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Optimal Inventory Control with Heterogeneous Suppliers
Zhongsheng Hua and Wei Zhang School of Management, University of Science & Technology of China, Hefei, Anhui 230026, China
[email protected] • [email protected]
Saif Benjaafar Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN 55455, USA
May 31, 2009
Abstract
We consider an inventory system with two heterogeneous suppliers differentiated by their fixed and variable costs as well as constraints on order sizes. Procuring from one of the supplier involves a high variable cost but negligible fixed cost while procuring from the other supplier involves a low variable cost but high fixed cost, as well as a constraint on the maximum order size. We show that the problem can be reduced to an equivalent problem with a single supplier. However, the corresponding cost is neither concave nor convex. Using the notion of quasi-convexity, we characterize the structure of the optimal policy and show that it can be specified in terms of multiple thresholds which determine when to order from each supplier and how much. In contrast to previous research, which does not consider order size constraints, we show that sourcing from a specific supplier can occur over multiple ranges of inventory levels and simultaneously with sourcing from other suppliers. We show how the results can be extended to systems with more than two suppliers and to order size constraints on all suppliers. We offer managerial insights into the benefit of dual-sourcing and how this benefit is affected by various supplier characteristics. In particular, we show that multi-sourcing is most valuable in the presence of order size constraints.
Key words: inventory systems, optimal control, multi-sourcing, stochastic dynamic programming, quasi-convexity, heuristics
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1. Introduction
We consider an integrated procurement and inventory control problem with two suppliers. Procuring from
the first supplier involves a high variable cost (cost per unit purchased) but negligible fixed cost.
Procuring from the second supplier involves a low variable cost but high fixed cost. The problem is
motivated by procurement decisions that many firms face when determining whether to source from a
local supplier whose prices per unit are high but with whom transaction costs (such as transportation
costs, management overhead costs, customs fees, and fees associated with foreign currency conversion,
among many others) are relatively negligible or to source from a global supplier whose prices are lower
but with whom transactions costs are significantly higher. Intuitively, we expect firms to prefer the local
supplier when order quantities are generally small and the global supplier when quantities are large. This
intuition perhaps explains the decision by some firms to source exclusively from either a local or a global
supplier. However, when demand is variable (leading order quantities to be also variable), there can be, as
we show in this paper, significant benefit to maintaining both suppliers and making decisions regarding
from whom to procure dynamically in each ordering period. Moreover, when there are limits on the
maximal order quantities that can be placed with either supplier (because of contractual agreements or
because of actual capacity limits), then there can also be significant benefit, as we also show in this paper,
to simultaneously source from both suppliers in some periods.
The setting we consider in this paper is of a firm that makes decisions over a planning horizon
consisting of discrete periods. The firm faces stochastic demand in each period, and depending on how
much inventory it has at the beginning of each period, places an order with one or both of the suppliers to
minimize its expected discounted cost over the planning horizon. Orders placed in one period from any of
the two suppliers can be used to satisfy demand from that period. Therefore, the supply leadtimes can be
considered as zero for both suppliers and the suppliers are differentiated primarily based on their fixed
and variable costs as well as their order size constraints.
The problem we consider is related to the growing literature dealing with inventory control with dual
(and in some cases multiple) suppliers. However, most of this literature considers systems where all
suppliers have negligible fixed costs but are differentiated instead based on their variable costs and their
leadtimes. The decision firms face is whether to source from the slow but cheap supplier or from the fast
but expensive supplier; see for example Alfredsson and Verrijdt (1999), Tagaras and Vlachos (2001),
Sethi et al. (2003), Feng et al. (2006), Veeraraghavan and Scheller-Wolf (2008), and the references
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therein. There is also literature that considers sourcing from multiple suppliers when some of the suppliers
are unreliable, either in terms of supply leadtimes or quantities delivered. In this case, firms may find it
optimal to diversify their supply sources to hedge against supply uncertainty; see for example Chen et al.
(2001), Tomlin (2006), Babich et al. (2007), Dada et al. (2007), Federgruen and Yang (2008), and Song
and Zipkin (2009), among others.
Surprisingly, very little of the existing inventory control literature deals with multi-sourcing when
suppliers are differentiated by their fixed versus variable costs, although such cost differences tend to
drive the sourcing decisions for many, if not most firms. A notable exception is the paper by Fox et al.
(2006) who consider a setting similar to ours with two suppliers, one with negligible fixed cost but high
variable costs and one with a positive fixed cost but low variable cost. However, in their case, there is no
order size constraint (the suppliers are un-capacitated). This leads to a relatively simple structure for the
optimal ordering policy. In particular, depending on the current inventory level, it is optimal to source in
each period exclusively from one of the suppliers but not the other. Characterizing the optimal policy is
simplified by the fact that the corresponding ordering cost function is concave. In our case, because of the
order size constraints, the optimal policy has a more complex structure. More importantly when the
starting inventory in a period is sufficiently small, it is possible that the optimal policy would specify
ordering from both suppliers. This significantly complicates the analysis as the ordering cost functions
become neither concave nor convex.
Closely related to Fox et al. (2006) is the earlier work of Porteus (1971, 1972). Porteus (1971)
considers an inventory control problem with concave ordering costs, which can arise from ordering from
multiple suppliers, and no order size constraints. He shows that the optimal policy is a generalized (s, S)
policy with multiple reorder points and multiple order-up-to levels. This structure is consistent with the
one shown by Fox et al. in the case of two suppliers. However, the treatment in Porteus is limited to
demand distributions that have a one-sided Polya density. The class of one-sided Polya densities is
somewhat difficult to characterize, but is known to include the exponential distribution and all finite
convolutions thereof. However, it does not include many common distributions, such as the normal and
truncated normal, beta, and most gamma distributions. Porteus (1972) shows that the same structure of
the optimal policy continues to hold when demand has the uniform distribution. Fox et al. (2006) rely, as
we also do in this paper, on a more general class of demand distributions. Namely, we assume, that
demand is strongly unimodal (or equivalently that the density function of the demand distribution is log-
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concave). This class of distributions is easier to characterize and includes any distribution whose density
functions φ satisfies the condition that log( ( ))φ ξ is concave. This condition is satisfied by the normal,
truncated normal, gamma for shape parameter 1α ≥ , beta for parameters p ≥ 1 and q ≥ 1, and uniform
distributions. Because Porteus (1971, 1972) does not assume constraints on order size, the optimal policy
in each period consists of ordering exclusively from a single supplier.
In this paper, we generalize the treatment in Fox et al. (2006) to include order size constraints on the
suppliers. We focus primarily on the case where there is an order size limit on ordering from the supplier
with the fixed cost and lower variable cost to which we refer as supplier L. We do so because this case
enables us to characterize the optimal policy under a broader set of conditions than the one where there
are order size constraints for both suppliers. In Section 5, we discuss the extent to which our results
extend to this more general case. Although less general, having order size constraint on supplier L allows
us to capture important effects that significantly alter the structure of the optimal policy. Also, even with
this simplifying assumption, the analysis is far from trivial. In particular the introduction of order size
constraints leads to an ordering cost function that is neither concave nor convex. This requires
methodology that is different from standard methodology in proving the structure of the optimal policy.
For example, most of the inventory literature relies on convexity of the optimal cost function (or a more
generalized notion of convexity, such as k-convexity) to prove structural results. In our case, our optimal
cost is neither convex nor k-convex. Instead we rely on the quasi-convexity of a component of the optimal
cost function. Our basic model with order size constraint on only one supplier can be viewed as
generalizing existing single supplier models with or without fixed costs and with or without capacity
constraints; see for example Scarf (1960), Federgruen and Zipkin (1986a, 1986b), Shaoxiang and
Lambrecht (1996), and Shaoxiang (2004).
The contributions of our paper are summarized below.
• Our paper is the first to consider an inventory control problem with dual suppliers where there is a
capacity constraint on one of the suppliers. As we show in our numerical results, dual-sourcing is
most valuable when such constraint is present.
• To our knowledge, our paper is among the first to consider an inventory control problem where the
ordering cost is neither concave nor convex (and the optimal cost function does not satisfy convexity
or other generalized forms of convexity, such as k-convexity).
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• The structure of the optimal policy which takes on the form of a multi-level threshold policy (four
threshold levels in the case of two suppliers) is different from known policies for other related
problems. In particular, in contrast to results obtained in the absence of order size constraint, sourcing
from a specific supplier can occur over multiple ranges of inventory levels and simultaneously with
sourcing from other suppliers.
• We offer a heuristic that simplifies the optimal policies when the inventory level is in a certain range
and show through extensive numerical results that the performance of the heuristic is within than one
percentage point of the optimal policy for all the cases tested.
• We show how our analysis can be extended under some conditions to systems with capacity
constraints on both suppliers and to systems with more than two suppliers.
• We offer managerial insights into the benefit of dual sourcing and how this benefit is affected by
various supplier characteristics.
The rest of the paper is organized as follows. In Section 2, we formulate the problem as a stochastic
dynamic program. In Section 3, we characterize the structure of the optimal policy. In Section 4, we
present a heuristic that simplifies the optimal policy for certain ranges of inventory levels, offer
supporting numerical evidence for the heuristic, and carry our numerical experiments to examine the
benefit of dual sourcing. In Section 5, we discuss how our treatment can be extended to systems with
multiple suppliers and systems with order size constraints on all suppliers. Section 6 offers a summary
and some concluding comments.
2. Problem Formulation
We consider an inventory control problem of a single item over a finite planning horizon consisting of n
discrete time periods. Demand in each period can be described by a non-negative random variable D.
Demand realizations in different periods are independently and identically distributed with probability
density functions that are strongly unimodal. As we mentioned previously, this class of distributions is
quite general and covers many of the commonly used distributions. In each period, and prior to demand
realization for that period, an order can be placed with one of two suppliers, supplier H and supplier L, or
with both suppliers. The purchase cost per unit from supplier H is cH while the purchase cost from
supplier L is cL, with cH > cL. That is, supplier H has a higher variable cost than supplier L. There is no
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fixed cost incurred from purchasing from supplier H, but there is a fixed cost to purchasing from supplier
L, which we denote by K, where K > 0. There is a limit on the order size that can be placed with supplier
L, which we denote by CL, where CL < ∞, but no limit on order sizes with supplier H. In section 5, we
show how the analysis can be extended to systems where there are order size constraints on both
suppliers. Orders placed in one period from any of the two suppliers can be used to satisfy demand from
that period, so that supply leadtimes can be viewed as zero.
Demand that cannot be fulfilled in one period is backlogged but incurs a unit backorder cost b per
period. Inventory carried from one period to the next incurs a unit inventory holding cost h. The objective
of the system manager is to minimize over the planning horizon the expected total discounted cost, where
the cost in each period is the sum of ordering cost and inventory holding and backordering costs. To avoid
speculative reasons for holding inventory, we assume that Hb c≥ and ,L Hh c cα+ ≥ where 0 1α< < is
the discount factor. At the beginning of a period t, t = 1, …, n, the firm observes the starting inventory
level xt, and places an order of size Htq with supplier H and L
tq with supplier L. Orders placed are then
delivered, bringing inventory level to t t ty x q= + where H Lt t tq q q= + and denotes the order quantity in
period t. Finally demand is realized, and available inventory is used to fulfill any backlogged demand and
part or all of the realized demand. Based on the realized demand and the available inventory, there can be
either leftover inventory carried to the next period or unfulfilled demand backlogged until the next period.
