convexity results for stochastic inventory networks
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Submitted to Management Sciencemanuscript
Convexity Results for Stochastic Inventory NetworksWoonghee Tim Huh
Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027,huh@ieor.columbia.edu
Ganesh JanakiramanStern School of Business, New York University, New York, NY 10012, gjanakir@stern.nyu.edu
In this paper, we establish the convexity of important cost functions in a general class of multi-echelon inven-
tory models. In particular, we first study an assembly system with a single finished product managed using
an echelon order-up-to policy. We show that the shortage penalty cost over any horizon is jointly convex
with respect to the base-stock levels and capacity levels. Our second result pertains to an arbitrary inventory
network, with multiple components, products, production stages and distribution locations, managed opti-
mally. We show that the cost-to-go function of the dynamic program is jointly convex in the inventory state
vector and the capacity vector for both the backorder and lost sales models. These convexity properties have
implications for developing algorithms for making optimal inventory and capacity decisions in such systems.
Key words : Inventory: multi-echelon, stochastic, base-stock policies; Dynamic programming; Convexity
History : This paper was submitted on September 6, 2006.
1. Introduction
In this paper, we study multi-echelon stochastic inventory networks under periodic review. We
show convexity properties of the objective functions in two types of important models. First,
we consider the class of order-up-to policies1 and show the convexity of the shortage costs with
respect to the order-up-to levels and capacity levels. This result is shown for assembly systems.
Secondly, we show that the minimum attainable cost in a stochastic inventory network is a convex
function of the starting inventory levels and capacity levels. By minimum attainable cost, we mean
the cost attained by the optimal policy computed from the dynamic program; this policy is not
necessarily an order-up-to policy. This result is shown for arbitrary acyclic directed inventory
networks, important examples of which are assembly systems, distribution systems or combinations
1 We use “order-up-to” and “base-stock” interchangeably in this paper.
1
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks2 Article submitted to Management Science; manuscript no.
of both. Our analysis considers both environments with backordering of excess demands as well as
environments in which excess demands are lost (we will refer to these as lost sales systems). Our
main technique is to use a result on the behavior of the optimal value of a convex program as a
function of the parameters of the program.
1.1. Motivation
In Section 2, we study an assembly system with one finished product, assuming that an echelon
order-up-to policy is used. Our main result is to show that the backorder or lost sales penalty cost
over any time horizon is jointly convex in the vector of base-stock levels and capacities.
We now motivate the usefulness of this result. Consider an assembly system with a single fin-
ished product, or a collection of independent assembly systems, each of which is dedicated to a
single finished product. Assume an echelon order-up-to policy is used to manage this system. Some
components required for this assembly are expensive, and there is a budget constraint on the total
amount that can be invested in these components. The problem of interest is that of allocating the
available budget across the inventory investments of individual components and across capacity
investments at individual operations or nodes. The inventory investment in a component is mea-
sured by the product of the unit value and the order-up-to level, representing the total amount
of money committed to the component in the system. (An alternate model where the inventory
investment is measured by the average amount of inventory on hand is intractable; see Section 2.4
for details.) The objective is to minimize the overall penalty costs, as measured by the average
amount on backorder in a period, or the average amount of lost sales in a period. This budget
allocation problem, discussed in Section 2.4, is shown to be a convex program, and we also discuss
algorithmic implications of this result. We note that Feigin (1998) has studied the related opti-
mization problem of minimizing inventory investment subject to service level constraints in the
context of managing a large assembly system for a personal computer manufacturer.
In Section 2.5, we discuss the problem of minimizing the sum of the expected holding and penalty
costs. It is well known that this function is not necessarily convex and finding optimal base-stock
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 3
levels is a computationally difficult problem. We show that this cost function can be written as
the sum of a convex function and a concave function, thereby implying that a special purpose
optimization technique (called DC optimization) can be employed.
In Section 3, we study an arbitrary inventory network that assembles and distributes multiple
products. We assume that an optimal policy is used for assembly, procurement and distribution in
each period, and consider a dynamic program that captures the inventory control problem for this
inventory network. We show that the cost-to-go function is jointly convex in the inventory state
vector and the capacity vector.
This result is useful from two perspectives, one theoretical and one practical, discussions of which
follow. From the theoretical perspective, the convexity of the cost-to-go function (with respect to
the inventory levels) is a fundamental and desirable property of an inventory control problem when
there are no fixed ordering or set-up costs. The interest in this property stems from the fact that
first-order conditions can be used to characterize the optimal policy either completely or partially.
While it is generally expected that most of these problems possess this property, researchers have
commonly proved this property for specific inventory models they study. To our knowledge, there
is no published result that establishes this convexity property for a large class of multi-echelon,
multi-product inventory models. Our result in Section 3 fills this gap. Also, from an intuitive aspect,
this result rigorously justifies the notion of decreasing marginal returns of additional capacity
investment at any operation.
From a practical perspective, although inventory systems are rarely managed using an optimal
policy due to computational difficulties, we argue below that the convexity result is valuable.
Usually, managers use heuristic policies that are easier to understand and implement but are
expected to be near optimal, in terms of cost performance. In such systems, capacity investment
decisions are made with the understanding that such a near optimal policy will be used once
any given capacity configuration is chosen. Given the state of the art today, simulation-based
optimization would be a good technique for deciding the best allocation of the capacity budget
to the different operations. See Padmos et al. (1999) for a discussion on i2 Technologies’ use of
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks4 Article submitted to Management Science; manuscript no.
simulation and optimization in their supply chain solutions. Our result on the convexity of the cost-
to-go function with respect to the capacity levels provides greater credibility to the use of gradient
descent methods, which are standard techniques used in efficient simulation-based optimization
procedures.
To reiterate the relevance of the models we study, we refer the reader to Lin et al. (2000) which
describes IBM’s Edelman-award winning work to develop an enterprise supply chain analysis tool
called AMT (Asset Management Tool). This tool has been used both at IBM and other companies
to study (among other things) issues central to our paper, namely inventory investments and
budgets in complex multi-echelon supply chains. Our paper supports the development of such
analytic decision-making tools by establishing the convexity of the cost function in a large class of
inventory optimization models since convexity guarantees that computationally efficient methods
can be used to make optimal decisions in these models.
1.2. Convexity Results in Inventory Theory
There has been substantial interest among researchers in both inventory theory and queuing theory
in proving convexity properties of important performance measures for the systems of interest.
Examples of such papers in inventory theory include Karush (1957), Zipkin (1986), Zhang (1998),
Downs et al. (2001), Janakiraman and Roundy (2004), and Johansen (2005). It is important to note
that all these papers study single stage inventory systems as compared to multi-echelon systems
studied here. Examples from queuing theory include the books by Stoyan (1983) and Shaked and
Shanthikumar (1994), and the papers by Harel and Zipkin (1987), Harel (1990), Fridgeirsdottir
and Chiu (2005), and Armony et al. (2005).
For deriving our main results, we use a property of the optimal value of a convex program when
the parameters of the convex program are perturbed. Similar results have been used earlier in
some other papers in the inventory theory literature. Harrison and Van Mieghem (1999) and Van
Mieghem and Rudi (2002) apply a linear programming perturbation property to a single-period
setting. Johansen (2005) uses this technique to show the convexity of the cost function with respect
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 5
to the base-stock level in a single stage inventory system with lost sales and Erlangian lead times.
Robinson (1990) applies a convex programming perturbation property to uncapacitated inventory
systems with transshipment; however, his approach does not generalize to capacitated systems.
These four papers consider single or two-echelon models. In our paper, we consider arbitrary multi-
echelon networks in a capacitated dynamic setting. Furthermore, this paper differs from the earlier
papers (with the exception of Johansen’s) in that we formulate a convex program whose feasible
region does not exactly correspond to a given inventory policy; rather, we frame a relaxation of the
inventory policy as a feasible region of a convex program, which is optimized by the given policy.
Our relaxation technique loosely resembles a standard approach in combinatorial optimization,
where one finds an optimal solution for an integer programming problem by relaxing it to a linear
program that has an integral optimal solution.
1.3. Preliminaries
The following property of convex programs is a fundamental driver of the results in this paper.
