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Abstract

We consider supplier diversification in an EOQ type inventory setting with multiple suppliers and binomial yields. We characterize the optimal policy for the model and show that, in this case, it does not pay to diversify, in contrast to previous results in the random yield literature.
Supplier Diversification under Binomial Yield
Mehmet Murat Fadıloğlu*
Emre Berk
Mustafa Çağrı Gürbüz
* Department of Industrial Engineering, Bilkent University, Ankara, Turkey
E-mail:mmurat @bilkent.edu.tr
Department of Management, Bilkent University, Ankara, Turkey
E-mail:eberk @bilkent.edu.tr
MIT-Zaragoza Logistics Center, Zaragoza, Spain
E-mail: mgurbuz@zlc.edu.es
February, 2008
We consider supplier diversification in an EOQ type inventory setting with multiple suppliers
under binomial yield and zero leadtimes. We characterize the optimal policy for the model and
show that, in this case, it does not pay to diversify and it is better to use a sole supplier. This
result constitutes a contrast with the previous ones in the literature assuming various yield
structures where there is benefit in supplier diversification due to risk pooling effect.
1. Introduction
In this paper, we analyze an inventory system with possible multiple suppliers under bino-
mial yield, characterize the optimal replenishment policy, and show that working with a single
supplier is always optimal.
There is a vast literature on inventory models facing yield uncertainty. Yano and Lee [16]
classify random yield models in the literature into four categories: Binomial yield, batch size in-
1
dependent-stochastically proportional yield, batch size dependent-stochastically proportional
yield and random capacity.
Binomial yield differs from the rest in the important aspect that it exhibits independence
across units in a given order whereas intra-batch correlation is implied in all other categories.
Mazzola et al. [11] provides the earliest work on an inventory model with binomial yield. For
other works in this category, see Beja [3], Grosfeld-Nir and Gerchak [8], Barad and Braha [2],
Sepheri et al. [14], and Gürbüz [10].
For a comprehensive review of individual works in the remaining categories, we refer the
reader to Yano and Lee [16], and mention only Erdem and Ozekici [5], Gerchak et al. [7], Gur-
nani et al. [9], Wang and Gerchak [15] and Erdem et al. [4] as recent works.
Most of the previous work has focused on inventory systems with a single supplier and do
not consider diversification issues. However, in practice buyers can work with multiple suppli-
ers, and thus may reduce variability of actual yield through diversification among suppliers. An-
upindi and Akella [1], Erdem and Ozekici [5], Gerchak and Parlar [6], Parlar and Wang [13]
consider diversification under different yield structures (excluding binomial yield), and Gürbüz
[10] under binomial yield. Erdem et al. [4] allow for more than two suppliers in the presence of
random capacity.
There is also evidence from practice that diversification is desirable for certain settings. A
good example is by Li & Fung, the supply chain integrator, which connects the fragmented sup-
plier market (Asia, Indian subcontinent, the Caribbean basin) to the large retailers in US and
Western Europe. When Li & Fung receives an order from a retailer, say for a 1000 units, depend-
ing on the characteristic of the product and the particular supply chain, they procure the item
from, say, 5 different suppliers (even within the same country), 200 from each (see [12]). How-
ever, in this paper we show that, under binomial supplier yield, working with a single supplier is
always optimal. Thus, not all sources of uncertainty can be remedied through diversification.
The rest of the paper is organized as follows: In Section 2, we describe the model. Sections 3
and 4 present the optimality results and the discussion. Section 5 concludes the paper with possi-
ble extensions of this work.
2
2. Model
We consider an EOQ type inventory model in which demand occurs continuously at a constant
rate D. The system is subject to an inventory holding cost h per unit held in inventory per unit
time. No backorders are allowed. The system is replenished by ordering from n () sup-
pliers where N denotes the total number of available suppliers. We assume that the replenish-
ments are instantaneous (i.e., zero lead time) and that each unit delivered by supplier i, where i =
1, 2, …, N, has a constant probability that it is of acceptable quality, p
nN
i, independent of the order
quantity, Qi. The system pays a unit purchasing cost ci for each unit ordered irrespective of its
quality. Later, the results are also extended to the case where the system pays only for the
“good” units. The “bad” units are discarded right at the delivery (and hence do not join the stock
and do not induce any holding cost). There is an ordering cost
(
)
1,..., N
KQ Q , which is a positive
function of the order quantities Qi. We assume that
(
)
1,..., N
KQ Q is nondecreasing in each order
quantity Qi.
