determining the optimal size of supply base with the consideration of risks of supply disruptions
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Int. J. Production Economics
Int. J. Production Economics 119 (2009) 122–135
0925-52
doi:10.1
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journal homepage: www.elsevier.com/locate/ijpe
Determining the optimal size of supply base with the considerationof risks of supply disruptions
Ashutosh Sarkar a,1, Pratap K.J. Mohapatra b,�
a Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi 221005, Indiab Department of Industrial Engineering & Management, Indian Institute of Technology, Kharagpur, West Bengal 721302, India
a r t i c l e i n f o
Article history:
Received 27 October 2006
Accepted 13 December 2008Available online 14 February 2009
Keywords:
Supply base
Supply risk
Supply disruption
73/$ - see front matter & 2009 Elsevier B.V. A
016/j.ijpe.2008.12.019
responding author. Tel.: +913222 283738; fax
ail addresses: [email protected] (A. Sa
rnet.in, [email protected] (P.K.J. Moh
l.: +91542 670 2776, mobile: +919451890065
a b s t r a c t
Determining the optimal size of the supply base has haunted managers for years.
A small supply base gives rise to the risk of supply disruption, whereas a large supply
base increases the fixed cost. In this paper, we consider the risks of supply disruption
due to occurrence of super, semi-super, and unique events in order to formulate a model
to determine the optimal size of supply base. We depict the model in a decision tree-like
structure and forward a tabular method of solution that obviates the need avoids
evaluation of a majority of non-optimal solutions and thus overcomes the problem of
dimensionality.
& 2009 Elsevier B.V. All rights reserved.
1. Introduction
Improved supplier practices and relationships provideenormous opportunities for cutting costs and improvingsupply chain performance. In order to achieve theircompetitive potential, organizations are increasinglydeveloping long-term partnerships with fewer reliablesuppliers. Rationalization of the supplier base is the firststep towards developing such partnerships. ‘‘Rationaliza-tion’’ and ‘‘reduction’’ of supplier base are the two termsthat practitioners as well as the academicians mostfrequently highlight to indicate the process of bringingthe size of a supply base to a rational level (Dowlatshahi,2000; Parker and Hartley, 1997; Sarkar and Mohapatra,2006; Swift, 1995). Sarkar and Mohapatra (2006) viewrationalization as a superset of the reduction process andpropose two steps to achieve this: (1) determination theoptimal size of supplier base and (2) determination ofconstituents of the supplier base. In this paper, we address
ll rights reserved.
: +913222 255303.
rkar), pratap@hijli.
apatra).
.
only the first issue of determining the optimal size of thesupply base.
Though not very frequent, events like earthquake,tsunami, flood, strikes, fire, and terrorist attacks havedisrupted inter-dependent supply chain resulting inhuge economic losses. Examples of such events that haveoccurred in the past are: 2002 US west coast port strike,terrorist attack on September 11, 2001, Hurricane Katrina in2005, fire in the supplier’s plant of Ericsson, earthquake inTaiwan in 1999 (Berger et al., 2004; Tang, 2006; Wu et al.,2006). It is desirable that firms consider the effect of suchdisruptive events while deciding the size of the supplybase. In this paper, we have addressed the problem ofdetermining the optimal size of the supply base from theperspective of supply risks due to such unforeseen events.
The organization of the paper is as follows: Section 2discusses issues and approaches relevant to the problemof determination of optimal size of supply base that arepresented in the current literature. Section 3 establishesthe relationship between supply risk and the replenish-ment lead time and discusses the relevant costs consid-ered in the formulation of the problem. Section 4 definesthe problem of determining the optimal size of supplybase and presents its mathematical formulation. Thesection also presents the theoretical background for the
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A. Sarkar, P.K.J. Mohapatra / Int. J. Production Economics 119 (2009) 122–135 123
development of a tabular method for solving the problem.Section 5 illustrates the use of the proposed method.Section 6 discusses the effect of varying the values ofmodel parameters of the model. Section 7 presentsconclusions and scope for further development.
2. Literature survey
2.1. Single versus multiple sourcing
There are many research papers that compare singlesourcing with dual or multiple sourcing (e.g., Ramaseshet al., 1991; Chiang and Benton, 1994; Mohebbi andPosner, 1998). These works are motivated by the Just-in-Time (JIT) philosophy that requires fewer suppliers havingshorter and reliable lead times. Mohebbi and Posner(1998) classified the literature on multiple sourcing intotwo groups. The first group focuses on the study of theimpact of order splitting on the lead time (e.g., Sculli andWu, 1981; Pan et al., 1991). In a multi-sourcing environ-ment, splitting of orders affects the effective lead time ofsupply. Sculli and Wu (1981) assumed normal lead timesfor a two-vendor sourcing environment and estimated themean and variance of the effective lead times. Pan et al.(1991) considered three different types of lead timedistribution, namely uniform, exponential and normal, toestimate the effective lead time in a multiple sourcingenvironment and argued that, in such an environment,order splitting is advantageous when the variability in thelead time is high.
The second group studies the total cost of followinga multiple sourcing strategy in an inventory system(e.g., Ramasesh et al., 1991; Mohebbi and Posner, 1998).Ramasesh et al. (1991) analyzed dual sourcing in thecontext of the ‘‘reorder point-order quantity’’ inventorysystem and compared it with single sourcing. Theyconcluded that when the uncertainty in the lead time ishigh and the ordering costs are low, the strategy of havingtwo suppliers is more cost effective compared to thestrategy of having a single supplier. Mohebbi and Posner(1998) analyzed an inventory system with stochasticdemand and lead times. They assumed that all backlogsare lost and included the cost of lost sales in the model.They concluded that multiple sourcing performs better interms of cost savings and service level compared to singlesourcing, except in the situation when one supplier ismuch more unreliable than the other.
The two groups of studies, discussed above, have usedthe information on lead time and total cost of inventorycriteria to compare between single sourcing and multiplesourcing. But they have not used the criterion of supplydisruption—the driver for multiple sourcing. Further,these studies assume the number of suppliers given andtake them as input to study its effect and do not advanceany method for finding the optimal supplier base.