Lemma 1. Let f and g be convex functions. Then, π(b) = min {f(x,b) | g(x,b) ≤ 0} is a
convex function of b.
The proof is straightforward and is omitted. Similar results are well known (see, for example,
Theorem 29.1 of Rockafellar (1970) or Section 5.6 of Boyd and Vandenberghe (2004)). In particular,
Lemma 1 implies the convexity of min{f(x) | g(x)≤ b} with respect to the right-hand side vector
b, for convex f and g.
2. Assembly Systems under Base-Stock Policies
In this section, we consider periodic-review assembly systems managed using echelon order-up-to
policies, and show that the backorder or lost-sales penalty cost function is convex with respect
to the base-stock levels and capacities. We describe our model and assumptions in Section 2.1.
Sections 2.2 and 2.3 contain the proofs of convexity for the backorder case and the lost sales
case, respectively. In Section 2.4, we consider the problem of minimizing the expected penalty cost
subject to budget constraints. In Section 2.5, we consider the problem of minimizing the sum of the
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks6 Article submitted to Management Science; manuscript no.
inventory holding and penalty costs. Our discussion on assembly systems under base-stock policies
is concluded in Section 2.6 with extensions and remarks.
2.1. Model
The assembly network we consider is an in-tree consisting of a set of nodes J and a set of arcs.
The network is an in-tree with exactly one sink node, which represents the finished product. This
sink node is labelled 1. For some pair of nodes j and k, there exists an arc j → k representing a
material flow from j to k. Each node j, except the sink node (node 1), has exactly one immediate
successor denoted by succ(j). We say node k is a descendant of j if there is a directed path from
j to k. We say j is a source node if j has no predecessor; otherwise, j is a non-source node.
We denote by τj the lead time for purchasing at j if j is a source node, or assembling at j if j is
a non-source node. (Our analysis allows for arc-dependent lead-times, but we use node-dependent
lead-times for simplicity of exposition.) We assume τj ≥ 1 for every non-sink node j, and τ1 = 0
for the sink node 1. The assumption of τ1 = 0 is without loss of generality by possibly introducing
a fictitious node. (When some other lead times are zero, an identical analysis can be used but it
involves more notation.) Without loss of generality, we assume one unit of j is used to make one
unit of succ(j).
With each node j, we associate two kinds of inventory variables. The first is the echelon-j
inventory level, which denotes the number of units of inventory at j plus the total amount of
inventory in its descendants, minus any possible backorders for the finished product at node 1.
The second is the echelon-j inventory position which equals the echelon-j inventory level plus the
number of outstanding orders for j.
The sequence of events within a period is the following. (1) At the beginning of a period t, the
outstanding order for echelon j placed in period t− τj arrives for each non-sink node j (i.e., j 6= 1).
This order is a purchase order if j is a source node, and an assembly order if j is a non-source
node. (2) New orders for node j are placed for each node j including node 1. These orders are
constrained from above by a capacity limit Cj for assembly or purchase, and also by the inventory
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 7
availability of subassemblies in immediate predecessors of j if j is a non-source node. (3) Since
τ1 = 0, the assemblies ordered by node 1 in this period are delivered. (4) The demand Dt for the
finished product is observed. (5) Demand is satisfied to the extent possible. Any excess demand is
either backordered (Section 2.2) or lost (Section 2.3). The starting inventory levels for period t+1
are updated.
Let xj,t denote the echelon-j inventory position before ordering (at the end of (1) in the above
sequence of events). Let yj,t (ej,t) denote the echelon-j inventory position (level) after ordering and
delivery, and before seeing demand (at the end of (3) above). Let zt represent the sales quantity,
denoting the number of finished product units committed to sale:
zt ={
Dt, in the backorder case;min{Dt, y1,t}, in the lost sales. (1)
(In the lost sales case, the above identity follows from the fact that y1,t = e1,t since τ1 = 0.) The
following identities are well known and are consequences of the definitions of echelon inventory
position and echelon inventory level:
xj,t = yj,t−1− zt−1 , and
ej,t = yj,t−τj− z[t− τj, t− 1] ,
where z[t1, t2] is the cumulative sales realized over the interval [t1, t2] if t2 ≥ t1 and zero if t2 < t1,
i.e., z[t1, t2] =∑t2
t=t1zt.
We assume throughout this section that the ordering policy under use is the echelon order-up-to
policy, which we will explain now. Let Sj be a target echelon-j inventory position, also called an
order-up-to or base-stock level. In any period t, the policy does not place an order for j if its
echelon inventory position xj,t exceeds Sj; otherwise, it orders enough to raise the echelon inventory
position to Sj if it is feasible to do so, or, orders the maximum amount permissible under the
capacity and material availability constraints if that is not feasible. In other words, the policy tries
to bring yj,t as close to Sj as possible in each period. Throughout Section 2, we impose the natural
condition that echelon base-stock levels satisfy Sj ≥ Sk for every j→ k.
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We will provide a formula that specifies the order quantity under this policy. We first state an
assumption about the starting state of the system. Since node 1 does not have any successor, we
define Ssucc(1) = 0 for simplicity of notation.
Assumption 1. At the beginning of period 1, there does not exist any backorder or any outstand-
ing purchase/assembly order anywhere in the system, and each node j ∈J has exactly Sj−Ssucc(j)
units available; i.e., ej,1 = Sj for all j.
Remark: Note that Assumption 1 is an assumption on the initial state of the inventory system,
which does not have any impact on the long-run average cost of the system. Moreover, the impact
of the initial state is limited to the first several periods, and its impact on the discounted cost
of the system is relatively minor when the planning horizon is sufficiently long and the discount
factor is close to 1. (At a cost of capital of 30% per annum and a review period of one week, the
discount factor is 0.995.)
We use ξk,t+1 to represent the material availability for xk,t+1. More precisely, let ξk,t+1 denote the
highest attainable echelon inventory position at node k after ordering in period t+1 if there is no
capacity constraint. We let ξk,t+1 =∞ for all t≥ 1 if k is a source node. Otherwise, if j → k is an
arc, then we cannot raise the echelon inventory position yk,t+1 of node k in period t+1 any higher
than ej,t+1, the echelon inventory level of node j after receiving the delivery due in that period.
Since no outstanding order arrives at node j during the first τj periods, Assumption 1 implies
ej,t+1 ={
Sj − z[1, t] , if t < τj ;yj,t−τj+1− z[t− τj +1, t] , if t≥ τj .
Thus,
ξk,t+1 ={
minj:j→k ej,t+1, if j is a non-source node∞, if j is a source node. (2)
Under Assumption 1, the echelon order-up-to policy can be described by the following recursive
formula:
yk,t+1 ={
Sk , for t = 0;min{Sk, yk,t− zt +Ck, ξk,t+1} , for t≥ 1. (3)
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Note that yk,t+1 in (3) is the minimum of three expressions, where the first expression is the echelon
order-up-to level Sk, the second expression imposes the ordering capacity constraint, and the third
expression enforces the material availability constraints.
The optimality of echelon base-stock policies has been shown for assembly systems by Rosling
(1989) when excess demand is backordered and there are no capacity constraints. In addition to
linear purchase/assembly costs and shortage costs, his model also includes linear holding costs.
In our cost model, we only consider the backorder or lost sales penalty costs incurred over a
finite horizon. The penalty cost b(·) is an increasing function in the number of units backordered or
lost at the end of each period, which is given by (Dt−y1,t)+. (Recall that y1,t is both the inventory
position and the inventory level at the sink node since τ1 = 0.) Thus, the objective function under
consideration is
T∑t=1
αtE[b((Dt− y1,t)+)] , (4)
where α∈ (0,1] is the discount factor. We assume b(·) is convex for the backorder case, and linear
for the lost sales case. In Sections 2.2 and 2.3, we show that (4) is jointly convex with respect to
the vector of base-stock and capacity levels {(Sj,Cj) | j ∈J }.