Because inventory level can be either positive or negative, we use the notation max(0, )t tx x+ = to refer to
physical inventory level and max(0, )t tx x− = − to refer to backorder level.
In what follows, we show that by redefining the ordering cost function, the above problem with dual
suppliers can be reduced to a problem with a single supplier with an appropriately specified ordering cost
function. First note that if 0 ( ),H Ltq C K c c≤ = − then it is optimal to order solely from supplier H and
the corresponding ordering cost is .Htc q If 0 ,t LC q C< ≤ then it is optimal to order solely from supplier L
and the corresponding ordering cost is .LtK c q+ On the other hand, if ,t Lq C> then it is optimal to order
CL units from supplier L and the remaining units from supplier H. Therefore, the corresponding ordering
cost is ( ) .L H HL tK c c C c q+ − + Consequently, we can view the dual-supplier inventory control problem
in any period t (t = 1, …, n) as a single-supplier inventory control problem where the ordering cost is
given by the following function:
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0
0
[0, ]
( ) ( , ]
( ) ( , ).
Ht t
Lt t t L
L H HL t t L
c q q C
OC q K c q q C C
K c c C c q q C
⎧ ∈⎪
= + ∈⎨⎪ + − + ∈ ∞⎩
(1)
Note that we assume that 0 ;LC C> otherwise, the problem degenerates to a classic single supplier
problem (supplier H) with no order size constraint.
Although we can reduce the dual-supplier problem to a single-supplier problem, the ordering cost
function ( )tOC q is neither convex nor concave (see Figure 1). This is what makes the dual-supplier
problem with order size constraint different from existing inventory problems and the analysis more
challenging.
Figure 1 - The ordering cost function
The problem of determining the optimal order quantity in each period can be formulated as a
stochastic dynamic program over a finite horizon consisting of n discrete periods (we number time
periods in reverse order so that period 1 is the last period in the planning horizon, while period n is the
first period). Let ( )nf x be the optimal expected discounted cost when there are n periods remaining and
the starting inventory level is x (without loss of generality, we define 0 ( ) 0)f x ≡ and define ( )ng y as
10 0 0( ) ( ) ( ) ( ) ( ) ( ) ( ) ,n ng y h y d b y d f y dξ φ ξ ξ ξ φ ξ ξ α ξ φ ξ ξ
∞ ∞ ∞+ −−= − + − + −∫ ∫ ∫ (2)
where φ is the probability density function of demand. The optimal expected discounted cost function
( )nf x can be easily shown to satisfy the following optimality equation
0C LC
K
qt
Ordering cost
slope Hc=
slope Lc=
slope Hc=
7
0
0
( ) [ , ]
( ) min ( ) ( , ]
( ) ( ) ( , ),
H Hn
L Ln n Ly x
L H H HL n L
c x g y c y y x x C
f x K c x g y c y y x C x C
K c c C c x g y c y y x C≥
⎧− + + ∈ +⎪
= − + + ∈ + +⎨⎪ + − − + + ∈ + ∞⎩
(3)
where the decision variable in period n is the order up to inventory level, y. Note that with our
reformulation of the ordering cost function, the single decision variable y is sufficient to fully characterize
not only the total order quantity, but also the order quantity from each supplier. This greatly simplifies the
dynamic programming formulation by reducing the multi-dimensional decision problem into one with a
single dimension.
3. The Structure of the Optimal Policy
In order to characterize the structure of the optimal policy we rely on the notion of quasi-convexity. A
function f(x) is quasi-convex if f (λx1 + (1− λ)x2 ) ≤ max{ f (x1), f (x2 )} for 0 ≤ λ ≤ 1. An implication of
this definition is that a quasi-convex function is either always increasing in x, always decreasing in x, or
there exists x* for which the function is decreasing if x ≤ x* and increasing if x ≥ x* (throughout this
paper, we use “increasing” and “decreasing” in the non-strict sense to mean “nondecreasing” and
“nonincreasing”, respectively). Note that a quasi-convex function cannot increase and then decrease.
The following lemma is important for characterizing the structure of the optimal policy.
LEMMA 1. The functions gnL ( y) = gn ( y) + cL y and gn
H ( y) = gn ( y) + cH y are quasi-convex for all 1n ≥ .
The proof of Lemma 1 (and of all subsequent results) can be found in the Appendix. Note that
proving that quasi-convexity holds for gnL ( y) and gn
H ( y) for all n is non-trivial, as quasi-convexity is not
necessarily preserved when common operators are applied. In particular, quasi-convexity is not preserved
for the minimum of two or more quasi-convex functions, a feature central to our problem.
Next, we show that the quasi-convexity of the functions gnL ( y) and gn
H ( y) implies a specific
structure for the optimal policy. Define arg min ( )L Ln y nS g y and arg min ( ).H H
n y nS g y Given that the
functions gnL ( y) and gn
H ( y) are quasi-convex, we have ( )Lng y and gn
H ( y) decreasing when Lny S≤ and
y ≤ SnH respectively, and increasing otherwise (it is straightforward to rule out the case of the functions
( )Lng y and gn
H ( y) being always increasing or always decreasing). Noting that the difference
gnH ( y) − gn
L ( y) = y(cH − cL ) ≥ 0 and is strictly increasing in y, it is easy to verify that SnL > Sn
H .
8
From the optimality equation, we can observe that determining the optimal order-up-to level in each
period involves a three way comparisons between three options. The first option involves ordering up to
;HnS the second option involves ordering up to ;L
nS and the third option involves not ordering at all. The
comparison is complicated by the order size constraint for supplier L. In the following theorem, we show
that the optimal policy can be fully characterized by four thresholds, 1ns , 2
ns , HnS and ,L
nS where 1 Hn n Ls S C− and 2
0H
n ns S C− . Note that since 0 ,LC C> we have 1 2 .H Ln n n ns s S S< < <
THEOREM 1. The optimal policy can be characterized by the four thresholds 1ns , 2
ns , HnS and L
nS as
follows:
(i) If 1nx s< , then it is optimal to order LC units from supplier L and the rest ( H
n LS C x− − ) units from
supplier H.
(ii) If 1 2n ns x s≤ < , then it is optimal to order up to L
nS (or as close as possible) from supplier L.
(iii) If 2 Hn ns x S≤ < , then it is optimal to either order up to H
nS from supplier H or to order up to LnS
(or as close as possible) from supplier L.
(iv) If H Ln nS x S≤ < , then it is optimal to either order up to L
nS (or as close as possible) from supplier
L, or to order nothing.
(v) If Lnx S≥ , then it is optimal to order nothing.
Theorem 1 states that when initial inventory is sufficiently small (case i) it is optimal to order
simultaneously from both suppliers. However, in all other cases, it is optimal to order exclusively from a
single supplier, or not to order at all, even when reaching the desired order-up-to level is not possible. In
particular, in case ii, it is optimal to order only from supplier L; in case iii, it is optimal to order
exclusively from either supplier L or supplier H; in case iv, it is optimal either to order exclusively from
supplier L or to order nothing; and in case v, it is optimal to order nothing. It is perhaps surprising to see
that in case iv it is never optimal to order from supplier H and that even in the case when it is not optimal
to order from supplier L (e.g., it is not optimal to order at least 0C units), it is preferable to order nothing
than to order from supplier H.
The structure of the optimal policy is graphically illustrated in Figure 2, where the line segments
E1F1G1H1 correspond to the expected discounted cost when the order-up-to level is LnS and ,L
nx S< and
the line segments E2F2G2H2 correspond to the expected discounted cost when the order-up-to level is HnS
and .Hnx S<
9
Figure 2 The structure of the optimal policy
In cases iii and iv, Theorem 1 does not fully characterize the optimal policy. Instead it specifies that
in deciding on the optimal ordering decision, it is optimal in each case to restrict consideration to only
two options. In case iii, these two options are either (1) to order up to HnS from supplier H or (2) to order
up to LnS (or as close as possible) from supplier L. In case iv, the two options are either (1) to order up to
LnS (or as close as possible) from supplier L or (2) to order nothing. In the remainder of this section, we
describe conditions under which it is possible to further characterize the optimal policy by specifying
which of the two options is optimal in each case.
In considering cases iii and iv, we differentiate between two scenarios L nC β≥ and ,L nC β< where
.L Hn n nS Sβ − We consider first the scenario where L nC β≥ . Let us define the following three functions
which are used in comparing the costs of the different options: 1 ( ) ( ) ( ) ,L L Ln n n nJ x g x g S K− −
2 ( ) ( ) ( ) ( ) ,H H L L H Ln n n n nJ x g S g S c c x K− − − − and 3 ( ) ( ) ( ) ( ) .H H L H L
n n n n LJ x g S g x C c c x K− + − − − First
note that, because ( )Lng x is quasi-convex, 1 ( )nJ x achieves the minimum at L
nS and is decreasing when
.Lnx S≤ It can also be directly verified that if L nC β≥ and ,nK γ≤ where ( ) ( )L H L L
n n n n ng S g Sγ − , 1 ( ) 0Hn nJ S ≥ and 1 ( ) 0.L
n nJ S < Therefore, there exists at least one solution to the equation 1 ( ) 0nJ x = and
we denote by 1ˆns one such solution. (Observe that when L nC β≥ and ,nK γ> 1 ( ) 0Hn nJ S < .) Second,
note that because ,H Lc c> 2 ( )nJ x is strictly decreasing in .x Moreover, if ,nK γ> then 2 2( ) 0n nJ s ≥ and 2 ( ) 0Hn nJ S < and there exists a unique solution 2 2ˆ [ , )H
n n ns s S∈ to the equation 2 ( ) 0.nJ x = Finally, note that
rewriting 3( )nJ x as 3 ( ) ( ) ( ) ( ) ,H H H H Ln n n n L LJ x g S g x C c c C K= − + + − − we can see that 3 ( )nJ x is
decreasing over 1[ , ),ns ∞ a result that follows from the fact that the function ( )Hn Lg x C+ is quasi-convex
and achieves its minimum at 1 .ns It can then be shown (see the Appendix) that when nK γ> and
Kx
( )ng x
LnSH
nSE1
G1
Cost
H1
slope Hc= −
slope Lc= −
slope Hc= −
2ns1
ns
F1
E2
F2G2
H2
10
,n L nCβ β ′≤ < where 2ˆ ,Ln n nS sβ ′ − there exists a solution 3 2ˆ [ , )H
n n ns s S∈ to the equation 3 ( ) 0.nJ x =
These observations lead immediately to the following theorem which completes the full characterization
of the optimal policy when .L nC β≥
THEOREM 2.
(1) If L nC β≥ and ,nK γ≤ then it is optimal to order up to LnS (or as close as possible) from supplier
L when 2 1ˆ[ , ),n nx s s∈ and not to order when 1ˆ[ , ).Ln nx s S∈
(2) If L nC β ′≥ and ,nK γ> then it is optimal to order up to LnS (or as close as possible) from supplier
L when 2 2ˆ[ , )n nx s s∈ , order up to HnS from supplier H when 2ˆ[ , ),H
n nx s S∈ and not to order when
[ , )H Ln nx S S∈ .
