant capital investment,
resistance to change, increased influence of IT departments,
inter-organizational incompatibility, and mismatch between
existing practices and roles of managers and new practices and
roles. There is a possibility that purchasing managers would feel
of some loss of control due to paring of the traditional work and
may feel overwhelmed by the requirement of strategic
negotiating, decision-making, and order allocation skills.
Moreover, given the fact there is simultaneous existence of two
types of purchasing roles: one for contractual purchase and the

other for market purchase, appropriate training of purchasing

144

 

 

professionals are to be undertaken to suit the skills with job

requirements in different procurement contexts.

6.3 Directions for future research

The thesis addresses several important issues in formulating long-term
sourcing strategy by proposing analytical models to account for buying firm’s
purchasing priorities, risk attitude and relative bargaining strength, and
competitive dynamics of the product-market as reflected in market price dynamics
and switching cost between arrangements. The models aid in generation of
normative insights and optimal sourcing plans for varied procurement contexts.
Several extensions can be considered for this research theme and some of them
are discussed below.

The models consider a disutility minimization perspective to express the
buying firrn’s purchasing priorities and risk attitude. It is assumed that a buying
firm’s utility function can be described as a single attribute utility function when
utilities of each transaction attributes can be specified objectively in terms of the
common denominator ‘cost’. The approach presumes that the utility functions of
non-cost attributes to be preferentially independent. However, in reality the
buying firm’s utility functions with respect to different attributes may not be
preferentially independent and additive. In such circumstances the appropriate
multi-criteria utility functions should be considered in formulating the models.
Accordingly, efforts may be undertaken to develop more realistic models while

incorporating more accurate preference structure of the buying firm.

145

 

 

The utility functions in the proposed models are described as a simple
power function of the expense for mathematical tractability. In reality, the
elicitation of the preference of the buying firm may result in a utility function,
which may be more complex. Thus as an extension of the base models, more
general single attribute utility functions may be considered for deriving
meaningful and more context specific insights.

The market price process is modeled as a continuous geometric Brownian
motion process. This process has been found to replicate the empirical price data
for commodity products reasonably well. However, there may be alternative price
processes, which may fit empirical data more closely particularly the ones with
sudden jumps. Investigations may be canied out for exploring optimal strategies
for such price processes.

The model in Chapter 5 assumes the switching costs to be proportional to
the magnitude of quantity readjustrnents between arrangements. However,
switching costs may be of varied types. In particular, these may have a fixed cost

component, which may be independent of the magnitude of adjustment. This
modification would provide more realistic insights on the impact of

switching/transaction cost on sourcing strategy.

The model with switching cost in Chapter 5 is implemented numerically.
This is because an analytical implementation of the model is difficult. However,
the accuracy of results is influenced by the granularity of discretization. The
illustrative numerical experiment on the hypothetical data is limited to 4 time

intervals and 11 discrete values for fractions sourced from market. Further

146

 

 

refinement of state space was difficult due to the exponential growth of
computational efforts with the increase in time steps and alternative values for
fractions sourced from market. Consequently, the results are unlikely to converge
with the continuous time results. Adoption of efficient computational algorithm
will enable to reduce search space and facilitate better convergence to more
precise continuous time results. Hence, efforts may be directed to implement a
more efficient computational algorithm.

The models in Chapters 3, 4 and 5 are based on a conceptual framework
that the sourcing arrangement of a firm are guided by the firm’s purchasing
priorities, product-market uncertainty, life cycle of the procured product, and
increase in market efficiency with the emergence of electronic market. Survey
and case study based research may be carried out for empirical validation of this
framework and for verifying the propositions generated from the application of
the analytical models on the PLC oriented data set. Results of such investigation
would enable the modification of the conceptual model and to revise the
analytical models for more realistic examination of the research theme.

Finally, the proposed models consider the operating context to be
unaffected by issues such as portfolio of products, demand uncertainty, capacity
limits, inventory costs, relationship development cost, possibility of quantity

discounts, size of supply base, contract price limits etc. From a very long-term
strategic perspective these issues may be ignored assuming that the unit price can
reflect the impact of these factors. However, in a short-medium term contexts

these issues may not be too small to ignore. Consequently, from a tactical and

147

 

 

 

Operational strategy perspective, research may be undertaken to develop models
that appropriately consider these issues for a closer representation of the

procurement contexts.

148

 

Appendix A

In the specified problem context, the objective can be described as: to find the

optimal fractions of units to source at open market price of P1(t) and at contract
price of P2 (t) to minimize the disutility of the procurement expenses X (t). Let
In (t) and n2 (t) be the respective fractions. The governing equations that

describe the situation are:

X0) ="1(t)Pr(t)+n2(t)P2(t) (I)
an) = m (t)dP1(t) + n (0sz (t) + A”1(01910) + A"2(t)1"2(t) (2)
1§"1(t)+"2(t) (3)

Let at any time‘t’,I—’1 =P1/P2. One can use a strategy of switching
completely from one source to another depending if Fl = 1+ (1 or 1—— a for a > O ,

the threshold level fixed in advance (i.e., nl changes from 1 to 0 if E =l+a ,
etc.). With this strategy at a time t of switch the savings would be:

A”1(t)1°1(t)+ A"2(t)1"2 (t) = a-Pz (t) (4)
To maximize the saving one has to choose ‘a’ as small as possible which naturally

leads to the limit of Za-Pz (tk) as a —> O , where {tk} are the switching times
k

between arrangements. It turns out that at the limit the following solution is

obtained:

0 if Fla)»
"1(t)={ , — (5)
1 1f 101mg

149

 

51..

 

n20) =1—n1(t) . (6)

X(t) =Pl(t)APz(t) _ (7)

am =n1(t)dP1(t)+n2(t)dP2(t)-P2(t)dL(t) (8)
T

2a - P201. )—a_—,—O-> jPz(t)dL(r) (9)

k 0

where, L(t) is known in stochastic analysis as the “local time” (Oksendal, 1996,

pp. 138) of E(t) at 1. It reflects the amount of time that P1 = P2 in the
planning horizon. It is interesting that L(t) is continuous, increasing fimction that
changes only when Pl = P2. In particular at the limit, X (t) is continuous even

though it has jumps when a >0. It has to be noted that the above solution is

theoretical in nature, since, L(t) is random and cannot be expressed explicitly.