2.2. The Backorder Case
In this section, we assume that the excess demand is backordered, i.e., zt = Dt, and show that the
discounted cost (4) is jointly convex in {(Sj,Cj) | j ∈ J }. We assume that the penalty b(·) in (4)
is convex. (We also refer to b(·) as the backorder cost.) In fact, we show that this convexity result
holds for any sample path of realized demands (Dt : t = 1, . . . , T ), not just the expectation. To prove
our result, we first show that the dynamics of the assembly system under an echelon order-up-to
policy can be captured by a mathematical program (MP-B) given below. In this formulation, the
objective function is a decreasing function of the inventory positions, and both the base-stock and
capacity vectors appear as right hand sides. Then, we appeal to Lemma 1 of Section 1.
Let b(y) be any decreasing function of the echelon inventory positions after ordering, where
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks10 Article submitted to Management Science; manuscript no.
y = (yj,t : j ∈ J , t = 1, . . . , T ). Fix any realization of demands (Dt : t = 1, . . . , T ), and consider the
following mathematical programming formulation:
(MP-B) miny
b(y)
s. t. yk,t ≤ Sk ∀ (k, t)
yk,t− yk,t−1 +Dt−1 ≤ Ck ∀ (k, t) s.t. t≥ 2
yk,t− yj,t−τj+D[t− τj, t− 1] ≤ 0 ∀ (j, k, t) s.t. j→ k and t≥ τj +1
yk,t +D[1, t− 1] ≤ Sj ∀ (j, k, t) s.t. j→ k and t≤ τj ,
where D[t1, t2] is the cumulative demand over the interval [t1, t2], i.e., D[t1, t2] =∑t2
t=t1Dt. The
decision variables in (MP-B) are (yk,t | k ∈J , t = 1, . . . , T ). We remark that the echelon order-up-
to policy described by (3) is a feasible solution to (MP-B). The following lemma shows that this
policy is, in fact, optimal in (MP-B).
Lemma 2. Suppose Assumption 1 holds. For a decreasing function b(·), an optimal solution to
(MP-B) is given by the recursive formula (3).
Proof: Let y = (yk,t | k ∈ J , t = 1, . . . , T ) be defined by the recursive formula (3). It is straight-
forward to show that y is a feasible solution to (MP-B). Now, let y′ = (y′k,t : k ∈ J , t = 1, . . . , T )
be any feasible solution to the above mathematical program (MP-B). We want to prove that the
objective values satisfy b(y)≤ b(y′). Since b(·) is a decreasing function, it suffices to show y≥ y′.
We will show that yk,t ≥ y′k,t holds for all k and t by induction on t. For the base case of t = 1,
this statement is trivially true for all k from y′k,1 ≤ yk,1 = Sk by (3). Let us now assume that the
statement is true for any small t∈ {1, . . . , T − 1} and prove the statement for t+1 also.
By the induction hypothesis, yk,t ≥ y′k,t holds for each k. Recall yk,1 = Sk holds for each k. Let
ξ′k,t+1 be defined as in (2) as a function of y′j,t−τj+1’s instead of yj,t−τj+1’s. It follows ξk,t+1 ≥ ξ′k,t+1.
From the recursive formula (3), we have
yk,t+1 = min{Sk, yk,t−Dt +Ck, ξk,t+1}
≥ min{Sk, y′k,t−Dt +Ck, ξ′k,t+1} . (5)
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 11
Note that y′ is a feasible solution of (MP-B), the constraints of which imply
y′k,t+1 ≤ Sk ∀ k
y′k,t+1 ≤ y′k,t−Dt +Ck ∀ k
y′k,t+1 ≤{
y′j,t−τj+1−D[t− τj +1, t] ∀ k such that j→ k if t≥ τj +1;Sj −D[1, t] ∀ k such that j→ k if t≤ τj.
It follows that the right-most expression of (5) is an upper bound on y′k,t+1. Thus, we conclude
yk,t+1 ≥ y′k,t+1 for each k ∈J , completing the induction step. 2
The main convexity result of this section is stated in the following theorem.
Theorem 1. Fix any realization of demands (Dt : t = 1, . . . , T ). Let b(·) be a convex increasing
function, and let α ∈ (0,1] be a discount factor. Under Assumption 1, the discounted backorder
cost function∑T
t=1 αt · b((Dt− y1,t)+) is jointly convex with respect to the vector of base-stock and
capacity levels {(Sj,Cj) | j ∈J }, where y solves (3).
Proof: Let
b(y) =T∑
t=1
αt · b((Dt− y1,t)+) .
Since b(·) is an increasing function, it is straightforward to show that b(·) is a decreasing function
of y. By Lemma 2, the echelon order-up-to policy gives an optimal solution to the mathematical
program (MP-B). Moreover, since b(·) is convex and increasing and (Dt− y1,t)+ is convex in y, we
obtain that b(·) is convex in y. Thus, (MP-B) has a convex objective function, and its constraints
are linear. Since the vector {(Sj,Cj) | j ∈J } appears only on the right side of the “≤” constraints,
the optimal value of (MP-B) is jointly convex in this vector by Lemma 1. This implies the result.
2
2.3. The Lost Sales Case
We now assume that the excess demand for the finished product is lost, i.e., zt = min{Dt, y1,t}.
The objective function (4) now represents the lost sales cost over a finite horizon. Here, we assume
that b(·) in (4) is linear, i.e., b(u) = b · u. We show the convexity of this function with respect to
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks12 Article submitted to Management Science; manuscript no.
the base-stock and capacity vectors. As in Section 2.2, we show this result for any sample path of
realized demands (Dt : t = 1, . . . , T ). Since the analysis is similar to that in the backorder case, our
discussion here will highlight the differences.
Under Assumption 1, the echelon-order-up-to policy can be characterized by equations (3) and
zt = min{Dt, y1,t}. Let l(u1, . . . , uT ) be a function of lost sales quantities such that it is piecewise
linear with decreasing slopes, i.e., l(u1, . . . , uT ) =∑T
t=1 λtut for some λt’s such that λ1 ≥ · · · ≥
λT ≥ 0. For example, we let λt = αt · b where b is a per-unit lost sales cost, and α ∈ (0,1] is a
discount factor. Fix any realization of demands (Dt : t = 1, . . . , T ), and consider a mathematical
programming formulation similar to (MP-B). In the new formulation, called (MP-L), we relax the
meaning of zt, and allow the sales quantity zt in each period t to be a decision variable. As a result,
both (yk,t | k ∈J , t = 1, . . . , T ) and (zt : t = 1, . . . , T ) are the decision variables in (MP-L).
(MP-L) miny,z
l(D1− z1, . . . ,DT − zT )
s. t. yk,t ≤ Sk ∀ (k, t)
yk,t− yk,t−1 + zt−1 ≤ Ck ∀ (k, t) s.t. t≥ 2
yk,t− yj,t−τj+ z[t− τj, t− 1] ≤ 0 ∀ (j, k, t) s.t. j→ k and t≥ τj +1
yk,t + z[1, t− 1] ≤ Sj ∀ (j, k, t) s.t. j→ k and t≤ τj
zt−Dt ≤ 0 ∀ t
zt− y1,t ≤ 0 ∀ t .
We remark that the first four sets of constraints of (MP-L) are similar to the constraints of (MP-B).
The last two remaining constraints correspond to zt = min{Dt, y1,t}, except that the equality is
replaced with two “≤” inequalities. In other words, the sales quantity zt is bounded above by the
available inventory at node 1 and the realized demand.
The feasible region of (MP-L) is a relaxation of the echelon order-up-to policy in terms of both the
order and sales quantities. However, this policy produces an optimal solution for (MP-L) as shown
in Lemma 3. There is no motivation either to order less than the maximum allowable quantity, or
to refuse a customer when a unit is available in inventory.
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 13
Lemma 3. Suppose Assumption 1 holds, and the objective function of (MP-L) is given by
l(u1, . . . , uT ) =T∑
t=1
λt ·ut
where λ1 ≥ · · · ≥ λT ≥ 0. Then, the vectors y = (yk,t | k ∈J , t = 1, . . . , T ) and z = (zt : t = 1, . . . , T )
given by the recursive formula (3) and zt = min{Dt, y1,t} form an optimal solution to (MP-L).
Proof: Clearly, y and z defined by (3) and zt = min{Dt, y1,t} form a feasible solution to (MP-L).