(3) If n L nCβ β ′≤ < and ,nK γ> then it is optimal to order LC units from supplier L when 2 3ˆ[ , ),n nx s s∈ order up to H
nS from supplier H when 3ˆ[ , ),Hn nx s S∈ and not to order when [ , ).H L
n nx S S∈
Next, we consider the scenario L nC β< . For case iii, Theorem 1 indicates that when 2 ,Hn ns x S≤ < it
is optimal to either order up to HnS from supplier H or to order LC units from supplier L (note that since
L nC β< , it is not possible to order up to LnS from supplier L when 2 ).H
n ns x S≤ < To decide which of
these two options is optimal, we compare the expected discounted costs of the two options using function 3( ).nJ x Note that 3 2 2 2( ) ( ) ( ) ( ) .H H L H L
n n n n n n L nJ s g S g s C c c s K= − + − − − Since 20 ,H
n ns S C= −
0)( H Lc CK c −= and ,H Ln L nS C S+ < we have 3 2
0( ) ( ) ( ) 0.L H L Hn n n n n n LJ s g S g S C C= − − + ≥ Because
function 3( )nJ x is decreasing, it can be directly observed that if 3 ( ) 0,Hn nJ S ≥ then it is optimal to order
LC units from supplier L. On the other hand, if 3 ( ) 0,Hn nJ S < since 3 2( ) 0,n nJ s ≥ then there exists a
solution 3 2ˆ [ , )Hn n ns s S∈ to equation 3 ( ) 0.nJ x = In this case, it is optimal to order LC units from supplier L
for 2 3ˆ ,n ns x s≤ < and order up to HnS from supplier H when 3ˆ .H
n ns x S≤ <
For case iv, Theorem 1 indicates that there are three possible courses of actions: ordering up to ,LnS
ordering LC , or not ordering at all. Determining which course of action is optimal involves evaluating the
function 1 ( ).nJ x Because function 1 ( )nJ x is decreasing and 1 ( ) 0Ln nJ S < , if 1 ( ) 0,L
n n LJ S C− < then 1 ( ) 0nJ x < for all [ , )L L
n L nx S C S∈ − and it is optimal not to order when .L Ln L nS C x S− ≤ < If
1 ( ) 0,Ln n LJ S C− ≥ then there exist a solution 1ˆ [ , )L L
n n L ns S C S∈ − to equation 1 ( ) 0.nJ x = Thus it is optimal
to order up to LnS from supplier L when 1ˆ ,L
n L nS C x s− ≤ < and not to order when 1ˆ .Ln ns x S≤ < These
results are summarized in the following theorem.
THEOREM 3. If L nC β< and the starting inventory level falls in the intervals 2[ , )Hn ns S and [ , ),L L
n L nS C S−
the optimal policy can be specified as follows.
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(1) If 3 ( ) 0,Hn nJ S < then it is optimal to order LC units from supplier L when 2 3ˆn ns x s≤ < and to order up
to HnS from supplier H when 3ˆ .H
n ns x S≤ < Otherwise (if 3 ( ) 0Hn nJ S ≥ ), then it is optimal to order LC
units from supplier L when 2 .Hn ns x S≤ <
(2) If 1 ( ) 0Ln n LJ S C− ≥ , then it is optimal to order up to L
nS from supplier L when 1ˆLn L nS C x s− ≤ < and
not to order when 1ˆ .Ln ns x S≤ < Otherwise (if 1 ( ) 0),L
n n LJ S C− < then it is optimal not to order when
.L Ln L nS C x S− ≤ <
Note that Theorem 3 leaves out the case when H Ln n LS x S C≤ < − . For this case, we need to compare
the expected discounted costs between the two options of ordering LC units from supplier L and not
ordering. This involves analyzing the function ( ) ( ) Ln L n Lg x C g x c C K+ − + + , which does not possess
simple monotonicity properties. In Section 4, we suggest the simple heuristic of always ordering LC units
from supplier L and provide numerical evidence that this leads to negligible increase in the optimal cost.
We conclude this section by noting that our results in Theorems 1-3 extend to the case of systems
with infinite horizon (a proof is included in Appendix B) as long as demand in any period has a finite
upper bound. In the infinite horizon case, policy parameters and function are defined in the same way,
except that all variables, parameters and functions are no longer indexed with the number of periods
remaining in the planning horizon, n. In particular, the optimal policy described in Theorem 1 is now
specified in terms of stationary thresholds 1,s 2 ,s HS and .LS The results in Theorems 1-3 also extend
to systems with discrete demand distribution, and the details are omitted for brevity.
4. Numerical Results and Managerial Insights
In this section, we provide numerical results to evaluate the effect of using the simple heuristic discussed
above. We also provide numerical results to evaluate the benefit of using the optimal policy, which
involves dual sourcing, to policies that allow only single sourcing.
4.1 Performance Evaluation of the Heuristic
We carried out extensive numerical experiments evaluating the impact on total cost of always ordering
LC units from supplier L when L nC β< and H Ln n LS x S C≤ < − . These results suggest that doing so leads
to negligible cost increases relative to the optimal policy. A set of representative results is shown in Table
1 (the results are for demand that has a truncated normal distribution over the interval [0, μ +3σ ], where
μ = 80; the reported costs are for the case of systems with infinite horizon). These numerical results are
12
supported by the fact that the relative cost difference between ordering LC unit from supplier L and not
ordering at all, as measured by the ratio ( ) ( )( )
( )
LL Lg x C c C K g xx
g xδ + + + −
=
is always bounded by the finite quantity max / ( ( ) ),L LLK g S c Cδ = + as shown in the following
proposition.
PROPOSITION 1. If LC β< , then max( )xδ δ≤ , for [ , )H LLx S S C∈ − .
Extensive numerical results suggest that this bound itself is relatively small. Representative results are
shown in Table 1.
Table 1 - Performance of the heuristic policy
(α = 0.90 and cL = 1)
Exp. # h cH b K CL σ δmax ×100% Percentage cost
difference from the optimal cost
1 1.5 2.75 2.75 70 41 40 2.762971 0.244003 2 1.0 2.20 2.20 48 41 40 2.388562 0.294156 3 1.5 2.75 13.75 70 41 40 2.293725 0.233025 4 1.5 2.75 2.75 70 41 80 2.207716 0.644327 5 1.0 2.20 11.00 48 41 40 2.102870 0.019830 6 1.0 2.20 2.20 48 41 80 1.906195 0.622328 7 0.5 1.65 1.65 26 41 40 1.844843 0.059021 8 1.5 2.75 2.75 70 41 120 1.747331 0.930964 9 1.5 2.75 13.75 70 41 80 1.654704 0.568072
10 1.0 2.20 2.20 48 41 120 1.647710 0.780632 11 1.0 2.20 11.00 48 41 80 1.601446 0.550179 12 1.5 2.75 27.5 70 41 80 1.576040 0.654966 13 0.5 1.65 1.65 26 41 80 1.515657 0.494385 14 1.0 2.20 22.00 48 41 80 1.416227 0.389954 15 1.5 2.75 13.75 70 41 120 1.385343 0.544209 16 1.5 2.75 2.75 35 21 40 1.376244 0.155502 17 0.5 1.65 8.25 26 41 80 1.271542 0.342136 18 0.5 1.65 1.65 26 41 120 1.241089 0.517867 19 1.0 2.20 11.00 48 41 120 1.221146 0.441302 20 0.5 1.65 16.50 26 41 80 1.220145 0.525824
13
4.2 The Benefit of the Optimal Policy
In this section, we present results from a numerical study that examines the benefit of using the optimal
policy, which can involve sourcing from multiple suppliers in the same period, against policies that
restrict sourcing to a single supplier. More specifically, we compare the optimal policy to two other
simpler policies that are common in practice and that have been studied previously in the literature. The
first policy, to which we refer as the fixed single-sourcing (SS) policy, requires the selection of a single
supplier at the beginning of the planning horizon and then sourcing exclusively from that single supplier
throughout the planning horizon. The second strategy, to which we refer as the dynamic single-sourcing
(DS) policy, requires sourcing from a single supplier in each period but allows for the supplier to change
from period to period. Under each policy, ordering decisions in each period are determined so as to
minimize the expected discount cost over the planning horizon. The optimal cost for each policy can be
obtained by formulating the corresponding problem as a stochastic dynamic program.
For the SS policy, in order to compute the optimal cost, we first compute the optimal cost associated
with using either supplier L or H exclusively over the entire planning horizon. If supplier L is used
exclusively, the optimal cost can be obtained as
[ , ]( ) min [ ( ) ( )],L
L L Ln y x x C nf x K y x c x g yδ∈ += − − + (4)
where
10( ) ( ) ( ) ( ) ,L L L
n ng y c y L y f y dα ξ φ ξ ξ∞
−= + + −∫
0 0( ) ( ) ( ) ( ) ( ) ,L y h y d b y dξ φ ξ ξ ξ φ ξ ξ
∞ ∞+ −= − + −∫ ∫
and δ is the indicator function with δ (u) = 1 if 0u > and ( ) 0uδ = otherwise. On the other hand, if
supplier H is used exclusively, the optimal cost is given by
( ) min [ ( )]H H Hn y x nf x c x g y≥= − + , (5)
where 10( ) ( ) ( ) ( ) .H H H
n ng y c y L y f y dα ξ φ ξ ξ∞
−= + + −∫ Then, the optimal costs in (4) and (5) are compared
and the supplier with the lower cost is selected. This means that the overall optimal cost under the SS
policy is given by SS ( ) min{ ( ), ( )}.L H
n n nf x f x f x= (6)
For the DS policy, the optimal cost is obtained by comparing in each period the cost of ordering from
supplier H to the cost of ordering from supplier L in that period (and then doing the same in subsequent
periods). This means that the optimal cost under the DS policy can be obtained as follows
14
DS DS DS[ , ]( ) min{inf [ ( ) ( )],inf [ ( ) ( ) ( )]},
L
H Ln y x n y x x C nf x c y x g y K y x c y x g yδ≥ ∈ += − + − + − + (7)
where DS DS10
( ) ( ) ( ) ( ) .n ng y L y f y dα ξ φ ξ ξ∞
−= + −∫ Note that under the DS policy, the sourcing decision is
revisited in each period so that different suppliers may be used in different periods. Note also that if a
decision is made to source from supplier L, then the maximum order quantity is CL. In the absence of the
order size constraints on supplier L, the DS policy is obviously optimal, as it is never preferable to split
orders between the two suppliers in the same period.
In practice, the SS policy is perhaps simpler to implement than either the optimal policy or the DS
policy. It is also perhaps consistent with common approaches to supplier selection, where typically
multiple suppliers are evaluated initially and then a single supplier is selected. The DS policy is more
complex and requires revisiting the supplier selection decision in each period. However, it is arguably
simpler than the optimal policy since in each period there is exclusive sourcing from one supplier.
Whether or not the more complex optimal policy can be justified depends on the additional cost savings
that could be obtained relative to these two simpler policies. The numerical results we present in this
section explore this question by examining the cost advantage of the optimal relative to these two policies
and the degree to which this advantage is sensitive to various problem parameters.
We carried out extensive numerical experiments where we compared the optimal policy against the SS
and DS policies for a wide range of problem parameter values and demand distributions. Representative
results are shown in Figures 3-5. To eliminate the effect of the length of the planning horizon, the results
shown are for problems with an infinite horizon (the infinite horizon problem is approximated by a finite
horizon problem with n periods, where n is sufficiently large so that the percentage difference in cost
between a problem with n periods and n-1 periods is less than 0.001). To eliminate the effect of initial
inventory, the results reported in Figures 3-5 are based on averages over all possible recurrent initial
inventory levels. Results in Figures 3-5 are for a discretized version of a truncated normal distribution
over the interval [0, Dmax], where Dmax= 3μ σ+ . We obtain results that are qualitatively similar when we
varied the assumptions regarding the demand distribution (data from these additional experiments is
available from the authors upon request). For each combination of problem parameter values, we obtain
the percentage cost difference ρSS between the optimal policy and the SS policy and the percentage cost
difference ρDS between the optimal policy and the DS policy, where
ρ i =
f i (x) − f (x)f i (x)
×100%, for i = SS and DS.