That is why the solution above is difficult to implement in practice.

150

 

 

 

Appendix B

Derivation Of Conditions On Minimum Duration Of Tactical
Planning Horizon For Sufficiency Of The Procured Units

The purchases from alternative arrangements and rebalance (i.e., control) process
to ensure sufficiency of units over time may be explained in terms of the
following diagram in discrete time. Let units procured (N), prices (P), and
expenses (X) at an instant (t=0,1,2), from open market arrangement (1), and

contract arrangement (2) be represented with subscripts ‘t ’, 1 and 2 respectively.

 

Time 0 1 2

Price (market) P10 P11 P12
(contract) P20 P21 P22

Units (market) N10 N11 N12
(contract) N 20 N 21 N 22

Therefore, expense at t = 0:
X0 =P10 N10+P20 N20 (10)

And, expense at t =1:

XO+dX0=(1910+dPlO)N10+(P20+dP20)N20=X1 (11)
:> X1: P11N10+P21N20 (12)
After rebalance, X1 =P,] N11+P21 N21 (13)

151

 

 

It is to be noted that the equilibrium condition at the time of rebalance is the
following: expense before reallocation of fractions between arrangements is same
as expense after reallocation of fractions at each instant. Thus, equating, (12) and

(13), the marginal rate of substitution (MRS) between two sources is:

 

dNll hi (14)
dNZl P11

The rebalance should be such that after rebalance, the total units are at least equal
to total number of units at time t = O.

In general terms at any time ‘ t ’ equation (10) and (11) can be written as:
X: = N121. +N2.P2. (15>

dX, = Nlthlt + N2th2t (16)
The differential equation for expense 'dX,‘ in (16) can be written as follows. If
‘ u ’ is the fraction allocated to market, then one can write (15) and (16) as:

X, = uX, + (1 —u)X, (17)

And, dXt= Nlthlt + NthPZt

dPlt dPZt
=uX,—+(1—u)X,— (18)
P1: P2:

Using equations (3.1) and (3.2), equation (18) can be written as:
<:> dX, = X,(ua1+(1—u)at2)dt+ufl1X,th (19)
The condition for sufficiency of units procured over the time horizon under

consideration may be derived as follows.

152

 

 

According to equation (19) the total procurement expense at ‘t’ while sourcing as
per the optimal fi'actions (e.g., u in open market, and I-u in contract
arrangements) across the two sourcing arrangements starting with an initial

expense of X0 will follow geometric Brownian motion and can be written as:

l
{a1u+a2(l-u)—%,B]2u2}t+j,61udW,
0

X, = Xoe (20)
Accordingly, the mean and variance of X , can be written as:

E(X,)=X0e{a1“+“2(l'“)}‘ (21)
Var(X, ) = Xgemwaz (1'“>}‘[e'312“2‘ — 1] (22)

Similarly, the open market unit price at ‘t’ will follow geometric Brownian

motion with initial price P10 and can be written as:

t
(cl—lawman
O

P it = PlOe (23)

Correspondingly, the mean and variance of P“ can be written as:

E<a.>=aoe“" <24)
2

Varmn = (Pm)2 ezal’ [e131 ‘ — 1] (25)

The contract purchase unit price at ‘t' will have a steady rate of change with
initial price P20 and can be written as:

a t
P2! = 20c 2 (26)

If the buying firm allocates fiaction ‘u’ of total expense to open market sourcing,

then, units procured from open market at ‘t’ can be written as:

153

 

 

1 t
X0 {a1u+(a'2 —a1)—a2u—%fl]2u2+§,612}t+,61(u—l)de,
N1: =— ue 0

P10

fl ue{(a1—az)u+(a2—al)+%,612(1—u2)}t—,B](1—u)W,

N1, = (27)
P10

The mean and variance of N1, can be written as:

E(Nlt) = Nloe{(a'2—a1)(1-“)+.312(1-u)}’ (28)

Var(N1t)=(N10)2 62{(a2-ar )(1-“)+1812(1-“)}t [eflrz(u-1)2t __1] (29)

Consequently, units procured from contractual arrangement at ‘t’ can be written

as:

 

_ _ __l_ 2 2
(1 1“) X0 e{(al a2 )u zfll U }t+fllu”/t (30)
P 20

N2t =

The mean and variance of N 2, can be written as:
E(NZt) = N20e(“l‘“2)“’ (31)

2 2
Var<N2, > = (N20)2 e"“1'“2)“‘ [e51 “ ’ -11 (32)
It may be noted that since a2 > or] (for positive contract premium requirement)

and variability of market prices increase with longer planning horizon or tactical

review period, the average units sourced from market ‘ N1, ’increases

continuously with 't', and the average units sourced from contract ‘ N2, ’

decreases continuously with 't'; however the variance of units sourced from both

154

 

 

the arrangements increase continuously with 't'. These patterns of mean and
variance of units sourced from alternative arrangements will have implications for

contract design.
The total units procured at instant ‘t’, N t = N1, + N 2t (33)

Setting,

dN - - -
27’ = 0 , for the W: that gives mrnrmum N t ;
t

1 2
:1- P {(a -0.' )+”.Bl it
1 n 20 e 2 l 2 ]

Wt :—
fll P10

(34)

The condition for number of units at any time ‘t’ to be more than initial units at t

= 0 with probability 1 is given by:

min(N1t +N2t)-N10 -N2020 (35)

Solving the above equation, after plugging the above value of W: to (27) and

(30), it is found:
.. P
t 2—22— (1—u)1nfl—1nP—1n(—l—) v0; (36)
fl] “(1 — u) PlO u

In (36), P is the proportion of units sourced from open market at t = 0, and t'is

the critical time instant at which W, renders the total units procured to be at least

equal to the initial (at, t = 0) units procured when the buyer allocates expenses

across market and contractual sourcing arrangements as per fractions [u, I—u].

Consequently, for t 2 ti equation (35) is satisfied with probability 1.

