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I am in need of help answering three questions that involve customer demand and inventory within the supply chain. I have attached the questions and the reading that covers it. if you have any questions or concerns please let me know.
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Please answer the following questions after reviewing the reading and
lessons for Week 2
1. How can firms cope with huge variability in customer demand?
2. What is the relationship between service and inventory levels?
3. What is the impact of lead time, and lead time variability, on inventory
levels?
Response must be 250 word mininum.
2
Risk Pooling in Business Logistics
This chapter outlines our research framework: It gives an overview of previous risk pooling
research (section 2.1), identifies ten main types of risk pooling and embeds them in the supply
chain, business logistics, and a value chain (section 2.2), which sets the structure for chapter 3,
defines risk pooling in business logistics (section 2.3), explains its mathematical and statistical
foundations (section 2.4) and examines five general features of risk pooling (section 2.5).
2.1
Business Logistics Risk Pooling Research
Risk pooling in business logistics is an active field of research with well over 600 publications so far. The number of publications in this area has steadily increased since its scarce beginnings in the 1960s, gained momentum especially in the 80s and 90s and reached its peak in
2003 as figure 2.1 shows. Perhaps driven by the economic downturn, research activity in 2009
was the highest after 2003. Most research focused on inventory pooling (centralization, the
square root law, portfolio effect, and inventory turnover curve), postponement and delayed
(product) differentiation, and transshipments and inventory sharing. Figure 2.1 is based on the
literature databases Business Source Complete, Business Source Premier, EconLit, and Regional Business News accessed via EBSCOhost® January 11, 2010.
After the concept borrowed from modern portfolio and insurance theory has been mainly applied to inventory centralization, meanwhile research focuses on other forms of risk
pooling, especially postponement and transshipments, and the coordination of risk pooling arrangements and cost and profit allocation through contracts31 and fair allocation
rules, schemes, or methods32 with game theory.
The academics dealing with risk pooling in business logistics are of different backgrounds (mathematics, operations research (OR), operations management (OM), management science, decision sciences, statistics, computer sciences, engineering, business
administration, management, economics, game theory, production, marketing, logistics/supply chain management, physical distribution, and transportation), orientation
(quantitative, analytical, modeling, simulation, empirical, and qualitative research), and
31
32
Dana and Spier (2001), Cheng et al. (2002), Cachon (2003), Wang et al. (2004), Bartholdi and
Kemahloğlu-Ziya (2005), Cachon and Lariviere (2005), Özen et al. (2008, 2010).
Gerchak and Gupta (1991), Robinson (1993), Hartman and Dror (1996), Anupindi et al. (2001), Lehrer
(2002), Granot and Sošić (2003), Ben-Zvi and Gerchak (2007), Dror and Hartman (2007), Wong et al.
(2007a), Dror et al. (2008). Robinson (1993), Raghunathan (2003), Kemahloğlu-Ziya (2004), Bartholdi
and Kemahloğlu-Ziya (2005), and Sošić (2006) consider allocations based on the Shapley (1953) value.
6
2 Risk Pooling in Business Logistics
prominence. Therefore they use inconsistent terms33, frameworks, and structures and
made our thorough literature review and definitions necessary. One might argue that due
to these differences this research must not be compared and only major research should be
cited. However, in contrast to previous research we would like to pursue a more holistic
approach to convey an integrated overview of business logistics risk pooling. Even less
prominent research can draw attention to important details and show directions for further
research. Some often cited risk pooling research has not been published in highly ranked
journals.34 Nevertheless most of the cited references are from the latter.
Most risk pooling research is quantitative and designs mathematical models of problems, develops solution methods (exact methods, algorithms, simulation methods, and heuristics), and determines solutions (optimal inventory control and risk pooling, e. g. transshipment, policies).
60
Inventory Pooling (150)
Postponement (105)
Transshipments (95)
50
Component Commonality (55)
Virtual Pooling (51)
Capacity Pooling (41)
Number of Publications
Product Substitution (40)
40
Order Splitting (37)
Centralized Ordering (27)
Product Pooling (21)
30
20
10
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
0
Year of Publication
Figure 2.1: Number of Publications on Risk Pooling in Business Logistics
33
34
Bender et al. (2004: 233) made similar observations for location theory.