15
(A.1) 2Hc = (B.1) 2Hc =
(A.2) 10K = (B.2) 10K =
Figure 3 - The impact of LC on the benefit of the optimal policy
( 80μ = , 1Lh c= = , 0.9α = , 20σ = and 3b = )
0%
5%
10%
15%
20%
25%
30%
0 20 40 60 80 100 120 140 160 180 200
10K= 20K= 30K=
LC
ρSS
0%
2%
4%
6%
8%
10%
12%
14%
16%
0 20 40 60 80 100 120 140 160 180 200
10K= 20K= 30K=
LC
ρDS
0%
5%
10%
15%
20%
25%
30%
0 20 40 60 80 100 120 140 160 180 200
ρSS
2Hc = 1.8Hc = 1.6Hc =
LC 0%
2%
4%
6%
8%
10%
12%
14%
16%
0 20 40 60 80 100 120 140 160 180 200
ρDS
2Hc = 1.8Hc = 1.6Hc =
LC
16
0%
5%
10%
15%
20%
25%
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2
0%
1%
2%
3%
4%
5%
6%
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2
(A.1) 80LC = (B.1) 80LC =
0%
5%
10%
15%
20%
25%
30%
35%
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2
0%
2%
4%
6%
8%
10%
12%
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2
(A.2) 10K = (B.2) 10K =
Figure 4 - The impact of /H Lc c on the benefit of the optimal policy
( 80μ = , 1Lh c= = , 0.9α = , 20σ = and 3b = )
10K = 20K = 30K =
/H Lc c
ρSS
10K= 20K= 30K=
ρDS
/H Lc c
ρSS
/H Lc c
70LC = 80LC = 90LC =
ρDS
/H Lc c
70LC = 80LC = 90LC =
17
0%
5%
10%
15%
20%
25%
30%
35%
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
0%
2%
4%
6%
8%
10%
12%
14%
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
(A.1) 2Hc = (B.1) 2Hc =
0%
5%
10%
15%
20%
25%
30%
35%
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
(A.2) 80LC = (B.2) 80LC =
Figure 5 - The impact of K on the benefit of the optimal policy
( 80μ = , 1Lh c= = , 0.9α = , 20σ = and 3b = )
2Hc = 1.8Hc = 1.6Hc =
K
ρSS
2Hc = 1.8Hc = 1.6Hc =
K
ρDS
K
70LC = 80LC = 90LC =
ρSS
K
ρDS
70LC = 80LC = 90LC =
18
Based on the numerical results, the following important observations can be made.
• The benefit of using the optimal policy can be significant, with percentage improvements in the
examples shown in Figure 3 of up to 28% relative to the SS policy and up to 14% relative to the DS
policy.
• The benefit of using the optimal policy is highest when the order size constraint parameter CL is in the
middle range (in the examples shown, this corresponds to CL falling in the range 0( , ]).C μ σ+
• There is little benefit to using the optimal policy when CL is either very small or very large. In fact,
when CL is sufficiently small ( 0 ),LC C< the problem degenerates into one where the SS policy is
optimal. Similarly, when CL is sufficiently large and the demand distribution has finite support (for
the examples shown when CL >μ +3σ), the problem degenerates into one where there DS policy is
optimal). In general, when CL is small, the additional benefit from using supplier L is limited since
only a limited amount can be ordered from that supplier. On the other hand when CL is large, the
benefit from splitting orders is small since in most cases it is possible to order solely from either
supplier H or L.
• The benefit of using the DS policy over the SS policy mirrors the benefit of using the optimal policy
over either the SS or DS policies. This suggests that, in the absence of order size constraints (as in
Fox et al. 2006), dynamically switching between suppliers is not particularly beneficial and that in
that case it is sufficient to choose one supplier and then stick with it throughout the planning horizon.
This result further motivates the importance of investigating multi-sourcing when there are order size
constraints, as we do in this paper.
• The effect of the variable costs on the benefit of using the optimal policy is not always monotonic. In
particular, if CL is relatively large compared with the mean of demand (for the examples shown when
CL μ≥ ), the benefit from using the optimal policy is highest when the ratio cH/cL is in the middle
range. When cH/cL is small, the benefit of using the optimal policy is small as the cost advantage of
sourcing from supplier L is small. When cH/cL is large, the benefit of using the optimal policy can be
smaller as the cost advantage of sourcing from supplier H is small (however, the benefit could still be
significant because of the constraint on how much could be ordered from supplier L).
• The effect of the fixed cost is also not monotonic, although it is generally decreasing. As the fixed
cost K increases, the benefit from using the optimal policy decreases as the cost advantage of using
supplier L decreases. In fact, when K > ( )H LLc c C− , it becomes optimal never to source from supplier
L. When the fixed cost K is small, the relative benefit of using the optimal policy depends on the
19
order size constraint CL. When CL is high the benefit of using the optimal policy is small as the cost
advantage of using supplier H is small. However, when this is not the case, the benefit could be
significant because of the limits on how much could be ordered from supplier L.
5. Extensions to Systems with Multiple Suppliers
In this section, we briefly discuss extension of our analysis to systems with more than two suppliers and
to systems for which there are fixed order size constraints on both suppliers. In particular, consider a
system with m ( 2m ≥ ) heterogeneous suppliers with fixed costs such that 1 1, , 0mK K − ≥K , 0,mK = and
variable costs 1 mc c b< < ≤L and finite order size constraints iC < ∞ for 1,..., .i m= The case we have
so far discussed corresponds to the special case where m = 2. Let C = Cii=1
m∑ . Also, let
( ) ( )i in ng y g y c y+ and arg min ( )i i
n y nS g y ( 1,..., ).i m= If we are able to show that the functions
( )ing y ( 1, , )i m= K are quasi-convex then that would imply a structure to the optimal policy similar to
the one we obtained for the two-supplier case. The following lemma provides a condition under which
this is the case.
LEMMA 2. For systems with multiple supplier, if the ordering cost function is continuous on [0, ]C , and
(1) 1 ,mh c cα+ ≥ (2) ,mC S≥ where max{arg min [ ( ) ]}m myS g y c y= + 1,
0( ) ( ) ( )g y f y dα ξ φ ξ ξ
∞= −∫
( )L y+ and [ , ]( ) min [ ( ) ( )],my x x Cf x g y c y x∈ += + − then functions ( )i
ng y are quasi-convex for all n ≥ 1
and all 1,..., .i m=
When the functions ( )ing y ( 1, , )i m= K are quasi-convex, the optimal policy can be characterized
similarly to the case with two suppliers. In particular, the optimal policy would again involve multiple
thresholds and multiple order-up-to levels, which can be characterized by carrying out a similar analysis
to the one we did for the two-supplier case.
For the important special case with two suppliers, suppliers L and H, with order size constraints on
both, LC and ,HC Lemma 2 can be further specified as follows.
COROLLARY 1. If , ( )H LL HC C K c c> − , H
L HC C S+ ≥ and L Hh c cα+ ≥ , then the optimal policy has
the structure described in Theorem 1, except that case i is replaced as follows: if 1nx s< , then it is optimal
1 Because functions ( )m
ng y and ( ) mng y c y+ may not be strictly monotonic, there may exist multiple solutions to
arg min ( )my ng y and/or arg min [ ( ) ]m
y ng y c y+ for n∀ . As we have shown in the proof of Lemma 2, the result would continue to hold by setting m
nS to be the largest solution to arg min [ ( ) ]my ng y c y+ .
20
to order up to HnS (or as close as possible) by ordering LC units from supplier L and the rest from
supplier H.
Intuitively, we would expect that having order size constraints on both suppliers increases the value of
using the optimal policy, say relative to the DS. This intuition is confirmed by numerical results, an
example of which is shown in Figure 6 where we show the percentage cost difference between the
optimal policy and the DS policy as we vary the maximum order size for both suppliers. As we can see,
the benefit from using the optimal policy can be significant, with over 70% improvement in some of the
examples shown.
0%
10%
20%
30%
40%
50%
60%
70%
80%
60 80 100 120 140 160 180 200
Figure 6 - The benefit of using the optimal policy when there are order size constraints on both suppliers ( 80μ = , 40σ = , L HC C C= = , 1h = , 3b = , 1Lc = , 2Hc = , 20K = and 0.90α = )
6. Conclusion
In this paper, we studied an inventory system with two heterogeneous suppliers differentiated by their
fixed and variable costs as well as their order size constraints. Procuring from one of the supplier involves
a high variable cost but negligible fixed cost while procuring from the second supplier involves a low
variable cost but high fixed cost, as well as a constraint on the maximum order size. We showed that the
problem can be reduced to an equivalent problem with a single supplier, although the corresponding cost
is neither concave nor convex. Using the notion of quasi-convexity, we were able to characterize the
structure of the optimal policy. In particular, we showed that the optimal policy is characterized by
multiple thresholds which determine when to order from each supplier and how much. In contrast to
previous research, which does not consider order size constraints, we showed that it can be optimal to
Perc
enta
ge b
enef
it
C
21
place orders with both suppliers in the same period. In fact, we showed that the presence of such
constraints is what makes dual sourcing particularly valuable. We extended the results obtained for two
suppliers to systems with multiple suppliers and order size constraints on all suppliers and provided
sufficient conditions under which the optimal policy continues to assume the same structure. The results
in this paper enrich the relatively limited literature on multi-sourcing when the suppliers are differentiated
by their cost structures and their capacities. The paper also enriches the literature on inventory with
general cost functions, including those that are neither concave nor convex. The paper also provides
insights into factors that may affect the value of multi-sourcing in practice.
There are several possible directions for future research. It would be of interest to consider systems,
where suppliers, in addition to being differentiated by their costs and capacities, are also differentiated by
their leadtimes. This could lead to unifying the existing (and more prevalent) literature that focuses on
multi-sourcing when suppliers are differentiated by their leadtimes with the less extensive literature on
multi-sourcing when suppliers differ in terms of their cost structures and capacities. It would also be of
interest to consider systems where suppliers, in addition to having different cost structures and capacities,
have varying yield, so that the amount delivered may be less than the amount ordered, or varying supply
failure probabilities, where failures correspond to an order placed with a supplier being significantly
delayed or cancelled.
22
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23
Appendix
This appendix has two parts: A and B. In part A, we present proofs for Lemmas 1-2, Theorems 1-3,
Proposition 1, and Corollary 1. In Appendix B, we extend our results to the case of systems with infinite
horizon.
Appendix A
Proof of Lemma 1
Denote by E the set of quasi-convex functions. We will use induction to prove that ( )Lng y ∈ E and
( )Hng y ∈ E for all 1.n ≥ First, note that the functions 1 ( )Lg y and 1 ( )Hg y are obviously quasi-convex.
