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Eﬃcient Optimization of the Dual-Index Policy Using Markov Chains

Joachim Arts1,∗

, Marcel van Vuuren2, Gudrun Kiesm¨uller3

1Eindhoven University of Technology, School of Industrial Engineering

P.O. Box 513, 5600 MB Eindhoven, the Netherlands

2Consultants in Quantitative Methods,

P.O. Box 414, 5600 AK Eindhoven, the Netherlands

3Christian-Albrechts-University at Kiel

Olshausenstr 40, 24098 Kiel, Germany

September 2, 2010

Abstract

We consider the inventory control of a single product in one location with two supply sources facing

stochastic demand. A premium is paid for each product ordered from the faster ‘emergency’ supply

source. Unsatisﬁed demand is backordered and ordering decisions are made periodically. The optimal

control policy for this system is known to be complex. For this reason we study a type of base-stock

policy known as the dual-index policy (DIP) as control mechanism for this inventory system. Under

this policy ordering decisions are based on a regular and an emergency inventory position and their

corresponding order-up-to-levels. Previous work on this policy assumes deterministic lead times and

uses simulation to ﬁnd the optimal order-up-to levels. We provide an alternate proof for the result that

separates the optimization of the DIP in two one-dimensional problems. An insight from this proof allows

us to generalize the model to accommodate stochastic regular lead times and provide an approximate

evaluation method based on limiting results so that optimization can be done without simulation. An

extensive numerical study shows that this approach yields excellent results for deterministic lead times

and good results for stochastic lead times.

Keywords: inventory, dual sourcing, dual-index policy, Markov Chain, approximation, lead times, D/G/c/c queue

1. Introduction

Research into inventory systems is mostly done under the assumption that only one supplier or

supply mode exists to procure, manufacture or ship goods. While many useful results have been

obtained under this assumption (e.g. News-vendor type results for many systems, see van Houtum

2006 for an overview), these models nevertheless omit an important aspect of many real inventory

systems, namely that inventories can be replenished in more than one way. For example, it is

common that one item can be procured from diﬀerent suppliers or manufactured in diﬀerent plants.

∗Corresponding author, E-mail: j.j.arts@tue.nl

1

Alternatively an item may be shipped over sea or by air (expediting). Even within the production

environment of a single plant the production lead time can be decreased by producing in overtime.

In all these examples there are multiple ways to replenish inventory with diﬀerent lead times and

costs.

The situations described in the previous subparagraph can be approached in roughly two ways.

The ﬁrst approach is to carefully select one of the suppliers/supply modes and then source all

inventory from that supplier/supply mode. We refer to the problem of making this decision as

vendor selection. The second approach is to use both suppliers/supply modes simultaneously. This

paper is concerned with the latter approach which we refer to as dual sourcing.

Suppliers are becoming more willing to oﬀer diﬀerent supply modes to their customers. Pac-

car parts in Eindhoven, for example, which handles spare-part logistics for DAF trucks, makes a

distinction between regular and emergency delivery modes for shipping parts to diﬀerent locations

throughout Europe. Another situation where multiple supply modes occur naturally is in remanu-

facturing systems. In this setting serviceable products can be produced from raw materials or by

remanufacturing returned items. These two modes of inventory replenishment are naturally asso-

ciated with diﬀerent costs and lead times. A similar situation also occurs in the inventory control

of spare parts. Spare parts are kept on stock so that a capital good can readily be made available

upon failure of a part. The failed part is then sent into normal or emergency repair with associated

diﬀerent lead times and costs.

In this paper we study a general model for the inventory control in dual sourcing systems.

We consider the inventory control of a single product in one location that is reviewed periodically

and has two supply sources with diﬀerent lead times. The lead times are assumed to be integer

multiples of the review period. The faster supply source will be referred to as the emergency

supplier while the slower supply source will be called the regular supplier. Units procured from the

emergency supplier incur additional per unit cost. Ordering from the regular channel may represent

manufacturing somewhere in Asia, while ordering through the emergency channel may represent

ordering from a more expensive local supplier. Other applications include, but are not limited to,

shipping goods by sea (‘regular’) or air (‘emergency’) freight and manufacturing with (‘emergency’)

or without (‘regular’) overtime. The problem we shall consider is the minimization of holding and

ordering costs subject to a service level constraint.

Models for the situations described above are diﬃcult to analyze. Under speciﬁc restrictive as-

sumptions, such as the assumption of a unit lead time diﬀerence, the analysis can become tractable.

2

For this latter model the optimal policy is known but the application area is very narrow. When

lead time diﬀerences are more than one period the optimal policy is known to be complex, diﬃcult

to implement and computationally hard to obtain (Whittmore & Saunders 1977, Feng et al. 2006a

and Feng et al. 2006b). In this paper we investigate exactly this context. For this reason we

consider a class of base-stock type policies and optimize within this class. Speciﬁcally we study the

dual-index policy (DIP) that has the attractive property of reducing to the optimal policy when

the lead time diﬀerence is only one period. This policy was originally proposed for remanufacturing

systems (Kiesm¨uller, 2003). It is easily implementable and has been shown to perform very close to

the optimal policy (Veeraraghavan & Scheller-Wolf 2008). Numerical studies have also shown that

in general it outperforms competing heuristic policies such as the simpler single-index policy and

constant order policy (Scheller-Wolf et al. 2003, Klosterhalfen et al. 2010). The good performance

of the DIP is also highlighted by Sheopuri et al. (2010) who benchmark their heuristic policies

against “the best available heuristic in the literature - the best dual index policy”. Until now the

DIP has resisted analytical or even approximate analytical optimization so researchers have had to

resort to simulation based procedures.

The DIP policy tracks two inventory positions: a regular inventory position (on-hand stock

+ all outstanding orders - backlog) and an emergency inventory position (on-hand stock + out-

standing orders that will arrive within the emergency lead time - backlog). In each period ordering

decisions are made to raise both inventory positions to their order-up-to-levels. Under this policy

the emergency inventory position can, and indeed often does, exceed its corresponding order-up-to-

level. This excess is called the overshoot and plays a central role in the analysis of the DIP. Despite

its relatively simple form, optimization of the DIP still requires substantial computational eﬀort

because it requires determining several overshoot distributions. In principle the overshoot distribu-

tion can be obtained exactly by solving a multidimensional discrete time Markov chain (DTMC).

However, this approach suﬀers from the curse of dimensionality and consequently the usual ap-

proach is to determine the overshoot distribution by simulation. Veeraraghavan & Scheller-Wolf

(2008) prove a separability result that drastically decreases the amount of simulation needed, but

the computational time remains substantial.

In this paper we study a model comparable to that of Veeraraghavan & Scheller-Wolf (2008).

The main diﬀerences are that we (i) consider the minimization of holding and purchasing costs

subject to a service level constraint and (ii) generalize the model by incorporating stochastic regular

lead times. The experience of the authors is that service level constraints are very common in

3

practice and are usually found in the form of service level agreements. The use of a service level

constraint in our model is convenient for managers who would like to incorporate service level

agreements in their decision making or who would like to assess the impact of diﬀerent service level

agreements on their operations. In any case Van Houtum & Zijm (2000) show that models with

penalty costs for disservice (e.g. backlogging cost) and service level constraints are related so that

if one problem can be solved, the other one can be too.

For our service level model the ability to determine the overshoot distribution remains key in

evaluating and optimizing the DIP. We provide an alternate proof of the aforementioned separa-

bility result for both deterministic and stochastic lead times. An insight from this proof is used

to construct a one-dimensional DTMC that describes the overshoot process. By approximating

the transition probabilities for this DTMC based on limiting results we obtain a computationally

eﬃcient optimization procedure.

This paper is organized as follows. In Section 2 we review the literature on dual sourcing and

position our results with respect to earlier work. We then present the model with deterministic

lead times in Section 3 and introduce the dual-index policy formally. In section 4 we analyze this

policy and give limiting results to easily ﬁnd approximately optimal settings. Next we generalize

our model to accommodate stochastic regular lead times in Sections 5 and 6. Section 7 provides

an extensive numerical study on the accuracy of our approximation. We give conclusions and

directions for further research in Section 8.