155

The corresponding value of critical W: is given by:

1 2
t. P {(a —a )+—,Bl }t*
W. =—1—1n[fle 2 1 2 ] (37)

t'B‘ P10

It is to be noted from (36) that if P10: P20 then t‘ = 0. Thus for such a situation,

the units procured at any time (t>0) would always be as much as the units

procured at t = 0.

However, for P10 at P20 and 0 <t < t'there would be a positive probability that,

N(t) may be less than N(0) fort < t. due to the dominance of stochastic influence
in short-term in a Brownian process. Accordingly, procuring as per the optimal
fractions might render the total units procured to be less than original units
procured (at t = 0), for very short time horizons where t <1. (i.e., very close to
initial time instant). During this period, at any instant ‘t ', the actual probability of
units procured to be more than original units procured at the initial point can be

computed by solving the condition:
Nit +N2t2 NO (38)
For this, the necessary condition becomes:

Re‘bWt +S.edW’ 21 (39)

“X0 [(aZ—al)(l-u)+%.,612(1—u2)]t
.e ,

 

where, R =

P

loeNO

1
(1 _ u)Xo [(a1 -a2)(l—u)—E. 2u2 ]t
= ———.e

S

9

156

 

b =fl1(1—u), and d = ,Blu.

Solving equation (38) for W, , it is found:

there exists W1 < W2 so that (39) holds if W, s W, or W, 2 W2.

Subsequently the probability for sufficiency of units can be computed as:

prob (N1, +N2,2 N0) = prob (W, S W1) + prob (W, 2 W2) (40)
It has been found through simulation study that the duration of likely shortage

(t < t‘ ), and corresponding shortage in units are very small to be of practical

significance for reasonable price and disutility parameter values.

157

 

Appendix C

Derivation Of Conditions For Sufficiency Of Units Procured At
Any Instant ‘t’ Over The Operating Horizon

The conditions on tactical planning horizon ‘t’ in Appendix B ensure that total
units procured are sufficient when the tactical planning horizon exceeds ‘t’. The
conditions for sufficiency of units at any instant without such a restriction of
minimum tactical planning horizon may be derived using Kuhn-Tucker conditions
as demonstrated below (Chang, 2003, pp. 241).

The optimization problem with condition on sufficiency of units can be stated by

modifying the equation (3.7) and (3.8) as the following:

Min{ x" +C,+a2xCx +u(a1 —a2)xC,, +%,Blzu2x2Cxx }= 0; for t < r and
u

x > O C(T,x) = S(x)
Subject to:

uX

Plt

 

1— X . . .
‘+( u) ’ 2D;whereDrs the units ofdemand at t1met=0.
P
2t

Using Lagrange-Multiplier technique the Lagrange can be written as:
_ n 1 2 2 2
L—x +C,+a2xCx+u(a1—a2)xCx+-§,61u x Cxx

“Xi + (1_u)Xt
Plt P2:

 

-/l[ —D] (41)

Using the Kuhn-Tucker conditions:

158

 

 

 

 

 

 

 

 

 

au’ = O
X X
:> x(at —a2>Cx + fltzux’Ca — 41 —’ ——-’—1 = 0 (42)
Plt 'P2t
ll 2 0 (43)
,t[“X’+(1’“)Xt -D]=O (44)
P1: P2t
and, “X’ +(1_“)X’—D_>_o (45)
Plt P2,
It may be observed that as per the sufficiency constraint, if
14X 1— X
‘+( u) ‘ —D > 0, then from (44), ,1 = Oand from (42):
P1: PZt
— C
= _ (a1 2 a2) x ; as in the unconstrained case in (3.9).
.61 xCxx
. . 14X 1— X . . . .
The condrtron —‘— + .(___3‘_)__t_ — D > 0 after srmplrficatron can be written as:
Plt P2t
P D — X
it > 2‘ 1 t 1 , if [i—;] is positive; (46)
P2tX1[___] Plt P2t
1t P2:
or,
P D - X
u < 2’ 1 ’ 1 ,if[—1———-—1—] is negative. (47)
PM, 1— - —1 Pu Pa -
Plt P2:

159

 

uX, + (1—u)X,

 

Altemately, if — D = 0, then from (42) and (43) it may be written

 

I’ll P2t
_ 2 2
that { (“1 “fixcx’ifl‘ “x C” =2 )20 (48)
[£_£]
I’ll P21

For (48) to hold good if, [—l——-l—] is positive, then the numerator of LHS
P P
1t 2t

should be positive.
Correspondingly, (a1 — a2 )xCx + ,612 usz x, 2 0

:> u 2 _(al “a2)Cx
.612an

 

; since CI,r is positive. (49)

Altemately, for (48) to hold good if, [i ——1—] is negative then the numerator
P P
2t

1t
of LHS should be negative.

Correspondingly, (a1 — a2 )xCx + fllzwcszx S 0

(a1 — 0'2 )Cx

 

 

 

:> u s — ; since, C is positive. (50)

fllzxcxx xx 1

. . uX 1— X . . . .
The condrtron ’ + ( u) t — D = 0 after srmplrficatron can be written as:
P1: PZt
“ ‘ 1 1
P21 Xt [_ _ —]
1t P2:

(01—02”): = 02 ‘5’]

Using (3.9) and (3.14), - 2 2
[31 XCxx 151 ("-1)

(51)

 

160

 

uX 1— X . .
Thus, for t + ( u) t 2 D to hold good, the necessary condrtrons are:

I’ll P2t

PZtD—Xt aZ‘al

 

u(x,t) = min{ } when, Pl: > P2,;

ptit[L_L] fl12(n-1)
1! P2!

or,

P D—X _
u(x,t) = max{ 2‘ t a2 a1

 

} when P > P

3 ’ 21 1t

ptit[L__l_] ,612(n —1)
1t P2t

161

(52)

(53)

Appendix D

Derivation For Continuous And Jump Adjustment Of Units

It is observed that for the specified price processes the units purchased from
alternative arrangements can change only in jumps when the prices across
alternative channels are equal. This finding is presented as a mathematical proof
to the following claim.