We show their optimality. Let y′ = (y′k,t : k ∈ J , t = 1, . . . , T ) and z′ = (z
′t : t = 1, . . . , T ) be any
feasible solution to (MP-L). We want to show l(D1− z1, . . . ,DT − zT )≤ l(D1− z′1, . . . ,DT − z′T ).
Since the target inventory levels Sj’s are fixed, the following results can be shown by induction
on t: (i) the cumulative sales quantity up to period t in the (y,z) system is at least as high as in
the (y′,z′) system, i.e., z[1, t]≥ z′[1, t]; (ii) the cumulative supply at each node k ∈J in the (y,z)
system is at least as high as in the (y′,z′) system, i.e., yk,t + z[1, t]≥ y′k,t + z′[1, t]. Since demand
is the sum of sales and lost sales and demand is common in both systems, it implies that the
cumulative lost sales in the (y,z) system is at most the cumulative lost sales in the (y′,z′) system,
i.e., for each t,
t∑s=1
(Ds− zs) ≤t∑
s=1
(Ds− z′s) .
Let λT+1 = 0. It follows from λ1 ≥ · · · ≥ λT ≥ 0 that
T∑t=1
λt · (Dt− zt) =T∑
t=1
(λt−λt+1)t∑
s=1
(Ds− zs)
≤T∑
t=1
(λt−λt+1)t∑
s=1
(Ds− z′s)
=T∑
t=1
λt · (Dt− z′t) .
Thus, we obtain the optimality of y and z. 2
The main result for the lost sales case is stated below.
Theorem 2. Fix any realization of demands (Dt : t = 1, . . . , T ). Let b≥ 0 be a per-unit lost sales
cost, and α ∈ (0,1] be the discount factor. Under Assumption 1, the discounted lost sales penalty
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks14 Article submitted to Management Science; manuscript no.
cost function∑T
t=1 αt · b · (Dt − zt)+ is jointly convex with respect to the vector of base-stock and
capacity levels {(Sj,Cj) | j ∈ J }, where zt = min{Dt, y1,t}, and y = (yk,t | k ∈ J , t = 1, . . . , T ) is
given by the recursive formula (3).
Proof: In the lost sales case, since the sales quantity in a period does not exceed demand, the
discounted penalty cost function can be written without the positive part operator. From b(u) = bu,
it follows
T∑t=1
αtb((Dt− zt)+) =T∑
t=1
αtb(Dt− zt) =T∑
t=1
(αtb) · (Dt− zt) .
Thus, by Lemma 3 and an argument similar to the proof of Theorem 1, we obtain the required
result. 2
2.4. Solving Budget Allocation Problems
In Sections 2.2 and 2.3, we have established that the expected penalty function is jointly convex
with respect to the vector of base-stock and capacity levels {(Sj,Cj) | j ∈ J }. In this section,
we consider the problem of allocating a given budget among the inventory investments in the
components and the capacity levels. We call this the budget allocation problem. We discuss the
backorder case first.
In this problem, the objective is to select S and C in order to minimize the expected value of
∑T
t=1 αt · b((Dt−y1,t)+). By Lemma 2, this objective is equivalent to minimizing ED[F (S,C | D)],
where F (S,C | D) is the optimal value of (MP-B) with b(y) =∑T
t=1 αt · b((Dt − y1,t)+). The
feasible region is
{ (S,C) | Sj ≥ Sk for every j→ k , and∑j∈J
βj ·Sj +∑j∈J
θj ·Cj ≤Λ } ,
where Λ is a given budget, βj ≥ 0 is the per-unit cost associated with the echelon-j base-stock
level, and θj ≥ 0 is the cost of capacity per unit at node j. We remark that the inventory cost is
associated with the order-up-to level as opposed to the average amount of inventory on-hand. If
βj is the value added at node j,∑
j∈J βj ·Sj represents the total amount of money committed to
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 15
system inventory.2 One way to interpret∑
j∈J βj ·Sj is that it represents the maximum amount of
capital tied to inventory at any given time. The alternate model in which the budget constraint
involves the average amounts of inventory on-hand, instead, is computationally intractable because
the feasible region above may not be a convex set.
Most of the analysis in this section holds when the inventory and capacity investment costs are
convex increasing functions rather than linear functions. With minor modifications, the budget
allocation problem can model situations in which certain capacity investments have already been
made, and the budget constraint is applicable for additional capacity investments. A special case of
the budget allocation problem is the case where all the capacities are fixed, and the budget applies
only to the inventory investments.
For the remainder of this subsection, we assume that the penalty cost function in (MP-B) is
linear with respect to the quantity of inventory shortage, i.e.,
b(y) =T∑
t=1
αt · b · (Dt− y1,t)+ .
Then, using a standard transformation, (MP-B) is equivalent to a linear program. We remark that
this linear program is a dual of a network transshipment problem.
We outline two computational approaches for solving the budget allocation problem. The first
approach is Sample Average Approximation (SAA). In SAA, the objective function is approximated
using a finite number of sample paths, and we solve the approximate problem optimally. (See
Shapiro (2003) for details.) Let D1,D2, . . . ,DN be a collection of sample paths, where N is the
number of sample paths. Then, SAA solves
minS
N∑n=1
wt(n) ·F (S,C | Dn) (6)
where wt(n) is an appropriate weight associated with sample path Dn, where wt(1) + wt(2) +
· · ·+ wt(N) = 1. Our result on the convexity of F ensures that (6) can be solved efficiently. In
particular, we can either use multiparametric linear programming methods (see Gal and Nedoma
2 We thank John Muckstadt for a discussion about this formulation.
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks16 Article submitted to Management Science; manuscript no.
(1972) and Borrelli et al. (2003)), or exploit the polyhedral convex structure of the cost function
(see Ruszczynski (1986) and Osborne (2001)).
The second computational approach is based on Infinitesimal Perturbation Analysis (IPA). If
ED[F (S,C |D)] is a differentiable function, under certain assumptions, this theory guarantees that
the expected value of the gradient of F (S,C | D) with respect to (S,C) is an unbiased estimate
of the gradient of ED[F (S,C | D)]. Furthermore, the estimate of this gradient from a sample path
is the vector of partial derivatives of F (S,C | D), which can typically be computed efficiently in
inventory systems (see, for example, Glasserman and Tayur (1995).) For more on the theory of
IPA, see Glasserman (1991).
In our case, however, for a given sample path D, F (S,C | D) is a piece-wise linear function of
(S,C), and thus not differentiable; therefore, ED[F (S,C | D)] is not differentiable without addi-
tional assumptions on D such as having a probability density function. In general, for non-smooth
functions, IPA gives only directional derivatives rather than a subgradient. (See, for example, Sec-
tion 3 of Robinson (1995).) We address the following two issues: (i) computing a subgradient of
F (S,C | D) for each sample D, and (ii) proving that the expected value of a sample subgradient
is a subgradient of ED[F (S,C | D)].
For (i), we consider the dual LP of (MP-B), which we call (MP-D). Let φ(MP-D)(u | S,C,D) denote
the dual objective function, where u is a vector of dual variables. This function is linear in (S,C).
Let u∗ be the optimal dual variables. Since (MP-B) is a linear program,5(S,C) φ(MP-D)(u∗ | S,C,D)
is a subgradient of F (S,C | D) at (S,C). (See Section 5.6 of Boyd and Vandenberghe (2004) and
Proposition 3.2 of Bemporad and Filippi (2005) for details.) We remark that dual variables u∗ can
be easily computed from the complementary slackness condition since we know the primal optimal
solution.
Now, consider (ii). For any sample path D, let ν(S,C | D) be any subgradient of F (S,C | D)
at S. For example, ν(S,C | D) =5(S,C) φ(MP-D)(u∗ | S,C,D). By definition of a subgradient,
F (S, C | D)−F (S,C | D) ≥ ν(S,C | D) · ((S, C)− (S,C)) for every (S, C).
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 17
Taking expectations on both sides with respect to D,
ED[F (S, C | D)]−ED[F (S,C | D)] ≥ ED[ν(S,C | D)] · ((S, C)− (S,C)) for every (S, C).