2.2. Supply risk
The quest for fostering long-term relationships with asingle supplier for encouraging supplier commitment to
quality and investment in newer technologies (Sedarageet al., 1999) ‘exposes the buying firm to a greater risk ofsupply interruptions’ (Burke et al., 2007). Tang and Tomlin(2008) observe that although sole sourcing reduces thecost of managing multiple suppliers and fosters bettersupplier relationships, it increases supply risk. Thefollowing incidences establishes the above: (1) Ericssonincurred an opportunity cost estimated at USD400 millionin potential revenue due a fire caused by lightening in thePhillips Semiconductor plant at Albuquerque, New Mexicowhich supplied a critical part to Ericsson (Latour, 2001;Tomlin, 2006); (2) Ford Motors had to suspend operationsfor several days after the famous 9/11 incidence;(3) Toyota’s net income dropped by USD300 million dueto the failure of its sole supplier that supplied 90% of itsbrake valves (Nishiguchi and Beaudet, 1998) and so on.These incidences have forced business managers to factorrisks into all business functions and processes (Cavinato,2004), meaning that risks should be considered whilemodeling business processes. Tang (2006) suggests thatimplementation of initiatives like agile and lean manufac-turing and outsourcing may give a firm cost advantage andincreased market share; however, it forces the supplychain to be vulnerable to supply interruptions. Oke andGopalakrishnan (2009), identified that a retail supply chainmay be disrupted by the following supply risks: (1) import,(2) climate, (3) man-made disasters, (4) natural disasters,socio-economic factors, and (5) loss of key suppliers. Asinbound supply affects the supply chain performance, anystudy on determining the supply base must consider theassociated risks. The occurrence of unforeseen events,mentioned above, may disrupt the inbound supply of asupply chain and they are required to be considered for anystudy on inbound supply. In fact, this realization led to aspurt in academic research that considers supply risks andevaluates its effect on business decisions (Berger et al.,2004; Ruiz-Torres and Mahmoodi, 2007; Wilson, 2007; Kulland Closs, 2008; Serel, 2008; Yu et al., 2009).
Zsidisin et al. (2004) recognized two distinct concepts-probability and impact—with the definition of inboundsupply risk. The first relates to the ‘‘measure of how oftena detrimental event that results in a loss occurs’’ and thesecond relates to the ‘‘significance of that loss to theorganization’’. Any model, meant for determination ofoptimal size of supply base, has to consider supply risksand its impact on the profitability of the purchasing firm.Considered from this point of view, Kraljic’s (1983)purchasing portfolio approach is of great relevance. Heclassified all procured products, into four portfolios basedon supply risk and profit impact. The classification givessufficient ‘‘insights relevant to supply costs and risks’’(Cavinato, 2004). We, for our purpose, define supply riskas ‘‘the probability that supply of an item will be affectedbecause of problems at the supplier’s end’’ and theresulting costs as its impact.
2.3. Previous research on optimal size of supply base
Studies that specifically deal with the problem ofdetermining the optimal size of the supply base are due
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to Agrawal and Nahmias (1997), Weber et al. (2000),Berger et al. (2004), Kauffman and Leszczyc (2005), andRuiz-Torres and Mahmoodi (2007). Agrawal and Nahmias(1997) formulated a profit maximization problem todetermine the optimal lot size and optimal number ofsuppliers. The model assumes that having more numberof suppliers reduces the yield uncertainty but increasesthe fixed cost associated with operating multiple suppli-ers. The model trades off the cost of yield with the fixedcost to determine the optimal size of supply base.
Kauffman and Leszczyc (2005) have used the conceptof buyer utility and decision-related costs to derive theoptimal choice set size for one-time- and repeat-purchasesituations. They have also used the data on actual bidprices and cost data from the industrial steel pipe marketto empirically arrive at the optimal size of the choiceset. Weber et al. (2000) used both multi-objectiveprogramming and data envelopment analysis techniqueto solve the problem. Considering that the biggestmotivation for having multiple suppliers is to preventcomplete disruption of supplies due to an unforeseennatural disasters (like earthquakes, cyclones, tsunami, andflood) and/or man-made disasters (like power grid fail-ures, strikes, and communal violence), the above twomodeling approaches are considered inadequate to deter-mine the size of the optimal supply base.
Berger et al. (2004) argued that supply disruption orinterruption of the inbound supply network can obstructthe functionality of the whole chain. In order to determinethe optimal size of the supply base, they consideredsupply risks-the risks posed by the occurrence ofcatastrophic events that leads to complete disruptionof inbound supply network. They classified these events as(1) ‘super events’ that can affect all suppliers simulta-neously and disrupt supplies from all the suppliers,exhibiting total effect (2) ‘semi-super events’ that affectonly a subset of suppliers, exhibiting regional effect, and(3) ‘unique events’ that affect a particular supplieruniquely, exhibiting local effect. The purchasing environ-ment determines the classification of an event as a super-,semi-super-, or unique event. For example, a cyclone in acoastal region may be termed as a super-event if allsuppliers of an item are located in this region and suppliesfrom this region fail. It may be leveled as semi-super-event
if a few but not all suppliers fail.Berger et al. (2004) considered the probabilities of
occurrence of super- and unique-events and used thedecision-tree approach to find the financial lossand operating cost of working with multiple suppliers.Ruiz-Torres and Mahmoodi (2007) extended the work ofBerger et al. (2004) by also considering states where theremay be partial loss associated with the failure of anyindividual supplier. They have also used the decision-treeapproach to determine the optimal number of suppliersfor the two cases: (1) when the individual supplier failureprobabilities due to unique-events are equal and (2) whenthey are not equal. However, both the works haveneglected the probability of occurrence of semi-super
events. In the present paper, we have considered all threetypes of events to determine the optimal size of thesupply base.
We assume that the semi-super-event is locationspecific. Occurrence of such an event disrupts all suppliersin a geographical location while it does not affectsuppliers in other locations. We also assume that allsuppliers in a location will have more or less similar kindof risks due to the occurrence of a unique-event, and, so,individual variation of supply risks for the suppliers canbe neglected for that location. Furthermore, a decisionbased on small variation in the probability of occurrenceof a unique-event for each individual supplier will beinappropriate, considering that precise information on allpotential suppliers (both the existing suppliers as well asthose who are not) may not be available and the existinginformation base needs to be updated from time to timefor use during the actual process of supply baserationalization. One has to consider individual variationsduring the actual rationalization of the supply base. Thus,for our case, the supply risk due to a unique-event
represents more the character of the supply market ratherthan the individual supplier, and it is a function of thenumber of suppliers engaged.
3. Preliminary parameters for modeling
3.1. Replenishment lead time
For the single-supplier case, if problems (labor,political, financial, technological, etc.) at the supplier’send cannot be solved, supplies never reach the buyer. Ifproblems can be solved in finite time, then the supply willbe delayed by a period that is function of the timerequired to solve the problem. The higher the supply risk,the longer is the delay in supply. Let, L and L0, respectively,denote the average replenishment lead time (average leadtime at normal times called here as the normal replen-ishment lead time) and the maximum replenishment leadtime in the case of a catastrophic event. Therefore, (L0�L)is the time required by a supplier to recover from suchcatastrophic events. Total number of shortage items(stockouts in terms of number of units) for this period isgiven by
S ¼ DðL0 � LÞ (1)
where D is the demand per unit time (assumed constant).
3.2. Relevant costs
In spite of the fact that multiple sourcing increases thefixed cost, the biggest motivation for having such astrategy is to avoid any emergency situation that maylead to a sudden, complete disruption of supplies.Mohebbi and Posner (1998) argued that multiple sourcingreduces the shortage costs but increases the orderingcosts. An increase in the number of suppliers increases theexpenditures for inviting quotations and loses the advan-tage of availing price discounts. Further, the increasemakes the decision more complex and so the time spentby the managers for taking a decision also goes up.Reducing the number of suppliers, on the otherhand, increases the supply risk, and the probability of a
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shortage. Shortage of an item may affect the production interms of stoppages, delay, inefficiency, and qualityproblems. These, two opposing types of cost primarilyinfluence the decision in optimal size determination of thesupply base.