Claim:

Let, X, >0 be the total procurement expenses of the firm across two sourcing
strategies at time t2 0; and, let ‘ ul (t) ’ be the fraction of units to purchase at open
market price (P1) and ‘ u,(t)’ be the fraction of units to purchase at long-term

contract price (P2) at time t to minimize expense disutility. Thus for continuous

expense:
Xt =“1(t)P1(t)+u2(t)P2(t) (54)
(“(0 = "1 (t)dP1(t) +u2(t)dP2(t) (55)
15 “10) + “2(t) (56)

and, ul and u 2 can be changed only at those times when Pl = P2.

Proof?

The market and contract price processes are defined as in (4.1) and (4.2).

In the following discussion the argument or subscript ‘t’ is dropped sometimes for

notational convenience.

Then (54)-(56) can be written as:

162

X = ulPl +u2P2, d/Y =u1dP1 +u2dP2 andlsu1+u2

It is to be noted that since, u1(t) + u2 (t) = l , at any instant Au, (t) =— Auz (t)

Let XzX/PZ, Pl =P1/P2,hence
dP1=1’1[(a1 -az)dt+/31th)]

(54) can be written as 2? = uIPI +u2=u1(Fl—1)+1

(55) can be written as dX = uldPl +(1— u1)dP2

Accordingly, dX— : d(X - P2'1)= Xd(P2'1) + 102—lax

P
= _ii;+ —l_[u1dP1 +(1-u1)dP2]
P2 P2 P2

=-[u1(FI—1)+11a2dt+,%lutdpl ”1‘4”le
2

After simplification:
= M (57)
Let the dynamics of units sourced from market arrangement be represented as:

dul (t) = K (t)dt + L(t)dW, + Aul (t) ,
where Au] (t) denote the instantaneous change at t.
Hence, the continuous change in ul is:
duf = K (t)dt + L(t)dW, ; (58)
where, K (t) and L(t) are the drift and standard deviation parameters, and W,

represents the Weiner process.

Since, X: u1(P1—l)+1

163

:> (12? = “ldPI + (Pl —1)du,c + dulc -dI—’1 + (Pl —1)Au1 + AuldFl (59)
Equating (57) and (59):

0 = (F, -1)duf + duf -dI5, + AuldF, +(r—>1 — 1)Au1 (60)
=(1_’1 -1)duf + duf final — a2>dt + 131th )1+AutdFt+(Pi —1)Aut

Since, X— is continuous one must have (P, — 1)Au1 =0, which makes,

Aul at 0 when, P, = l ; this implies contract price equals market price.

Thus (60) becomes: (7’1 -1)duf + duf -I—’1[(a, —a2)dt + p,dW)]+Au1dF1 (61)

Further, since, PI is continuous Auld P1: 0.

Hence, (61) becomes:

0= (E —1)duf+duf-Ft[<a1—az)dt+/31th)i (62)
Using (5 8) in (62), and ignoring the higher order terms of dt and terms with
dW,dt :

o = (7’1 —1)[K(t)dt + L(t)dW, ] + L(t)}ipldt (63)
It may be observed that, since W, is random RHS of (63) can be zero only when

both K (t) and L(t) are zero.

Thus, dulc is zero, and correspondingly, for restriction u1(t)+u2 (t) =1, and

continuous expense X, , the fraction of re-adjustments across alternative

arrangements can be made in jumps (lumps) at discrete time points only when
prices across channel are equal. However, for such adjustment, the expense

remains unchanged at the instant. In light of above proof, it is apparent that the

164

 

fraction of units derived in (4.14) and (4.17) for a planning horizon (0,t) remains
optimal only if prices do not become equal at any instant during the planning
horizon. And, if prices become equal at an instant during the planning horizon,
then it would be prudent to revise the optimal fiaction at that instant using (4.14)
and (4.17) for the remaining part of planning horizon. Thus in a given sourcing

context, the sourcing strategy (in terms of optimal fraction u1(t) and uz (t))

would benefit from revision (when prices are equal) over time for the remaining
part of planning horizon. This implies the buying firm may have to renegotiate the
contract as per optimal fractions from time to time by matching the contract price
with competitive market price.

For given initial prices (P1(0),P2 (0)), the expected time at which the prices

become equal can be computed as below.

Let tbe the earliest time at which the prices are equal.

. 2

_A W
Then, P1(0)e(al 2 ”+51 '=P2(0)e"2’.
int—3:2?)
Then, E(z'): 1( ) 2 ;where r=inf{t:P,(t)=P2(t)} (64)

 

1
a-a+—
(212)

165

Appendix E

Computer Code To Obtain The Optimal Sourcing Strategy With
Proportional Switching Cost

/*
* FILE: optim.c
*/

#include <stdlib.h>
#include <stdio.h>
#include <math.h>

FILE *inFile, *outFile;

void computePenultimateS(int node, double laml, double lam2, double price[],
double uParent, double *3);
double a(double price, double laml, double lam2);
double b(double price);
double c(double price, double laml, double 1arn2);
void findUC(int noOfLevel, int origLevel, double qutParentN, double laml ,
double lam2,
double price[], double u[], double expectUp, double

expectDown, double *CO);
void findOthC_7Node(int noUNode, double delU, int noOfLevel, int origLevel,
double qutParentN, double laml, double lam2,

double price[], double expectUp, double
expectDown, double *COptO, double *UOptO);
void findOthC_3Node(int noUNode, double delU, int noOflevel, int origLevel,
double qutParentN, double laml, double lam2,

double price[], double expectUp, double
expectDown, double *COptO, double *UOptO);
void findOthC_1Node(int noUNode, double delU, int noOflxvel, int origLevel,
double qutParentN, double laml, double lam2,

double price[], double expectUp, double
expectDown, double *COptO, double *UOptO); .
void findReducedPrice(int noOfLevel, int j, double price[], double
reducedPrice[]);
void findPenultimateUltimateU(int noOfLevel, double laml , double lam2, double
price[], double u[]);

void main(){

166

int noOfLevel = 0;

double alphal = 0.0;

double alpha2 = 0.0;

double beta = 0.0;

double delt = 0.0;

double p0 = 0.0;

double price[32] = {0.0};

int start = 0, end = 0, i = 0,

j = 0, totalNodes = 0, parentN = 0;

double up = 0.0;

double down = 0.0;

double expl = 0.0, exp2 = 0.0;