We employ the following EOQ type ordering policy: The inventory level is monitored con-
tinuously and an order of size Qi is placed at supplier i whenever inventory level hits zero.
Clearly there is no incentive to order prior to reaching zero inventory level because of the zero
lead time assumption. The objective is to minimize the total long-run average cost, which con-
sists of ordering, purchasing and holding components. All the order quantities have to be opti-
mized simultaneously in order to characterize the optimal policy parameters.
The reader should note that the number of acceptable units from supplier i is the sum of Qi
independent Bernouilli random variables and thereby has binomial distribution with parameters
pi and Qi. We call this quantity the effective received from supplier i and denote it as Ri. The
number of the effectives received from different suppliers are independent. The sum of these
quantities from all suppliers is the total effective received R where
1
N
i
i
R
R
=
=
. (1)
We can easily compute the expected values of R and of R2 as
11
[] [ ]
NN
i
ii
ii
E
RER p
==
==
∑∑
Q
and (2)
3
22
111
2
1111
2
11
[] []2 [][]
() []2 [][ ]
(1 ) .
NNN
iij
iiji
NNNN
ii i
iiiji
NN
iii ii
ii
ER ER ER ER
VarR E R ER ER
ppQ pQ
===+
====+
==
=+
=++
⎛⎞
=−+
⎜⎟
⎝⎠
∑∑
∑∑
∑∑
j
(3)
Note that the expected total effective exhibits similar structure with the models incorporating
stochastically proportional yield instead of binomial. Yet, the variance of the effective received
is different in the sense that it is proportional to the order quantities in contrast to the stochasti-
cally proportional yield models in which the variance is proportional to the square of the order
quantities. This characteristic of the binomial yield eliminates the need for diversification in or-
der to obtain risk pooling effect as shown in the rest of the manuscript.
The inventory level of the described inventory system is a stochastic process. The process
regenerates itself every time the inventory level hits zero. Thus, one can obtain the long-run av-
erage cost for the system using the renewal-reward analysis. The replenishment cycle is defined
as the time elapsed between two consecutive instants the inventory level hits zero. The system
orders from its suppliers once per cycle. Even though the problem setting is based on discrete
inventory units, we treat the units as continuous variables for analytical convenience as is cus-
tomary in the previous yield literature ([11], [22], and [29]).
The expected cycle length is
[][E CycleLength E R D]
=
(4)
and the expected cycle cost is
()
2
1
1
[ ] ,..., [ ]
2
N
Nii
i
h
E
CycleCost K Q Q cQ E R
D
=
=++
. (5)
Finally, due to renewal-reward theorem, the long-run average cost for the system, denoted as CR,
can be obtained from
[
[]
]
E
CycleCost
CR
Cycle Length
=. (6)
4
3. Optimality Results
Define adjusted unit cost for supplier i as
(
1
2
i
i
i
ch
)
i
A
C
pD
=+ −p
0
. (7)
Lemma 1: Let i and j be the indices for any two suppliers such that , let
,
ij
QQ>
(
)
iji
ppQ Q−<Δ
j
<
, and let
()
1,..., N
KQ Q K
=
. If the order quantity from supplier j is de-
creased by , and the order quantity from supplier i is increased by , then the resulting
increase in the long-run average cost is
Δ)/( ij pp
()
(
[]
ii
ii ji jj
jj
j
Qq i j
Qq pp Qq
Qq
Dp
CR CR AC AC
ER
=
=+ Δ =
=−Δ
)
=−Δ
, (8)
where the notation ij
Qa
Qb
Exp =
=
denotes the same expression Exp when is set to a and
i
Q
j
Q is set
to b.