Therefore, ED[ν(S,C | D)] is a subgradient of E[F (S,C | D)] at (S,C). Now, having established
both (i) and (ii), standard subgradient methods can be used with a sample average of ν(S,C | D)
to find the optimal inventory and capacity investment decisions.
The analysis of this subsection has been carried out for the backorder case by considering (MP-
B). However, all the results of this section also hold for (MP-L) in the lost sales case. The only
difference is that the dual of (MP-L) is no longer a network transshipment problem. The two
computational approaches are also applicable for the lost sales case.
2.5. Sum of Holding and Penalty Costs as the Objective Function
The objective function in Section 2.2 or 2.3 (and also in Section 2.4) only includes the penalty cost,
and our analysis there is not applicable if the holding cost is included in the objective function.
Our proofs are based on the property that order-up-to policies are optimal for the mathematical
programming relaxations (MP-B) and (MP-L) for any sample path of demand. However, if holding
costs are included in the objective function, the optimal solutions for these relaxations may not be
order-up-to policies. Therefore, the proofs of Theorems 1 and 2 fail. In fact, it is easy to construct
simple examples where the sum of the expected holding and penalty costs is not convex with
respect to the order-up-to levels.
We note that the problem of minimizing the sum of holding and penalty costs within the class
of echelon order-up-to policies has been studied by Glasserman and Tayur (1995). They apply IPA
in the gradient-descent framework in order to find a vector of base-stock levels that are locally
optimal. Their analysis is restricted to the backorder case only, and no attempt was made to study
the convexity or quasi-convexity of the objective function.
In Sections 2.5.2 and 2.5.3, we show that the cost function (sum of holding and penalty costs)
can be expressed as the sum of a convex function and a concave function. This is useful since
specialized algorithms have been developed for the minimization of such functions (see Section
2.5.4 for details).
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks18 Article submitted to Management Science; manuscript no.
2.5.1. Holding Cost Formulation Let Hj be the installation holding cost at node j. Define
hj = Hj −∑
k∈J : k→j
Hk ,
i.e., hj is the echelon-j holding cost per unit per period. We assume hj ≥ 0. This cost is charged
at the end of a period, say period t, on the total number of units of physical inventory in echelon
j. This quantity is the sum of (i) the physical inventory in echelon-j except node 1, and (ii) the
physical inventory at node 1 after sales, i.e.,
(ej,t− y1,t) + [y1,t− zt]+ = (ej,t− zt) + [zt− y1,t]+ (7)
where ej,t is the echelon-j inventory level at the beginning of period t after receiving the delivery
due in that period; y1,t equals e1,t since the lead time at node 1 is zero; and zt is the sales quantity
in period t defined in (1).
Throughout Section 2.5, F (S1, S2 | D) refers to the discounted sum of holding and shortage costs
along a sample path D.
2.5.2. The Backorder Case In the backorder case, the physical echelon-j inventory given in
(7) can be written as
ej,t−Dt + [Dt− y1,t]+ .
Thus, it is easy to see that the expected holding and shortage costs satisfy
F (S,C | D) =T∑
t=1
∑j
αt ·hj · (ej,t−Dt) +T∑
t=1
∑j
αt ·hj · [Dt− y1,t]+
+T∑
t=1
αt · b((Dt− y1,t)+) .
Theorem 1 establishes the convexity of the second and third terms with respect to S. We will now
show that the first term is concave.
Theorem 3. Assume excess demand is backordered. Fix any realization of demands (Dt : t =
1, . . . , T ). Under Assumption 1, the function∑T
t=1
∑j αt · hj · (ej,t −Dt) is jointly concave with
respect to the vector of base-stock and capacity levels {(Sj,Cj) | j ∈J }.
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The proof is similar to that of Theorem 1 and is given in the appendix. Thus, we conclude that
E[F (S1, S2 | D)] is the sum of a convex and a concave function.
2.5.3. The Lost Sales Case In the lost sales case, since zt = min{Dt, y1,t}, (7) can also be
written as
ej,t− zt = (ej,t + z[1, t− 1]−Dt)− z[1, t] +Dt
= (ej,t + z[1, t− 1]−Dt)+t∑
s=1
(Ds− y1,s)+−D[1, t− 1] .
Thus, it follows that the expected holding and shortage costs satsify
F (S,C | D) =T∑
t=1
∑j
αt ·hj · (ej,t− zt) +T∑
t=1
αt · b · (Dt− y1,t)+
=T∑
t=1
∑j
αt ·hj · (ej,t + z[1, t− 1]−Dt)
+T∑
t=1
(αt · b+(αt +αt+1 + · · ·+αT ) ·∑
j
hj) · (Dt− y1,t)+
−T∑
t=1
∑j
αt ·hj ·D[1, t− 1] .
From Theorem 2, the second term above is convex. The third term is a constant. The following
result shows that the first term is concave. The proof of Theorem 4 is similar to the proofs of
Theorems 2 and 3, and is omitted.
Theorem 4. Assume excess demand is lost. Fix any realization of demands (Dt : t = 1, . . . , T ).
Under Assumption 1, the function∑T
t=1
∑j αt · hj · (ej,t + z[1, t− 1]−Dt) is jointly concave with
respect to the vector of base-stock and capacity levels {(Sj,Cj) | j ∈J }.
Thus, F (S,C | D) is the sum of a convex function and a concave function.
2.5.4. DC Minimization Consider the problem of minimizing the expected holding and
penalty costs in an assembly system within the class of base-stock policies by determining the base-
stock levels. Sections 2.5.2 and 2.5.3 imply that the sum of the expected holding and penalty costs,
ED[F (S,C | D)], can be expressed as the sum of a concave function and a convex function. This is
equivalent to minimizing the difference of two convex functions (DC functions). Such minimization
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks20 Article submitted to Management Science; manuscript no.
problems, also known as DC programs, have been extensively studied in the global optimization
literature, e.g., Chapter X of Horst and Tuy (1993) and Chapter 4 of Horst et al. (2000). In par-
ticular, when the feasible region is a polytope, a special mechanism, called a prismatic branch and
bound algorithm, is well suited for such problems. This algorithm is applicable to our problem
since the feasible region of order-up-to vectors is a polyhedron {S | Sj ≥ Sk for every j→ k}.
2.6. Extensions and Remarks
2.6.1. Infinite Planning Horizon and Continuous Time Our main results in this section
are stated for the finite horizon discounted cost model under periodic review. However, Theorems
1 and 2 can be easily extended to the infinite horizon discounted cost case. In case of the infinite
horizon average cost, these convexity results hold even without the assumption on the initial state
(Assumption 1). The proofs are similar to those used in Section 5 of Janakiraman and Roundy
(2004).
The discrete-time demand model in this section can be replaced by a continuous-time model
where demands arise individually or in batches, with arbitrary inter-arrival distribution. All the
results continue to hold under the assumption that orders are placed only at arrival epochs. We
briefly sketch the proof here. Under a given order-up-to policy, fix a realization of demand arrivals
and batch quantities, and consider the corresponding mathematical programming relaxation, where
t is now the index for the arrival epochs. The constraints of this formulation should be adjusted,
properly accounting for lead times. The discount factor and the backorder cost used in the objective
function should be appropriately defined based on the interarrival times on the given sample path
of arrivals.
2.6.2. Cyclic Demand We consider the case where demand is cyclic, and the base-stock levels
depend on the seasonality within a cycle. Kapuscinski and Tayur (1998) analyze a capacitated
single-echelon inventory system with cyclic demand and backorders. They consider both holding
and backorder costs, and Property 8 in their paper states that under the order-up-to policy the
infinite horizon average cost is convex with respect to the seasonality-dependent base-stock levels.
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This statement, if true, should imply that the the expected average backorder cost is also convex
by setting the holding cost parameter to zero. In the appendix, we provide an example to show
that this statement is untrue, and show that both the backorder cost and the total cost may fail
to be convex with respect to base-stock levels. Interestingly, we are able to show that the holding
cost, however, is convex with respect to the vector of seasonality-dependent base-stock levels for
this single-echelon model. (Please see the appendix for a formal statement of this result and the
proof.)
2.6.3. Non-Stationarity of Demand In Sections 2.2 and 2.3, we do not make any station-
arity assumption on demand, which can be cyclic (as in Section 2.6.2), correlated or non-identical.