2. Literature review

Minner (2003) provides a review of the literature pertaining to many diﬀerent issues surrounding

multiple supply sources. Broadly speaking the research in multiple sourcing is divided into the

strategic approach, which studies issues such as exchange rate volatility, risk management and

vendor selection, and the operational approach that mainly studies the inventory control of such

systems. Among the diﬀerent perspectives we focus on operational/tactical control of multiple

sourcing systems. One body of research focusses on the number of supply sources as a decision

variable and usually assumes that diﬀerent sources are identical. In these situations replenishment

orders are split among the diﬀerent supply sources and optimal order splitting is the object of study.

Another body of research considers situations with two (or more) suppliers that have diﬀerent

lead times. Replenishing inventory from the faster supplier incurs additional cost. This paper

4

contributes to this body of research. As Minner (2003) provides an excellent review of research

up to around 2001 we will discuss key results from before that time only brieﬂy. Then we discuss

relevant research since that time.

Early research focusses on the structure of the optimal policy for periodic inventory systems with

dual sourcing. Barankin (1961) considers the single period problem with instantaneous emergency

delivery and a regular lead time of one period. Fukuda (1964) formulates the problem as one of

negotiable lead time for the inﬁnite horizon case and gives an analytical derivation of the optimal

policy by discounted dynamic programming. He considers a system that operates in discrete time,

and has two suppliers whose lead times are deterministic and diﬀer by exactly one period. Sethi

et al. (2003) extend Fukuda’s (1964) model with ﬁxed ordering costs, demand forecast updates

and show that the optimal policy is of the (s, S)-type. Yazlali & Erhun (2009) extend Fukuda’s

(1964) model with minimum and maximum capacity requirements for both suppliers and derive

the optimal policy. Scheller-Wolf & Tayur (2009) add order bands and state dependent demand

to Fukuda’s model and derive the optimal policy. The assumption that the lead times of both

suppliers diﬀer by only one period is crucial to obtaining optimal policies with a simple structure.

In 1977 Whittmore & Saunders and more recently Feng et al. (2006a, 2006b) showed that in the

optimal policy ordering decisions depend on the entire vector of outstanding orders for general lead

time diﬀerences. Thus the optimal policy is complex and not of the base-stock type when the lead

time diﬀerence is more than one period.

Despite the fact that the optimal policy for general lead time diﬀerences has been known to be

complex since 1977, the focus on good policies with a simpler structure is rather recent. Scheller-

Wolf et al. (2003) consider the same setting as Whittmore & Saunders (1977) and propose the

single-index policy under which ordering for both the emergency and regular supplier are based on

a single state parameter: the inventory position. This policy is simple and can easily be optimized

when demand distributions are mixtures of Erlangian distributions. When the lead time diﬀerence

is one period the single-index policy also reduces to the optimal policy. Kiesm¨uller (2003) proposes

the use of a policy that tracks two inventory positions associated with diﬀerent lead times in the

context of a remanufacturing system. The key idea here is that the decision on the amount to order

at the emergency supplier should not be based on information about orders that will arrive after

this order. Veeraraghavan & Scheller-Wolf (2008) study this policy in the context of two supplier

models. They provide the aforementioned separability result for deterministic lead times. This

separability result separates the optimization of the DIP, which is a two-dimensional optimization

5

problem, to two one-dimensional optimization problems.

Sheopuri et al. (2010) study generalizations of the dual-index policy and relate these generaliza-

tions to the lost sales inventory problem. The generalization of the dual-index policy comes at the

expense of policy structure in that the generalized policy is no longer of the base-stock type for both

regular and emergency orders. These policies outperform the dual-index policy by on average 1.1%

in their computational study, and require roughly the same amount of computational eﬀort. They

also consider a type of policy that is base-stock in the sense that each period exactly the demand of

the previous period is ordered but the allocation between the regular and the emergency supplier is

heuristic and cannot be parameterized. In a computational study they show that the performance

of these policies is similar to the performance of the dual-index policy but less computationally

burdensome compared to the simulation-optimization procedure they use to ﬁnd a good dual-index

policy. We present an eﬃcient procedure to optimize the dual-index policy that can also be used

to optimize the generalized dual-index policy that Sheopuri et al. (2010) propose.

A completely diﬀerent policy for this problem setting is the standing order or constant order

policy. In this policy the regular supplier delivers a ﬁxed quantity every period while the emergency

supplier may be controlled using various types of policies. This type of policy was ﬁrst studied by

Rosenshine & Obee (1976). Recent contributions in this area are Chiang (2007), who derives the

optimal policy structure given that the regular order quantity is ﬁxed and Allon & van Mieghem

(2008), who approximate the related Tailored Base Surge policy using Brownian motion.

A closely related problem is the expedition of orders after they have entered the pipeline. Lawson

& Porteus (2001) study this problem in a serial multi-echelon periodic review context. They show

that a type of base-stock policy, called a “top down base-stock policy” is optimal when orders can

be expedited and delayed at will in the entire supply chain. Zhou and Chao (2010) consider a

model similar to that of Lawson and Porteus (2001) and derive the optimal policy when lead time

diﬀerences between regular and expedited shipping are again restricted to one period. They also

provide newsvendor bounds on policy parameters. Gallego et al. (2007) study a single stock-point

in continuous time with the possibility of expediting existing orders and derive the optimal policy

under the assumption of Poisson demand.

In assemble-to-order systems, where lack of a single component may render the system unable

to ﬁll an order, expediting is often also included in the model (Plambeck & Ward 2007, Hoen et al.

2010). In this setting expediting is usually assumed to have zero lead time except for one period

problems (Fu et al. 2010). Arslan et al. (2001) provide analytic models for when and how to

6

expedite in make-to-order systems. Thus this model does not include inventory.

All literature in dual sourcing assumes deterministic lead times except for Song & Zipkin (2009)

and Gaukler et al. (2008). Song and Zipkin study a model of a stock-point facing Poisson demand

operating in continuous time. They assume a (S−1, S )-type ordering policy and show how to

model this system as a network of queues with one or more overﬂow bypasses. Gaukler et al.

(2008) also consider a single stock-point operating in continuous time and propose a policy based

on the classical (Q, R)-policy. They show how to ﬁnd optimal parameter settings under a set of

speciﬁc assumptions.

The setting we consider is similar to the settings in Fukuda (1964), Whittmore & Saunders

(1977) and Veeraraghavan & Scheller-Wolf (2008). Our two most important contributions are (i)

the development of an eﬃcient approximation for the overshoot distribution so that optimization of

the DIP becomes computationally more feasible and (ii) the incorporation of stochastic lead times

in the periodic review setting.

3. Model with deterministic lead times

Our model is similar to the model studied by Veeraraghavan & Scheller-Wolf (2008). We consider

the inventory control of a single product in one location with two supply sources facing stochastic

demand. The purchase price for regular (emergency) units is cr(ce). Note that since in the long

run all demand must be ordered, incurring at least crper unit, we are only interested in those

units for which we pay more than cr. Thus we may say a premium c=ce−cris paid for each

product ordered from the faster ‘emergency’ supply source, and ignore the constant of expected

demand times cr. Unsatisﬁed demand is backordered and ordering decisions are made periodically.

Without loss of generality we assume the length of a review period to be one. Demand per period

is a sequence of non-negative i.i.d. discrete random variables {Dn}with na period index. We

assume that Pr(D > 0) >0 and Pr(D < ∞) = 1. The net inventory (stock on-hand - backlog)

at the beginning of period nwill be denoted In. Any on-hand stock I+

nat the beginning of a

period nincurs a holding cost of hper SKU. (We use the standard notations x+= max(0, x) and

x−= max(0,−x)). We denote the backlog at the beginning of a period Bn=I−

n. Orders placed

at the regular (emergency) channel arrive after a deterministic lead time lr(le) and we assume

l:= lr−le,l≥1. Lead times are assumed to be an integer multiple of the review period. The

regular (emergency) order placed in period nis denoted Qr

n(Qe

n). Later (in Section 5) we will

7

Figure 1: Graphical representation of model with deterministic regular lead times

relax the assumption that lris deterministic. A schematic representation of the situation described

above is given in Figure 3.