3.3. Item criticality
Kraljic (1983) has shown that both supply risk andcorresponding impact on profit of the organization varywith the nature of item being purchased. With the samelevel of supply risk, the cost or impact of a shortage onthe organization will vary from item to item. We recognizethat it is the criticality of an item that defines the resultingimpact of a supply disruption on the profitability of thebusiness. The word ‘criticality’ that we refer here has adifferent meaning from that used by Kraljic. Kraljic used itto refer to those groups of items/products whose risks andprofit impact are high. However, we use the word‘criticality’ in a more general sense meaning to be acharacteristic of the product. The less critical the product,the less is its impact on the profit of the organization inthe case of supply disruption.
Throughout the paper, we have assumed the case ofMRO (maintenance, repairs and operations) itemsalthough our approach is also applicable to the case ofraw materials. MRO items may be used in differentmachines for different purposes. Non-availability of theitem may lead to: (1) a complete stoppage of machine,(2) a partial monetary loss due to quality and productivityproblems, or (3) no effect at all. The first case is morecommon for repair items whereas the latter two cases arecommon for maintenance and operations items. The profitimpact or the cost associated with the non-availability of
GeographicalPlLocation
Stock...
.
.
.
.
.
S11
S21
S31
S41
S51
S12
S22
S32
S1K
S2KS3K
S4K
Fig. 1. Schematic represen
the item depends on the criticality of the item. The higherthe criticality, the more the profit impact.
4. The model
Organizations procure large number of items havingdifferent characteristics and uses. They are normallysourced from different sets of suppliers. The proposedmodel is applicable for rationalizing the supply base of asingle item only. For rationalizing the whole supply baseof the organization, the model has to be appliedindividually to each item. Items may also be prioritizedusing various portfolio models available in the literature.We exclude the issue of prioritization from our study.
4.1. Probabilities of supplier failures
We have considered the special case of MRO itemswhere a single item may be used in different machines.Fig. 1 shows a typical description of the problem where anitem, used by m different machines, is being sourced fromsuppliers at K different locations. The objective here is todetermine the number of suppliers at each location, andtherefore the total number of suppliers, who shouldconstitute the supply base, so that monetary loss owingto supply disruption due to occurrence of unforeseencatastrophic events is minimized.
We define the following:
P* probability of occurrence of a super-event caus-ing all suppliers to fail
P��k probability of a localized semi-super-event caus-ing all suppliers at location k, (k ¼ 1,y,K), to fail
ant
Machine 1
.
.
.
.
.
.
.
.
Machine 2
Machine 3
Machine m
tation of the model.
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rjk probability of a unique-event causing supplier j,(j ¼ 1,y,Jk) at location k to fail
Jk number of suppliers at location k
ik number of suppliers chosen from location k
Fig. 2 is a decision tree-like representation, named hereas the probability tree, of the alternative ways in whichsupply disruption can take place. Chance nodes (O)indicate the occurrence of disjoint and collectivelyexhaustive events. A branch, emanating from such achance node, indicates the occurrence of an event and islabeled by its probability of occurrence. The node denotedas a square (&) indicates the alternative branches tolocation-specific and supplier-specific events. The nodedenoted with symbol � indicates that the events aremultiplicative in nature. The symbol is used in two placesin the diagram to show: (1) the joint probability ofsuppliers at all locations failing due to simultaneousoccurrence of semi-super-events and (2) the joint prob-ability of all suppliers failing due to the simultaneousoccurrence of unique events. From Fig. 2, various prob-abilities can be easily derived. We refer P(.) to representthe probability of occurrence of an event.
We develop the expressions for computing differentprobabilities by making use of Fig. 2:
P (all suppliers at location k fail because of a semi-
super-event only)
¼ ð1� P�ÞP��k
P (all suppliers at location k fail because of either asuper-event or a semi-super-event)
¼ ½P�� þ ½ð1� P�ÞP��k �
Super event
Suppliers
No super event
*P
Location 1**
kP.
Location k
Location K
Semi-super-event
No Semi-superevent
*1 P−**1 kP−.
.
.
.
.
.
.
.
.
.
.
.
.
.
Supplier
Supplier
j
Fig. 2. Possible outcomes for supp
P (all suppliers at all locations fail because of either asuper-event or a semi-super-event)
¼ ½P�� þ ½ð1� P�ÞP��1 P��2 . . . P��k �
P (a supplier j at location k fails because of a unique-
event)
¼ ½1� fP� þ ð1� P�ÞP��k g�rjk
P (all suppliers at location k fails because of either asuper-event or a semi-super-event or a unique event)
¼ ½P�� þ ½ð1� P�ÞP��k � þ ½1� fP� þ ð1� P�ÞP��k g�YJk
j¼1
rjk
24
35
¼ ½P�� þ ½ð1� P�ÞP��k � þ ð1� P�Þð1� P��k ÞYJk
j¼1
rjk
24
35 (2)
P (all suppliers at all locations fail because of either asuper-event or a semi-super-event or a unique event)
¼ ½P�� þ ð1� P�ÞYKk¼1
P��k
" #þ ð1� P�Þ
YKk¼1
ð1� P��k ÞYJk
j¼1
rjk
8<:
9=;
24
35
(3)
Assuming all supplier failure probabilities (due to unique-
events) to be equal, r1k ¼ r2k ¼ � � � � � � ¼ rJkk ¼ rk, we canrewrite the above-stated equation as
¼ ½P�� þ ð1� P�ÞYKk¼1
P��k
" #þ ð1� P�Þ
YKk¼1
fð1� P��k ÞrJk
k g
" #
(4)
Like Berger et al. (2004), when we do not consider theprobability of occurrence of a semi-super-event, Eq. (4)
*P
All suppliers fail because of super-event
∏=
K
kkP
1
**
All suppliers fail becauseof unique events
All suppliers fail because( ) ∏
=
−K
kkPP
1
***1
of Semi-super-events
Unique event
No supplier fails
∏=
kJ
jjk
1
ρjkρ
⊗
⊗
1
.
.
.( ) ( )∏ ∏
= =
−−K
k
J
jjkk
k
PP1 1
*** 11 ρ.
No Unique event....
kJ
( ) ( )∏ −−K
kPP *** 11
liers and their probabilities.
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reduces to
¼ ½P�� þ ð1� P�ÞYKk¼1
rJk
k
" #(5)
Eq. (5) is the same as that obtained by Berger et al.(2004) who considered only the occurrence of super- andunique-events and neglected the occurrence of semi-super
events.
4.2. The cost function and analysis
In the light of the discussions made in Sections 3.1 and3.3, we consider the cost associated with the criticality ofan item as the loss of output per unit of the item in thecase of its non-availability.