Proof: Using (4), we compute the expected cycle length under the new order quantities as
()
(
)
(
)
()
,
1
[]
[]
ii ji
jj
ii
jj
ll i i j i j j
lij
Qq pp
Qq
N
ll
lQq
Qq
pQ p q p p p q
E CycleLength D
pQ E CycleLength
D
=+ Δ
=−Δ
==
=
+
+
=
==
Δ
(9)
and observe that the expected cycle length and the expected total effective received remain the
same. Then using (5) and (9) we compute the change in the expected cycle cost as
5
()
()
()
()
()
()
()
()
() ()
[] []
(1 ) (1 )
2
(1 ) (1 )
2
(1 ) (1 )
22
ii
ii ji jj
jj
Qq
Qq pp Qq
Qq
ii j i i j j j
iiiji ij jj j
iji j j i j j
j
i
ji j
ij
E CycleCost E CycleCost
cq p p q cq q
h
p
pq pp q p pq q
Dh
cpp c pp pp
D
c
chh
pp p
pD pD
=
=+ Δ =
=−Δ
=+ Δ+ −Δ
+−+Δ+Δ
Δ+ΔΔ
⎛⎞
⎛⎞
=++
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠
Δ
⎜⎟
. (10)
Finally by bringing together (6), (9), and (10) we obtain (8). The reader should note that this
analysis is only valid when
()
iji
ppQ Q−<Δ
j
<, since the order quantities always need to be
positive. If one of the order quantities becomes zero, one would need to add the change in the
ordering costs as well.
When the ordering cost is constant, Lemma 1 suggests that it is better to order more from a
supplier with a lower adjusted unit cost while reducing the order quantity from a supplier with a
higher adjusted unit cost such that the total expected effective received is kept constant. Thus,
by switching the order quantities in a proper fashion to a “cheaper” supplier we can always de-
crease the long-run average cost for the inventory system. If the adjusted unit costs of two sup-
pliers are the same, no improvement (nor worsening) is possible by the kind of switch described.
It is easy to recognize that the first component of the adjusted unit cost, ii
cp, is the purchas-
ing cost per good unit; whereas the second component,
(
)
(
)
21
i
hD p, can be interpreted as the
extra holding cost due to the variability caused by the unreliable nature of the ordered units. The
latter disappears as the supplier becomes more reliable, i.e., as pi tends to one.
Theorem: Define 1
*min i
iN
A
CA
≤≤
=C
and
{
}
*|1 , *
i
SiiNACAC=≤ = . The following three
results hold for three different ordering cost structures:
(i) Consider the setting when
(
)
1,..., N
KQ Q K
=
. The ordering quantity vector(s)
, satisfying the following two conditions are optimal:
(
1,..., N
QQ
)
6
1. For , ,
*iS0
i
Q=
2.
*
2
ii
iS
KD
pQ h
=
. (11)
(ii) Consider the setting when where
()
1
1
,..., 1{ 0}
N
Ni
i
KQ Q K Q
=
⎛⎞
=
⎜⎟
⎝⎠
>(.)
K
is a positive
monotonically increasing function and 1 is the indicator function that returns 1 if
and returns 0 otherwise. The ordering quantity vector(s)
{ 0}
i
Q>0
i
Q>
(
)
1,..., N
QQ
, satisfying the following
three conditions are optimal:
1.
1
1{ 0} 1
N
i
iQ
=
>=
2. For some *jS
,hDK
p
Q
j
j
(.)21
=,
3. For ,
{1,..., }\{ }iNj0
i
Q
=
.
(iii) Consider the setting when
()
1
1
,..., 1{ 0}
N
Nii
i
KQ Q K k Q
=
=
+
>
where and all
, 1. Define
0K
0
i
k>iN≤≤
()
2
i
ii
ii
KkD hpQ
CR AC D
pQ ii
+
=++
, (12)
()
*2
ii
CR K k Dh AC D=++
i
, (13)
and
i
{
}
*|1 , *
ii
SiiNCRCR=≤ = . The ordering quantity vector(s)
(
)
1,..., N
QQ
, satisfying the
following three conditions are optimal.
1.
1
1{ 0} 1
N
i
iQ
=
>=
2. For some
i
*jS,
(
)
2
1j
jj
KkD
Qph
+
=,
3. For ,
{1,..., }\{ }iNj0
i
Q
=
.
7
Proof: (i) From Lemma 1, the long-run average cost can be continuously improved as order
quantities are properly shifted (i.e., keeping the total expected effective received constant) from a
supplier with higher unit adjusted cost to a supplier with lower unit adjusted cost until it hits
zero. Thus, it is not optimal to order from any supplier not in S*, consisting of the supplier(s)
with the lowest unit adjusted cost.