// double u[32] = {0.0};

double uParent = 0.0;

double laml = 0.0;

double lam2 = 0.0;

double sNode = 0.0;

double high = 0.0, low = 0.0;

double marketC = 0.0, contractC = 0.0, penaltyC = 0.0, upChildC = 0.0,
downChildC = 0.0;

int upChild = 0, downChild = 0;

double expectUp = 0.0, expectDown = 0.0;

double C[32] = {0.0};

inti0=0,i1 =0,i2=0,i3=0,i4=0,i5=0,i6=0;

int noUNode = 0;

double delU = 0.0;

double minCO = 1000000000;

double uMin[32] = {0.0};

 

double COptO = 0.0, UOptO = 0.0;
double COpt[32] = {0.0};

double UOpt[32] = {0.0};

double qutParentN = 0.0;
double reducedPrice[32] = {0.0};

// FILE *inFile, *outFile;

inFile = fopen("input.txt", "r");
outFile = fopen("output.txt", "w");

/* User Inputs

printf("No of levels: ");
scanf("%d", &noOfLevel); .
printf("Alphal: ");

167

scanf("%lf", &alphal);
printf("Alpha2: ");
scanf("%lf“, &alpha2);
printf("Beta: ");
scanf("%lf', &beta);
printf("Lambdal: ");
scanf("%lf“, &larnl);
printf("LambdaZ: ");
scanf("%lf", &larn2);
printf("delt: ");
scanf("%lf“, &delt);
printf("pOI ");
scanf("%lf', &pO);
printf("expectUp: ");
scanf("%lf", &expectUp);
printf("expectDown: ");
scanf("%lf“, &expectDown);
printf("noUNode: ");
scanf("%d", &noUNode);

 

*/
fscanf(inFile, "%d %lf %lf %lf %lf %lf %lf %lf %lf %lf %d",
&noOfLevel, &alphal, &alpha2, &beta, &larnl, &larn2,
&delt, &p0, &expectUp, &expectDown,
&noUNode);

/* Verify user input */

fprintf(outFile, "No of levels: %d Alpha]: %f Alpha2: %f Beta: %f
Lambdal: %f Lambda2: %f delt: %f p0: %f expectUp: %f expectDown: %f
noUNode: %d\n",

noOflxvel, alphal, alpha2, beta, laml, lam2, delt, p0, expectUp,
expectDown, noUNode);

/* Compute price array */

expl =(alpha1-alpha2) * delt;
exp2 = beta * sqrt(delt);

// printf("expl: %lf exp2: %lfln", expl, exp2);

up = exp(expl + exp2);
down = exp(exp1 - exp2);

168

 

fprintf(outFile,"up: %lf down: %lfln", up, down);
price[O] = p0;
for (i= 0; i < noOflevel; i++){

start = pow(2, i) - 1;

end = 2 * start;

for (j = start;j < end-1'1 ;j++){

price[2*j+l] = price[i] * up;
price[2*j+2] = price[j] * down;

}
totalNodes = pow(2, noOfl.evel+1) - l;

//fprintf(outFile,"\n**** PRICE VALUES ****\n");

//for (i= 0; i < totalNodes; ii-t)
// fprintf(outF i1e,"price[%d]: %If \n", i, price[i]);

delU = 1.0/(noUNode-1);

for (i = 0; i < noOflaevel-l; i++){
start = pow(2, i) - 1;
end = 2 * start;

for (j = start;j < end+1§ji+){

fprintf(outFile "1|!*****************************\n");
3

fprintf(outFile,"**** OUTPUT for NODE: %d ****\n",

fprintf(outFile,"********"'*"'*******************\n");

if (i > 0){
parentN = (j -1 )/2;
qutParentN = UOpt[parentN];
findReducedPrice(noOfLevel-i, j, price,

reducedPrice);

}
if(i = 0)

169

 

findOthC_7Node(noUNode, delU, noOfLevel-i, i,
qutParentN, laml, lam2, price, expectUp, expectDown, &COptO, &UOptO);
else if ( 1 == 1)
findOthC_3Node(noUNode, delU, noOfl.evel-i, i,
qutParentN, laml, lam2, reducedPrice, expectUp, expectDown, &COptO,
&UOptO);
else if (i = 2)
findOthC_1Node(noUNode, delU, noOfLevel-i, i,
qutParentN, laml, lam2, reducedPrice, expectUp, expectDown, &COptO,
&UOptO);

COth] = COptO;
UOpt[j] = UOptO;

}

findPenultimateUltimateU(noOfl.evel, laml, lam2, price, UOpt);

3

fprintf(outFile,"!Hhr*IIIIHIIN!*********************\n")

fprintf(outFile,"**** OPTIMAL U values ******\n");

fprintf(OUtFile "****It*************************\n")-
a a

for (i= 0; i < totalNodes; iH)
fprintf(outFile,"UOpt[%d]: %If \n", i, UOpt[i]);

}

void computePenultimateS(int node, double laml, double lam2, double price[],
double uParent, double *s){

double liml = 0.0, lim2 = 0.0;

double high = 0.0, low = 0.0, upPrice = 0.0, downPrice = 0.0, nodePrice =
0.0;

int upChild = 0, downChild = 0;

double aUp = 0.0, aDown = 0.0,.aNode = 0.0, bNode = 0.0, cUp = 0.0,
cDown = 0.0, bDown = 0.0, bUp = 0.0;

double den2 = 0.0, num2 = 0.0, num3 = 0.0, numll = 0.0, num = 0.0, den
= 0.0, sNode = 0.0;

high =(1. +lam2)/(1. -laml);
low = (l. - lam2)/(l. +lam1);

upChild = 2 * node + l;
downChild = 2 * node + 2;

upPrice = price[upChild];
downPrice = price[downChild];

170

nodePrice = price[node];

/"‘ Case 1 */

if (upPrice > high && downPrice > high){
/* Compute common expressions */
aUp = a(upPrice, laml, lam2);
aDown = a(downPrice, laml, lam2);
den2 = aUp * aUp + aDown * aDown;
num2 = aUp + aDown;
num3 = uParent * den2;
aNode = a(nodePrice, laml, 1arn2);
bNode = b(nodePrice);
numll = 2. * (bNode * uParent + l);
/* Check for +ve S */
num = numll * (aNode + bNode) + num2 + num3;

den = (2. * (aNode + bNode) * (aNode + bNode) + den2);

sNode = —num/den;

if (sNode > 0.0){
/* +ve S valid */
*s = sNode;