For example Eppen and Schrage (1981).
7
2 Risk Pooling in Business Logistics
2.2
Placing Risk Pooling in the Supply Chain,
Business Logistics, and a Value Chain
Business logistics35 comprises the holistic, market-conform, and efficient planning, organization, handling, and control of all material, product, and information flows from the supplier to
the company, within the company, and from the company to the customer36 and back (reverse
logistics37).
As figure 2.2 shows, the efficient and effective material, product, and information flow
is impaired by inter alia demand and lead time uncertainty. Risk pooling methods can mitigate these uncertainties. They can be implemented at or between the various supply chain
members (suppliers, manufacturers, wholesalers, distribution centers, central warehouses,
delivery or regional warehouses, retailers, stores, and customers).
Thoroughly reviewing the literature referred to in section 2.1, we identified ten risk
pooling methods: capacity pooling, central ordering, component commonality, inventory pooling, order splitting, postponement, product pooling, product substitution,
transshipments, and virtual pooling. We will define and characterize them in detail in
chapter 3.
They can be implemented everywhere along the supply chain. Component commonality
rather concerns production. In general capacity and inventory pooling and postponement
may pool inventories or capacities of or for different locations. Component commonality,
postponement, product pooling, and product substitution may refer to products or their
components. Transshipments and virtual pooling deal with product, material, and information flows between supply chain members within an echelon or across echelons, and central ordering and order splitting with procurement between supply chain members.
In our opinion, all mentioned risk pooling methods but order splitting can reduce demand uncertainty, order splitting only lead time uncertainty, component commonality,
capacity pooling, inventory pooling, product pooling, and product substitution only de-
35
36
37
Business logistics (management) is difficult to distinguish from supply chain management (SCM). The
terms are often used synonymously, although logistics is majorly seen as a part of SCM and not vice versa as in earlier literature (Ballou 2004b: 4, 6f., Larson and Halldórsson 2004, CSCMP 2010). According
to Kotzab (2000) the German business logistics conception already comprised a holistic management
along the whole value creation chain before it adopted the English term SCM. Gabler’s business enzyclopedia defines SCM as building and administrating integrated logistics chains (material and information
flows) from raw materials production via processing to end consumers (Alisch et al. 2004: 2870).
Wegner (1996: 8f.).
See e. g. Richter (1996a, 1996b, 1997), Richter and Dobos (1999, 2003a, 2003b), Dobos and Richter
(2000), Richter and Sombrutzki (2000), Richter and Weber (2001), Richter and Gobsch (2005), Gobsch
(2007).
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2 Risk Pooling in Business Logistics
mand uncertainty. Transshipments and virtual pooling, postponement, and central ordering
may dampen both uncertainties. Concerning order splitting and component commonality
there are dissenting individual opinions, which we will elaborate on in section 4.4.
Various economic conditions have been found to favor the different risk pooling methods. We will focus on these favorable conditions in chapter 4. They flow into a Risk
Pooling Decision Support Tool that helps to determine suitable risk pooling methods for
specific conditions (section 4.4). This tool is applied to determine adequate risk pooling
methods for a German paper merchant wholesaler for coping with its demand and lead
time uncertainty in chapter 5.
Demand Uncertainty
Suppliers
Manufacturers
Distributors
Retailers
Customers
Lead Time Uncertainty
Economic
Conditions
Material, product, and information flow
Risk pooling methods
Figure 2.2: Placing Risk Pooling in Business Logistics
Supply chain member names based on Chopra and Meindl (2007: 5).
Logistics may be considered as a cross-organizational (figure 2.2) and at every member of the
above supply chain a cross-departmental coordination function across all divisions, especially
storage, transportation, procurement, production of goods and services (including R&D,
recycling, and remanufacturing), and sales and distribution (including order processing, recovery, return, and disposal)38 in the value chain in figure 2.3.