Assume that the item in question is used in m differentmachines with the mean failure time of the item at the lthmachine being tl. If, tð¼ max½tl : l 2 f1;2; . . . ;mg�Þ, is thelongest failure time, then by the time this machine fails,the machine, l, will fail t/tl times on an average. Let Cl
(rupees/unit of the item) be the monetary loss due toresulting inefficiency, delay, or other related problems inproduction caused due to the non-availability of the itemat machine l, then the monetary loss (per unit of the itemper unit time), CS, is given as
CS ¼1
mt
Xm
l¼1
t
tl
� �cl ¼
1
m
Xm
l¼1
1
tl
� �cl
We refer to CS as cost of criticality. CS represents themonetary loss (per unit item per unit time) when there isa complete failure of the supplier network and the item isnot available. However, we now calculate the effectivecost to the buyer when the supply network completelyfails due to a catastrophic event.
The number of units short during the period of failurecan be estimated from the information on demand andthe duration of time required for recovering from acatastrophic event failing the suppliers to supply theproducts.
Number of units short during the period ðL0 � LÞ ¼ DðL0 � LÞ
Total shortage cost during the period of supply failure
CL ¼ CSðL0� LÞDðL0 � LÞ
We refer to CL as the cost of risks. Here, CL represents thetotal monetary loss during the period of supply disrup-tion. The costs of risks multiplied with the totalprobability that the supply network will fail completelywill give us an estimate of the cost to the buyer in theevent of a catastrophic event.
Thus, the total cost of engaging n suppliers, f(n) is thesum of cost of operating n suppliers and costs due toshortages occurred when all suppliers fail to make theirsupplies.
f ðnÞ ¼ CðnÞ þ CL P� þ ð1� P�ÞYKk¼1
P��k
"
þ ð1� P�ÞYKk¼1
ð1� P��k ÞYKk¼1
rikk
( )#
where C(n) is the cost of operating n suppliers andn ¼
PKi¼1ik.
The problem, thus, is to find the optimum number ofsuppliers n that minimizes the sum of the cost ofoperating the suppliers and the cost of shortage dueto supply disruption. A complete list of symbols used inthis paper is given in Appendix A.
4.3. Implications of the optimal solutions
In this section we make a theoretical analysis ofsituations when to engage more number of suppliers.
Theorem 1. If n suppliers are chosen from a single location,such that npik, k 2 ð1;2;3; . . . ;KÞ, then location k will be
preferred to others when ð1� P��k Þð1� rnk Þ is the minimum.
The proof of the above theorem is given in Appendix B.From the above theorem it can be concluded that thevalues of P��k and rk influence the choice of a location. If,for two locations, rk ¼ rkþ1, then P�� determines thechoice of location from where the suppliers are to bepicked.
For a given location k, n suppliers will be preferred tosingle supplier only when
f ðnÞ � f ð1Þo0
Therefore,
CðnÞ þ CL½P�þ ð1� P�ÞP��k þ ð1� P�Þð1� P��k Þr
nk � � cð1Þ
� CL½P�þ ð1� P�ÞP��k þ ð1� P�Þð1� P��k Þrk�o0
Or,
CðnÞ � Cð1Þ
CLoð1� P�Þð1� P��k Þðrk � r
nkÞ (7)
Berger et al. (2004) referred to the left-hand side ofinequality (7) as a ‘critical ratio’. If the right-hand side isless than the ‘critical ratio’, then a single supplier will bepreferred to multiple suppliers. When the ‘cost of risk’increases, the ‘critical ratio’ decreases keeping all othervariables constant suggesting that more suppliers are tobe involved to keep the total costs minimum. It is alsoevident from inequality (7) that as P* increases, the right-hand side decreases, implicitly meaning that the advan-tage of having multiple suppliers reduces graduallywith an increase in the values of P*. The same cannotbe interpreted for P��k based on only inequality (7) as thecondition is true only for the case when all suppliers arechosen from a single location. Increase in the probabilityof unique-event, rk, demands engagement of more numberof suppliers if lnðrkÞo½lnðnÞ=ðn� 1Þ��. At lower levels of rk
the marginal benefit of engaging one more supplier isrelatively more than at higher levels.
Similarly, for the choice of n suppliers to be economic-ally justified compared to the choice of n�1 suppliers, thefollowing condition has to be satisfied
CðnÞ � Cð1Þ
CLoð1� P�Þð1� P��k Þr
n�1k ð1� rkÞ (8)
Like Berger et al. (2004), we have also considered that theoperating cost, C(n), has two components: (1) a fixedcomponent, a and (2) a variable component, b, that varies
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with the size of the supplier base, n. Therefore, CðnÞ ¼
aþ bðnÞ and inequality (8) can be rewritten as
b
CLoð1� P�Þð1� P��k Þr
n�1k ð1� rkÞ (9)
From inequality (9) it can be concluded that the existingsupplier base could be reduced as long as the followingcondition holds:
n41þ lnfb=½CLð1� P�Þð1� P��k Þð1� rkÞ�g=ðln rÞ (10)
Inequality (9) can also be extended to multiple locationsand can be rewritten as for the case when choosing n+1suppliers is superior to choosing n suppliers
b
CLoð1� P�Þ
Yu
k¼1
ð1� P��k Þrnk
k ð1� ruÞ (11)
Location 1
Location 2
Location 2
Location 2
0S
0S
0S
0S
1S
2S
1S1S
2S
2S
1S
2S
Fig. 3. Decision tree representation for thre
where nk ðPu
k¼1nk ¼ nÞ represent the number of supplierschosen from location k, (k ¼ 1,2,y,u) and the n+1thsupplier is chosen from location u.
4.4. Solution procedure
Berger et al. (2004) used the decision-tree approach toobtain a solution for the problem considering only super-
event and unique-event. However, in our formulation of theproblem we have also considered the semi-super-event.The decision tree approach, when used in the formulationof the problem with semi-super-event results in unma-nageable number of branches. For example, Fig. 3 showsthe decision tree for a problem of three locations withonly two suppliers in each location. The decision makerhave three choices for each location: (1) no supplier
0 suppliers(not permitted)
Location 3
Location 3
Location 3
Location 3
Location 3
Location 3
Location 3
Location 3
Location 3
1S 1 supplier
0S2S
2 suppliers1 supplier
1S 2 suppliers
0S2S
3 suppliers2 suppliers
1S 3 suppliers
0S2S 4 suppliers
1 supplier
1S 2 suppliers
0S2S 3 suppliers
2 suppliers
1S 3 suppliers
0S2S 4 suppliers
3 suppliers
1S 4 suppliers
2S 5 suppliers
0S 2 suppliers
1S 3 suppliers
0S2S 4 suppliers
3 suppliers
1S4 suppliers
0S2S 5 suppliers
4 suppliers
1S5 suppliers
2S 6 suppliers
e locations with two suppliers each.