Also note that if functions ( )Lng y and ( )H
ng y are quasi-convex, then we have: (1) ( )ng y is decreasing
on ( , ]LnS−∞ because ( ) ( )L L
n ng y g y c y= + is decreasing on ( , ];LnS−∞ (2) L H
n nS S> and 0LnS ≥
because (a) the difference gnH ( y) − gn
L ( y) = y(cH − cL ) ≥ 0 and is strictly increasing in y and (b) ( )Lng y is
strictly decreasing ( Lb c> ) on ( ,0].−∞
We next prove that 1( )Lng y+ ∈ E and 1( ) ;H
ng y+ ∈ E we do so in three parts. In Part 1, we prove the
monotonicity of ( ) Hnf x c x+ on [0, );∞ in Part 2, we prove the monotonicity of ( )nf x on ( ,0];−∞ and
finally in Part 3, we prove the quasi-convexity of functions 1( )Lng y+ and 1( ).H
ng y+ First, define
0
1[ , ]( ) inf ( ) ,H H
n x x C nF x g y c x+= −
0
2[ , ]( ) inf ( ) ,
L
L Ln x C x C nF x g y c x K+ += − + 3
[ , )( ) inf ( ) ( ) ,L
H H L Hn x C n LF x g y c x c c C K+ ∞= − + − + then
from equation (3) we have 1 2 3( ) min[ ( ), ( ), ( )].n n n nf x F x F x F x=
Part 1: To show the monotonicity of ( ) ,Hnf x c x+ we investigate the monotonicity of functions
1( ) ,HnF x c x+ 2 ( ) H
nF x c x+ and 3( ) .HnF x c x+
(i) Since 0
1[ , ]( ) inf ( ),H H
n x x C nF x c x g y++ = 1( ) HnF x c x+ is increasing when 0[ , ).H
nx S C∈ − ∞ In
addition, we have 10( ) ( )H H
n nF x c x g x C+ = + when 0( , ].Hnx S C∈ −∞ −
(ii) Since 0
2[ , ]( ) inf ( ) ( ) ,
L
H L H Ln x C x C nF x c x g y c c x K+ ++ = + − + 2 ( ) H
nF x c x+ is increasing when
.Ln Lx S C≥ − When ,L
n Lx S C< − we have 2 ( ) ( ) ( ) ,H H L Hn n L LF x c x g x C c c C K+ = + + − + which
is also increasing if .Hn Lx S C≥ − Therefore 2 ( ) H
nF x c x+ is increasing on [ , ).Hn LS C− ∞ In
24
addition, we have 20( ) min[ ( ), ( ) ( ) ]H H L H L
n n n LF x c x g x C g x C c c x K+ ≤ + + + − + for ,x∀ and 2 ( ) ( ) ( )H L H L
n n LF x c x g x C c c x K+ = + + − + when .Hn Lx S C< −
(iii) Since 3[ , )( ) inf ( ) ( ) ,
L
H H L Hn x C n LF x c x g y c c C K+ ∞+ = + − + 3 ( ) H
nF x c x+ is increasing for .x∀ In
addition, we have 3 ( ) ( ) ( ) .H L H Ln n LF x c x g x C c c x K+ ≤ + + − +
By combining (i), (ii) and (iii), we obtain:
a) when ,Hn Lx S C< − 3( ) ( )H H
n nf x c x F x c x+ = + is increasing;
b) when 0 ,H Hn L nS C x S C− ≤ < − 2 3( ) min[ ( ) , ( ) ]H H H
n n nf x c x F x c x F x c x+ = + + is increasing;
c) when 0 ,Hnx S C≥ − 1 2 3( ) min[ ( ) , ( ) , ( ) ]H H H H
n n n nf x c x F x c x F x c x F x c x+ = + + + is increasing.
In summary, ( ) Hnf x c x+ is increasing for all x. Obviously ( ) H
nf x c x+ is increasing on [0, ).∞
Part 2: To verify the monotonicity of ( )nf x we similarly investigate the monotonicity of 1( ),nF x 2 ( )nF x
and 3 ( ).nF x (i) For
0
1[ , ]( ) inf ( ) ,H H
n x x C nF x g y c x+= − we have
a) when ,Hnx S≤
0[ , ]inf ( )Hx x C ng y+ is decreasing, and thus 1( )nF x is decreasing;
b) when ,H Ln nS x S< ≤ 1( ) ( )n nF x g x= is also decreasing.
Hence, 1( )nF x is decreasing on ( , ].LnS−∞ In addition, we have 1
0( ) ( )H Hn nF x g x C c x≤ + − for all x.
(ii) For 0
2[ , ]( ) inf ( ) ,
L
L Ln x C x C nF x g y c x K+ += − + it can be observed that
0[ , ]inf ( )L
Lx C x C ng y+ + is decreasing
when 0.Lnx S C≤ −
Hence 2 ( )nF x is decreasing on 0( , ].LnS C−∞ − In addition, we have 2
0( ) ( )H Hn nF x g x C c x= + −
1( )nF x≥ when 0 ,L Ln nS C x S− < ≤ and 2 ( ) ( )L L
n n LF x g x C c x≤ + − K+ when 0.Lnx S C≤ −
(iii) For 3[ , )( ) inf ( ) ( ) ,
L
H H L Hn x C n LF x g y c x c c C K+ ∞= − + − + we have
a) when ,Hn Lx S C≤ − [ , )inf ( ) ( ),
L
H H Hx C n n ng y g S+ ∞ = so 3 ( )nF x is decreasing;
b) when ,H Ln L n LS C x S C− < ≤ − 3 ( ) ( ) L
n n L LF x g x C c C K= + + + is decreasing.
Hence, 3 ( )nF x is decreasing on ( , ].Ln LS C−∞ − In addition, since 2 ( ) ( )L L
n n LF x g x C c x K≤ + − +
( ) Ln L Lg x C c C K= + + + for ,x∀ and 3 ( ) ( ) L
n n L LF x g x C c C K= + + + when [ , ],L Ln L nx S C S∈ − we
have 3 2( ) ( )n nF x F x≥ for [ , ].L Ln L nx S C S∀ ∈ −
By combining (i), (ii) and (iii), we have:
a) when 0 ,L Ln nS C x S− < ≤ 1( ) ( )n nf x F x= is decreasing;
b) when 0 ,L Ln L nS C x S C− < ≤ − 1 2( ) min[ ( ), ( )]n n nf x F x F x= is decreasing;
c) when ,Ln Lx S C≤ − 1 2 3( ) min[ ( ), ( ), ( )]n n n nf x F x F x F x= is decreasing.
25
In summary, ( )nf x is decreasing on ( , ].LnS−∞ Since 0,L
nS ≥ ( )nf x is decreasing on ( ,0].−∞
Part 3: Rewrite functions 1( )Lng y+ and 1( )H
ng y+ as follows:
1 10( ) ( ) ( ) ( ) ,L L L
n ng y c E R y dξ ξ φ ξ ξ∞
+ += + −∫
1 10( ) ( ) ( ) ( ) ,H H H
n ng y c E R y dξ ξ φ ξ ξ∞
+ += + −∫
where E represents expectation, 1( ) ( ),L Ln nR z hz bz c z f zα+ −
+ = + + + and 1( )H HnR z hz bz c z+ −
+ = + +
( ).nf zα+ Since
1
( ) ( ) 0( )
( ) [ ( ) ] 0,
LnL
n L H Hn
b c z f z zR z
h c c z f z c z z
α
α α+
⎧− − + ≤⎪= ⎨+ − + + >⎪⎩
and
1
( ) ( ) 0( )
( ) [ ( ) ] 0,
HnH
n H H Hn
b c z f z zR z
h c c z f z c z z
α
α α+
⎧− − + ≤⎪= ⎨+ − + + >⎪⎩
( ) Hnf x c x+ is increasing on [0, )∞ (Part 1), and ( )nf x is decreasing on ( ,0]−∞ (Part 2), 1( )L
nR z+ and
1( )HnR z+ are decreasing on ( ,0]−∞ and increasing on [0, )∞ when ,L Hh c cα+ ≥ i.e., 1( )L
nR z+ ∈E and
1( ) .HnR z+ ∈E
Since the property of quasi-convexity is preserved for integral convolutions, provided that the demand
distribution is strongly unimodal (Ibragimov 1956, Dharmadhikari and Joag-Dev 1988), we have
1( )Lng y+ ∈E and 1( ) .H
ng y+ ∈E □
Proof of Theorem 1
Define * 1 2 3( ) arg min [ ( ), ( ), ( )].n y n n ny x F x F x F x We partition the beginning inventory levels into five
regions, and discuss the corresponding optimal order-up-to level, * ( ),ny x in five cases.
Case i: When 1 ,nx s< i.e., ( , ),Hn LS x C∈ + ∞ we have
10 0 0( ) ( ) ( ) ,H H H
n n nF x g x C c x g x C c C= + − = + + 2
0 0 0( ) ( ) ( ) ( ) ,L L L Hn n L L n nF x g x C c C K g x C c x K g x C c C= + + + ≤ + − + = + + and 3 ( ) ( ) ( ) ( ) .H H H L H L
n n n L n L LF x g S c x c c C K g x C c C K= − + − + ≤ + + +
Hence 3 2 1( ) ( ) ( ),n n nF x F x F x≤ ≤ and * 3( ) arg min ( ) Hn y n ny x F x S= = if 1 .nx s<
Case ii: When 1 2 ,n ns x s≤ < i.e., 0( , ],Hn LS x C x C∈ + + we have
26
10 0( ) ( ) ,H
n nF x g x C c C= + + 3 ( ) ( ) .Ln n L LF x g x C c C K= + + +
There are two subcases:
Subcase 1. If ( , ),Ln LS x C∈ + ∞ then min( , ) ,L
n L LS x C x C+ = + and 2 1( ) ( ) ( ).L
n n L L nF x g x C c C K F x= + + + ≤
Subcase 2: If 0( , ],Ln Lx CS x C∈ + + then min( , ) ,L L
n L nS x C S+ = and 2 1 3( ) ( ) min[ ( ), ( )].L L L
n n n n nF x g S c x K F x F x= − + ≤
Hence * 2( ) arg min ( ) min( , )Ln y n n Ly x F x S x C= = + if 1 2.n ns x s≤ <
Case iii: When 2 ,Hn ns x S≤ < i.e., 0( , ],H
nS x x C∈ + we have 1
0 0( ) ( ) min[ ( ) , ( )],H H H Hn n n n nF x g S c x g x C c C g x= − ≤ + + 3 ( ) ( ) .L
n n L LF x g x C c C K= + + +
There are three subcases:
Subcase 1: If ( , ),Ln LS x C∈ + ∞ then min( , ) ,L
n L LS x C x C+ = + and 2 3( ) ( ) ( ).L
n n L L nF x g x C c C K F x= + + + =
Subcase 2: If 0( , ],Ln Lx CS x C∈ + + then min( , ) ,L L
n L nS x C S+ = and 2 3( ) ( ) ( ).L L L
n n n nF x g S c x K F x= − + ≤
Subcase 3: If 0( , ),LnS x x C∈ + then min( , ) ,L L
n L nS x C S+ = and 3 2 1
0 0 0 0( ) ( ) ( ) ( ) ( ).L Hn n n n nF x F x g x C c C K g x C c C F x≥ = + + + = + + ≥
Please note that in Subcase 3, we have ( ) ( ) ( )L L L H L H H H Hn n n n n ng S c x K g S c x g S c x− + ≥ − ≥ − because (a)
0(0, )LnS x C− ∈ and (b) ( ) ( )H H H
n n ng S g y≤ for all y.