}

else{

/* +ve S invalid; check for -ve S */
/* Check for +ve S */

num = numll * (bNode - aNode) + num2 + num3;
den = (2. * (bNode - aNode) * (bNode - aNode) + den2);
sNode = -num/den;
if (sNode < 0.0){
/* -ve S valid */

*s = sNode;

}

else {

171

/* both +ve S and -ve invalid; s = 0 */
*s = 0;

l
/* Case 2 */
if (upPrice > high && low < downPrice && downPrice < high){
/* Compute common expressions */
aUp = a(upPrice, laml, lam2);
bDown = b(downPrice);
den2 = aUp * aUp + bDown * bDown;
num2 = aUp + bDown;
num3 = uParent * den2;
aNode = a(nodePrice, laml, 1arn2);
bNode = b(nodePrice);
numll = 2. * (bNode * uParent + 1);
/* Check for +ve S */
num = numll "' (aNode + bNode) + num2 + num3;
den = (2. * (aNode + bNode) * (aNode + bNode) + den2);
sNode = -num/den;
if (sNode > 0.0){
/* +ve S valid */
*s = sNode;
else{
/"‘ +ve S invalid; check for -ve S */
/* Check for +ve S */
num = numll * (bNode - aNode) + num2 + num3;
den = (2. * (bNode - aNode) '1‘ (bNode - aNode) + den2);
sNode = mum/den;

if (sNode < 0.0){

172

/* -ve S valid */

*s = sNode;

}

else{
/* both +ve S and -ve invalid; s = 0 */
*s = 0;

}
/* Case 3 */

if (low < upPrice && upPrice < high && low < downPrice &&
downPrice < high){

/* Compute common expressions */
bUp = b(upPrice);
bDown = b(downPrice);

den2 = bUp * bUp + bDown * bDown;
num2 = bUp + bDown;
num3 = uParent * den2;

aNode = a(nodePrice, laml, lam2);
bNode = b(nodePrice);
numll = 2. * (bNode * uParent + l);
/* Check for +ve S */
num = numll * (aNode + bNode) + num2 + num3;
den = (2. "‘ (aNode + bNode) * (aNode + bNode) + den2);
sNode = -nurn/den;
if (sNode > 0.0){
/* +ve S valid */

*s = sNode;

}

else{
/* +ve S invalid; check for -ve S */
/* Check for +ve S */

num = numll * (bNode - aNode) + num2 + num3;

173

 

den = (2. * (bNode - aNode) * (bNode - aNode) + den2);

sNode = ~num/den;

if (sNode < 0.0){
/* -ve S valid */
*s = sNode;
}
else{
/* both +ve S and -ve invalid; s = 0 */
*s = 0;
}
}
}
/* Case 4 */

if (low < upPrice && upPrice < high && downPrice < low){
/* Compute common expressions */
bUp = b(upPrice);
aDown = a(downPrice, laml, lam2);
cDown = c(downPrice, laml, lam2);
den2 = bUp * bUp + aDown * aDown;
num2 = bUp - aDown "‘ cDown;
num3 = uParent * den2;
aNode = a(nodePrice, laml, lam2);
bNode = b(nodePrice);
numll = 2. * (bNode * uParent + 1);
/* Check for +ve S */
num = numll * (aNode + bNode) + num2 + num3;
den = (2. * (aNode + bNode) * (aNode + bNode) + den2);
sNode = -num/den;
if (sNode > 0.0){
/* +ve S valid */

*s = sNode;

else{

174

/* +ve S invalid; check for -ve S */
/* Check for +ve S */

num = numll * (bNode - aNode) + num2 + num3;
den = (2. * (bNode - aNode) * (bNode - aNode) + den2);
sNode = mum/den;
if (sNode < 0.0){
/* -ve S valid */

*s = sNode;

}

else{
/* both +ve S and -ve invalid; s = O */
*s = O;

}

/* Case 5 */

if (upPrice < low && downPrice < low){
/* Compute common expressions */
aUp = a(upPrice, laml, lam2);
aDown = a(downPrice, laml, 1arn2);
cDown = c(downPrice, laml, lam2);
cUp = c(upPrice, laml, 1arn2);
den2 = aUp * aUp + aDown * aDown;
num2 = cUp * aUp + cDown "' aDown;
num3 = uParent * den2;
aNode = a(nodePrice, laml, 1arn2);
bNode = b(nodePrice);
numll = 2. * (bNode * uParent + l);
/* Check for +ve S */
num = numll * (aNode + bNode) - num2 + num3;

den = (2. * (aNode + bNode) ‘1' (aNode + bNode) + den2);

sNode = mum/den;

175

_ L.

if (sNode > 0.0){
/* +ve S valid */
*s = sNode;
}
else{
/* +ve S invalid; check for -ve S */
/* Check for +ve S */

num = numll * (bNode - aNode) - num2 + num3;
den = (2. * (bNode - aNode) * (bNode - aNode) + den2);
sNode = -num/den;

if (sNode < 0.0){
/* -ve S valid */
*s = sNode;

}

else{
/* both +ve S and -ve invalid; s = 0 */
*s = 0;

}
/* Case 6 */
if (upPrice > high && downPrice < low){

/"‘ Compute common expressions */
aUp = a(upPrice, laml, lam2);
aDown = a(downPrice, laml, 1arn2);
cDown = c(downPrice, laml, lam2);
cUp = c(upPrice, laml, 1arn2);

den2 = aUp * aUp + aDown * aDown;
num2 = aUp - cDown * aDown;

num3 = uParent * den2;

aNode = a(nodePrice, laml, lam2);

bNode = b(nodePrice);
numll = 2. * (bNode '1‘ uParent + 1);