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chosen from a location (0S), (2) only one supplier ischosen (1S), and (3) the two suppliers from that locationare chosen (2S). Considering that at least one supplierfrom a location is to be selected, the total number ofdecision alternatives that are to be evaluated is 26( ¼ 33
�1). The number of decision alternatives growsgeometrically as the number of suppliers in a locationincreases and as the number of locations increases. Wepropose an alternative method to solve the problem—amethod that is both elegant and simple.
It can be observed that the problem of determining theoptimal size of the supply base is combinatorial in natureand finding solution to the problem is computationallycumbersome. The proposed solution method is based onthe following theorems which help to reduce the solutionspace considerably and make the solution computation-ally very simple.
Theorem 2. If selecting n suppliers from a set of u locations
is advantageous compared to selecting n+1 suppliers from
the same set of locations then the former will be more
economic compared to selecting n+2 suppliers chosen from
the same locations.
Theorem 3. If n suppliers are to be chosen from a set of K
locations, with at least one from each location then it is
always economically advantageous to choose as many
suppliers as possible from the location that has the minimum
of unique-event probabilities.
The proofs of the above theorems are given in AppendixB. While searching for solutions to the problem ofdetermining optimal size of supply base, two questionshave to be answered: (1) What should be the total numberof suppliers to be engaged? (2) How these chosensuppliers will be distributed among the various locations?The above two theorems help in efficiently answering thequestions. Theorem 2 restricts the engagement of morenumber of suppliers from a set of locations once the totalcost f(n) starts increasing, thus limiting the number ofevaluations for higher values of n. Theorem 3 allows us toevaluate only the combination that selects as manysuppliers as possible from the location for which rk isthe least. Thus, Theorems 2 and 3 help in reducing thesolution space for the problem considerably. The use ofthe above theorems helps in devising a simple tabularmethod of solution for hand computations. The method ismuch like the tabular method often used in dynamicprogramming.
The problem of selecting n suppliers from K locations iscombinatorial. When all suppliers are chosen from r
locations, rpK, the number of ways the location can bechosen is K Cr . Thus, if we have to choose n suppliers fromK locations, then we have q different combinations oflocations from where these n suppliers can be chosenwhere,
q ¼ K C1 þK C2 þ � � � � � � þ
K CK
When r is fixed, there are a number of ways in whichsuppliers can be selected from these r locations. Theproposed tabular method evaluates each alternative foreach combination of locations separately and helps find
the optimal number of suppliers to be chosen from eachlocation to minimize the total cost, f(n). For example,given five locations if we have to evaluate for variousalternative values of n, there will be five separate tables.The first table evaluates the various alternative values of n
for each location, the second table for all possiblecombinations of two locations, the third for threelocations, and so on. The minimum possible value of n
in a table will be equal to r, the number of locationsconsidered in that table, i.e., at least one supplier will bechosen from each location for a combination. For a givenvalue of n, all possible alternatives for the number ofsuppliers that can be chosen from each location areconsidered for evaluation. Let the variables n1, n2, n3, andn4 represent the various alternative values for how these n
suppliers may be selected from the first, second, third, andfourth locations, respectively. For the same value of n thevalues of n1, n2, n3, and n4 may have many alternatives. Forsuch cases, the alternative combinations are arranged insuch a manner that the combination for which morenumber of suppliers are assigned to the columns on theleft are placed ahead of others.
Based on Theorem 2 it can be argued that total costs fora column in the table has to be calculated to get a value ofn such that f ðnÞ4f ðn� 1Þ for that column. Thus, for acolumn there is no need for calculating the total costbeyond a value of n for which the condition f ðnÞ4f ðn� 1Þis satisfied. For a given value of n, the number of ways inwhich n can be distributed among the locations is large.However, if we arrange the locations sequentially inincreasing order of their unique-event probabilities, thenTheorem 3 allows us to retain only those rows that choosemaximum possible number of suppliers from the firstlocation while discarding all other alternatives for thegiven value of n. Thus, if we assume that the unique-event
probabilities are related as r1or2or3or4, then manynon-optimal rows can be deleted. This leads to significantsaving in computational time. The optimum solution isthe minimum of all the best solutions obtained in theindividual tables.
The proposed solution procedure starts with thegeneration of the truncated tables for selecting n suppliersfrom rðr ¼ 1;2; . . . ;KÞ different locations. The step by stepprocedure for obtaining the solution for the problem isgiven below:
Step 1: For a table (suppliers are to be selected from r
locations), calculate the total cost f rðnÞ (representsthe total cost for the corresponding values of n and ni. ni
ði ¼ 1;2; . . . ; rÞ represents the number of suppliers selectedfrom the ith location) corresponding to the first cell of thefirst column.
Step 2: Calculate the total cost f rðnÞ corresponding tothe next cell for the present column.
Step 3: If the most recently calculated cost associatedwith the present column is more than the previous costfor the same column, then stop and go to Step 4; else goeither to Step 2 if all cells for the present column have notbeen evaluated or to Step 4 otherwise.
Step 4: If the Table is completed (there is no empty cellleft) find the minimum of all costs, f �r ðnÞ, calculated for thecorresponding table and then go to Step 5; else go to 6.
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Step 5: If all Tables have been evaluated, then go to Step7; else go to Step 1.
Step 6: Move to the next column. Calculate the totalcost f rðnÞ, and then go to Step 2.
Step 7: Find the solution to the problem f �ðnÞ ¼
MINðf �r ðnÞÞ; r 2 f1;2; . . . ;Kg where, the correspondingvalues for ni represent the number of suppliers that areto be engaged from ith location.
5. Illustrative example
We take the following parameter values for illustratingthe application of the method:
Table 1Decision alternatives when supp
only.
n n1 Location A Loc
1 1 2131.71 216
2 2 1163.56 116
3 3 1153.92 115
Table 2Decision alternatives when suppl
n n1 n2 L
2 1 1 4
3 2 1 4
3 1 2 4
4 3 1 4
4 1 3 4
4 2 2 4
5 4 1 5
5 1 4 5
5 3 2 5
5 2 3 5
6 5 1 5
6 1 5 5
6 4 2 5
6 2 4 5
6 3 3 5
Demand D
¼ 2 stock keeping units per dayLead time (normal time) L
¼ 7 daysLead time (time due to a
catastrophic event)
L
0 ¼ 30 daysFixed component of C(n) a
¼ 50 (Rupees)Variable component of C(n) b
¼ 20 (Rupees/supplier)Super-event probability P
* ¼ 0.01Semi-super-event probabilities P
��1 ¼ P��2 ¼ P��3 ¼ P��4 ¼ 0:02Unique-event probabilities r
11 ¼ r21 ¼ r31 ¼ � � � � � � ¼ ri11¼ 0:031r
12 ¼ r22 ¼ r32 ¼ � � � � � � ¼ ri21¼ 0:030r
13 ¼ r23 ¼ r33 ¼ � � � � � � ¼ ri31¼ 0:033r
14 ¼ r24 ¼ r34 ¼ � � � � � � ¼ ri41¼ 0:033Number of locations K
¼ 4The costs associated with the criticality of the item andthe mean time between failure of the item at eachmachine are assumed as under:
liers are chosen from a single location
ation B Location C Location D
5.67 2199.62 2233.58
5.63 1167.77 1169.98
4.01 1154.11 1154.22
iers are chosen from two locations.
ocation A:B Location A:C Loca
84.81 485.81 486.