38
Delfmann (2000: 323f.), Ballou (2004b: 9ff., 27, 29), Kuhlang and Matyas (2005: 659f.), Grün et al.
(2009: 15, 305), Wannenwetsch (2009: 21f.).
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2 Risk Pooling in Business Logistics
The value chain is a management concept that was developed by Porter (1985: 33ff.)
and describes a company as a collection of activities. These activities create value, use resources, and are linked in processes. In our value chain, the main value activities are procurement, production, and sales and distribution, which are supported by the value activities transportation and storage. Value activities are “technologically and economically distinct activities [… a company] performs to do business”39.
Inventory pooling (IP) mainly pertains to storage, virtual pooling (VP) via dropshipping and cross-filling and transshipments (TS) to transportation, centralized ordering
(CO) and order splitting (OS) to procurement, capacity pooling (CP), component commonality (CC), and (form) postponement (PM) to production, and product pooling (PP) and
product substitution (PS) to sales. However, VP, extending a location’s inventories to other
locations’ ones, is also related to storage and sales, transportation and sales or service capacities may be pooled, any logistics decision may be postponed, and PP and PS may be
applied in production as well. This not mutually exclusive and exhaustive classification is
reflected in figure 2.3 and the next chapter’s structure. We structured our review according
to the value chain approach, because we focus on the impact of risk pooling on the five
mentioned logistics value activities.
Risk pooling helps a company to cope with demand and/or lead time uncertainty and
thus to carry out these value activities at a lower cost for a given service level, a higher
service level for a given cost, or a combination of both40. Thus it may increase expected
profit41 by reducing expected costs and/or increasing expected revenue.
It may allow a company to win a competitive advantage over its competitors by effectively combining Porter’s (2008: 75) competition strategies of differentiation and cost leadership, e. g. in mass customization enabled by postponement42.
39
40
41
42
Porter (2008: 75)
Cf. Chopra and Meindl (2007: 336), Cachon and Terwiesch (2009: 325, 350).
Porter’s (1985: 38) value chain considers margin instead of profit. “Margin is the difference between total
value and the collective cost of performing the value activities”.
For example Feitzinger and Lee (1997).
10
2 Risk Pooling in Business Logistics
Transportation
CP, PM, TS, VP
Procurement
CO, OS, PM
Production
CC, CP, PM, PP,
PS
Sales &
Distribution
CP, PM, PP,
PS, VP
Profit
Storage
IP, PM, VP
Figure 2.3: Important Value Activities Using Risk Pooling Methods
2.3 Defining Risk Pooling
The literature offers various definitions of and confusion about the terms variability, variance,
or volatility43, uncertainty44, and risk45. Lead time and demand uncertainty may arise from
lead time and demand variability or incomplete knowledge46. “Uncertainty is the inability to
determine the true state of affairs of a system”47. “Uncertainty caused by variability is a result
of inherent fluctuations or differences in the quantity of concern. More precisely, variability
occurs when the quantity of concern is not a specific value but rather a population of values”48.
Lead time and demand uncertainty may lead to economic risk49, the possibility50 of a negative
deviation from expected values or desired targets51. The corporate target is expected profit
(figure 2.3), the difference of expected revenue and expected cost.52 The possibility of a positive deviation from an expected value constitutes a chance.53
Despite the costs risk pooling entails54, it may reduce variability and thus uncertainty
and expected (ordering, inventory holding, stockout, and backorder) costs55 and/or increase
expected revenue (product availability, fill rate, service level)56 and thus expected profit57.
43
44
45
46
47
48
49
50
51
52
53
54
Hubbard (2009: 84f.). “Outside of finance, volatility may not necessarily entail risk—this excludes considering volatility alone as synonymous with risk” (Hubbard 2009: 91).
Knight (2005: 19ff.), Haimes (2009: 265ff.), Hubbard (2010: 49f.).
Wagner (1997: 51), Knight (2005: 19ff.), Hubbard (2009: 79ff., 2010: 49f.).