/* Check for +ve S */

176

num = numll * (aNode + bNode) + num2 + num3;
den = (2. * (aNode + bNode) * (aNode + bNode) + den2);
sNode = -num/den;

if (sNode > 0.0){
/* +ve S valid */
*s = sNode;
}
else{
/* +ve S invalid; check for -ve S */
/* Check for +ve S */

num = numll * (bNode - aNode) + num2 + num3;
den = (2. * (bNode - aNode) * (bNode - aNode) + den2);
sNode = -num/den;

if (sNode < 0.0){
/* -ve S valid */
*s = sNode;
}
else{
/* both +ve S and -ve invalid; s = 0 */
*s = 0;

}

double a(double price, double laml, double lam2){
renu'n (larnl * price + lam2);

}

double b(double price){
return (price - 1.);

}

double c(double price, double laml, double lam2){
return (price + a(price, laml, lam2));
}

177

void findUC(int noOfLevel, int origLevel, double qutParentN, double laml,
double lam2,

double price[], double u[], double expectUp, double
expectDown, double *C0){

int start = 0, end = 0, i = 0,
j = 0, parentN = 0;

double exp] = 0.0, exp2 = 0.0;

// double u[32] = {0.0};

double uParent = 0.0;

double sNode = 0.0;

double high = 0.0, low = 0.0;

double marketC = 0.0, contractC = 0.0, penaltyC = 0.0, upChildC = 0.0,
downChildC = 0.0;

int upChild = 0, downChild = 0;

// double expectUp = 0.0, expectDown = 0.0;

double C[32] = {0.0};

int totalNodes = 0;

/* Compute u values at penultimate and ultimate level */
findPenultimateU1timateU(noOfl.evel, laml, lam2, price, 11);
totalNodes = pow(2, noOfl.evel+l) - 1;
//fprintf(outFile,"\n**** U VALUES (0-30) ****\n");
//for (i= 0; i < totalNodes; iH)
// fprintf(outFile,"%lf \ ", u[i]);
//fprintf(outFile,"\n");
/* C values */
for (i = noOfl..evel; i >= 0; i--){
start = pow(2, i) - 1;
end = 2 * start;
for (j = start;j < end+1;j'H'){
marketC = u[j] * price[j];
contractC = l. - u[j];

/* Except for the first layer, all other layers have parent */

if(i 2= 0){

178

parentN = (j -1)/2;
penaltyC = (laml * price[j] + lam2) * fabs (u[j] -
u[parentN]);
}
else if( i = 0){
if (origLevel = 0)
penaltyC = 0.0;
else {
// parentN = origParent; .
penaltyC =(lam1 "' price[j] + lam2) * fabs
(um - qutParentN);

}

/* Except for the last layer, all other layers have children */
if ( i != noOfLevel){

upChild=2 *j + 1;

downChild = 2 * j + 2;

upChildC = expectUp * C[upChild];

downChildC = expectDown * C[downChild];

 

}
else{
upChildC = 0.0;
downChildC = 0.0;
}

CD] = (marketC + contractC + penaltyC) * (marketC +

contractC + penaltyC)
+ upChildC + downChildC;
// fprintf(outFile,"marketC: %lf contractC: %lf penaltyC:
%lf upChildC: %lf downChildc: %lf C[%d]: %lfln",
// marketC, contractC, penaltyC, upChildC,

downChildC, j, cm);

}
}

//fprintf(outFile,"\n**** C VALUES (30 - O) ****\n");
//for (i = totalNodes - 1; i >= 0; i--)

// fprintf(outFile,"%lf \t", C[i]);
//fprintf(outFile,"\n");

*CO = C[O];

179

void findOthC_7Node(int noUNode, double delU, int noOfLevel, int origLevel,
double qutParentN, double laml , double lam2,

double price[], double expectUp, double
expectDown, double *COptO, double *UOpt0){

inti0=0,il =0,i2=0,i3=0,i4=0,i5=0,i6=0;
// int noUNode = 0;

// double delU = 0.0;

// double minCO = 1000000000;

double u[32] = {0.0};

double C0 = 0.0;

int totalNodes = 0, i = 0;
double uMin[32] = {0.0};
totalNodes = pow(2, noOfLevel+l) - l;

*COptO = 1000000000;
/* Loops start here for U-discretizations */

for (i0 = 0; i0 < noUNode; i0++){
u[O] = i0 * delU;
for (i1 = 0; i1 < noUNode; il++){
u[l]=i1* delU;
for (i2 = 0; i2 < noUNode; iZ++){
u[2] = i2 "' delU;
for (13 = 0; i3 < noUNode; i3++){
u[3] = i3 * delU;
for (i4 = 0; i4 < noUNode; i4++){
u[4] = i4 * delU;
for (i5 = 0; i5 < noUNode; i5++){
u[5] = i5 * delU;
for (i6 = 0; i6 < noUNode;
i6++){
u[6] = i6 * delU;

findUC(noOfLevel,
origLevel, qutParentN, laml, lam2, price, u, expectUp, expectDown, &CO);

// fprintf(outFile,"%lf
\n", C0);

// fprintf(outFile,"\n");

/* Find the min C0 */

180

if (C0 < *COpt0){
*COptO = C0;
*UOptO =
u[0];
for (i= 0; i <
totalNodes; i++)

uMin[i] = u[i];

}

/* MinCO and u-set for the parameter combo */

//fprintf(outFile, "\n **** Optimum Values ****\n");
//fprintf(outFile, "minCO: %lf\n", *COptO);

//fprintf(outFile, "minUO: %lf\n", *UOptO);
//fprintf(outFile,"\n**** OPTIMUM U VALUES (0-30) ****\n");

for (i= 0; i < totalNodes; i++)
fprintf(outFile,"%lf \t", uMin[i]);
fprintf(outFile,"\n");

void findOthC_3Node(int noUNode, double delU, int noOflxvel, int origLevel,
double qutParentN, double laml, double lam2,

double price[], double expectUp, double
expectDown, double *COptO, double *UOpt0){

inti0=0,il =0,i2=0,i3=0,i4=0,i5=0,i6=0;
// int noUNode = 0;

// double delU = 0.0;

// double minCO = 1000000000;

double u[l6] = {0.0};

double C0 = 0.0;

int totalNodes = 0, i = 0;
double uMin[32] = {0.0};

181

 

totalNodes = pow(2, noOfLevel+1) - 1;
*COptO = 1000000000;