74.79 474.82 474.
74.82 474.88 474.
93.89 493.89 493.
93.89 493.89 493.
93.89 493.89 493.
13.86 513.86 513.
13.86 513.86 513.
13.86 513.86 513.
13.86 513.86 513.
33.86 533.86 533.
33.86 533.86 533.
33.86 533.86 533.
33.86 533.86 533.
33.86 533.86 533.
tion A:D L
81 4
85 4
95 4
89 4
90 4
89 4
86 5
86 5
86 5
86 5
86 5
86 5
86 5
86 5
86 5
ocation B:C
86.87
74.88
74.92
93.89
93.89
93.89
13.86
13.86
13.86
13.86
33.86
33.86
33.86
33.86
33.86
Lo
48
4
4
49
49
49
5
5
5
5
53
53
53
53
53
cation B:D
7.90
74.92
74.98
3.89
3.90
3.89
13.86
13.86
13.86
13.86
3.86
3.86
3.86
3.86
3.86
Lo
48
47
47
49
49
49
51
51
51
51
53
53
53
53
53
Machine
1
Machine
2
Machine
3
Machine
4
Machine
5
Cl (Rs./ unit
item)
856
645 1991 939 2146t1 (days)
56 67 21 43 90t/t1
1.607 1.343 4.286 2.093 1Therefore, cost of criticality Cs ¼ 33.081 (Rupees/unittime/unit of the item)
Cost of risk CL ¼ 33.081�23�2�23 ¼ 35,000 (Rupees)
In the example, there are four locations (K ¼ 4). Wearrange these locations in order of their unique-event
probabilities and label them as location A, B, C, and D.The semi-super event probabilities and the uniqueprobabilities for these locations are as follows:
Location A
Location B Location C Location DSemi-super event
0.020 0.020 0.020 0.020Unique-event
0.030 0.031 0.032 0.033As there are four locations, the first table will have 4(¼ 4C1) columns and all suppliers are to be chosen from asingle location only; the second table will have 6 (¼ 4C2)columns with all suppliers to be chosen from any twolocations; the third table will have 4 (¼ 4C3) columns withall suppliers to be chosen from any three locations andfourth table will have one (¼ 4C4) column with allsuppliers to be chosen from any four locations. We havealso assumed that a maximum of three suppliers can bechosen from each location. The variables n1,n2, n3, and n4
represent the number of suppliers chosen from the first,second, third, and the fourth location, respectively. Whenall suppliers are selected from one location (Table 1), themaximum value of n can be three, because we have
cation C:D
9.00
4.98
5.02
3.90
3.90
3.90
3.86
3.86
3.86
3.86
3.86
3.86
3.86
3.86
3.86
ARTICLE IN PRESS
Table 3Truncated table when suppliers are chosen from two locations.
n n1 N2 Location A:B Location A:C Location A:D Location B:C Location B:D Location C:D
2 1 1 484.81 485.81 486.81 486.87 487.90 489.00
3 2 1 474.79 474.82 474.85 474.88 474.92 474.98
4 3 1 493.89 493.89 493.89 493.89 493.89 493.90
5 4 1 513.86 513.86 513.86 513.86 513.86 513.86
6 5 1 533.86 533.86 533.86 533.86 533.86 533.86
Fig. 4. Numerical example solved using spreadsheets.
A. Sarkar, P.K.J. Mohapatra / Int. J. Production Economics 119 (2009) 122–135 131
assumed every location to have three suppliers. When twolocations are selected (Table 2), a maximum of sixsuppliers can be chosen. It is not necessary to computetotal costs for each case, however. Following the solutionprocedure as explained in the previous section, wecompute the total cost for each location starting fromthe lowest value of n. We can stop whenever a highervalue of the total cost is obtained. Thereafter we proceedto the next location and adopt the same procedure. Thetruncated table for r ¼ 1 will remain same for theexample, but for r ¼ 2 the size of the original table givenin Table 2 will be reduced drastically as shown in Table 3.Comparing Table 2 with Table 3, we see that the numberof rows in Table 3 have reduced to 3 from 15 in Table2—an enormous saving in the amount of computation.
The solution procedure is simple enough to allowspreadsheets to be used to solve a problem. Fig. 4 showsthe solution to the problem outlined above usingspreadsheets. The first six rows of the spreadsheet showthe given values of the input parameters of the modelsuch as probability of occurrence of super, semi-super andunique events, and the cost of risks. The rows 6–11 showthe solution table when suppliers are chosen from eachlocation. Similarly, rows 12–19, 20–26, and 27–32 showthe truncated solution table when suppliers are chosen
from two, three and four locations with at least onesupplier from each location. The figures that are high-lighted are the minimum cost solution for the correspond-ing tables. The minimum of all the best values from eachtable is the optimal solution to the problem that is shownin rows 33–35 with a heading ‘Optimal Solution is’. Theoptimal solution to the problem is obtained as n ¼ 3, withone supplier chosen from each of the locations, A, B, and C.
6. Sensitivity analysis
We also carry out sensitivity testing to see the effect ofchange in values of model parameters on the optimalsolution. In this case we assume the problem to have fivelocations, with a maximum of three suppliers at eachlocation. The base parameter values are as under:
P� ¼ 0:010
P��1 ¼ P��2 ¼ P��3 ¼ 0:030
P��4 ¼ P��5 ¼ 0:020
r1 ¼ r11 ¼ r21 ¼ r31 ¼ r41 ¼ r51 ¼ 0:030
ARTICLE IN PRESS
Table 5
(a) Effect of variation of P��1 and (b) effect of variation of P��k .