Cf. Haimes (2009: 265). For a detailed treatment of the confusion about the terms uncertainty and variability please refer to Haimes (2009: 265ff.).
Haimes (2009: 265).
Haimes (2009: 266).
Bowersox et al. (1986: 58), Delfmann (1999: 195), Pishchulov (2008: 17).
Wagner (1997: 51).
Cf. e. g. Wagner (1997: 51), Köhne (2007: 321).
Wagner (1997: 52).
Wagner (1997: 51).
Kim and Benjaafar (2002: 16), Cachon and Terwiesch (2009: 328).
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2 Risk Pooling in Business Logistics
We define risk pooling in business logistics as consolidating individual variabilities
(measured with the standard deviation58) of demand and/or lead time in order to reduce the total variability59 they form and thus uncertainty and risk60 (the possibility
of not achieving business objectives61). The individual variabilities are consolidated by
aggregating62 demands63 (demand pooling64) and/or lead times65 (lead time pooling66).
Consolidating and aggregating mean “combining several different elements […] into a
whole”67.
As individual variabilities68 and not individual risks69 are pooled, the term risk pooling
is misleading. Nonetheless, we use it, because it is conventional.
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
Eppen (1979), Chen and Lin (1989), Tagaras (1989), Tagaras and Cohen (1992: 1080f.), Evers (1996:
114, 1997: 71f.), Cherikh (2000: 755), Eynan and Fouque (2003: 704), Kemahloğlu-Ziya (2004), Bartholdi and Kemahloğlu-Ziya (2005), Wong (2005), Jiang et al. (2006: 25), Thomas and Tyworth (2006:
253), Chopra and Meindl (2007: 324ff.), Pishchulov (2008: 8, 17f.), Schmitt et al. (2008a: 14, 20), Simchi-Levi et al. (2008: 48), Cachon and Terwiesch (2009: 325, 331, 344, 350).
Krishnan and Rao (1965), Tagaras (1989), Tagaras and Cohen (1992: 1080f.), Evers (1996: 111, 114,
1997: 71f., 1999: 122), Eynan (1999), Cherikh (2000: 755), Ballou and Burnetas (2000, 2003), Xu et al.
(2003), Ballou (2004b: 385-389), Wong (2005), Kroll (2006), Reyes and Meade (2006), Chopra and
Meindl (2007: 324ff.), Cachon and Terwiesch (2009: 325, 329, 344, 350).
Anupindi and Bassok (1999), Cherikh (2000: 755), Lin et al. (2001a), Eynan and Fouque (2003: 704, 707,
2005: 98), Kemahloğlu-Ziya (2004), Bartholdi and Kemahloğlu-Ziya (2005), Özen et al. (2005), Chopra
and Meindl (2007: 324ff.), Simchi-Levi et al. (2008: 234-238), Cachon and Terwiesch (2009: 331), Yang
and Schrage (2009: 837).
Sussams (1986: 8), Romano (2006: 320). “The standard deviation is the most commonly used and the
most important measure of variability” (Gravetter and Wallnau 2008: 109). Zinn et al. (1989: 2) and Chopra and Meindl (2007: 307) consider the standard deviation a measure of uncertainty. Cachon and Terwiesch (2009: 331f., 282f.) and Chopra and Meindl (2007: 307) use the derived coefficient of variation
(standard deviation divided by mean) as a measure for demand variability or uncertainty.
Pooling independent random variables does not change total variability measured with the variance. Of
course, one could argue that the measuring unit of the variance is squared and therefore difficult to interpret and that the standard deviation and not the variance is used to calculate safety stock. Pooling variabilities measured with the range may even increase total variability.
Cf. e. g. Chopra and Meindl (2007: 336).
Cf. e. g. Pishchulov (2008: 18).
Wagner (1997: 51).
Cf. e. g. Anupindi et al. (2006: 167), Chopra and Meindl (2007: 336).