// fprintf(outFile,"\rr**** U VALUES (0-30) & C0 ****\n\n");
/* Loops start here for U-discretizations */

for (10 = 0; i0 < noUNode; i0++){
u[0] = i0 * delU;
for (i1 = 0;i1< noUNode;i1++){
u[1]=i1* delU;
for (i2 = 0; i2 < noUNode; i2++){
u[2] = i2 * delU;

findUC(noOfLevel, origLevel, qutParentN, laml,
lam2, price, u, expectUp, expectDown, &CO);

//fpnntf(outFire,"\n**** C VALUES (30 - 0)

****\n");
//for (i = totalNodes - 1; i >= 0; i--)
// fprintf(outFile,"%lf \ ", C[i]);
//fprintf(outFi1e,"\n");
//fprintf(outFile,"%lf \n", C[0]);
//fprintf(outFile,"\n");
/* Find the min C0 */
if (C0 < *COpt0){
*COptO = C0;
*UOptO = u[O];
for (i= 0; i < totalNodes; i++)
uMin[i] = u[i];
}
}
}
}

/"' MinCO and u-set for the parameter combo */

//fprintf(outFile, "\n **"'* Optimum Values ****\n");
//fprintf(outFile, "minCO: %lf\n", *COptO);

//fprintf(outFile, "minUO: %lf\n", *UOptO);
//fprintf(outFile,"\n**** OPTIMUM U VALUES (030) ****\n");

for (i= 0; i < totalNodes; i++)

182

 

fprintf(outFile,"%lf \t", uMin[i]);
fprintf(outFile,"\n");
}

void findOthC_1Node(int noUNode, double delU, int noOfLevel, int origLevel,
double qutParentN, double laml, double lam2,

double price[], double expectUp, double
expectDown, double *COptO, double *UOpt0){

inti0=0,i1=0,i2=0,i3=0,i4=0,i5=0,i6=0;
// int noUNode = 0;

// double delU = 0.0;

// double minCO = 1000000000;

double u[8] = {0.0};

double C0 = 0.0;

int totalNodes = 0, i = 0;
double uMin[32] = {0.0};
totalNodes = pow(2, noOfl..evel+1) - 1;

*COptO = 1000000000;

// fprintf(outFile,"\n**** U VALUES (0-30) & C0 ****\n\n");
/* Loops start here for U-discretizations */

for (i0 = 0; i0 < noUNode; i0++){
u[0] = i0 "‘ delU;

findUC(noOfl.evel, origLevel, qutParentN, laml, lam2, price, u,
expectUp, expectDown, &CO);

//fprintf(outFile,"\n**** C VALUES (3O - 0) ****\n");

//for (i = totalNodes - l; i >= 0; i--)
// fprintf(outFile,"%lf \t", C[i]);
//fprintf(outFile,"\n");

//fprintf(outFile,"%lf \n", C[0]);
//fprintf(outFile,"\n");
/* Find the min C0 */

if (C0 < *COptO){
*COptO = C0;
*UOptO = u[O];
for (i= 0; i < totalNodes; i++)

183

}

uMin[i] = u[i];

}
}
/* MinCO and u-set for the parameter combo */
//fprintf(outFile, "\n **** Optimum Values ****\n");
//fprintf(outFile, "minCO: %lf\n", *COptO);
//fprintf(outFile, "minUO: %lf\n", *UOptO);
//fprintf(outFile,"\n**** OPTIMUM U VALUES (0-30) ****\n");

for (i= 0; i < totalNodes; i++)
fprintf(outFile,"%lf \t", uMin[i]);
fprintf(outFile,"\n");

void findReducedPrice(int noOfl.evel, int j, double price[], double
reducedPrice[]) {

}

int totalNodes = 0, i = 0, k = 0;
int arr[32] = {0};

totalNodes = pow(2, noOfLevel) - 1;

reducedPrice[O] = price[j];
arr[0] =j;

for (i = O; i < totalNodes; i++){
k = arr[i];
arr[2*i+l] = 2*k+l;
arr[2*i+2] = 2*k+2;
reducedPrice[2*i+l] = price[2*k+1];
reducedPrice[2*i+2] = price[2*k+2];
}
//fprintf(outFile,"\n**** REDUCED PRICE VALUES ****\n");

//for (i= 0; i < 2*totalNodes+1; i++)
// fprintf(outFile,"price[%d]: %If \n", i, reducedPrice[i]);

void findPenultimateUltirnateU(int noOfl.eve1, double laml, double lam2, double
price[], double u[]){

int start = 0, end = 0,
j = 0, parentN = 0;
double uParent = 0.0;
double sNode = 0.0;
double high = 0.0, low = 0.0;

184

 

start = pow(2, noOfLevel-l) - 1;
end = 2 * start;

// fprintf(outFile,"\n**** penultimate U VALUES (unnorrnalized)
****\n");
for (j = start;j < end+1 ;j++){
parentN = (j -1)/2;
uParent = u[parentN];
computePenultimateS(j, laml, 1arn2, price, uParent, &sNode);
/* Find u at node */
u[j] = uParent + sNode;

// fprintf(outFile,"uParent: %lf sNode: %lf u[%d]: %lf \n", uParent,
sNode, j, uU]);

/* check and limit 11 value */

if (u[j] < 0.0)
u[j] = 0.0;
if(u[j] > 1.0)
u[j] = 1.0;
}

/* Compute u and C values at ultimate level */

start = pow(2, noOfLevel) - 1;
end = 2 * start;

high =(1. +1am2)/(1. - laml);
low = (1. - lam2)/(1. +lam1);

// fprintf(outFile,"high: %lf low: %lf\n", high, low);

for (j = start;j < end+1 ;j++){

if (price[j] < low)
u[j] = 1.0;

else if (price[j] > high)
u[j] = 0.0;

else{
parentN = (j -1)/2;
uParent = u[parentN];
ufi] = uParent;

185

 

BIBLIOGRAPHY

Anupindi, R and Bassok, Y. 1999. Supply contracts with quantity commitments
and stochastic demand. Quantitative Models for Supply Chain
Management, Kluwer Academic Publishers. MA, USA.

Aytekin, U. and Birge, J. R. 2004. Optimal investment and production across
markets with stochastic exchange rates. Working Paper, Northwestern
University.

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