P��1 P��2 P��3 P��4 P��5 n n1 n2 n3 n4 n5
(a)
0 0.03 0.03 0.02 0.02 3 3 0 0 0 0
0.001 0.03 0.03 0.02 0.02 3 1 1 1 0 0
0.01 0.03 0.03 0.02 0.02 3 1 1 0 1 0
0.05 0.03 0.03 0.02 0.02 3 1 1 0 1 0
0.1 0.03 0.03 0.02 0.02 3 0 1 0 1 1
0.5 0.03 0.03 0.02 0.02 3 0 1 0 1 1
0.7 0.03 0.03 0.02 0.02 3 0 1 0 1 1
0.8 0.03 0.03 0.02 0.02 3 0 1 0 1 1
(b)
0 0 0 0 0 3 3 0 0 0 0
0.001 0.001 0.001 0.001 0.001 3 2 1 0 0 0
0.01 0.01 0.01 0.01 0.01 3 1 1 1 0 0
0.1 0.1 0.1 0.1 0.1 4 1 1 1 1 0
0.3 0.3 0.3 0.3 0.3 5 1 1 1 1 1
0.5 0.5 0.5 0.5 0.5 5 1 1 1 1 1
0.7 0.7 0.7 0.7 0.7 5 1 1 1 1 1
0.9 0.9 0.9 0.9 0.9 5 1 1 1 1 1
Table 6(a) Effect of variation of r1 and (b) effect of variation of rk.
r1 r2 r3 r4 r5 n n1 n2 n3 n4 n5
(a)
0 0.031 0.032 0.033 0.034 3 1 0 0 1 1
0.001 0.031 0.032 0.033 0.034 3 1 0 0 1 1
0.01 0.031 0.032 0.033 0.034 3 1 0 0 1 1
0.05 0.031 0.032 0.033 0.034 3 0 1 0 1 1
0.1 0.031 0.032 0.033 0.034 3 0 1 0 1 1
0.3 0.031 0.032 0.033 0.034 3 0 1 0 1 1
0.5 0.031 0.032 0.033 0.034 3 0 1 0 1 1
0.8 0.031 0.032 0.033 0.034 3 0 1 0 1 1
(b)
0 0 0 0 0 3 0 1 0 1 1
0.001 0.001 0.001 0.001 0.001 2 0 0 0 1 1
0.01 0.01 0.01 0.01 0.01 2 0 0 0 1 1
0.1 0.1 0.1 0.1 0.1 4 2 0 0 1 1
4 1 0 0 2 1
4 1 0 0 1 2
4 0 2 0 1 1
4 0 1 0 2 1
4 0 1 0 1 2
4 0 0 2 1 1
4 0 0 1 2 1
4 0 0 1 1 2
0.3 0.3 0.3 0.3 0.3 6 2 1 1 1 1
6 1 2 1 1 1
6 1 1 2 1 1
6 1 1 1 2 1
6 1 1 1 1 2
Table 7Effect of variation of CL.
CL N n1 n2 n3 n4 n5
200 1 0 0 0 1 0
500 2 1 0 0 1 0
1000 2 1 0 0 1 0
5000 2 1 0 0 1 0
20 000 3 1 1 0 1 0
35 000 3 1 1 0 1 0
50 000 3 1 1 0 1 0
Table 4Effect of variation of P*.
P* n n1 n2 n3 N4 n5
0.001 3 1 0 0 1 1
0.01 3 1 0 0 1 1
0.05 3 1 0 0 1 1
0.1 3 1 0 0 1 1
0.5 3 1 0 0 1 1
0.7 2 0 0 0 1 1
0.8 2 0 0 0 1 1
A. Sarkar, P.K.J. Mohapatra / Int. J. Production Economics 119 (2009) 122–135132
r2 ¼ r12 ¼ r22 ¼ r32 ¼ r42 ¼ r52 ¼ 0:031
r3 ¼ r13 ¼ r23 ¼ r33 ¼ r43 ¼ r53 ¼ 0:032
r4 ¼ r14 ¼ r24 ¼ r34 ¼ r44 ¼ r54 ¼ 0:033
r5 ¼ r15 ¼ r25 ¼ r35 ¼ r45 ¼ r55 ¼ 0:035
CL ¼ 35;000 ðRupeesÞ
a ¼ 50 ðRupeesÞ b ¼ 20 ðRupees=supplierÞ
To carry out the sensitivity tests, one parameter value hasbeen changed (except for Table 5(b) and 6(b)) whilekeeping other parameters at their base values. Table 4shows the effect of varying P*, super-event probability, onthe optimal size of supplier base. The highlighted row inthe table is the optimal solution for the base problem.It can be seen in the table that as the probability ofoccurrence of super-event increases, the motivation forhaving more number of suppliers diminishes. This isobvious as a result of inequality (8).
Table 5(a) and (b) show the effect of varying the semi-super-event probabilities P��k (k ¼ 1,2,y,5). Table 5(a)shows the effect of changing P��1 at location ‘1’ only, whilekeeping the values of all other parameters at base values,whereas Table 5(b) shows the effect on optimal solution
when the probabilities of occurrence at all locations arechanged. As shown in Table 5(a), changes in the value ofP��1 do not affect the optimal size of supplier base;however, they do affect the attractiveness of location ‘1’.Another important observation that can be made from theresults is that except for the condition P��1 ¼ 0 it is a betterstrategy to choose suppliers from as many numbers oflocations as possible. When all the probabilities ofoccurrence of semi-super-event are changed simulta-neously, as shown in Table 5(b), the optimal size ofsupplier base varies. It is always a better strategy toengage suppliers from as many locations as possible. FromEq. (4) it can be observed that for a given value of n
choosing suppliers from as many locations reduces thetotal probability of supply disruption. Increase in thevalue of P��k increases the total probability of supplierfailure and so more suppliers have to be engaged in suchcases. However, an increase in the number of suppliers
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also increases the fixed cost. Thus, the advantage of havingmore number of suppliers ceases beyond a P��k value of 0.3.The same is shown in Table 5(b).
A similar effect can be observed (Table 6(a)) when onlyr1 is changed while keeping the value of all otherparameters at their base level. Change in r1 has no effecton the optimal size of the supply base; however, theattractiveness of location ‘1’ depends on the value of r1. Asthe value of r1 increases, the attractiveness of location ‘1’diminishes. Table 6(b) shows the effect on optimal size ofthe supply base when the probabilities of occurrence ofunique-events at all locations are varied. The emptyspaces in Table 6(b) indicate that the same values arebeing continued. It can be seen that the optimal size of thesupplier base also goes up with the increase in the value ofprobability of occurrence of unique-events at all locations.
The variation in ‘cost of risks’, CL does influence theoptimal size of supplier base; however, it has no effect onthe choice of location. This fact is evident in Table 7.
7. Conclusions
Organizations, in today’s competitive world, havefocused on their core competencies and depend onsuppliers for other required products and services. Thisstrategy has brought about a complete transformation inthe practice of buyer-supplier relationships. It is almostagreed universally that for a sustainable buyer-supplierrelationship, the existing supplier base needs to betrimmed to a manageable size. We have addressedthe problem of determining the optimal size of supplierbase considering possible disruption of supply due tounforeseen events.
The decision-tree approach has been used in theliterature to determine optimal size of supplier base forthe cases of two types of supply risks resulting from super-
events and unique-events. A major contribution of ourwork is that the problem has been addressed considering,in addition to the above two types of supply risks, thepossibility of occurrence of semi-super-events. When allthe three types of supply risks are considered, thedecision-tree approach results in an unmanageable num-ber of decision alternatives. In order to avoid this problem,we have developed a simple, yet novel, method todetermine the optimal size of supplier base. The methoduses tables to evaluate all possible decision alternatives.The dimensionality problem in the use of tables isovercome by arranging the locations in the increasingorder of the probabilities of occurrence of unique-events.This arrangement helps in excluding a large number ofsub-optimal decision alternatives from evaluation andthus saving computational efforts. From a practitioner’spoint of view this tool is very attractive since spreadsheetscan be used to find the optimal solution.