Our proofs of convexity hold for every sample path of demand. The analysis of these sections,
however, assumes a stationary base-stock vector and a stationary capacity vector. When demand
is non-stationary, it may prove useful to allow period-dependent base-stock and capacity levels.
In this case, the convexity results in Theorems 1 and 2 hold with respect to {(Sj,t,Cj,t) | j ∈
J , t = 1, . . . , T} provided that base-stock levels Sj,t are increasing in t for each j, that is, within
{(S,C) | Sj,t ≤ Sj,t+1 for every j ∈J and t = 1,2, . . . , T − 1}.
2.6.4. Convexity of the Penalty Function In Section 2.3 on the lost sales model, we assume
that the penalty cost function b(·) must be linear, and its slopes must be decreasing in t. This is
in contrast with Section 2.2 on the backorder model, where b(·) is any period-dependent convex
increasing function. In the lost sales case, when the penalty cost function is non-linear, we show,
by an example in the appendix, that the expected penalty cost may fail to be convex.
2.6.5. Cost Dominance under Convex Ordering The convexity results of Sections 2.2
and 2.3 can be used to show that the expected shortage cost is increasing with respect to demand
variability. We say D is less than D in convex order if E[φ(D)]≤E[φ(D)] for any convex function
φ :<→<. This is usually denoted by D≤cx D in the stochastic comparison literature (see Shaked
and Shanthikumar (1994)).
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks22 Article submitted to Management Science; manuscript no.
Theorem 5. Let {Dt} be identically and independently distributed with Dt ∼D. Similarly, let
{Dt} be identically and independently distributed with Dt ∼ D. Assume that {Dt} is independent
of {Dt}. Also, assume D ≤cx D. Suppose Assumption 1 holds. Consider the penalty cost function
used in Theorems 1 and 2:∑T
t=1 αt · b((Dt−y1,t)+) in the backorder case and∑T
t=1 αt · b · (Dt−zt)+
in the lost sales case. Then,
ED[F (S,C | D)] ≤ ED[F (S,C | D)]
holds where F (S,C | D) is the optimal value of (MP-B) in the backorder case or (MP-L) in the
lost sales case.
Using the mathematical programs (MP-B) of Section 2.2 and (MP-L) of Section 2.3, we can see
that this theorem is a direct consequence of Lemma 1 and the convex ordering assumption. The
proof is in the appendix.
3. Convexity in Dynamic Programs
In this section, we study a class of dynamic programs representing a large number of inventory
problems without fixed costs. In particular, we examine the convexity properties of the cost-to-go
function with respect to the inventory state and capacities. In this section, we assume that the
optimal policy is used to manage the inventory system; this policy need not be an echelon order-up-
to policy. We consider arbitrary supply networks in which multiple products can be assembled and
distributed to multiple outlets; moreover, multiple supply sources with different cost and lead time
attributes for procuring the same component may be available. Multiple products processed on
a common capacitated resource are also allowed. Multiple customer classes with different penalty
cost functions are also allowed. We allow the external demand to occur at every node. We do not
describe in detail how each of these features is modeled, in the interest of space; instead, we provide
an abstract, general framework that encompasses these features.
In Section 3.1, we describe our general model and the dynamic programming formulation. The
convexity result for the backorder case is contained in Section 3.2. The lost sales case is discussed in
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Section 3.3. In Section 3.4, we establish the relationship between variability in demand distributions
and the optimal costs.
3.1. Model and the Main Convexity Result
We model the supply network as an acyclic graph consisting of a set of nodes J and a set of directed
arcs. A node represents a combination of a product and a location, and an arc represents a trans-
formation of inventory such as assembly, procurement and distribution. The inventory physically
available at each node of the supply network in period t is denoted by vector xt = (xj,t | j ∈ J )
in this section. (In Section 2, we used x to denote an echelon inventory vector.) We denote by qt
the vector representing assembly, procurement and distribution quantities in period t. We call qt
the action vector. The choice of these quantities represented by qt in each period is restricted by
the capacity and materials availability constraints. Each component of qt corresponds to an arc
of the network. We allow the possibility of external demands at all nodes. Let Dt represent the
random vector of demands at all nodes in the network in period t. Let Ct be a deterministic vector
of capacities. We assume that the demand vectors are independent through time, but they need
not be identical. (For simplicity of notation, we drop the subscript t from Dt and Ct for most of
this section.)
We use an auxiliary vector z = z(x,q,D) called the sales vector. The dimension of z is the same
as that of D, and each component of z corresponds to the number of units committed to sale at
each node. The value of z depends on x and q as well as the realization of demand D. The exact
determination of z also depends on whether excess demand is backordered or lost: z(x,q,D) = D
in the backorder case, and z is a part of the decision vector in the lost sales case. For the lost
sales case, we explicitly assume, unlike Section 2, that the sales quantities are decision variables.
(Theorem 8 and the ensuing discussion consider the case where customer demand must be satisfied
to the maximum extent possible.) We denote the single-period cost function by G(x,q,z,D). Let
γ(x,q) = ED[ G(x, q, z(x,q,D), D) ] (8)
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks24 Article submitted to Management Science; manuscript no.
be the expected single-period cost. (Again, γ and G are allowed to depend on the period index;
however, their subscripts are suppressed for simplicity.)
We are now ready to present the dynamic program (DP). Let υt(x,q,C) denote the expected
discounted cost incurred in [t, T ] assuming that (1) the state in period t is x, (2) the action in
period t is q, and (3) an optimal policy is followed from period t+1 onwards. Let ft(x,C) denote
the value of υt(x,q,C) when the optimal choice of q is made in period t. For α∈ [0,1], let
υt(x,q,C) = γ(x,q) + αED[ ft+1(A · (x,q,−z(x,q,D)), C) ] , (9)
where A is a non-negative3 matrix that transforms the current inventory vector, action vector and
sales into the inventory vector of the next period. Then, the DP recursion is given by
(DP) ft(x, C) = minq
υt(x,q,C)
s. t. B · (x,q,C) ≤ 0
q ≥ 0 .
We let fT+1(x,C) = 0, where T is the planning horizon. Matrix B represents the material and
capacity constraints.
The above DP recursion could model various inventory control models by defining A and B
appropriately. (See the introductory paragraph of Section 3.)
3.2. The Backorder Case
In this section, we consider the backorder case where z(x,q,D) = D, and show the convexity of ft.
We make the following assumption:
Assumption 2. The single-period cost function G(x,q,D,D) is jointly convex in x, q, and D.
This assumption is satisfied in most inventory models, the most notable exceptions being those
with fixed setup costs. The following theorem shows the convexity of the cost-to-go functions
with respect to the inventory state and capacities. Typically, in the literature, a specific inventory
3 By a “non-negative” matrix, we mean a matrix all elements of which are non-negative.
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 25
model is studied and the convexity property with respect to the inventory state is established; our
result generalizes such results to include capacities and arbitrary inventory networks. The proof of
Theorem 6 follows standard arguments and Lemma 1.
Theorem 6. In the backorder model, under Assumption 2, ft in (DP) is convex in (x,C) for
each t = 1, . . . , T .
Proof: We prove this result by induction. The result is trivially true for t = T + 1. We now prove
the result for t by assuming the result for t+1.
We first claim υt is convex with respect to (x,q,C). Recall, from (8) and (9),
υt(x,q,C) = γ(x,q) + αED[ ft+1(A · (x,q,−D), C) ] ,
where γ(x,q) = ED[ G(x, q, D, D) ]. The convexity of υt follows from the convexity of G and
ft+1, and the fact that convexity is preserved under linear transformations and expectations.
Now, the convexity of ft follows from the convexity of υt and Lemma 1. 2
We make a few comments on Theorem 6. (1) This theorem formalizes the notion of decreasing
marginal value of capacity in a model where decisions are made optimally. (2) This theorem differs
from Theorem 1 of Section 2.2 in the following sense. Theorem 1 establishes the convexity of
the backorder cost with respect to capacities and order-up-to levels when base-stock policies are
used; Theorem 6 establishes the convexity of a general cost function with respect to capacities and
inventory levels when an optimal policy is used. (3) In many inventory models, the convexity of
the cost-to-go function ft with respect to the state vector x is often the first step in the analysis of
optimal policies. Theorem 6 provides a simple and unified proof of the convexity of this function
in a wide range of inventory models. (4) Note that Theorem 6 and its proof also hold for the
case where the demand vectors are correlated across time. In that case, the cost function will also
depend on historical demand information.