Using (3) and (5), we now have
()
2
**
[] 1
22
i
iii
iS iS
i
chh
ECycleCost K p p Q pQ
pD D
∈∈
⎛⎞
=+ + − +
⎜⎟
⎝⎠
⎝⎠
∑∑
ii
. (14)
Since the suppliers in S* all have the same adjusted unit cost, AC*, we have
2
**
[]*
2
ii ii
iS iS
h
E
CycleCost K AC pQ pQ
D
∈∈
=+ +
⎝⎠
∑∑
. (15)
Similarly (4) becomes
*
[]
ii
iS
p
Q
E CycleLength D
=
. (16)
Hence,
*
*
*2ii
iS
ii
iS
KD h
CR AC D pQ
pQ
=++
⎝⎠
. (17)
Noting the canonical EOQ type cost structure, we obtain (11).
(ii) From Lemma 1 and the fact that the fixed ordering cost is monotonically increasing in
the number of suppliers with positive order quantities, long-run average cost is improved by set-
ting equal to zero the order quantities for suppliers not in S*. For any given value of the ex-
pected total effective received (
*
ii
iS
p
Q
), long-run average cost is minimized when
, which implies operating with only one of the supplier(s) in S*. The result fol-
lows from univariate optimization.
1
1{ 0} 1
N
i
iQ
=
>=
(iii) Consider the minimization of long-run average cost with respect to any two suppliers i
and j such that and
,
ij
QQ>0ji
A
CAC
, while keeping constant . If , ,
l
Ql iji
kk
j
<
, by
Lemma 1 long-run average cost is improved by setting 0
j
Q
=
. If , the best solution is
obtained when either or , since the interior points are suboptimal due to Lemma 1.
i
kkj
0
j
Q=0
i
Q=
8
Hence, for any subset of two suppliers, it is optimal to work only with one of them. Proceeding
in the same fashion with all possible supplier subsets of size two, it is shown that it is optimal to
work with a single supplier.
The average cost expression using single supplier i can be obtained by modifying (17) as
()
2
i
ii
ii
ii
K
kD hpQ
CR AC D
pQ
+
=++
. (18)
The reader should note that this expression is just like the one in the canonical EOQ model with
the exception of replacing Q with ii
p
Q. Thereby, modifying the result for the EOQ model the
optimal average cost with supplier i yields
()
*2
ii
CR K k Dh AC D=++
i
. (19)
Thus, the optimal supplier to work with is found by
{
}
1
*argmin *
i
iN
iC
≤≤
=R
(20)
and the optimal order quantity from this supplier is
()
*
*
*
2
1
*i
ii
KkD
Qph
+
=. (21)
Remark 1. If there are no supplier specific minor costs (as in i and ii), the suppliers with ad-
justed unit costs higher than the minimum available adjusted unit cost should never be used. In
the presence of such costs (as in iii), it may be possible to exploit the trade-off between the or-
dering cost component and the remaining cost components and, work with the supplier that pro-
vides the best cost rate combination, but not necessarily has the lowest individual components.
Remark 2. Consider the case when the fixed ordering cost is a constant. If the cardinality of
the set S*, i.e., the number of suppliers with the minimum adjusted unit cost, is greater than one,
then the optimality conditions given in the above theorem do not yield a unique optimal solution.
Infinitely many solutions lie over an optimal simplex which is the intersection of the positive
quadrant and the hyperplane defined by a constant value of the expected total effective received.
However, when we have the fixed ordering cost as strictly increasing with the number of suppli-
ers used, i.e., given that , then the set of optimal solutions is characterized by the
vertices of the aforementioned simplex.
n
KK>mnm>
9
4. Discussion
A key observation from the analysis above is that if suppliers are subject to binomial yield
uncertainty to use a single supplier is always optimal. Although there may be other optimal con-
figurations in cases where we are indifferent between suppliers, in practice an exact equality of
adjusted unit prices is unlikely. This result constitutes a contrast to the conventional wisdom in
the area of random yield. We provide this managerial finding in the following.
Corollary. (i) In the presence of constant fixed ordering cost, zero leadtimes and binomial
yield, diversification does not bring additional cost benefit over using a single supplier.
(ii) When fixed ordering costs are monotonically increasing in the number of suppliers used
or include supplier specific minor ordering costs, diversification is never optimal under binomial
yields and zero leadtimes.