As control mechanism for this inventory system we study the dual-index policy (DIP), deﬁned

by two parameters (Se, Sr), which operates as follows. At the beginning of each period nwe review

the emergency inventory position

I P e

n=In+Pn−l

i=n−lrQr

i+Pn−1

i=n−leQe

i(1)

and if necessary place an emergency order Qe

nto raise the emergency inventory position to its

order-up-to-level Se,

Qe

n= (Se−I P e

n)+.(2)

After placing the emergency order we inspect the regular inventory position which includes the

emergency order just placed

I P r

n=In+Pn−1

i=n−lrQr

i+Pn

i=n−leQe

i=I P e

n+Qe

n+Pn−1

i=n+1−lQr

i(3)

and place a regular order Qr

nto raise the regular inventory position to its order-up-to-level Sr,

Qr

n=Sr−I P r

n.(4)

After ordering, shipments are received and demand for the period is satisﬁed or backordered if

there is no stock available. Thus within a period nthe sequence of events can be summarized as

follows: (1) review the on-hand inventory and incur holding costs hI+

n; (2) review the emergency

inventory position and place an emergency order Qe

n; emergency ordering costs are incurred as

cQe

n; (3) review the regular inventory position and place a regular order Qr

n; (4) receive shipments

Qe

n−leand Qr

n−lr; (5) demand Dnoccurs and is satisﬁed except for possible back-orders Bn. Note

that the emergency inventory position under this policy can, and indeed often does, exceed the

emergency order-up-to-level Se. The amount by which the emergency inventory position exceeds

8

the emergency order-up-to-level is called the overshoot. After ordering the emergency inventory

position is given by Se+Onwhere On∈ {0,1, ..., Sr−Se}is the overshoot and satisﬁes

On=I P e

n+Qe

n−Se= (I P e

n−Se)+.(5)

Determining the stationary distribution of the overshoot Owill play a key role in evaluating the

performance of a given policy (Sr, Se).

Our objective is the minimization of the long run average cost subject to a modiﬁed ﬁll-rate

constraint. The modiﬁed ﬁll-rate is deﬁned as

γ= 1 −E[B]/E[D].(6)

The modiﬁed ﬁll-rate is closely related to the regular ﬁll-rate often denoted β. As opposed to

the regular ﬁll-rate, orders staying in backlog multiple time periods aﬀect the modiﬁed ﬁll-rate

negatively for the entire time they are backlogged. Van Houtum and Zijm (2000) show that γ

service level models are related to backlogging costs for fairly general inventory systems. They also

provide a discussion of diﬀerent service levels in inventory systems. When service-levels are high,

the modiﬁed ﬁll-rate is a very tight lower bound of the regular ﬁll-rate (Van Houtum 2006).

The average costs related to our problem are the costs of emergency ordering and holding costs

given by

C(Se, Sr) = hE[I+] + cE[Qe].(7)

We are now in a position to formulate the optimization problem P:

(P) min C(Se, Sr)

s.t. γ(Se, Sr)≥γ0

Se, Sr∈Z.

(8)

Here γ0denotes the target service level. This problem is a non-linear integer programming problem

(NLIP). The integrality constraint on Seand Sris the consequence of the discrete nature of demand.

Note that continuous demand distributions can also be used, but discretization has to be applied.

If the integrality constraint on the decision variables are relaxed fractional values, i.e. randomized

solutions may become optimal. In practice however fractional solutions are awkward and our

optimization method will require integrality of the involved decision variables. An overview of all

introduced notations and other notations that will be introduced in later sections is given in Table

3.

9

notation description

AnAmount of ordered products that will not arrive within the emergency lead time

in period nafter ordering (:= Pn

i=n+1−lQr

i)

BnBacklog in period n

cPremium to buy one product at the emergency supplier

C(Se, Sr) Average holding and incremental ordering costs for policy (Se, Sr)

γModiﬁed ﬁll rate

γ0Target modiﬁed ﬁll-rate

DnDemand in period n, random variable

∆ Diﬀerence between regular and emergency order-up-to-level (:= Sr−Se)

hInventory holding cost per period per SKU

InThe net inventory (on-hand stock - backlog) at the beginning of period n

IP r

nRegular inventory position at the beginning of period nafter ordering at the emergency supplier

IP e

nEmergency inventory position at the beginning of period nbefore ordering

leReplenishment lead time for emergency orders

lrReplenishment lead time (deterministic) for regular orders

lDiﬀerence between regular and emergency replenishment lead time (:= lr−le)

nPeriod index

OnOvershoot in period n(:= (I P e

n−Se)+)

SrRegular order-up-to-level

SeEmergency order-up-to-level

Qr

nRegular order quantity placed in period n

Qe

nEmergency order quantity placed in period n

Table 1: Summary of notations

4. Analysis of model with deterministic lead times

This section is organized as follows. In section 4.1 we present our separability result and show how it

can be exploited to ﬁnd the optimal DIP if the overshoot distribution can be determined. In section

4.2 we present an exact one-dimensional Discrete Time Markov Chain (DTMC) that describes the

overshoot. Following this, in section 4.3 we provide approximations for the transition probabilities

such that this DTMC can be eﬃciently utilized to approximate the overshoot distribution.

Throughout the analyses in this paper for any random variable Xnwe deﬁne the stationary ex-

pectation and distribution as E[X] = limn→∞ 1

nPn

i=1 Xnand Pr(X≤x) = limn→∞ 1

nPn

i=1 I{Xn≤

x}where I{x}is the indicator function of the event x. Whenever we drop the index of a random

variable we are referring to the stationary random variables with mean and distribution deﬁned

above. Additionally we denote the k-fold convolution of a random variable Xas X(k)and the

squared coeﬃcient of variance of a random variable Xas c2

X:= Var[X]

E2[X].

10

4.1 Optimization

In our analysis we shall see that the diﬀerence between Srand Seplays an important role. Therefore,

we deﬁne ∆ := Sr−Se. This deﬁnition allows for the speciﬁcation of a DIP as either (Se, Sr) or

(Se, Se+ ∆). For the analysis it will be more convenient to consider Seand ∆ as decision variables.

In this section we show how to ﬁnd the optimal Sefor ﬁxed ∆. This allows for a simple search

procedure over ∆ to ﬁnd the optimal DIP.

First we investigate an interesting property of the DIP. Consider the pipeline stock that will

not arrive within the emergency lead time and denote this quantity Anin period nafter ordering:

An=Pn

i=n+1−lQr

i.(9)

Since we will consider a more general model in section 5, the proofs of the next three lemma’s will

be deferred to section 6.

Lemma 4.1. (Key functional relation) Consider the dual-index policy for a system with determin-

istic lead times and Anas deﬁned in equation (9). Suppose that IP r

k≤Srfor some k∈N0. Then

for all n≥kthe dual-index policy ensures that the following identity holds

∆ = On+An.(10)

The intuition behind this lemma is simple. After ordering the regular inventory position is

Sr, but it also equals the emergency inventory position (Se+On) plus all outstanding orders not

included in the emergency inventory position (An). Thus Se+ ∆ = Se+On+An. Lemma 4.1

essentially states that Anand Onare complements so that any knowledge regarding Animplies

knowledge regarding On. The identity ∆ = On+Analso completely describes the operation of

the DIP as is evident from the proof. Before establishing our separability result we need one more

lemma which is originally due to Veeraraghavan & Scheller Wolf (2008).