3.3. The Lost Sales Case
We consider the lost sales case, and again show the convexity of ft. In the inventory network of
Section 3, external demands may occur at all nodes. The sales vector z(x,q,D) in each period
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is decided after demand in the period is realized. In particular, in period t, let z(x,q,D) be the
minimizer of the cost-to-go function given the state x, action q, and the realized demand D:
(SALES) minz
G(x,q,z,D) + α · ft+1(A · (x,q,−z), C)
s. t. z ≤ x
z ≤ D
z ≥ 0 .
Notice that this formulation allows deliberate withholding of inventory due to speculative reasons.
In general, inventory can be withheld from customers at a certain node when there are other
nodes that can be served by this node in subsequent periods. This can occur even when the cost
parameters and demand distributions are stationary over time.
Assumption 3. The single-period cost function G(x,q,z,D) is jointly convex in (x,q,z,D).
Theorem 7. In the lost sales model, under Assumption 3, ft in (DP) is convex in (x,C) for
each t = 1, . . . , T .
Proof: From (8), (9) and (SALES), notice that υt(x,q,C) is the expectation of the optimal value
of (SALES) where the expectation is taken over D. Lemma 1 implies the convexity of υt. The proof
of this theorem is now identical to the proof of Theorem 6. 2
We now make an important observation about (SALES) under an assumption on the cost function
G.
Assumption 4. The single-period cost function G(x,q,z,D) satisfies the following:
(a) G(x,q,z,D) is given by
G(x, q, z, D) = T(q) + h · (x− z)+ + b · (D− z)+ ,
where T is a convex increasing function representing ordering, assembly and distribution costs,
and h and b are non-negative vectors that represent the holding cost parameters and the lost sales
penalty cost parameters, respectively.
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 27
(b) T, h and b are stationary across time.
The vectors (x− z)+ and (D− z)+ represent ending inventory levels and excess demands. It
is easy to see that Assumption 4 (a) implies Assumption 3. The following theorem shows that
inventory is never withheld from customers at sink nodes.
Theorem 8. Under Assumption 4, there exists an optimal solution zt to (SALES) in period t
such that zj,t = min{xj,t,Dj,t} for each sink node j, i.e., a node for which there does not exist any
k such that j→ k is an arc in the inventory network.
Proof: We take a sample-path approach by fixing a sequence of realized demands (Ds|s = t, . . . , T ).
Let z∗t be the optimal choice of zt in (SALES), and suppose that there exists a sink node j such
that z∗j,t 6= min(xj,t,Dj,t) occurs.
We compare two systems described below. Both xt and qt are input parameters. For the first
system, we implement the sales decision given by z∗t in period t, and then follow the optimal
decision corresponding to ft+1(A · (xt,qt,−z∗t ), C) from period t+1 to periods T .
We construct the second system. Observe that both z∗j < xj,t and z∗j < Dj,t hold. For the ordering
decisions, let the second system order the same quantity as the first system at each node in each
period. For the sales decision in period s = t, t + 1, . . . , T , set the sales quantity at node j to
zj,s = min{Dj,s, xj,s}, where xj,s denotes the amount of inventory available at node j in period s in
the second system. The sales decisions at all nodes other than j are the same as the first system.
The following statements comparing the two systems are easy to verify for every period. (1) For
every arc in the network, the flows (quantities transferred) in both systems are identical. (2) The
sales at every node other than j in both systems are identical. (3) The cumulative sales quantity at
node j in the second system is no smaller than that in the first system. These statements together
imply that the ordering, holding and lost sales costs at all nodes other than j are identical in both
systems. As for node j, the ordering costs are identical whereas the holding and lost sales costs in
the second system are at most the corresponding costs in the first system.
We repeat the above construction to establish the required result. 2
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks28 Article submitted to Management Science; manuscript no.
The following theorem is an important result directly implied by Theorem 7 and Theorem 8.
Theorem 9. Let demands arise only at sink nodes, i.e., Dj,t = 0 for all t and all nodes j such
that j has at least one successor. Let z(x,q,D) = min(x,D). Under Assumption 4, ft(x,C) is
convex for all x, C, and t.
In most lost sales inventory models (for example, Morton (1969) and Moses and Seshadri (2000)),
customer demands only occur at sink nodes. In such models, it is a standard practice for researchers
to explicitly assume that the customer demand should be satisfied to the maximum extent possi-
ble, i.e., z(x,q,D) = min(x,D). With this equality constraint, it is difficult to directly prove the
convexity of υt since z(·) is not a linear transformation. Our approach above circumvents this
difficulty, in the case where demands occur only at sink nodes, by treating the sales quantity as
a decision variable, and then showing that the optimal sales decision satisfies the condition that
z(x,q,D) = min(x,D). Thus, the convexity result of Theorem 7 holds even in models in which sink
nodes are explicitly constrained to meet as much customer demand as possible.
To our knowledge, the only studies of the dynamic program in a lost sales inventory model (with
positive lead times) are due to Morton (1969) and Janakiraman (2002). The former showed the
convexity of the cost-to-go function with respect to the state vector for a single stage, lost sales
inventory model with arbitrary lead times while the latter shows the result for a two-stage serial
system with unit lead times. Their proofs are quite involved because of the non-linear transforma-
tion mentioned above. Morton’s convexity result was re-derived in a simpler way and extended to
other single stage models with lost sales by Zipkin (2006) by introducing a decision variable for the
number of units sold in a period. Our analysis in this section is a generalization of Zipkin’s result
for single stage systems to arbitrary supply networks.
The four comments at the end of Section 3.2 about the convexity of the cost-to-go function are
also applicable here.
3.4. Cost Dominance under Convex Ordering
In this section, we establish that the cost-to-go function, evaluated at any state, is increasing with
respect to demand variability. (A similar result for single stage inventory systems with lost sales
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 29
has been shown by Zipkin (2006).)
We now make the dependence of the demand distribution and the single period cost functions
on the period index explicit. Let Dt and Dt be the demand vectors in period t in two systems.
Assume Dt ≤cx Dt holds for every t, that is, EDt [φ(Dt)] ≤ EDt[φ(Dt)] for every convex function
φ(·). We use tilde to denote costs associated with the system with demand distributions {Dt}.
Theorem 10. Assume demands are independent across time. Suppose Dt ≤cx Dt for all t. Under
Assumption 2 for the backorder case and under Assumption 3 for the lost sales case, ft(x,C) ≤
ft(x,C) holds for any t, x, and C.
The proof, which can be found in the appendix, is inductive and the key ideas required for it are
those that we have used for earlier proofs.
4. Conclusions
In this paper, we have established two convexity results in stochastic inventory networks. In an
assembly system, managed using an echelon order-up-to policy, the penalty cost is convex with
respect to the base-stock levels and the capacity levels. In an arbitrary acyclic directed inventory
network, managed using an optimal policy, the cost-to-go function of the dynamic program is
convex with respect to the inventory state vector and capacity levels. Our results are valid for both
the backorder and lost sales cases.
Furthermore, in the former system, we have outlined computational approaches for solving the
budget allocation problem. We have also shown that the sum of holding and penalty costs is a
difference of two convex functions, and thus DC minimization techniques become applicable. In
both systems, we have established cost dominance properties under convex ordering of demand
distributions.
Acknowledgement
The convex programming idea in Section 2 was motivated by a suggestion of an anonymous referee
for an earlier paper of one of the authors; that paper, however, was restricted to a single stage
inventory system with lost sales.
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks30 Article submitted to Management Science; manuscript no.
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Online Appendix
This section contains proofs and examples that were omitted in the main document.
Proof of Theorem 3 in Section 2.5.2
Consider a mathematical program, say (MP-B2), which is identical to (MP-B) except that the
objective function is replaced by
maxy
h(y)
where h is any increasing function. As in Lemma 2, it can be shown that an optimal solution to
(MP-B2) is given by the recursive formula (3).