Hence * 1 2( ) arg min [ ( ), ( )] arg min{ ( ) , [min( , )] }H H H L L Ln y n n n n n n Ly x F x F x g S c x g S x C c x K= = − + − + if
2 .Hn ns x S≤ <
Case iv: When ,H Ln nS x S≤ < i.e., ( , ],H
nS x∈ −∞ we have 1
0 0( ) ( ) ( ) ,Hn n nF x g x g x C c C= ≤ + + 3 ( ) ( ) .L
n n L LF x g x C c C K= + + +
There are three subcases:
Subcase 1: If ( , ),Ln LS x C∈ + ∞ then min( , ) ,L
n L LS x C x C+ = + and 2 3( ) ( ) ( ).L
n n L L nF x g x C c C K F x= + + + =
Subcase 2: If 0[ , ],Ln LS x C x C∈ + + then min( , ) ,L L
n L nS x C S+ = and 2 3( ) ( ) ( ).L L L
n n n nF x g S c x K F x= − + ≤
27
Subcase 3: If 0( , ),LnS x x C∈ + then min( , ) ,L L
n L nS x C S+ = and 3 2 1
0 0( ) ( ) ( ) ( ).Hn n n nF x g x C c C F x F x≥ + + = ≥
Note that in Subcase 3, we have ( ) ( ) ( ) ( )L L L H L H H Hn n n n n ng S c x K g S c x g x c x g x− + ≥ − ≥ − = because (a)
0(0, )LnS x C− ∈ and (b) ( )H
ng x is increasing when SnH ≤ x < Sn
L . Hence * 1 2( ) arg min [ ( ), ( )]n y n ny x F x F x=
arg min{ ( ), [min( , )] }L L Ln n n Lg x g S x C c x K= + − + if Sn
H ≤ x < SnL .
Case v: When ,Lnx S≥ i.e., ( , ],L
nS x∈ −∞ we have 1
0 0( ) ( ) ( ) ,Hn n nF x g x g x C c C= ≤ + + 2
0 0 0 0( ) ( ) ( ) ( ) ,L H Ln n n n L LF x g x C c C K g x C c C g x C c C K= + + + = + + ≤ + + + 3 ( ) ( ) .L
n n L LF x g x C c C K= + + +
Hence, 1 2 3( ) ( ) ( )n n nF x F x F x≤ ≤ and * 1( ) arg min ( )n y ny x F x x= = if x ≥ SnL . □
Proof of Theorem 2
For L nC β≥ and K ≤ γ n , we partition the region 2[ , )Ln ns S into three sub-regions, i.e., 2[ , ),H
n ns S 1ˆ[ , )Hn nS s
and 1ˆ[ , ),Ln ns S and then discuss the corresponding optimal order-up-to levels in three cases.
Case 1: When 2[ , ),Hn nx s S∈ we have
1( ) ( ) ,H H Hn n nF x g S c x= −
2 ( ) ( ) ( ) ( )
( )
L L H H H L Ln L n L L n L
n L L L Ln n n L
g x C c x K g x C c x c c C K if x S CF x
g S c x K if x S C
⎧ + − + = + − − − + < −⎪= ⎨− + ≥ −⎪⎩
3( ) ( ).Ln L L ng x C c C K F x≤ + + + =
Since ( )Hn Lg x C+ is increasing on 1 2[ , ) [ , )H
n n ns s S∞ ⊃ and ,H Lc c> we have 1 2( ) ( )n nF x F x− is
decreasing when 2[ , ).Hn nx s S∈ Because 1 2 1( ) ( ) ( ) 0,H H H
n n n n n n nF S F S J S Kγ− = = − ≥ we have 1 2( ) ( )n nF x F x≥ when 2[ , ).H
n nx s S∈ Hence * 2( ) arg min ( ) min( , )Ln y n n Ly x F x S x C= = + if 2[ , ).H
n nx s S∈
Case 2: When 1ˆ[ , ),Hn nx S s∈ we have
1( ) ( ),n nF x g x= 2 3( ) ( ) ( ) ( ).L L L L
n n n n L L nF x g S c x K g x C c C K F x= − + ≤ + + + =
28
Since 1 2 1( ) ( ) ( )n n nF x F x J x− = is decreasing when 1ˆ[ , ),Hn nx S s∈ and 1 1 2 1 1 1ˆ ˆ ˆ( ) ( ) ( ) 0,n n n n n nF s F s J s− = = we
have 1 2( ) ( )n nF x F x≥ when 1ˆ[ , ).Hn nx S s∈ Hence * 2( ) arg min ( ) L
n y n ny x F x S= = if 1ˆ[ , ).Hn nx S s∈ This
optimal order-up-to level can also be rewritten as * ( ) min( , )Ln n Ly x S x C= + because .L H
L n nC S S≥ −
By combining Cases 1 and 2, we have * ( ) min( , )Ln n Ly x S x C= + when 2 1ˆ .n ns x s< ≤
Case 3: When 1ˆ[ , ),Ln nx s S∈ we have
10 0( ) ( ) ( ) ,H
n n nF x g x g x C c C= ≤ + +
02
0 0 0
( ) ( )
( )
L L L Ln n n
n H Ln n
g S c x K if x S CF x
g x C c C if x S C
⎧ − + ≤ −⎪= ⎨+ + > −⎪⎩
3( ) ( ).Ln L L ng x C c C K F x≤ + + + =
Since 1 2 1( ) ( ) ( )n n nF x F x J x− = is decreasing when 0 ,Lnx S C≤ − and 1 1 2 1ˆ ˆ( ) ( ) 0,n n n nF s F s− = we have
1 2( ) ( )n nF x F x≤ when 1ˆ[ , ).Ln nx s S∈ Hence * 1( ) arg min ( )n y ny x F x x= = if 1ˆ[ , ).L
n nx s S∈
For L nC β ′≥ and K > γ n , we similarly partition the region 2[ , )Ln ns S into three sub-regions, i.e., 2 2ˆ[ , ),n ns s
2ˆ[ , )Hn ns S and [ , ),H L
n nS S and then discuss the corresponding optimal order-up-to levels in three cases.
Case 1: When 2 2ˆ[ , ),n nx s s∈ we have 1( ) ( ) ,H H H
n n nF x g S c x= −
2 ( ) ( ) ( ) ( )
( )
L L H H H L Ln L n L L n L
n L L L Ln n n L
g x C c x K g x C c x c c C K if x S CF x
g S c x K if x S C
⎧ + − + = + − − − + < −⎪= ⎨− + ≥ −⎪⎩
3( ) ( )Ln L L ng x C c C K F x≤ + + + = .
Since ( )Hn Lg x C+ is increasing on 1 2 2ˆ[ , ) [ , )n n ns s s∞ ⊃ and ,H Lc c> we have 1 2( ) ( )n nF x F x− is decreasing
when 2 2ˆ[ , ).n nx s s∈ Because 1 2 2 2 2 2ˆ ˆ ˆ( ) ( ) ( ) 0,n n n n n nF s F s J s− = = we have 1 2( ) ( )n nF x F x≥ when 2 2ˆ[ , ).n nx s s∈
Hence * 2( ) arg min ( ) min( , )Ln y n n Ly x F x S x C= = + if 2 2ˆ[ , ).n nx s s∈
Case 2: When 2ˆ[ , ),Hn nx s S∈ we have
10 0 0( ) ( ) ( ) ( ) ,H H H H H H
n n n n nF x g S c x g x C c x g x C c C= − ≤ + − = + +
02
0 0 0
( ) ( )
( )
L L L Ln n n
n H Ln n
g S c x K if x S CF x
g x C c C if x S C
⎧ − + ≤ −⎪= ⎨+ + > −⎪⎩
3( ) ( ).Ln L L ng x C c C K F x≤ + + + =
29
Since 1 2 2( ) ( ) ( )n n nF x F x J x− = is decreasing when 0Lnx S C≤ − and 2 2ˆ( ) 0,n nJ s = we have 1 2( ) ( )n nF x F x≤
when 2ˆ[ , ).Hn nx s S∈ Hence * 1( ) arg min ( ) H
n y n ny x F x S= = if 2ˆ[ , ).Hn nx s S∈
Case 3: When [ , ),H Ln nx S S∈ we have
10 0 0( ) ( ) ( ) ( ) ,H H H
n n n nF x g x g x C c x g x C c C= ≤ + − = + +
02
0 0 0
( ) ( )
( )
L L L Ln n n
n H Ln n
g S c x K if x S CF x
g x C c C if x S C
⎧ − + ≤ −⎪= ⎨+ + > −⎪⎩
3( ) ( ).Ln L L ng x C c C K F x≤ + + + =
Since ( )Lng x is decreasing when 0 ,L
nx S C≤ − 1 2( ) ( )n nF x F x− is decreasing. From 1 2( ) ( ) 0,H H
n n n n nF S F S Kγ− = − < we have 1 2( ) ( )n nF x F x≤ when x ∈[SnH ,Sn
L ). Hence * 1( ) arg min ( )n y ny x F x x= = if [ , ).H L
n nx S S∈
For nK γ> and βn ≤ CL < ′βn , since the condition L nC β ′< can be rewritten as 2ˆ ,Ln L nS C s− > we have
3 2 2 2ˆ( ) ( ) ( ) ( )( ) ( ) ( ) 0.L H H L L H L L Ln n L n n n n n L n n L n nJ S C g S g S c c S C K J S C J s− = − − − − − = − < = Because
L nC β≥ , 3 20( ) ( ) ( ) 0.L H L H
n n n n n n LJ s g S g S C C= − − + ≥ So there exists an 3 2 2ˆ [ , ) [ , )L Hn n n L n ns s S C s S∈ − ⊆
satisfying 3 3ˆ( ) 0.n nJ s = We can then use the two parameters 3ˆns and HnS to partition the beginning
inventory interval 2[ , )Ln ns S into three sub-intervals, i.e., 2 3ˆ[ , ),n ns s 3ˆ[ , )H
n ns S and [SnH ,Sn
L ), and discuss the
corresponding optimal order-up-to levels in three cases.
Case 1: When 2 3ˆ[ , ),n nx s s∈ we have 1( ) ( ) ,H H H
n n nF x g S c x= − 2 ( ) ( ) ( ) ( )L L H H H L
n n L n L LF x g x C c x K g x C c x c c C K= + − + = + − − − + 3( ) ( ).L
n L L ng x C c C K F x= + + + =
Since ( )Hn Lg x C+ is increasing on 1 2 3ˆ[ , ) [ , ),n n ns s s∞ ⊃ we have that 1 2( ) ( )n nF x F x− is decreasing when
2 3ˆ[ , ).n nx s s∈ Because 1 3 2 3 3 3ˆ ˆ ˆ( ) ( ) ( ) 0,n n n n n nF s F s J s− = = we have 1 2( ) ( )n nF x F x≥ when 2 3ˆ[ , ).n nx s s∈ Hence * 2( ) arg min ( )n y n Ly x F x x C= = + if 2 3ˆ[ , ).n nx s s∈
Case 2: When 3ˆ[ , ),Hn nx s S∈ we have
1( ) ( ) ,H H Hn n nF x g S c x= −
2 ( ) ( ) ( ) ( )
( )
L L H H H L Ln L n L L n L
n L L L Ln n n L
g x C c x K g x C c x c c C K if x S CF x
g S c x K if x S C
⎧ + − + = + − − − + < −⎪= ⎨− + ≥ −⎪⎩
30
3( ) ( ).Ln L L ng x C c C K F x≤ + + + =
Since ( )Hn Lg x C+ is increasing on 1 3ˆ[ , ) [ , )H
n n ns s S∞ ⊃ and ,H Lc c> we have 1 2( ) ( )n nF x F x− is
decreasing when 3ˆ[ , ).Hn nx s S∈ Because 1 3 2 3 3 3ˆ ˆ ˆ( ) ( ) ( ) 0,n n n n n nF s F s J s− = = we have 1 2( ) ( )n nF x F x≤ when
3ˆ[ , ).Hn nx s S∈ Hence * 1( ) arg min ( ) H
n y n ny x F x S= = if 3ˆ[ , ).Hn nx s S∈
Case 3: When [ , ),H Ln nx S S∈ we have
10 0( ) ( ) ( ) ,H
n n nF x g x g x C c C= ≤ + +
02
0 0 0
( ) ( )
( )
L L L Ln n n
n H Ln n
g S c x K if x S CF x
g x C c C if x S C
⎧ − + ≤ −⎪= ⎨+ + > −⎪⎩
3( ) ( ).Ln L L ng x C c C K F x≤ + + + =
Similar to Case 3 of (2), we can prove that 1 2( ) ( )n nF x F x≤ when [ , ).H Ln nx S S∈ Hence
* 1( ) arg min ( )n y ny x F x x= = if [ , ).H Ln nx S S∈ □
Proof of Theorem 3
There are three cases to consider, corresponding to the three intervals for starting inventory level.