Diversification has been found to be beneficial due the risk pooling effects under yield struc-
tures investigated in the literature (e.g. stochastically proportional, all or nothing, random capac-
ity). The underlying mathematical reason behind this phenomenon is that when we sum random
variables, their variability, best manifested in their coefficient of variation, decreases.
In the case of binomial yield, the reason that diversification is not beneficial is that one can
obtain variability reduction just by increasing the order quantities without resorting to additional
suppliers. The critical fact to be observed is that the binomial distribution is itself a sum of Ber-
nouilli distributions. As the order quantity from a supplier, i.e., number of trials in a binomial
experiment increases, we sum up Bernouilli random variables and benefit from variability reduc-
tion. Thus, instead of obtaining risk pooling effect via supplier diversification, one can obtain it
by simply augmenting the order from the “cheapest” supplier.
The three fixed ordering cost structures investigated herein are commonplace and correspond
to realistic settings. For instance, the case in which the fixed ordering cost as an increasing func-
tion of the number of suppliers used, models the administrative burden due to the management
effort in maintaining a large supplier base. Our finding is consistent with the recent efforts in
many Industries in reducing the supplier base. The fixed ordering cost structure with supplier
specific minor costs is found when there are fixed components of freight costs in addition to ad-
ministrative efforts, or when foreign suppliers are used with customs clearance costs.
10
5. Extensions
There are a number of possible extensions to the model herein. We discuss these next.
One of the assumptions of the model, which states that the purchasing cost is paid per item
received irrespective of its quality, can be modified with a simple adjustment in the results. For
example, if we assume that we only pay the purchasing cost, ci, per “good” item received, all the
derived results are valid once the adjusted unit cost (given in (7)) is redefined as
(
1
2
ii i
h
)
A
Cc p
D
=+ . (22)
A similar variation in the model is to allow the holding costs to depend on the purchasing
price of the inventory. In this circumstance the inventory holding cost for one unit supplied from
supplier i per unit time can be expressed as . If we assume that inventories originat-
ing from different suppliers are consumed simultaneously such that they all deplete at the same
moment in time, i.e., when the inventory level drops to zero, the derived results are again valid
the adjusted unit cost (given in
ˆ
i
hhhc=+ i
(7)) is redefined as
(
1
2
ii
i
i
ch
)
i
A
C
pD
=+ −p
. (23)
A final variation we would like to talk about is the incorporation of supplier capacities to the
model. In the case that the capacities do not bind the optimal solution, the previous optimal solu-
tions are still valid. Otherwise, we can only present a solution for the case that the ordering cost
is constant irrespective of the number of suppliers used, i.e., n
KK
=
. In this case, one would
have to sort the suppliers according to their adjusted unit costs and order from the “cheapest sup-
plier” up to its capacity and then continue filling the supplier capacities until (11) is satisfied. If
it cannot be satisfied even when all available supplier capacities are used then the process stops
and ordering to full capacity is the optimal policy. If the most general cost structures along with
capacity constraints are assumed then the problem becomes combinatorial in nature and there is
little that can be done except for trying all possible supplier combinations.
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12
... There exist several early review papers that cover the extensive work in this area (e.g., Yano and Lee, 1995;Wright and Mehrez, 1998;Mula et al., 2006), which has grown further by addressing different decisions, such as risk mitigation and sharing (Zare et al., 2019), contract design and pricing under random yield within the past decade. Main streams of work have been on supplier diversification, including the case of dual sourcing, and supplier selection strategies (Gerchak and Parlar, 1990;Parlar and Wang, 1993;Anupindi and Akella, 1993;Fadiloglu et al., 2008;Tomlin, 2009;Babich et al., 2012;Schmitt and Snyder, 2012;He and Zhao, 2012;Yan et al., 2012;Li, 2017;Saputro et al., 2021;He et al., 2018;Dong et al., 2021), joint pricing and ordering decisions (Li and Zheng, 2006;Chao et al., 2016;He et al., 2015;Kumar et al., 2018;Golmohammadi and Hassini, 2020)), as well as on the determination of optimal control policies some of which considered all-or-nothing supply uncertainty (e. g., Parlar et al., 1995;Güllü et al., 1997;Atasoy et al., 2012;Mohammadivojdan et al., 2021). Below, we summarize work assuming supplier yield uncertainty in more detail due to their relevance to our work. ...