Lemma 4.2. (Recursions for On,Qe

nand Qr

n)Consider the model with deterministic lead times

operated by the dual-index policy. The overshoot On, emergency order quantity Qe

nand regular

order quantity Qr

nsatisfy the following recursions:

On+1 = (On−Dn+Qr

n+1−l)+,(11)

Qe

n+1 = (Dn−On−Qr

n+1−l)+,(12)

Qr

n+1 =Dn−Qe

n+1.(13)

11

The recursions (11)-(13) are quite intuitive. Equation (11) describes that the overshoot dimin-

ishes each period with the demand and increases with the regular order that enters the information

horizon of the emergency inventory position. The emergency order quantity can also be thought of

as the ‘undershoot’, i.e., Qe

n= (Se−I P e

n)+from which relation (12) follows. Relation (13) follows

from the property that in each period the total order amount equals the demand in the previous

period. With these results we now establish the separability result, part of which also appears

as proposition 4.1 in Veeraraghavan & Scheller-Wolf (2008). We remark again that our proof is

diﬀerent.

Lemma 4.3. (Separability result) Consider the model with deterministic lead times operated by

the dual-index policy. The distributions of O,Qrand Qedepend on Srand Seonly through their

diﬀerence ∆ = Sr−Se.

Let us deﬁne O∆as the stationary random variable Ofor a given ∆. Lemma 4.3 can be exploited

to obtain the optimal DIP for ﬁxed ∆.

Theorem 4.4. (On the optimal choice for Se)Consider the dual-index policy for the control of our

model with deterministic lead times. For ﬁxed ∆the optimal Seis the smallest integer that satisﬁes

the following inequality:

∆

X

k=0

ED(le+1) −Se−k+Pr(O∆=k)≤(1 −γ0)E(D).(14)

Proof. As a consequence of lemma 4.3 the cost term related to emergency ordering, cE[Qe], becomes

a ﬁxed constant when ∆ is ﬁxed. Thus, for ﬁxed ∆ the relevant cost function is given by e

C(Se) =

hE[I+] and the problem reduces to a one-dimensional optimization problem we shall call Q.

(Q) min e

C(Se)

s.t. γ(Se, Se+ ∆) ≥γ0

Se∈Z.

(15)

Now by the identity γ= 1 −(E[B]/E[D]) the service level constraint can be modiﬁed into a

constraint on E[B]. The expected backlog can be found by conditioning on the emergency inventory

position after ordering, using that demand is an i.i.d. sequence and recalling that by lemma 4.3

the distribution of Ois already ﬁxed:

E[B] =

∆

X

k=0

ED(le+1) −Se−k+Pr(O∆=k)≤(1 −γ0)E[D].(16)

12

The objective function

hE[I+] =

∆

X

k=0

ESe+k−D(le+1)+Pr(O∆=k) (17)

is non-decreasing in Seas can easily be shown by recalling that probabilities are non-negative and

using ﬁnite diﬀerences. This implies that the smallest integer Sethat satisﬁes inequality (16) is the

optimal solution to Q, which completes the proof.

Remark It is also easy to show that E[B] is a non-increasing function of Se. Thus the optimal Se

given ∆ can easily be found using a simple method such as a bisection search.

The above result provides a simple way to ﬁnd the optimal DIP if the distribution of Oand

E[Qe] can be determined for a ﬁxed ∆. If this can be done one may simply perform a search

procedure over ∆ to ﬁnd the globally optimal DIP. To evaluate the cost term cE[Qe] for the

objective function of problem Pwe note that the ﬁrst moment of Ocompletely determines the ﬁrst

moment of Qethrough the relations E[Qr] = E[A]

l=∆−E[O]

land E[D] = E[Qr] + E[Qe]. Thus from

the distribution of Oit is easy to determine the cost term cE[Qe]. In the next two subsections we

describe a one-dimensional Discrete Time Markov Chain (DTMC) that describes the overshoot.

Moreover, we provide approximations for its transition probabilities which enable the overshoot to

be approximated eﬃciently.

4.2 A one-dimensional Markov Chain for the Overshoot

Lemma 4.1 gives insight into the behavior of On. Instead of studying Onwe may study Anwhich

has a straightforward physical interpretation as the pipeline stock that will not arrive within the

short lead time le.Anobeys the following recurrence relation:

An+1 = ∆ −On+1

= ∆ −∆−Dn−Pn

i=n+2−lQr

i+

= ∆ −∆−Dn−An+Qr

n+1−l+

= min(∆, An−Qr

n+1−l+Dn).

(18)

The ﬁrst equality follows from lemma 4.1, the second by simultaneous substitution of equations (11)

and (10). The third equality follows again by substituting equation (10) while the forth is a standard

identity. In principle Ancan be modeled by a DTMC. To construct this DTMC for Anhowever,

we would need to store the last lregular order quantities in the state information. This leads to

13

an l-dimensional Markov Chain. From equation (18) we retrieve that Qr

n∈ {0,1, ..., ∆}and so

this DTMC would have P∆

k=0 Pk

x1=0 Pk−x1

x2=0 ···Pk−Pl−1

i=1 xi

xl=0 k

x1,x2,...,xlstates. It is computationally

infeasible to ﬁnd the equilibrium distribution of this DTMC for most practical instances. To remedy

this we study a more compact DTMC with only one dimension and ∆ + 1 states.

Observe that the recurrence relation (18) completely deﬁnes a Markov Chain for Anif the

probability mass functions of Dand {Qr

n+1−l|An}are known. This DTMC is deﬁned by the

transition probabilities pij = Pr (An+1 =j|An=i) that can be obtained by distinguishing the

cases j < ∆ and j= ∆. First consider the case j < ∆, we have

pij = Pr(An+1 =j|An=i)

= Pr(An−Qr

n+1−l+Dn=j|An=i)

=Pj

k=0 Pr(Qr

n+1−l=An+Dn−j|An=i, Dn=k) Pr(Dn=k)

=Pj

k=0 Pr(Qr

n+1−l=i+k−j|An=i) Pr(D=k).

(19)

The case j= ∆ is very similar:

pi∆= Pr(An+1 = ∆|An=i)

= Pr(An−Qr

n+1−l+Dn≥∆|An=i)

=Pi

k=0 Pr(Dn≥∆ + Qr

n+1−l−An|An=i, Qr

n+1−l=k) Pr(Qr

n+1−l=k|An=i)

=Pi

k=0 Pr(Qr

n+1−l=k|An=i) Pr(D≥∆ + k−i).

(20)

Now we organize these transition probabilities in the transition matrix P:

P=

p00 . . . p0∆

.

.

.....

.

.

p∆0 . . . p∆∆

.(21)

If we let π(x) denote Pr(A=x), π= [π(0), . . . , π(∆)] and e= [1,1,...,1]T, then the stationary

distribution πcan be found by solving the set of linear equations

πP =π, πe = 1.(22)

The distribution of Dis assumed to be known, but the distribution of {Qr

n+1−l|An}is in fact

unknown. In the next subsection we construct an approximation for this distribution based on

limiting results so that the introduced one-dimensional DTMC can be used to approximate the

overshoot distribution.

14

4.3 Approximations for the transition probabilities

To determine the transition probabilities in the DTMC of the previous section we need the proba-

bility mass functions of Dand {Qr

n+1−l|An}. The latter can be approximated using the following

(limiting) result.

Proposition 4.5. The following statements hold:

(i) As ∆→ ∞,Pr(Qr

n+1 =x)→Pr(Dn=x).

(ii) As ∆→ ∞,Pr Qr

n+1−l=x|An=y→Pr Dn+1−l=x|Pn

i=n+1−lDi=y.

(iii) For ∆=1,Pr Qr

n+1−l=x|An=y= Pr Dn+1−l=x|Pn

i=n+1−lDi=y.