Considering the lack of precise information about thesupply market, accurately estimating the supply risks interms of numerical values is a difficult exercise. To makesuch estimates, knowledge and experience of individualshaving an in-depth exposure to the supply market have tobe utilized. One interesting finding of the work is that
although all the probability estimates affect the choice ofnumber of suppliers to be engaged, the probabilities ofsemi-super events and unique-events determine the choiceof locations. The parameter sensitivity tests carried out onthe model indicate plausible results. The tests show that itis always a better strategy to have suppliers from as manylocations as possible, and from practical considerationsthis can be viewed as an important conclusion.
One of the basic assumptions the model made is thatthe probabilities of occurrence of unique events for alocation are equal. An extension of this work can be torelax this assumption and develop a solution methodol-ogy for the resulting problem. The costs that have beenconsidered in determining the optimal size of supplierbase are: (1) cost of operating multiple suppliers and(2) cost of risks. The formulation of the problem can befurther developed considering the cost of supplier devel-opment. Further, statistics of extreme events could beused to estimate the probabilities of the occurrence ofvarious events considered in the model to determineoptimal size of supplier base.
Appendix A. List of symbols
S total number of items short during the period ofsupplier disruption
L0 maximum time by which the scheduled ship-ment is delayed after a supply disruptionoccurred due to a catastrophic event
L normal replenishment time of an itemD demand of the item per unit timeP* probability of occurrence of a super event
causing all suppliers to failP��k probability of a localized semi-super event caus-
ing all suppliers at location k to failrjk probability of a unique event causing supplier j at
location k to failJk maximum number of suppliers available at
location k
ik number of suppliers chosen from location k
rk ¼ r1k ¼ r2k ¼ � � � ¼ rJlk
K total number of location where all the suppliersare distributed
m maximum number of machines using a commonmachine part
tl mean failure time of the item at the lth machinecl monetary loss (Rs./unit time) due to non-avail-
ability of an item at machine k
CS cost of criticalityCL cost of risksn total number of suppliers chosen from all
locations. ¼PK
k¼1ikC(n) fixed cost of operating n suppliersf(n) total cost of engaging n suppliersfk(n) total cost of engaging n suppliers if all n
suppliers are chosen from location k
r number of locations from where all the n
suppliers are chosen and rpK
a fixed component of C(n)b variable component of C(n) that varies with n
ARTICLE IN PRESS
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Appendix B. Proof of theorems
Theorem 1. If n suppliers are chosen from a single location,such that npik, k 2 ð1;2;3; . . . ;KÞ, then location k will be
preferred to others when ð1� P��k Þð1� rnkÞ is the minimum.
Proof. Suppliers at location k will be preferred tosuppliers at location, k+1, only when
f kðnÞ � f kþ1ðnÞo0
where fk(n) is the cost of selecting n suppliers fromlocation k.
Therefore,
CL½P�þ ð1� P�ÞP��k þ ð1� P�Þð1� P��k Þr
nk �
� CL½P�þ ð1� P�ÞP��kþ1 þ ð1� P�Þð1� P��kþ1Þr
nkþ1�o0
Or,
ð1� P�Þ½P��k þ ð1� P��k Þrnk � P��kþ1 � ð1� P��kþ1Þr
nkþ1�o0,
if r1k ¼ r2k ¼ � � � � � � ¼ rikk ¼ rk
As the occurrence of a ‘super’ event does not influence the
decision about the choice of location, we can assume the
value of P* as given and constant. Therefore,
½P��k þ ð1� P��k Þrnk � P��kþ1 � ð1� P��kþ1Þr
nkþ1�o0
Or,
ð1� P��k Þð1� rnkÞ4ð1� P��kþ1Þð1� r
nkþ1Þ (A.1)
It is obvious from the above inequality that all locations
can be ordered based on the value of ð1� P��k Þð1� rnk Þ if
ð1� P��k Þð1� rnkÞað1� P��kþ1Þð1� rn
kþ1Þ. From this ordered
sequence, the supplier having the least value of
ð1� P��k Þð1� rnkÞ will be preferred to all other suppliers. &
Theorem 2. If selecting n suppliers from a set of u locations
is advantageous compared to selecting n+1 suppliers from
the same set of locations then the former will be more
economic compared to selecting n+2 suppliers chosen from
the same locations.
Proof. If the selection of n suppliers from a set of u
locations is economically more advantageous compared tochoosing n+1 suppliers then from inequality (11), we get
b
CL4ð1� P�Þ
Yu
k¼1
ð1� P��k Þrnk
k ð1� ruÞ (A.2)
assuming that n+1th supplier is chosen from location u.
Similarly assuming that both n+1th and n+2th supplier
are chosen from location u. The selection of n+2 suppliers
is economically more advantageous compared to selecting
n+1 suppliers from the same set of locations if the
following condition holds:
b
CLoð1� P�Þ
Yu
k¼1
ð1� P��k Þrnk
k ð1� ruÞru (A.3)
However, sinceruo1, one can derive the following con-
clusion from (A.3):
b
CLoð1� P�Þ
Yu
k¼1
ð1� P��k Þrnk
k ð1� ruÞru
Thus, if for same set of locations, selecting n suppliers is
economically advantageous compared to selecting n+1
suppliers then it will be also economically advantageous
compared to selecting n+2 suppliers from the same set of
locations. &
Theorem 3. If n suppliers are to be chosen from a set of K
locations, with at least one from each location then it is
always economically advantageous to choose as many
suppliers as possible from the location that has the minimum
of ‘unique-event’ probabilities.
Proof. Let us consider that n suppliers are to be chosenfrom u locations then the total cost function f(n) can begiven as
f ðnÞ ¼ cðnÞ þ CL½fP�þ P��1 � � � P
��u
þ ð1� P�Þð1� P��1 Þ � � � ð1� P��u Þrn1
1 � � �rnuu g�
where n1,n2,y,nu represent the number of supplierschosen from the location 1,2,y,u, respectively.
Say, we relocate a supplier earlier chosen from location
1 to location u then the total cost f 0ðnÞ can be given as
f 0ðnÞ ¼ cðnÞ þ CL½fP�þ P��1 � � � P
��u
þ ð1� P�Þð1� P��1 Þ � � � ð1� P��u Þrn1�11 � � �rnuþ1
u g�
If the latter alternative has to be advantageous than the
former, the following condition has to be satisfied.
f ðnÞ � f 0ðnÞ40
or
ð1� P�Þð1� P��1 Þ � � � ð1� P��u Þrn1
1 � � �rnuu
� ð1� P�Þð1� P��1 Þ � � � ð1� P��u Þrn1�11 � � �rnuþ1
u 40
or
ð1� P�Þð1� P��1 Þ . . . ð1� P��u Þrn1
1 . . .rnuu 1�
ru
r1
� �40
or
ru
r1
o1
The above condition will be violated only if ru4r1. &
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