Gerchak and Mossman (1992: 804), Swaminathan (2001: 131), Hillier (2002b: 570), Randall et al. (2002:
56), Gerchak and He (2003: 1027), Özer (2003: 269), Chopra and Sodhi (2004: 55, 60), Nahmias (2005:
334), Anupindi et al. (2006: 167, 187), Çömez et al. (2006), Romano (2006: 320), Chopra and Meindl
(2007: 177), Pishchulov (2008: 8, 18, 26), Simchi-Levi et al. (2008: 48, 196, 281, 348), Yu et al.
(2008: 1), Yang and Schrage (2009: 837), Bidgoli (2010: 209).
Evers (1997: 55, 57, 1999: 121f.), Benjaafar and Kim (2001), Benjaafar et al. (2004a: 1442, 2004b: 91),
Chopra and Sodhi (2004: 59ff.), Tomlin and Wang (2005: 37), Gürbüz et al. (2007: 302), Van Mieghem
(2007: 1270f.), Ganesh et al. (2008: 1134), Cachon and Terwiesch (2009: 332), Wanke and Saliby (2009:
690), Yang and Schrage (2009: 837).
Thomas and Tyworth (2006: 254).
Evers (1999: 121f.), Cachon and Terwiesch (2009: 336).
Soanes and Hawker (2008).
Gerchak and He (2003: 1028).
The business logistics risk pooling literature finds “the risk related to the uncertainty” of individual demands (Zinn 1990: 12), risk “over uncertainty in customer demand” (Anupindi and Bassok 1999: 187),
standard deviations (Zinn 1990: 13) or variances of individual demands (Zinn 1990: 16), risk “over demand uncertainty” (Weng 1999: 75) or risk “over random supply lead time” (Weng 1999: 82), “inventory
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2 Risk Pooling in Business Logistics
Among others Hempel (1970: 654), Wacker (2004: 630), and the references they give
make requirements for a “good”70 definition. To our knowledge previous attempts to define risk pooling do not satisfy them. They merely describe its causes71, effects72, or aim73,
only target demand pooling74, and equate risk pooling with inventory pooling75 and the
square root law76. Moreover they do neither define nor differentiate between variability77,
uncertainty, and risk. The business logistics risk pooling literature states risk pooling reduces
variability78, “lead-time variability”79, lead time demand variability80, demand variability81, demand variation82, variation83, “demand variance”84, variance of delivery time85, “the variance of the retailers’ net inventory processes”86, “the mean and
variance of cycle stock”87,
uncertainty88, demand uncertainty89, “the uncertainty the firm faces”90, “the effect
of uncertainty”91,
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
risks” (Van Hoek 2001: 174, Pishchulov 2008: 27), “inventory risk” (Kemahlioğlu-Ziya 2004: 76), “forecast risk” (Chopra and Sodhi 2004: 58, 60), risk (Schwarz 1989: 830, 832, McGavin et al. 1993: 1093,
Chopra and Sodhi 2004: 58, 60f., Simchi-Levi et al. 2008: 357, Yang and Schrage 2009: 837), risks (Aviv
and Federgruen 2001a: 514, Gerchak and He 2003: 1027), “risk over the outside-supplier leadtime”
(Schwarz 1989: 828, McGavin et al. 1993: 1093), “lead-time risk” (Thomas and Tyworth 2006: 245f.,
2007: 169), “lead-time uncertainty” (Evers 1997: 70, Thomas and Tyworth 2006: 245), “lead-time fluctuations” (Evers 1997: 70), “demand uncertainty” (Evers 1997: 70), uncertainty (Evers 1994: 51, Rabinovich and Evers 2003a: 206), or “quantity and timing uncertainty” (Collier 1982: 1303) is or are pooled.
“A ‘good’ [formal conceptual] definition […] is a concise, clear verbal expression of a unique concept
that can be used for strict empirical testing” (Wacker 2004: 631). Hempel (1970: 654) requires inclusivity, exclusivity, differentiability, clarity, communicability, consistency, and parsimony.
Nahmias (2005: 334), Anupindi et al. (2006: 168), Romano (2006: 320), Simchi-Levi et al. (2008: 48),
Wisner et al. (2009: 513), Bidgoli (2010: 209).