Notice that
ej,t =
Sj −D[1, t− 1], if t≤ τj;y1,t, if j = 1 and t > τj;yj,t−τj
−D[t− τj, t− 1], for j ≥ 2 and t > τj.
Now, let
h(y) =T∑
t=1
∑j∈J
αt ·hj · (ej,t−Dt)
=∑j∈J
τj∑t=1
αt ·hj · (Sj −D[1, t]) +T∑
t=τ1+1
αt ·h1 · (y1,t−Dt)
+∑
j∈J\{1}
T∑t=τj+1
αt ·hj · (yj,t−τj−D[t− τj, t]) ,
which is an increasing and linear function of y. The result now follows from Lemma 1. 2
Example for Section 2.6.2
Suppose that demand is cyclic with a cycle length of 2 periods, and it is deterministically 40 in odd
periods and 20 in even periods. Ordering capacity is infinite, replenishment is instantaneous, and
the holding and backorder costs are $1 and $2 per unit per period, respectively. Let (Sodd, Seven)
be the vector of base-stock levels. We fix Seven at 10, and vary Sodd.
• If Sodd = 40, then there is no shortage in odd periods, and 10 units of shortage in even periods,
incurring a backorder cost of $10 · 2 = 20 per cycle and zero holding cost.
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks34 Article submitted to Management Science; manuscript no.
• If Sodd = 50, then 10 units of inventory are carried over from an odd period to the next (even)
period, in which no additional replenishment is made since Seven = 10. Thus, the cost per cycle is
$10 · 2 = 20 for backordering and $10 · 1 = 10 for holding. The total cost per cycle is $30.
• If Sodd = 60, then 20 units are carried over from an odd period to an even period, and there
is no shortage, and the holding cost is $20 · 1 = 20 per cycle.
Thus, the backorder costs are 20, 20 and 0, and the total costs are 20, 30 and 20. Thus, neither of
these costs is convex with respect to Sodd. Thus, it follows that the infinite horizon average cost is
not convex with respect to Sodd.
Convexity of Holding Costs in the Cyclic Demand Model of Section 2.6.2
Let each cycle consist of L periods, l = 0, . . . ,L−1. Let [t]L = t (mod L); then period t is the [t]L’th
period in a cycle. Let Sl be the base-stock level used in the l’th period of each cycle. We assume
the lead time is zero for simplicity. (It is easy to verify the result for positive lead times also.)
Let OHt(S0, . . . , SL−1) denote the inventory on hand at the end of period t for a given vector of
base-stock levels.
Lemma 4. Consider a capacitated single-echelon inventory system with cyclic demand, back-
orders and instantaneous replenishment. Assume the system starts period 1 with max{Sl : l =
0, . . . ,L− 1} units of inventory on hand. Let Dt = 0 for all t≤ 0. Then, the following statements
hold for every t≥ 1 and for every sample path of demands.
(a) OHt(S0, . . . , SL−1) is given by
max{0, S[t−L+1]L −D[t−L+1, t], . . . , S[t−1]L −D[t− 1, t], S[t]L −D[t, t]
}.
(b) Both OHt(S0, . . . , SL−1) and∑t
t′=1 OHt′(S0, . . . , SL−1) are convex functions.
Proof: Statement (a) implies (b) since the maximum of a set of linear functions is a convex
function. Statement (a) can be verified using a straightforward induction argument. 2
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 35
Example for Section 2.6.4
The following example shows that the conclusion of Theorem 2 may fail, i.e., the penalty cost may
not be convex with respect to the base-stock levels. Consider a two-echelon system with a lower
stage lead time of τ1 = 0 and an upper stage lead time of τ2 = 1. Demand is deterministically 3 in
each period. Let the lost sales penalty cost function be convex and piecewise linear, with a slope of
$0/unit from 0 to 1, and a slope of $1/unit from 1 onwards, i.e., b(u) = [u− 1]+. We assume that
the initial inventory satisfies Assumption 1 for each of the following base-stock levels:
• (S1, S2) = (2,3). In the first period, the after-delivery inventory level in the sink node is y1,1 = 2,
and thus the quantity of lost sales is 3−2 = 1 unit. In the second period, y1,2 = 1, and the quantity
of lost sales 3− 1 = 2 units. Thus, the two-period total lost sales cost is $1.
• (S1, S2) = (3,4). In the first period, there is no lost sales. The after-delivery inventory level at
the sink node in the second period is y1,2 = 1, and a lost sales of 3− 1 = 2 units occurs. The total
lost sales cost is $1.
• (S1, S2) = (4,5). There is no lost sales in the first period. In the second period, the quantity
of lost sales is 3− 2 = 1 units. The total lost sales cost is $0.
Thus, the above example shows that when b() is convex but not linear, the two-period lost sales
cost is not jointly convex with respect to (S1, S2), i.e., the conclusion of Theorem 2 does not hold.
Proof of Theorem 5 in Section 2.6.5
First we prove the following result.
Lemma 5. Let {Dt} be identically and independently distributed with Dt ∼ D. Similarly, let
{Dt} be identically and independently distributed with Dt ∼ D. Assume that {Dt} is independent
of {Dt}. Also, assume D≤cx D. Then,
E[φ(D1, . . . ,DT )] ≤ E[φ(D1, . . . , DT )]
holds for any convex φ :<T →<.
Proof: We prove for the case T = 2. Observe
E[φ(D1,D2)] = E[E[φ(D1,D2)|D1]] ≤ E[E[φ(D1, D2)|D1]]
Huh and Janakiraman: Convexity Results for Stochastic Inventory Networks36 Article submitted to Management Science; manuscript no.
= E[φ(D1, D2)]
= E[E[φ(D1, D1)|D2]] ≤ E[E[φ(D1, D1)|D2]]
= E[φ(D1, D2)] .
The proof for T > 2 is similar. 2
Now we prove Theorem 5. From Lemma 5, it is sufficient to show that F (S,C | D) is convex
in D. In the backlogging case, consider (MP-B) of Section 2.2. The objective function is convex
with respect to Dt’s, and the constraints can be re-written such that the right side expressions are
linear with respect to Dt’s. Thus, by Lemma 1, F (S,C | D) is convex in D. The lost sales case can
be addressed in a similar manner using (MP-L). 2
Proof of Theorem 10 in Section 3.4
The result holds trivially for t = T + 1. We assume the result for t + 1, and proceed by induction
to prove the result for t. It suffices to prove υt(x,q,C)≤ υt(x,q,C) since this inequality implies
ft(x,C)≤ ft(x,C) by (DP).
We prove the claim for the backorder case and the lost sales case separately. For the backorder
case, the convexity of Gt(x,q,D,D) with respect to D and the convex ordering between Dt and
Dt imply
γt(x,q) = EDt [Gt(x,q,Dt,Dt)] ≤ EDt[Gt(x,q, Dt, Dt)] = γt(x,q) .
Also, since z(x,q,Dt) = Dt holds for the backorder case,
EDt [ ft+1(A · (x,q,−Dt), C) ] ≤ EDt [ ft+1(A · (x,q,−Dt), C) ]
≤ EDt[ ft+1(A · (x,q,−Dt), C) ] .
The first inequality above follows from the induction hypothesis. The second inequality follows
from the convex ordering since A is a linear operator and ft+1 is convex. Summing up these results,
we obtain, from (9),
υt(x,q,C) = γt(x,q) + αEDt [ ft+1(A · (x,q,−Dt), C) ]
Huh and Janakiraman: Convexity Results for Stochastic Inventory NetworksArticle submitted to Management Science; manuscript no. 37
≤ γt(x,q) + αEDt[ ft+1(A · (x,q,−Dt), C) ]
= υt(x,q,C) ,
completing the proof of the above claim for the backorder case.
For the lost sales case, consider (SALES) in Section 3.3. By Lemma 1, the optimal value of
(SALES) is convex with respect to Dt. Note that υt(x,q,C) is the expectation of the optimal value
of (SALES), where the expectation is taken with respect to Dt. The remainder of the proof follows
the argument above for the backorder case. 2
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