Case 1: When 2[ , ),Hn nx s S∈ we have
1( ) ( ) ,H H Hn n nF x g S c x= − 2 ( ) ( ) ( ) ( )L L H H H L
n n L n L LF x g x C c x K g x C c x c c C K= + − + = + − − − + 3( ) ( ).L
n L L ng x C c C K F x= + + + =
To compare 1( )nF x and 2 ( ),nF x we have: (i) 1 2 3( ) ( ) ( )n n nF x F x J x− = is decreasing because
( )Hn Lg x C+ is increasing on 1 2[ , ) [ , );H
n n ns s S∞ ⊃ (ii) 3 20( ) ( ) ( ) 0.L H L H
n n n n n n LJ s g S g S C C= − − + ≥ So the
optimal order-up-to level in this case depends on the sign of 3 ( ) :Hn nJ S
(1) If 3 ( ) 0,Hn nJ S ≥ then 1 2( ) ( )n nF x F x≥ and * 2( ) arg min ( )n y n Ly x F x x C= = + when 2[ , ).H
n nx s S∈
(2) If 3 ( ) 0,Hn nJ S < then there exists a 3 2ˆ [ , )H
n n ns s S∈ satisfying 3 3ˆ( ) 0.n nJ s = When 2 3ˆ ,n ns x s≤ < * 2( ) arg min ( ) ;n y n Ly x F x x C= = + when 3ˆ ,H
n ns x S≤ ≤ * 1( ) arg min ( ) .Hn y n ny x F x S= =
Case 2: When 0[ , ),L Ln L nx S C S C∈ − − we have
1( ) ( ) ( ),H Hn n nF x g x c x g x= − = 2 3( ) ( ) ( ) ( ).L L L L
n n n n L L nF x g S c x K g x C c C K F x= − + ≤ + + + =
31
To compare 1( )nF x and 2 ( ),nF x we have: (i) Fn1(x) − Fn
2 (x) = Jn1(x) is decreasing; (ii) 1
0( )Ln nJ S C−
0( ) ( ) 0.L L L Ln n n ng S C g S= − − ≤ So the optimal order-up-to level in this case depends on the sign of
1 ( ) :Ln n LJ S C−
(1) If 1 ( ) 0,Ln n LJ S C− < then 1 2( ) ( ),n nF x F x≤ and * 1( ) arg min ( )n y ny x F x x= = when
0[ , ).L Ln L nx S C S C∈ − −
(2) If 1 ( ) 0,Ln n LJ S C− ≥ then there exists a 1
0ˆ [ , )L Ln n L ns S C S C∈ − − satisfying 1 1ˆ( ) 0.n nJ s = When
1ˆ ,Ln L nS C x s− ≤ < * 2( ) arg min ( ) ;L
n y n ny x F x S= = when 10ˆ ,L
n ns x S C≤ < − * 1( ) arg min ( ) .n y ny x F x x= =
Case 3: When 0[ , ),L Ln nx S C S∈ − we have
10 0 0 0( ) ( ) ( ) ( ) ,H L
n n n nF x g x g x C c C g x C c C K= ≤ + + = + + + 2 3
0 0( ) ( ) ( ) ( ).L Ln n n L L nF x g x C c C K g x C c C K F x= + + + ≤ + + + =
Thus Fn1(x) ≤ Fn
2 (x) ≤ Fn3(x) and * 1( ) arg min ( )n y ny x F x x= = when Sn
L − C0 ≤ x < SnL . □
Proof of Proposition 1
Since function ( ) ( )L Lg y g y c y= + ∈ E arrive its minimum at LS , we have
g(x) ≥ g(x + CL ) + cLCL , (A.1)
for [ , ).H LLx S S C∈ − Because ( )Lg y is decreasing on ( , ],LS−∞ ( )g y is decreasing on ( , ],LS−∞ and
( ) ( ),LLg x C g S+ ≥ (A.2)
for [ , ).H LLx S S C∀ ∈ − Combining equation (A.1) and (A.2) gives
( ) ( )LL Lg x C c C K g x K+ + + − ≤ and ( ) ( ) 0.L L
Lg x g S c C≥ + >
Thus we have ( ) ( ) .
( ) ( )
LL L
L LL
g x C c C K g x Kg x g S c C
+ + + −≤
+ □
Proof of Lemma 2
Let 0 0
( ) ( ) ( ) ( ) ( ) ,L y h y d b y dξ φ ξ ξ ξ φ ξ ξ∞ ∞+ −= − + −∫ ∫ and denote by OC the ordering cost function of
the inventory system with multiple suppliers. We prove this lemma in two parts: in part 1, we prove that
there exists an upper bound on ;mnS in part 2, we prove the quasi-convexity of the functions ( )i
ng y
( 1,..., )i m= for all n ≥ 1.
32
Part 1: Consider the inventory system with single capacitated supplier, in which the supplier has the same variable and fixed costs as supplier m, and has a capacity constraint .C The optimal expected discounted cost function of this inventory system is
( ) min [ ( ) ( )],mn x y x C nf x c y x g y≤ ≤ += − + (A.3)
where 10( ) ( ) ( ) ( )n ng y L y f y dα ξ φ ξ ξ
∞
−= + −∫ and 0 ( ) 0.f x ≡ Noting that the optimal expected
discounted cost function of the inventory system with multiple suppliers is ( ) min [ ( ) ( )],n x y x C nf x OC y x g y≤ ≤ += − + (A.4)
where 10( ) ( ) ( ) ( )n ng y L y f y dα ξ φ ξ ξ
∞
−= + −∫ and 0 ( ) 0.f x ≡ We first prove ( ) ( )n ndf x dx df x dx≥ for
1n∀ ≥ and x∀ by induction.
It is easily verified that 1 1( ) ( )df x dx df x dx≥ for ,x∀ because (1) 1 1( ) ( ) ( ),g y g y L y= = (2) function
OC is continuous, and (3) ( ) .mdOC q dq c≤ If ( ) ( )n ndf x dx df x dx≥ for ,x∀ then from equations
(A.3) and (A.4), we have 1 1( ) ( )n ndg y dy dg y dy+ +≥ for .y∀ Because function OC is continuous and
( ) ,mdOC q dq c≤ we have 1 1( ) ( )n ndf x dx df x dx+ +≥ for .x∀ Note that ( )nf x and ( )ng y may be non-
differentiable at some points. At these points, we use the left-side derivatives instead.
Let max{arg min ( ( ) )}.m mn y nS g y c y= + Since function ( ) m
ng y c y+ achieves its minimum at ,mnS
we have
10
( )( ) ( ) 0.m mn n
mn
y S y S
df ydL y d cdy dy
ξα φ ξ ξ
∞−
= =
−+ + =∫
Suppose for any 1n ≥ , m mn nS S> , then
10
( )( ) ( ) 0.m mn n
mn
y S y S
df ydL y d cdy dy
ξα φ ξ ξ
∞−
= =
−+ + >∫ (A.5)
We have proved that ( ) ( )n ndf x dx df x dx≥ for 1n∀ ≥ and x∀ , from equation (A.5) we have
10
( )( ) ( ) 0.m mn n
mn
y S y S
df ydL y d cdy dy
ξα φ ξ ξ∞
−
= =
−+ + >∫
This is in contradiction to the definition of .mnS So we have m m
n nS S≤ for 1.n∀ ≥
From Federgruen and Zipkin (1986b), { }mnS is an ascending sequence and lim ,m m
n nS S→∞ = thus we
have m mnS S≤ for 1.n∀ ≥
Part 2: We prove ( )ing y ∈ E ( 1,..., )i m= for 1n∀ ≥ by induction.
33
It is obvious that 1 ( )ig y ∈ E ( 1,..., ).i m= Suppose ( )ing y ∈ E ( 1,..., ),i m= similar to the proof of
Lemma 1, we can prove that: 1) ( ) mnf x c x+ is increasing on [ , ),m
nS C− ∞ and 2) ( )nf x is decreasing on 1( , ].nS−∞ Since m m
nS S≤ for 1n∀ ≥ (Part 1), ( ) mnf x c x+ is increasing on [0, )∞ if .mS C≤ In
addition, because 1 ( )ng y is strictly decreasing ( 1b c> ) on ( ,0],−∞ we have 1 0.nS ≥ From 2), we know
that ( )nf x is decreasing on ( ,0].−∞
Define 1( )inR z+ ( 1,..., )i m= as follows:
1
( ) ( ) 0( )
( ) ( ( ) ) 0.
ini
n i m mn
b c z f z zR z
h c c z f z c z z
α
α α+
⎧− − + ≤⎪⎨
+ − + + >⎪⎩
When ib c≥ ( 1, , )i m= K and 1 ,mh c cα+ ≥ it is obvious that 1( )inR z+ ( 1,..., )i m= is decreasing on
( ,0],−∞ and increasing on [0, ).∞ That is, 1( )inR z+ ∈ E ( 1,..., ).i m=
Rewrite the functions 1( )ing y+ ( 1,..., )i m= as 1 10
( ) ( ) ( ) ( ) .i i in ng y c E R y dξ ξ φ ξ ξ
∞
+ += + −∫ Since the
property of quasi-convexity is preserved for integral convolutions, provided that the demand distribution is strongly unimodal (Ibragimov 1956, Dharmadhikari and Joag-Dev 1988), we have 1( )i
ng y+ ∈E
( 1,..., ).i m= □
Proof of Corollary 1
For the inventory system with two capacitated suppliers, according to Lemma 2 we have
( ), ( )L Hn ng y g y ∈E if , ( ),H L
L HC C K c c> − HL HC C S+ ≥ and .L Hh c cα+ ≥ When the two suppliers
have capacity constraints LC and ,HC the maximal order quantity is limited at .L HC C+ By checking the
optimal order-up-to levels of the five cases of Theorem 1, we find that this limitation has no impact on
cases ii to v because the optimal order quantity will never be more than LC in these cases. However, this
limitation has impact on cases i. When 1 ,nx s< i.e., ( , ),H
n LS x C∈ + ∞ we have 1
0 0 0( ) ( ) ( ) ,H H Hn n nF x g x C c x g x C c C= + − = + + 2
0 0( ) ( ) ( ) ,L Hn n L L nF x g x C c C K g x C c C= + + + ≤ + +
3 ( ) ( ) ( )
( ) ( )
H H H L H Hn n L n L H
n H H L H Hn L H L n L H
g S c x c c C K if S x C CF x
g x C C c x c c C K if S x C C
⎧ − + − + ≤ + +⎪= ⎨+ + − + − + > + +⎪⎩
( ) .Ln L Lg x C c C K≤ + + +
Hence, 3 2 1( ) ( ) ( ),n n nF x F x F x≤ ≤ and * 3( ) arg min ( ) min( , )Hn y n n L Hy x F x S x C C= = + + if x < sn
1 . □
34
Appendix B Systems with Infinite Horizon
In this part, we will show that there exists stationary policy for the inventory system with two suppliers,
and the limit of any convergent subsequence of the optimal policies in finite horizon is the optimal policy
for the infinite horizon problem. In the infinite horizon problem, variables, parameters and function are
defined in the same ways as those in the finite horizon problem, except that they are no longer indexed
with the number of period; demand D is assumed to be supported on [0, Dmax], where maxD is a positive
real number.