Gerchak and He (2003: 1027), Özer (2003: 269), Romano (2006: 320), Simchi-Levi et al. (2008: 48),
Cachon and Terwiesch (2009: 325, 350), Bidgoli (2010: 209).
Chopra and Meindl (2007: 212), Cachon and Terwiesch (2009: 321).
Flaks (1967: 266), Gerchak and Mossman (1992: 804), Gerchak and He (2003: 1027), Özer (2003: 269),
Nahmias (2005: 334), Anupindi et al. (2006: 187), Chopra and Meindl (2007: 212).
Anupindi et al. (2006: 168).
Wisner et al. (2009: 513).
Tallon (1993: 192, 199f.) equates variability with uncertainty.
Eynan and Fouque (2005: 91), Anupindi et al. (2006: 167).
Thomas and Tyworth (2007: 171).
Benjaafar and Kim (2001).
Flaks (1967: 266), Zinn (1990: 11), Randall et al. (2002: 56), Eynan and Fouque (2003: 705, 2005: 91),
Romano (2006: 320), Thomas and Tyworth (2006: 255, 2007: 188), Chopra and Meindl (2007: 212),
Pishchulov (2008: 18), Simchi-Levi et al. (2008: 48), Cachon and Terwiesch (2009: 325, 331), Wisner et
al. (2009: 513), Bidgoli (2010: 209).
Eynan and Fouque (2005: 91), Nahmias (2005: 334).
Evers (1999: 133).
Kemahlioğlu-Ziya (2004: 71).
Masters (1980: 71), Ihde (2001: 33).
Schwarz (1989: 830).
Thomas and Tyworth (2006: 247f.)
Pishchulov (2008: 22), Simchi-Levi et al. (2008: 281).
Eynan and Fouque (2003: 714), Pishchulov (2008: 17).
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2 Risk Pooling in Business Logistics
“the risk associated to the variability”92, the “impact of individual risks”93, “risks
associated with forecasting errors and inventory mismanagement”94, and “inventory
risk”95.
Risk pooling is described “to hedge uncertainty so that the firm is in a better position to mitigate the consequence of uncertainty”96, “enables one to avoid […] uncertainty”97, or “removes
some of the uncertainty involved in planning stock levels”98.
It is also referred to as “statistical economies of scale”99, “portfolio efficiencies”100,
“Pooling Efficiency trough Aggregation” or “Principle of Aggregation”101, and “IMPACT
OF AGGREGATION ON SAFETY INVENTORY”102.
Risk pooling is also considered a form of operational hedging. “Hedging is the action
of a decision maker to mitigate a particular risk exposure. Operational hedging is risk mitigation using operational instruments”103, e. g. pure diversification or demand pooling104.
For more on operational hedging please refer to Boyabatli and Toktay (2004) and Van
Mieghem (2008: 313-350).
2.4
Explaining Risk Pooling
Risk pooling can be shown e. g. for inventory or location pooling: Let a single product be
stocked at n separate locations. Demand for this product is a normal random variable105 xi with
known mean µi and standard deviation106 σi for each location i = 1, …, n. The standard devia90
Cachon and Terwiesch (2009: 321).
Özer (2003: 269).
92
Risk pooling “is applied to portfolio theory in finance here [sic!] the risk associated to the variability in
the return from individual stocks is diluted when an investor keeps a portfolio of stocks“ (Zinn 1990: 12).
93
Dilts (2005: 23).
94
Yang and Schrage (2009: 837).
95
Chopra and Sodhi (2004: 59).
96
Cachon and Terwiesch (2009: 321).
97
Pishchulov (2008: 26) remarks this for risk pooling through delayed differentiation.
98
Jackson and Muckstadt (1989: 2).
99
Eppen (1979: 498), Eppen and Schrage (1981: 52), Evers (1994: 51), Özer (2003: 269), Rabinovich and
Evers (2003a: 206).
100
Eppen and Schrage (1981: 52).
101
Anupindi et al. (2006: 187, 189).
