convexity results for stochastic inventory networks

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,[email protected]

Ganesh JanakiramanStern School of Business, New York University, New York, NY 10012, [email protected]

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


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


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


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


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.


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)


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


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)


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


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.


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


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


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.


(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).


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).


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 }.


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


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.


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)).


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


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)


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.


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


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.


(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


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


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.


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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.


• 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


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]]


= 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) ]


≤ 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

convexity results for stochastic inventory networks

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