Proof. Part (i) and (ii) are special cases of proposition 6.5; we defer the proof to there. Part (iii)

holds trivially under the conditioning An= 0. For the conditioning An= 1 we need only show

that Pr(Qr

n+1−l= 1|An= 1) = Pr(Dn+1−l= 1|Pn

i=n+1−lDi= 1) because Pr(Qr

n+1−l= 0|An= 1)

is the complement of Pr(Qr

n+1−l= 1|An= 1). Recall the deﬁnition of Anas the sum of lregular

orders. When An= 1, there is exactly one order of one SKU in the pipeline An. Since ∆ is

an upperbound to the number of items in Anthis one item blocks other items from entering the

pipeline as it travels through. Hence this one item can be any of the regular orders included in An

with probability 1/l, i.e., Pr(Qr

n+1−l= 0|An= 1) = 1/l. Now we have

Pr Dn+1−l= 1|Pn

i=n+1−lDi= 1=Pr(D= 1) Pr D(l−1) = 0

Pr(D(l)= 1)

=Pr(D= 1) Pr(D= 0)l−1

lPr(D= 1) Pr(D= 0)l−1(23)

= 1/l.

But 1/l = Pr(Qr

n+1−l= 0|An= 1) as required.

Intuitively parts (i) and (ii) of proposition 4.5 are obvious because ∆ = ∞corresponds to single

sourcing with the regular supplier, in which case Qr

n+1 =Dn. Parts (ii) and (iii) of proposition 4.5

suggest that Pr Dn+1−l=x|Pn

i=n+1−lDi=ycan be used to approximate Pr Qr

n+1−l=x|An=y

as this approximation is exact for extremely small ∆ (∆ = 1) and extremely large ∆ (∆ → ∞).

Thus an approximation for PrQr

n+1−l=x|An=yis given by

Pr Qr

n+1−l=x|An=y≈Pr Dn+1−l=x|Pn

i=n+1−lDi=y(24)

=Pr(D=x) Pr D(l−1) =y−x

Pr D(l)=y.

15

Using this approximation for Pr Qr

n+1−l=x|An=ywe can compute an approximation for Pr(A=

x) by solving the set of linear equations (22). Then by using relation (10) we obtain an approxi-

mation for the distribution of Oas Pr(O=x) = Pr(A= ∆ −x).

Remark Note that the above approximation is also exact when l= 1, because in this situation

An=Qr

n+1−l=Qr

n. Since the DIP is the optimal policy for l= 1 this approach yields the globally

optimal policy whenever l= 1.

Numerical experiments indicate that this approximation works well in a wide range and closely

approximates both the ﬁrst two moments of Oand the shape of its distribution. The shape of

the overshoot distribution for four typical examples are given in Figure 4.3, where the overshoot

distribution as determined by simulation is shown in conjunction with the approximation based on

the above analysis. The demand distributions in these examples are either mixtures of geometric or

mixtures of negative binomial distributions ﬁtted on the moments given. The ﬁtting procedure is

due to Adan et al. (1996) and details are provided in the appendix. Results of a detailed numerical

study are presented in Section 7.

5. Model with stochastic lead times

The model we consider is identical to the model described in section 3 with one exception: now we

assume that Lr

nis a stochastic integer and that the lower bound of its support is at least le+ 1.

This ensures that the emergency supply mode is always faster than the regular supply mode; this

makes physical sense and lends tractability to the model. We only assume the regular supply mode

to be stochastic because the stochasticity for the emergency supply mode is usually negligible. For

example the emergency supply mode may represents overnight shipping using a logistics service

provider such as FedEx. To facilitate analysis we deﬁne the random variable Lnas

Ln:= Lr

n−le.(25)

The support of Lis constituted by the positive integers N. We can think of Lr

nas consisting of a

deterministic part leand a stochastic part Ln. We will assume that {Ln}is an i.i.d. sequence and

Pr(L=ν) = qν. This implies that order crossover is possible and places us in a setting similar to

that of Robinson et al. (2001). Further we let lnand lr

ndenote realizations of the random variables

Lnand Lr

n.

16

Figure 2: Overshoot distributions for a few illustrative cases as determined by simulation and the

corresponding Markov Chain approximations

Inventory positions are now deﬁned using set notation. Let Xnbe the set of all period indices

such that at the beginning of period nbefore ordering, the regular orders from these periods have

not yet arrived in stock,

Xn={k|k≥n−lr

k, k < n}.

Additionally let Ynbe the set of all period indices such that at the beginning of period nbefore

ordering the regular orders from these periods have not yet arrived in stock but will do so within

the emergency lead time:

Yn={k|k≥n−lr

k, k ≤n−lk}.

Using these sets we can again deﬁne the emergency and regular inventory positions as

I P e

n=In+Pi∈YnQr

i+Pn−1

i=n−leQe

i(26)

17

and

I P r

n=In+Pi∈XnQr

i+Pn

i=n−leQe

i=I P e

n+Qe

n+Pi∈Xn\YnQr

i.1(27)

Notice that these deﬁnitions reduce to the earlier deﬁnitions in case of deterministic regular

lead times. We also remark that we do not necessarily need to know the realizations of Lr

nup to

time nfor the inventory positions to be well deﬁned. The only necessary information needed is

to know in real time when the order from period kwill arrive within the emergency lead time le,

i.e., we need to know when k=n−lk. In essence the random variable Lr

nconsists of a random

component Lnand a deterministic component le. We assume that the random component becomes

known before or at the time the remaining lead time of a regular order is le.

There are multiple ways for this information to become available in practice. First we may

know what the regular lead time will be as soon as we place an order. Second we may know when

a regular order will arrive within the emergency lead time because this time is naturally associated

with known events such as a shipment harboring at the port. In the context of manufacturing in

overtime or other use of ﬂexible capacity this information may be available by simple inspection of

the job ﬂoor.

Ordering decisions are still given by equations (2) and (4) and the overshoot still satisﬁes the

original deﬁnition in equation (5). We now proceed to analyze the DIP when regular lead times

are stochastic.

6. Analysis of model with stochastic lead times

Our analysis will proceed along the same lines as the analysis for deterministic regular lead times,

i.e., we show how to ﬁnd the optimal DIP for ﬁxed ∆ (Section 6.1) and provide a one dimensional

DTMC that describes the overshoot (Section 6.2). We provide approximations for the transition

probabilities of this DTMC in Section 6.3.

6.1 Optimization

Let us turn again to the amount of pipeline stock that will not arrive within the emergency lead

time, An. Let Unbe the set of all period indices such that in period nafter ordering the regular

orders from these periods will not arrive within the emergency lead time:

Un={k|k≥n−lk+ 1, k ≤n}.(28)

1A\Bis the set Aminus B

18

Now the deﬁnition of Ancan be written as

An=Pi∈UnQr

i.(29)

Lemma 6.1. (Key functional relation) Consider the model with stochastic lead times operated by

the dual-index policy and Anas deﬁned in equation (29). Suppose that IP r

k≤Srfor some k∈N0.

Then for all n≥kthe dual-index policy ensures that the following identity holds

∆ = On+An.(30)

Proof. Reconsider the regular inventory position as given in equation (27),

I P r

n=I P e

n+Qe

n+Pi∈Xn\YnQr

i.(31)

Now we substitute the deﬁnition of the overshoot (from equation (5)) and add Qr

nto both sides of

this equation,

I P r

n+Qr

n=Se+On+Pi∈Xn\Yn∪{n}Qr

i.(32)

By supposition I P r

n≤Srso Qr

n=Sr−I P r

nand the left-hand side of (32) becomes Sr. When we

take a closer look at the set over which the sum in (32) runs it is straightforward to verify that

Un=Xn\Yn∪ {n}so that we can substitute the deﬁnition of Anto obtain

Sr=Se+On+An.(33)

Rearrangement and substitution of the identity ∆ = Sr−Seyields the result.

Lemma 6.1 is a direct generalization of lemma 4.1 and essentially states that Anand Onare

direct compliments also in the presence of stochastic regular lead times. Note also that lemma 6.1

holds for all ergodic stochastic processes {Ln}n∈N0, not just i.i.d. sequences.