102
Chopra and Meindl (2007: 318).
103
Van Mieghem (2007: 1270).
104
Van Mieghem (2007: 1270f.).
105
A random variable is a variable that takes its values (realizations) with certain probabilities respectively
whose values are assigned to certain probability densities (Alisch et al. 2004: 3454).
106
If (the empirical distribution of) demand is forecast (Thonemann 2005: 255f.), σi is the standard deviation
of the distribution of the forecast error in formula (2.1) for calculating safety stock (Caron and Marchet
1996: 239, Pfohl 2004a: 114, Thonemann 2005: 255f., Chopra and Meindl 2007: 306). An estimate of
expected demand (forecast value) is ordered to satisfy the expected value of demand and safety stock is
91
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2 Risk Pooling in Business Logistics
tion σi is a measure of dispersion of individual values of the random variable xi around the
mean µi for every entity i and therefore a measure of xi’s variability107.
If each location just satisfies its own demand, location i needs to hold an amount of
safety stock that allows it to hedge against the demand uncertainty associated with xi. Let
the optimal safety stock at location i in accordance with the newsboy model108 be
(2.1)
,
where z is the safety factor that corresponds to a certain target service level. Therefore,
the total safety stock across all locations is
∑
.
(2.2)
If all inventory holding is centralized at one location, this location needs to serve the total demand
∑
.
(2.3)
The individual demands are aggregated across all locations. Safety stock in the centralized system is
(2.4)
,
where
∑
2∑
109
∑
(2.5)
is the standard deviation of x and ρij is the correlation coefficient of the value of the random
variable for locations i and j. It can be formally shown that the aggregated variability (standard
deviation of total demand σa) is less than or equal to the sum of the individual variabilities
built up as protection against the forecast error, which is at least as high as the demand uncertainty or
standard deviation of demand (personal correspondence with Professor Ulrich W. Thonemann, University
of Cologne, in 2008), and not against uncertainty in demand (Thonemann 2005: 255f.). A high safety
stock is needed, if the (standard deviation of the) forecast error is high. The size of demand fluctuations is
irrelevant (Thonemann 2005: 257). If the distribution of demand is known, σi is the standard deviation of
demand (Zinn et al. 1989: 4) and safety stock is held to hedge against uncertainty in demand. The higher
the uncertainty in demand, the higher is the safety stock. The standard deviation of demand is zero and no
safety stock is needed, if there is no demand uncertainty (Thonemann 2005: 238) and no lead time uncertainty either. Nonetheless, some companies forecast demand, but wrongly use the standard deviation of
demand instead of the standard deviation of the forecast error in calculating safety stock (Korovessi and
Linninger 2006: 489f.).
107
Gravetter and Wallnau (2008: 109).
108
See e. g. Thonemann (2005: 220), Cachon and Terwiesch (2009: 235).
109
Cf. Mood et al. (1974: 178), Zinn et al. (1989: 5), Jorion (2009: 43). This expression is also written
∑
2∑
∑
(Eppen 1979: 500) or
∑
2∑
∑
(cf.
Weisstein 2010). “The double summation sign ∑ ∑ indicates that all possible combinations of i and j
should be included in calculating the total value” (Moyer et al. 1992: 222), where j is larger than i.
15
2 Risk Pooling in Business Logistics
(sum of standard deviations of demand at the n locations σ) because of the subadditivity property of the square root of non-negative real numbers110:
∑
∑
2∑
∑
.
(2.6)
Therefore the safety stock in the centralized system is less than or equal to the one in the
decentralized one
∑
∑
2∑
∑
.
(2.7)
Inequality (2.6) is a special case of the known Minkowski inequality for p = 2. It is always correct, if the variances exist, therefore also for the Poisson and Binomial distribution.111
Hence, the standard deviation of the aggregate demand is lower than or equal to the sum
of the standard deviations of the individual demands. Consequently, inventory pooling or
centralization at a single location can reduce the amount of safety stock necessary to ensure a given service level. The reduction in safety stock depends on the correlations between xi, i = 1, …, n. Inventory pooling does not always reduce safety stock due to the
less-than-or-equal sign.