... The case of p = 1 represents a perfectly reliable supplier who always delivers the quantity ordered. Several other studies considering supplier unreliability use a similar binomial yield model (see, for example, Mazzola et al. (1987), Barad and Braha (1996) and Fadiloglu et al. (2008)). More specifically, we model the quantity delivered in period k, Y k (u k ), as a binomial random variable with ...
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We consider the real-life problem of a coach bus manufacturer located in Turkey, facing the problem of setting ordering quantities for a part procured from an unreliable supplier, where the number of items delivered is binomially distributed with an unknown yield parameter, p. We use the well-defined finite-horizon planning context with deterministic demand per period, purchasing, holding, and shortage costs to investigate the effectiveness of a fill-rate based approximate learning scheme in comparison to an exact Bayesian learning scheme, where observations on the supplier's delivery performance are used to update the assumed distribution of p. We formulate the exact optimal learning problem as a Bayes-adaptive Markov decision process and solve the corresponding finite horizon stochastic dynamic program to provide insights on the value of online learning in comparison to the unrealistic perfect information (PI) and no information (NT) benchmarks. We contrast the performance of the so-called Bayesian Updating (BU) policy to other practical approaches such as using an assumed/guessed value of p and implementing a constant safety stock. Noting the significant value of learning, we finally study the effectiveness of an approximate learning formulation that does not enjoy the asymptotic consistency and convergence properties but involves much lower computational burden, and demonstrate its confounding performance, at times beating the BU policy with exact Bayesian updates.
... However, when an EOQ-like inventory control system is used, Fadıloğlu et al. (2008) show that the sole sourcing is optimal when the supplier's yield is binomial. The result is generalized to include the cases with more general cost structure by Tajbakhsh et al. (2010), and Yan and Wang (2013) give a further generalization which shows that the sole sourcing is optimal under general random yield. ...
... Hence it is optimal to choose a supplier which gives the minimum cost rate among the suppliers. □ Example : One of the most widely studied cases is the binomial random yield (Fadıloğlu et al., 2008;Tajbakhsh et al., 2010). In this case, the random yield follows a binomial distribution with parameter p which denotes the probability of a unit being good. ...
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Though the supplier diversification is considered as a vital tool to mitigate the risk due to supply chain disruptions, there are results which show the optimality of the sole sourcing. This paper further generalizes the results to show that the sole sourcing is optimal under very mild conditions. Discussion on why the sole sourcing is optimal is given with the insight on the value of supplier diversification.
... The fact that buyers end up effectively paying for the long-term contracted supplier capacity and its cost rather than the short-term delivered output is because of the high specific investment incurred by suppliers and inability for the buyer to switch to other suppliers once the supplier relationship is established ( [16]). Such an assumption is not uncommon in the random yield literature (see [4], [7], [11] and [8]). ...
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Supply chain management is becoming an increasingly important issue, especially when in most industries the cost of materials purchased comprises 40-60% of the total sales revenue. Despite the benefits cited for single sourcing in the popular literature, there is enough evidence of industries having two/three sources for most parts. In this paper we address the operational issue of quantity allocation between two uncertain suppliers and its effects on the inventory policies of the buyer. Based on the type of delivery contract a buyer has with the suppliers, we suggest three models for the supply process. Model I is a one-delivery contract with all of the order quantity delivered either in the current period with probability beta, or in the next period with probability 1 - beta. Model II is also a one-delivery contract with a random fraction of the order quantity delivered in the current period; the portion of the order quantity not delivered is cancelled. Model III is similar to Model Il with the remaining quantity delivered in the next period. We derive the optimal ordering policies that minimize the total ordering, holding and penalty costs with backlogging. We show that the optimal ordering policy in period n for each of these models is as follows: for x greater-than-or-equal-to u(n)BAR order nothing; for v(n)BAR less-than-or-equal-to X < u(n)BAR, use only one supplier; and for x < v(n)BAR, order from both suppliers. For the limiting case in the single period version of Model I, we derive conditions under which one would continue ordering from one or the other or both suppliers. For Model II, we give, sufficient conditions for not using the second (more expensive) supplier when the demand and yield distributions have some special form. For the single period version of Models II and III with equal marginal ordering costs we show that the optimal order quantities follow a ratio rule when demand is exponential and yields are either normal or gamma distributed.