Now we introduce Vnthe set of period indices such that at the beginning of period nafter

ordering the regular orders from these periods will enter the information horizon of the emergency

inventory position in period n+ 1,

Vn={k|k=n−lk+ 1}.(34)

We emphasize that the sets Xnand Ynare deﬁned before ordering while Unand Vnare deﬁned

after ordering. As before we now turn our attention to recursions for On,Qe

nand Qr

nand then

establish our separability result.

19

Lemma 6.2. (Recursions for On,Qe

nand Qr

n)Consider the model with stochastic lead times as

deﬁned in Section 5. The overshoot On, emergency and regular order quantities satisfy the following

recursions:

On+1 =On−Dn+Pi∈VnQr

i+,(35)

Qe

n+1 =Dn−On−Pi∈VnQr

i+,(36)

Qr

n+1 =Dn−Qe

n+1.(37)

Proof. The emergency inventory position satisﬁes

I P e

n+1 =I P e

n+Qe

n−Dn+Pi∈VnQr

i

=Se+On−Dn+Pi∈VnQr

i.

(38)

Rewriting the deﬁnition of the overshoot (equation (5)) we obtain

On+1 = (I P e

n+1 −Se)+

= (Se+On−Dn+Pi∈VnQr

i−Se)+

= (On−Dn+Pi∈VnQr

i)+.

(39)

Similarly for the emergency order quantity we have by rewriting (2):

Qe

n+1 = (Se−I P e

n+1)+

= (Dn−On−Pi∈VnQr

i)+.

(40)

The identity Qr

n+1 =Dn−Qe

n+1−lfollows immediately from the fact that the DIP ensures that in

each period the total amount ordered equals demand from the previous period.

With these results we can prove the same separability result that was shown to hold for deter-

ministic regular lead times.

Lemma 6.3. (Separability result) Consider the model with stochastic lead times as deﬁned in

Section 5. The distributions of Oand Qeand Qrdepend on Srand Seonly through their diﬀerence

∆ = Sr−Se.

Proof. Recall the recursions in lemma 6.2. To make these equations independent of the start-

ing conditions we substitute the identity for Onin lemma 6.1. This substitution also makes the

operation of the DIP explicit:

On+1 =∆−Dn−Pi∈Un\VnQr

i+,(41)

Qe

n+1 =Dn+Pi∈Un\VnQr

i−∆+,(42)

Qr

n+1 =Dn−Qe

n+1.(43)

20

For the summation Pi∈Un\VnQr

iwe read 0 whenever Un\Vn=∅. These recursions completely

determine the stochastic processes {On},{Qr

n}and {Qe

n}once the stochastic sequences {Dn},

and {Ln}have been speciﬁed. Since the stochastic processes {On}and {Qe

n}and {Qr

n}can be

described completely using Srand Seonly through their diﬀerence, it follows that their stationary

distributions are functions of Srand Seonly through their diﬀerence.

Remark In establishing lemma 6.3 we did not require that either {Dn}or {Ln}are i.i.d. sequences.

In principle the stationary overshoot distribution is well deﬁned when ∆ is ﬁxed for all ergodic

processes {Dn}and {Ln}such that Dn∈N0and Ln∈N0for all n∈N0. We do use that {Dn}

and {Ln}are i.i.d. in sections 6.2 and 6.3 to construct an eﬃcient approximation for Pr(O=x).

However the distribution of O,Qeor Qrcan be determined by simulation for more general processes

{Dn}and/or {Ln}.

Let us deﬁne O∆as the stationary random variable Ofor a given ∆. Lemma 6.3 leads to the

following theorem on the optimal choice for Sefor ﬁxed ∆

Theorem 6.4. (On the optimal choice for Se)Consider the model with stochastic lead times as

deﬁned in Section 5. For ﬁxed ∆the optimal Seis the smallest integer that satisﬁes the following

inequality

∆

X

k=0

ED(le+1) −Se−k+Pr(O∆=k)≤(1 −γ0)E(D).(44)

Proof. The proof is analogous to the proof of Theorem 4.4 and therefore omitted.

The optimal DIP for the system with stochastic lead times can also be found by a search

procedure over ∆. To ﬁnd the cost term cE[Qe] for the objective function of problem Pin this more

general situation, we make use of the identities E[Qr] = ∆−E[O]

E[L]and E[D] = E[Qr] + E[Qe]. In the

next two sections we describe a one-dimensional DTMC and transition probability approximations

for our generalized model.

6.2 A one-dimensional Markov Chain for the overshoot

As was the case for the model with deterministic lead times, lemma 6.1 allows us to study Anto

ﬁnd the distribution of O.Anstill has the appealing physical interpretation as the pipeline stock

21

that will not arrive within the short lead time leand obeys the following recurrence relation

An+1 = ∆ −On+1

= ∆ −∆−Dn−Pi∈Un\VnQr

i+

= ∆ −∆−Dn−An+Pi∈VnQr

i+

= min ∆, An−Pi∈VnQr

i+Dn

(45)

It is evident from the model with discrete lead times that an exact DTMC for this problem suﬀers

even more from the curse of dimensionality. For this reason we again turn our attention to a one-

dimensional DTMC that can be constructed in a manner analogous to that in Section 4.2. This

DTMC is given by the transition probabilities pij = Pr (An+1 =j|An=i):

pij =

Pj

k=0 Pr Pi∈VnQr

i=i+k−j|An=iPr(D=k),if j < ∆;

Pi

k=0 Pr Pi∈VnQr

i=k|An=iPr(D≥∆ + k−i),if j= ∆.

(46)

To make this one-dimensional DTMC of use, it remains to ﬁnd the distribution of Pi∈VnQr

i|An

or an approximation thereof. This will be done in the next subsection.

6.3 Approximations for the transition probabilities

To determine the transition probabilities in the DTMC of the previous section we need the proba-

bility mass functions of Dand Pi∈VnQr

i|An. The latter can be approximated using the following

limiting result.

Proposition 6.5. The following statements hold

(i) As ∆→ ∞,Pr(Qr

n+1 =x)→Pr(Dn=x)

(ii) As ∆→ ∞,Pr Pi∈VnQr

i=x|An=y→Pr Pn

i=n−|Vn|+1 Di=x|Pn

i=n−|Un|+1 Di=y

Proof. We rewrite equation (45) to

An+1 = min ∆, An−Pi∈VnQr

i+Dn

= min ∆,Pi∈UnQr

i−Pi∈VnQr

i+Dn

= min ∆,Pi∈Un\Vn∪{n+1}Qr

i−Qr

n+1 +Dn

= min ∆, An+1 −Qr

n+1 +Dn.

(47)

Now if we let ∆ → ∞ and recall the condition Pr(D < ∞) = 1 we immediately retrieve part (i) of

the proposition. Part (i) also implies that Dn

dist

=Qr

n+1 for all nwhen ∆ → ∞, where dist

= denotes

22

equality in distribution. Now since {Dn}is i.i.d., so is {Qr

n}when ∆ → ∞. This implies that as

∆→ ∞, the distribution of Pi∈VnQr

i

dist

=Qr(|Vn|)dist

=D(|Vn|)and An

dist

=Qr(|Un|)dist

=D(|Un|). From

this part (ii) immediately follows.

Remark When considering deterministic lead times we already showed in proposition 4.5 that the

approximation we propose is exact also for ∆ = 1. For stochastic Lnthis is no longer the case.

The numerical results in Section 7 reﬂect this fact.

Part (ii) of proposition 4.5 suggests that Pr Pn

i=n−|Vn|+1 Di=x|Pn

i=n−|Un|+1 Di=ycan be

used to approximate Pr Pi∈VnQr

i=x|An=y. The computation of this approximation is however

not straightforward because it requires knowledge of the random variables |Un|and |Vn|which in

turn depend on the process {Ln}. Indeed for the computation of this probability we digress to

study the joint stationary distribution of |Un|and |Vn|when Lnis assumed to be a sequence of i.i.d

random variables with ﬁnite support. In principle one may study the joint distribution of |Un|and

|Vn|for diﬀerent lead time processes {Ln}.