If there exists a function, ( ),r x which is said to be the optimal expected discounted cost function over
the long run for the infinite horizon system with a beginning inventory level ,x then ( )r x should satisfy
the following functional equation: max
0( ) min[ ( ) ( ) ( ) ( ) ],
D
y xr x OC y x L y r y dα ξ φ ξ ξ
≥= − + + −∫ (B.1)
where max max
0 0( ) ( ) ( ) ( ) ( ) .
D DL y h y d b y dξ φ ξ ξ ξ φ ξ ξ+ −= − + −∫ ∫
The optimal policy for the infinite horizon system exists only when the optimal solution to equation
(B.1) exists. Before proving the existence of ( ),r x we first give some properties about the optimal policy
for the finite horizon system when demand is supported on [0, Dmax].
LEMMA B1. Parameters HnS and L
nS of the optimal policy for the finite horizon system satisfy 1 1[ [( ) ( )], [( ) ( )]],H H H H
nS b c b h b c c b hα− −∈ Φ − + Φ + − + 1 1[ [( ) ( )], [( ) ( )]],L L H L
nS b c b h b c c b hα− −∈ Φ − + Φ + − +
for all 1,n ≥ 1−Φ is the inverse cumulative distribution function of demand.
PROOF. From the definitions of HnS and L
nS we have
max 10
( ) ( )( ) ( )( ) ,H H H Hn n n n
DH Hn n
y S y S y S y S
dg y df ydL y dL yc d cdy dy dy dy
ξα φ ξ ξ α−
= = = =
−− = = + ≥ −∫
max 10
( ) ( )( ) ( )( ) ,L L L Ln n n n
DL Hn n
y S y S y S y S
dg y df ydL y dL yc d cdy dy dy dy
ξα φ ξ ξ α−
= = = =
−− = = + ≥ −∫
where the inequalities are valid because ( ) Hnf x c x+ is increasing for 1n∀ ≥ (please refer to the proof of
Lemma 1). Note that ( )nf x and ( )ng y may be non-differentiable at some points. At these points, we use
the left-side derivative instead. Then we have
35
( ) ,Hn
H H
y S
dL y c cdy
α=
≤ − ( ) .
Ln
H L
y S
dL y c cdy
α=
≤ −
Since 0
( ) ( ) ( ) ,ydL y b h d b
dyφ ξ ξ= + −∫ we have 1[( ) ( )],H H H
nS b c c b hα−≤ Φ + − + and
1[( ) ( )]L H LnS b c c b hα−≤ Φ + − + for 1.n∀ ≥
We next prove 1L LnS S≥ and 1
H HnS S≥ for 1n∀ ≥ by induction. Suppose 1
L LnS S≥ and 1 .H H
nS S≥ If
1 1L LnS S+ < or 1 1 ,H H
nS S+ < then we have
1
( ) 0Ln
n
y S
df ydy
ξ
+=
−≤ or
1
( ) 0,Hn
n
y S
df ydy
ξ
+=
−≤
because ( )nf x is decreasing when ( , ]Lnx S∈ −∞ (please refer to the proof of Lemma 1), and
1 1H L Ln nS S S+ < ≤ or 1 1 .L L L
n nS S S+ < ≤
Since ( )L y is strictly convex, and 1 1L LnS S+ < or 1 1 ,H H
nS S+ < we have
1
( )Ln
L
y S
dL y cdy
+=
< − or 1
( ) .Hn
H
y S
dL y cdy
+=
< −
This is in contradiction to the definitions of 1LnS + or 1.
HnS + Thus we have 1
L LnS S≥ and 1
H HnS S≥ for
1.n∀ ≥
Obviously we have 11 [( ) ( )]L LS b c b h−= Φ − + and 1
1 [( ) ( )].H HS b c b h−= Φ − + For 1,n∀ ≥ we have 1 1[ [( ) ( )], [( ) ( )]],H H H H
nS b c b h b c c b hα− −∈ Φ − + Φ + − + 1 1[ [( ) ( )], [( ) ( )]].L L H L
nS b c b h b c c b hα− −∈ Φ − + Φ + − + □
It can be directly observed from Lemma B1 that max, [0, ].H Ln nS S D∈ The following theorem describes
the monotonicity and uniform convergence of the function sequences, { ( )}nf x and { ( )},ng y and the
continuity of their limits.
THEOREM B1. { ( )}nf x and { ( )}ng y are both ascending sequences for any given x , and uniformly
convergent for any finite ,x lim ( ) ( ),n nf x f x→∞ = lim ( ) ( ),n ng y g y→∞ = ( )f x and ( )g y are
continuous.
PROOF. Obviously we have 1 0( ) 0 ( ).f x f x≥ = Suppose that 1( ) ( )n nf x f x−≥ for .x∀ Then max
1 0( ) min [ ( ) ( ) ( ) ( ) ]
D
n y x nf x OC y x L y f y dα ξ φ ξ ξ+ ≥= − + + −∫
36
max
10min [ ( ) ( ) ( ) ( ) ] ( ).
D
y x n nOC y x L y f y d f xα ξ φ ξ ξ≥ −≥ − + + − =∫
i.e., { ( )}nf x is an ascending sequence. From equation (2), we know that { ( )}ng y is also an ascending
sequence.
Since max* * *1 1 1 10( ) [ ( ) ] [ ( )] [ ( ) ] ( )
D
n n n n nf x OC y x x L y x f y x dα ξ φ ξ ξ+ + + += − + + −∫ max* * *
0[ ( ) ] [ ( )] [ ( ) ] ( ) ,
D
n n n nOC y x x L y x f y x dα ξ φ ξ ξ≤ − + + −∫
we have max * *1 10
0 ( ) ( ) { [ ( ) ] [ ( ) ]} ( ) .D
n n n n n nf x f x f y x f y x dα ξ ξ φ ξ ξ+ −≤ − ≤ − − −∫
From Theorem 1, we know that * ( ) Ln ny x S≤ when ,L
nx S< and * ( )ny x x= if .Lnx S≥ Denote by
(0, )M ∈ ∞ a large positive number, and maxmax( , ).M M D′ = Since max ,LnS D≤ for 1n∀ ≥ and
[ , ],x M M∀ ∈ − we have *max( ) [ , ].ny x M D Mξ ′− ∈ − − Hence,
max * *[ , ] 1 [ , ] 10
max | ( ) ( ) | max | [ ( ) ] [ ( ) ] | ( )D
M M n n M M n n n nf x f x f y x f y x dα ξ ξ φ ξ ξ− + − −− ≤ − − −∫
max max[ , ] 1 [ , ] 1max | ( ) ( ) | max ( ).nM D M n n M nD Mf x f x f xα α′ ′− − − − −≤ − ≤ ≤L
According to Federgruen and Zipkin (1986a), there are positive constants A and B such that
( ) | |L x A B x≤ + for all x. Thus we have 1 1( ) ( ) ( ) | |,f x g x L x A B x≤ = ≤ + and
[ , ] 1 maxmax | ( ) ( ) | max[ ( ), ].nM M n nf x f x A B M nD A BMα− + ′− ≤ + + +
Because 1
max
max
max{ [ ( 1) ], }lim 1,max[ ( ), ]
n
n n
A B M n D A BMA B M nD A BM
α αα
+
→∞
′+ + + += <
′+ + +
11[ ( ) ( )]n nn
f x f x∞+=
−∑ is convergent for [ , ].x M M∀ ∈ − So ( )nf x is uniformly convergent if x lies in a
finite interval. According to the relationship between ( )ng y and ( ),nf x ( )ng y is also uniformly
convergent if x lies in a finite interval. Since functions ( )nf x and ( )ng y are continuous, ( )f x and ( )g y
are also continuous. □
For all solutions to equation (B.1), we have ( ) ( )nf x r x≤ for , .x n∀ Thus we have ( ) ( )f x r x≤ for
.x∀ Therefore, ( )f x is the optimal expected discounted cost function for the infinite horizon system.
The following theorem describes the optimal policy for the infinite horizon system.
THEOREM B2. The optimal policy for the infinite horizon system can be fully characterized by four
thresholds, 1,s 2 ,s HS and LS , and has the same structure as shown in Theorem 1, where 1 ,H
Ls S C= − 20 ,Hs S C= − HS and LS are the limits of any convergent subsequences of { }H
nS and
{ },LnS respectively.
37
PROOF. From Lemma B1 we know that { }HnS and { }L
nS are both bounded. So there exist convergent
subsequences { }l
LnS and { }
l
HnS whose limits are LS and .HS According to the uniform convergence of
{ ( )}ng y and the continuity of ( ),g y we have
0 | ( ) ( ) | | ( ) ( ) | | ( ) ( ) | 0,l l l l l l
L L L L L Ln n n n n ng S g S g S g S g S g S≤ − ≤ − + − → ,l → ∞
0 | ( ) ( ) | | ( ) ( ) | | ( ) ( ) | 0,l l l l l l
H H H H H Hn n n n n ng S g S g S g S g S g S≤ − ≤ − + − → ,l → ∞
i.e., lim ( ) ( )l l
L Ll n ng S g S→∞ = and lim ( ) ( ).
l l
H Hl n ng S g S→∞ =
From the definition of parameters LnS and ,H
nS we have ( ) ( )l l l
L L Ln n ng S g y≤ and ( ) ( )
l l l
H H Hn n ng S g y≤ for
.y∀ Taking limit on both sides, we have
( ) ( )L L Lg S g y≤ and ( ) ( ),H H Hg S g y≤ for ,y∀
i.e., arg min ( )L LyS g y= and arg min ( ).H H
yS g y= □
Theorem B2 indicates that the optimal policy for infinite horizon system can be viewed as the
convergent form of the optimal policy over finite horizon. Similarly, the results described in Theorems 2
and 3 can be correspondingly extended to the infinite horizon system.
Acknowledgements
This research was supported by the National Natural Science Foundation of China under Grants No.
70725001 and 70821001.
References
Dharmadhikari, S., K. Joag-Dev. 1988. Unimodality, Convexity and Applications. Academic Press Series
in Probability and Mathematical Statistics. Academic Press, Boston, MA.
Federgruen, A., P. Zipkin. 1986a. An inventory model with limited production capacity and uncertain
demands I. The average-cost criterion. Math. Oper. Res. 11 193-207.
Federgruen, A., P. Zipkin. 1986b. An inventory model with limited production capacity and uncertain
demands II. The discounted-cost criterion. Math. Oper. Res. 11 208-215.
Ibragimov, I. A. 1956. On the composition of unimodal distributions. Theoret. Probab. Appl. 1 255-266.
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