Yet, the sum of the individual variabilities (standard deviations) only equals the total
aggregated variability (the square root of the sum of the individual variances plus two
times the covariance of the random variable’s value for two entities i and j) in two cases:
(1)
The random variables xi are perfectly positively correlated (the coefficient of
correlation ρij equals 1,
2∑
∑
i, j):
∑
∑
∑
112
(2.8)
∑
(2)
∑
1
∑
∑
.
Random variables xi cannot mutually balance their fluctuations, if at least
n-1 σi equal zero: If n-1 σi equal zero, (2.6) becomes
(2.9)
for this single non-zero σi. If all σi are zero, (2.6) becomes
110
111
112
Gaukler (2007).
Minkowski (1896), Abramowitz and Stegun (1972: 11), Bauer (1974: 72), Gradshteyn and Ryzhik (2007:
1061).
Cf. Moyer et al. (1992: 222). This expression is also written
∑
∑
∑
,
(cf.
Jorion 2009: 43).
16
2 Risk Pooling in Business Logistics
0
(2.10)
.
Apart from this, for independent (the correlation coefficient ρij is equal to 0,
i, j) nor-
mally distributed random variables risk pooling leads to variability reduction:
∑
∑
.
(2.11)
The highest possible variability reduction is achieved, if there are negative correlations which make the second term under the square root equal to minus the first term113 in
(2.6).
Some authors114 give the impression that risk pooling always reduces total variability or
enables to reduce inventory, although this must not be the case as shown above.
Likewise industry and academia often assume that inventory pooling, a type of risk
pooling, always is beneficial, i. e. it either reduces costs or increases profits, and that the
value of inventory pooling increases with increasing variability of demand.115
Kemahlioğlu-Ziya (2004: 40) states this was only always correct for normally distributed
demand such as in Eppen (1979) or Eppen and Schrage (1981). She neglects though that
this is not correct for perfectly positively correlated demands and if at least n-1 σi equal
zero.
Furthermore, for uncertain demand and certain conditions more willingness to substitute
may not lead to higher expected profits or “lower optimal total inventory”116 for full117 and
partial substitution or risk pooling118.
113
Cf. Eppen (1979: 500).
Tallon (1993: 186) imprecisely remarks, “Mathematically, the square root of the sum of the variances is
less than the sum of the individual standard deviations”. Likewise, Anupindi et al. (2006: 191) inaccurately state “the inventory benefits” from physical centralization “result from the statistical principle
called the principle of aggregation, which states that the standard deviation of the sum of random variables is less than the sum of the individual standard deviations”. Gaukler (2007) uses the less-than-orequal sign, but inconsistently says the standard deviation of the aggregate demand was lower than (not
lower than or equal to) the sum of the standard deviations of the individual demands. Although Gaukler
(2007) states his remarks were not intended to be a rigorous derivation of this concept, they should be
consistent. “It is well known that the pooled demand has a lower standard deviation than the sum of standard deviations of individual demands. Thus, the safety stock as well as inventory holding and shortage
costs are lower when products are more substitutable” (Ganesh et al. 2008: 1134). “Inventory pooling
represents a strategy of consolidating inventories and aggregating stochastic demands, enabling reductions in inventory holding and shortage costs” (Pishchulov 2008: 8). “Risk pooling suggests that demand
variability is reduced if one aggregates demand across locations” (Romano 2006: 320, Simchi-Levi et al.
2008: 48). “The aggregation of demand stemming from risk pooling leads to reduction in demand variability, and thus a decrease in safety stock and average inventory” (Simchi-Levi et al. 2008: 50). “Centralizing inventory reduces both safety stock and average inventory in the system” (Simchi-Levi et al. 2008:
51). Finally, inventory pooling does not automatically reduce inventory, but it may allow to reduce the
inventory necessary to provide a given service level.
115
Kemahlioğlu-Ziya (2004: 40).
116
Yang and Schra