Let Kndenote the number of orders in the pipeline that will not arrive within the emergency

lead time in period nafter ordering,

Kn=|Un|.(48)

Further let Λndenote the number of orders that are about to enter the information horizon of the

emergency inventory position,

Λn=|Vn|.(49)

We wish to determine the joint stationary distribution of these two quantities Pr(K=κ∩Λ =

λ). We do this recursively. Recall that the distribution of Lis given by qν= Pr(L=ν), ν∈

{1,2, ..., Lmax}. Further we deﬁne

ϕκ,λ,ν = Pr(K=κ∩Λ = λ|orders were placed the last νperiods only (not before)),(50)

where we allow for orders of size 0 (which may occur if demand in a certain period is zero).

Obviously, this deﬁnition means that the distribution needed is given by

Pr(K=κ∩Λ = λ) = ϕκ,λ,Lmax := ϕκ,λ,(51)

since orders that were placed more than Lmax periods ago cannot belong to the sets Unor Vn. The

probabilities ϕκ,λ,ν can be computed recursively as follows:

ϕκ,λ,ν =ϕκ−1,λ−1,ν−1qν+ϕκ−1,λ,ν −1·PLmax

m=ν+1 qm+ϕκ,λ,ν−1·Pν−1

m=1 qm.(52)

23

The initial probabilities are straightforwardly seen to be

ϕ1,0,1=PLmax

m=2 qm, ϕ1,1,1=q1, ϕκ,λ,1= 0 otherwise.(53)

This concludes our derivation of the joint stationary distribution of |Un|and |Vn|.

Remark The process Kncan also be thought of as the number of customers in a discrete time

D/G/Lmax/Lmax-queue. Each period na customer arrives (order is placed) and that customer

immediately enters service for a random time Ln(order stays in the set Ufor Lnperiods). Thus

this D/G/c/c-queue has the special property that the service distribution has a ﬁnite support

on {1, ..., Lmax}while the interarrival time is 1. In general the evaluation of the steady state

distribution of D/G/c/c-queues cannot be done in polynomial time if it can be done at all. For

this speciﬁc case the evaluation can be done in O(L4

max) time. To see this, note that the number

of times we compute recursion (52) including the initialization before we obtain ϕκ,λ is given by

PLmax+1

i=2 Pi

x=2 x=PLmax+1

i=2 i2+i−2

2

=1

2PLmax+1

i=2 i2+1

2PLmax+1

i=2 i−Lmax

=1

6L3

max +L2

max +5

6Lmax.

(54)

Since computing recursion (52) can be done in O(Lmax) time, the overal complexity is O(L4

max).

Now that the joint distribution of Kand Λ is known, we can compute the approximation for

Pr Pi∈VnQr

i=x|An=yby conditioning on the values of |Un|and |Vn|:

Pr Pi∈VnQr

i=x|An=y≈Pr Pn

i=n−Λ+1 Di=x|Pn

i=n−K+1 Di=y

=PLmax

κ=1 Pκ

λ=1 Pr(Λ = λ|K=κ) Pr K=κ|Pn

i=n−K+1 Di=y

×Pr Pn

n−λ+1 Di|Pn

i=n−κ+1 Di=y

=PLmax

κ=1 Pκ

λ=1 Pr(Λ = λ|K=κ) Pr K=κ|Pn

i=n−K+1 Di=y

×Pr D(λ)=xPr D(κ−λ)=y−x

Pr D(κ)=y.(55)

In expression (55) the probability Pr(K=κ|Pn

i=n−K+1 Di=y) is obtained by applying Bayes’

theorem:

Pr K=κ|Pn

i=n−K+1 Di=y=Pr D(K)=y|K=κPr(K=κ)

Pr D(K)=y

=Pr D(κ)=yPr(K=κ)

PLmax

z=1 Pr D(z)=yPr(K=z),(56)

24

while the probabilities Pr(Λ = λ|K=κ) are easily obtained from ϕκ,λ, the joint density of Λ and

K.

Using this approximation for Pr Pi∈VnQr

i=x|An=ywe can compute an approximation for

Pr(A=x) by ﬁnding the equilibrium distribution of the DTMC for An. Then by using relation

(30) we obtain an approximation for the distribution of Oas Pr(O=x) = Pr(A= ∆ −x).

7. Numerical results

In this section we report on a numerical study to test the accuracy of the Markov Chain approxi-

mation that we propose. To this end a test bed of 1680 instances of problem Pwas created that is

a full factorial design of the parameter settings summarized in Table 7. The demand distributions

we used are mixtures of either two negative binomial or two geometric distributions. As such they

are discrete phase type distributions. These distributions were ﬁtted on the ﬁrst two moments

using the procedure suggested by Adan et al. (1996). For convenience we have included this ﬁtting

procedure in the appendix. The diﬀerent types of distributions for Lare deﬁned in Table 7.

Parameter settings

E[D] 25

c2

D

1

4,1

2,1,3

2,2

le1,2

E[L] 4,8,12

h1

c10,20,30,40

γ00.95,0.98

Distribution type of LU1, U2, S1, S2, LS, RS, DET

Table 2: Test-bed of problem instances P

For each instance we performed the optimization by simulation. This was done by using recur-

sion (45) to determine the overshoot distribution. After a warm-up of 100 periods, recursion (45)

was computed until the width of 95% conﬁdence intervals for E(O) and σ(O) was less than 1% of

the respective point estimates. Then we applied theorem 6.4 to ﬁnd the DIP (Ssim

e, Ssim

r) to be

optimal with respect to simulation.

We also performed the optimization with our approach involving approximate Markov Chains,

which found the DIP (SMC

e, SM C

r) to be optimal with respect to the Markov chain approxima-

tion. Note that in literature optimization of the dual-index policy is always done using simulation

25

Pr(L=x)

Distribution Type \xE[L]−2E[L]−1E[L]E[L]+1 E[L]+2

U1 (uniform1) 0 1

3

1

3

1

30

U2 (uniform2) 1

5

1

5

1

5

1

5

1

5

S1 (symmetric1) 0 1

4

2

4

1

40

S2 (symmetric2) 1

10

2

10

4

10

2

10

1

10

LS (left skewed) 0 4

10

3

10

2

10

1

10

RS (right skewed) 1

10

2

10

3

10

4

10 0

DET (deterministic) 0 0 1 0 0

Table 3: Distribution types for L

(Kiesm¨uller, 2003, Scheller-Wolf et al. 2003, Veeraraghavan & Scheller-Wolf, 2008, Sheopuri et al.,

2010, Klosterhalfen et al. 2010) and so the solutions obtained through the simulation procedure

are the best solutions currently known. All computations were coded in MATLAB and executed

on a PC with an Intel-M 1.6 GHz processor with 768 MB of RAM.

Since both methods determine the total cost function and modiﬁed ﬁll-rate with diﬀerent meth-

ods, they are not directly comparable. Therefore, we evaluated the total cost and modiﬁed ﬁll-rate

for both solutions using simulation. The simulation was run such that the width of 95%-conﬁdence

interval for cost was less than 1% of the point estimate. As measures of optimality for the ap-

proximate Markov chain approach, we considered the relative deviation from the optimal DIP with

respect to simulation

∆C=Csim SMC

r, SM C

e−Csim Ssim

r, Ssim

e

Csim (Ssim

r, Ssim

e)·100%

and the absolute deviation from the target modiﬁed ﬁll-rate

δγ=γsim SMC

r, SM C

e−γ0.

Figure 7 shows a scatter-plot of ∆Cversus δγfor the 240 problem instances with deterministic

lead times. Note that solutions that are optimal with respect to simulation would lie on the non-

negative x-axis. Thus solutions in the ﬁrst quadrant outperform simulation solutions in that they

have lower costs while still meeting/exceeding the required service level. Solutions in the third

quadrant are dominated by simulation solutions while solutions in the second and fourth quadrant

are either more costly or do not meet the required service level. Note however that all service

level deviations are within 1% which is also the simulation tolerance and that negative relative cost

26

Figure 3: Quality of approximate optima with respect to simulation for deterministic lead times

deviations are also within the simulation tolerance. Thus the performance of the solutions found

are not statistically distinguishable from the simulation solutions.

Figure 7 shows a scatter-plot of ∆Cversus δγfor the 1440 problem instances with stochastic

lead times. For stochastic lead times there is a tendency to ﬁnd solutions that are more expensive

than simulation solutions at an increase in service relative to the target level. An explanation of the

superior performance of the approximation for deterministic lead times over stochastic lead times

is to be found in the fact that the approximation for deterministic lead times is based on limiting

results for ∆ → ∞ and ∆ = 1, while for stochastic lead times they are based on limiting results

only for ∆ → ∞. However most solutions for instances with stochastic lead times are still very

close to the origin so that the approximation is usually very tight.

Since no instances diﬀered signiﬁcantly from the required service level, we investigated what

typiﬁed instances with a large relative cost deviation from the simulation optimum. To this end we

tabulated average minimum and maximum relative cost deviations from simulation solutions (∆C)

for the entire test bed for all problem parameters in Table 4.

We see that the approximation improves as demand variability increases. This is a convenient

property because dual sourcing is a way to buﬀer demand variability. Our approximation also

27

Figure 4: Quality of approximate optima with respect to simulation for stochastic lead times

becomes more accurate when the emergency lead time increases. This is in line with expectation

because holding cost can also be written as hE[(Se+O−D(le+1) )+], from which we see that

the demand distribution (which we know exactly) inﬂuences holding cost more when leis large.

Accuracy also increases when the expedition premium goes up. This is because expediting becomes

less attractive when cgoes up, so that ∆ becomes larger and our approximation works better. That

our approximation becomes less accurate as E[L] increases can be explained again by inspecting the

holding cost hE[(Se+O−D(le+1))+]. The contribution of O, which we know only approximately,

compared to D(le+1) becomes smaller when E[L] decreases. This is because ∆ (and therefore also O)

increases with E[K] = E[L] (by Little’s law). The target service level and diﬀerent non-stochastic

distributions for Ldo not inﬂuence accuracy much. We do see as before that the approximation

performs much better when lead times are deterministic.

Computational times for our approximation are also much shorter than for the simulation based

procedure. For this test-bed the optimization method based on the approximation was on average

70 times faster than the simulation based method for problem instances with deterministic lead

times and on average 35 times faster for problem instances with stochastic lead times. For the

instances with deterministic lead times the optimization time of our procedure was always within 3

28

Relative cost deviation from simulation optimum [∆C]

c2

Davg. min max

1

42.13 -5.36 6.51

1

21.05 -2.78 4.30

1 0.32 -3.25 2.97

3

20.40 -1.10 2.07

2 0.06 -2.37 1.52

leavg. min max

1 0.80 -3.25 6.51

2 0.76 -5.36 6.36

cavg. min max

10 1.35 -5.36 6.40

20 0.84 -2.06 6.51

30 0.59 -3.25 5.95

40 0.35 -4.10 5.95

γ0avg. min max

0.95 0.79 -4.10 6.51

0.98 0.77 -5.36 6.01

E[L] avg. min max

4 0.03 -5.36 4.72

8 0.95 -2.37 6.09

12 1.37 -1.56 6.51

Distribution type avg. min max

U1 0.88 -3.25 5.27

U2 0.98 -4.10 5.86

S1 1.01 -2.78 6.40

S2 1.13 -2.36 6.26

LS 1.06 -1.23 6.51

RS 0.90 -5.36 5.42

DET -0.48 -2.37 1.48

Total avg. min max

0.78 -5.36 6.51

Table 4: Quality of approximate solutions for diﬀerent problem parameters

seconds while the computation time for the simulation approach was between 72 and 111 seconds.

For instances with stochastic lead times our approach always had a computation time within 15

seconds while the simulation based approach had a computation time between 98 and 183 seconds.

29

8. Conclusion and directions for future research

In this paper we presented two models. The ﬁrst model deals with the dual-index policy for a

single stage dual sourcing inventory system facing stochastic demand with deterministic lead times

controlled by the dual-index policy. Our main contributions here are to (i) provide an alternate

and insightful proof of the separability result that reduces the optimization of the DIP to two

one-dimensional optimization problems and (ii) provide an approximate evaluation method of the

dual-index policy using Markov Chains based on limiting results that does not require simulation,

thus making optimization more eﬃcient.

The second model we presented was a generalization of the ﬁrst by allowing regular lead times

to be stochastic. In this situation we (i) deﬁned a dual-index policy with mild informational

requirements on the realizations of regular lead times; (ii) proved that the same separability result

holds as for the model with deterministic lead time and (iii) developed an approximate evaluation

method using Markov Chains based on limiting results again making optimization much more

eﬃcient.

In an extensive numerical study we showed that the approximations we suggest perform very

well in ﬁnding a close to optimal dual-index policy and are faster by at least an order of magnitude.

The research in this paper can be extended in several important ways. The most obvious and

possibly useful extension is to deﬁne and analyze the dual-index policy for multi-echelon inventory

systems. Consider for example a serial supply chain. Clark & Scarf (1960) showed that base-stock

policies are optimal for this system and that the optimal base-stock levels can be obtained by

successively solving newsvendor equations. This decomposition result relies on the fact that all

stock points in a serial supply chain face the same demand process. When the most downstream

stock point is the only stock point with two sources, this property is retained. In that case ﬁnding

the optimal echelon-DIP should be a straightforward task using the results in this paper.

When stock points other than the most downstream stock point have two sources the property

that each stock point essentially faces the same demand process is not preserved, because some

of the demand is ordered via the second source. Inventory control for this type of system is an

interesting new research direction. Perhaps a model closer to that of Lawson & Porteus (2000)

where there are diﬀerent shipping modes between stock points could lead to tractability here.

Other research directions may generalize/extend the model to accommodate stochastic emer-

gency lead times, capacitated order quantities and ﬁxed emergency ordering costs.

30

Acknowledgements

The authors thank the associate editor and two anonymous referees for their helpful comments

and suggestions. They also thank Jan van Doremalen and Geert-Jan van Houtum for helpful

discussions.

Most of this work was done while the ﬁrst author was doing an internship at consultants in

quantitative methods.

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Appendix: Fitting procedure of Adan et al. (1996)

Here we describe a procedure for ﬁtting discrete distributions on the ﬁrst two moment of a discrete

random variable that is due to Adan et al. (1996).

Let Xbe a random variable on N∪ {0}with mean µXand squared coeﬃcient of variation c2

X

and deﬁne a=c2

X−1/µX. Then the discrete random variable Ymatches the ﬁrst two moments of

Xif it is chosen as follows:

33

•If −1

k≤a≤−1

k+1 for some k∈N, then Yis a mixture of two binomial random variables such

that:

Y=

BI N (k, p),w.p. q;

BI N (k+ 1, p),w.p. 1 −q.

where

q=1+a(1+k)+√−ak(1+k)−k

1+a, p =µX

k+1−q

•If a= 0, then Yis a Poisson random variable with mean µX,Y=P ois(µX).

•If 1

k+1 ≤a≤1

kfor some k∈N, then Yis a mixture of two negative binomial random

variables such that:

Y=

NegBin(k, p),w.p. q;

NegBin(k+ 1, p),w.p. 1 −q.

where

q=(1+k)a−√(1+k)(1−ak)

1+a, p =µX

k+1−q−µX

•If a≥1, then Yis a mixture of two geometric random variables such that:

Y=

Geo(p1),w.p. q;

Geo(p2),w.p. 1 −q.

where

p1=µX(1+a+√a2−1)

2+µX(1+a+√a2−1) , p2=µX(1+a−√a2−1)

2+µX(1+a−√a2−1) , q =1

1+a+√a2−1

34