(R,Q) Policy with Lead Time Options — Offered

by a manufacturer / transporter to a retailer

Metin Cakanyildirim ∗

Sirong LuoSchool of Management

The University of Texas at Dallas

Abstract

We study lead time options in a manufacturer and a retailer supply chain where the retailer uses the(R, Q) inventory policy: the retailer places an order to the manufacturer who uses a transporter to deliverthe order a lead time later. The manufacturer promises to expedite or postpone the delivery (using itsown means or different transportation options) by a certain amount of time if the retailer makes sucha request. Consequently, the retailer has an option to modify the lead time by using most up-to-datedemand information. However, the retailer does not modify the order quantity. We establish the optimallead time policy and provide approximations for the critical levels associated with it. We also discussthat R is much more sensitive to lead times than Q, and primarily focus on finding optimal R. Towardsthat, a cost approximation which yields unimodular costs in R is provided. Finally, the effect of lead timeoptions on supply chain performance is illustrated with a numerical study.

Keywords: Lead Time Options, (R,Q) Policy, Supply Chain Management, Approximations.

∗For correspondence: metin@utdallas.edu

1 Introduction

In uncertain environments, flexibility to adapt previously made decisions to recently observed outcomes is

always valuable. A special case of this general concept has found vast use in supply chains (SC) such as

quantity flexibility [24] or buyback contracts [16]. However, most SC contracts have dealt with quantity

flexibility while totally ignoring lead time flexibility as far as we know. Lead time flexibility refers to an

arrangement between the retailer and the transporter or the manufacturer, where delivery lead times are not

firm when orders are placed but they evolve into firm(er) times. This evolution is controlled by the retailer

as long as it is within limits posed by the transporter/manufacturer. The retailer monitors its demand

and requests lead times accordingly; high demand observations lead to shorter lead times, e.g. expediting

deliveries via using a faster transportation mode or via rushing the the production at the manufacturer or

buying a portion of the production lot from a spot market. Conversely, low demand observations suggest

using a slower transportation mode or postponing deliveries from the manufacturer where inventory holding

costs are lower. For example, aircraft manufacturers (e.g. Boeing) provide delivery lead time flexibility

to its customers (e.g. Delta airlines). Airlines place aircraft orders to Boeing without firm delivery (lead)

times, timing is made firmer with expediting or postponing deliveries as airlines observe more passenger

demand. Also, cargo transporters (e.g. Delta) provide transportation time flexibility which leads to delivery

time flexibility for any deliveries made with these transporters.

Under uncertainty, flexibility provides a hedge that distributes or reduces the risk. In the well known

example of quantity flexibility contracts the manufacturer and the supplier share the risk by promising the

availability and purchase of a certain amount of goods. This set up gives retailer some leverage to counter

against extreme demands which are unlikely but possible. It is possible to provide a similar hedging mech-

anism to retailer with flexible lead times. Our objective is to study this hedging mechanism, its benefits for

the retailer and for the SC.

We believe that flexible lead times can be more acceptable to a manufacturer because manufacturer’s

sales (order quantities for retailer) are never altered, unlike quantity flexibility contracts. Especially, the

manufacturers, who wish to stabilize their demand with a long term relationship with the retailer, appre-

ciate the constant lot sizes. Moreover, our flexible lead time model studies costs in continuous review over

a long horizon and hence departs from the established SC literature which focuses mostly on newsvendor

type models.

The purpose of our work is threefold: (1) To further the properties of the (R,Q) inventory model with

regard to lead times. Although lead time management is extremely important in competitive environments

and is conceptually emphasized by operations management strategies (e.g. just in times, time based compe-

tition and quick response), their study is limited perhaps because of technical difficulties which are carefully

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handled here. (2) To develop a flexible lead time model which can improve the performance of the whole

SC even when the order quantity is fixed for the sake of the manufacturer. Most of the paper is devoted to

measure the benefits of the flexible lead times for the retailer. However, it also takes a larger SC perspective

and studies how lead time flexibility can be provided and at what cost. (3) To obtain important management

insights about how the key cost/demand parameters affect the performance of the SC. The paper finally

discusses when the lead time flexibility improves the SC performance.

The paper is organized as follows. In next section, relevant literature is briefly surveyed. Simple prop-

erties of (R, Q) policy and how these properties behave with variable lead times are discussed in Section 3.

Most proofs are presented in the appendix. In Section 4, we develop the inventory costs for the flexible lead

time model and introduce the optimal lead time policy. Section 5 deals with the optimization of decision

variables; mostly R and briefly Q. In section 6, we provide simple approximations for the critical levels

used in defining the optimal lead time policy. Section 7 discusses how lead time flexibility can be provided

via transportation and manufacturing. Section 8 and 9 respectively include our numerical experiments with

managerial insights and a brief conclusion.

2 Literature Survey

In operations management literature, there are many models studying the concept of flexibility in various

contexts. Most of these focus on quantity flexibility to deal with the demand uncertainty. [24] analyzes a

quantity flexibility contract for a supplier and a retailer SC. In general, the retailer’s incentive to induce

the supplier to produce more can cause SC inefficiency. To counter that, [24] suggests lower limits on re-

tailer’s purchase and supplier’s supply quantities. By correctly setting the whole sale price as a function

of these limits, the quantity flexibility contract is shown to induce the supplier to produce (order) the SC

optimal. [2] discusses a capacity flexibility contract called “Pay to Delay”, which comes from semiconductor

and commodity manufacturers. It analyzes the optimal decision for order size (called “take” quantity) and

the reserved capacity (called “pay” quantity), and the impact of information and cost parameters on the

decisions. [1] summarizes the existing quantity flexibility contracts in a supplier and a retailer system. It

provides the general conditions for these type contracts to coordinate the SC.

On the other hand, lead time management is another important issue in supply chain management. [21]

investigates how the stochastic lead time affects the behavior of a single product continuous-time inventory

system facing compounded Poisson demand and attempting to keep the inventory at a base stock level. It

assumes that the supplier’s production and distribution systems deal with orders one by one so orders never

cross each other. The results show that the stochastically larger lead times require larger optimal base stocks

and stochastically more variable lead times lead to higher optimal average costs. [9] discusses the benefits

of sharing demand information in a SC consisting of a single supplier and a retailer. It finds instances that

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sharing demand information can stochastically increase (or decrease) the lead time, therefore, increase (or

decrease) the retailer’s expected total cost.

In manufacturing industries, it is well accepted that shorter lead time is always good for an inventory

system’s performance because shorter lead time decreases instances of shortages and reduces obsolescence

risk. [13] discusses the quick response which is used in industry to reduce the lead time, their result show

that when service level is less than 0.5, both the manufacturer and the retailer are better off. However,

when service level is more than 0.5, only retailer is better off but manufacturer is worse off from the lead

time reduction. It also explores the service, whole sale price, and volume commitment to make the quick

response pareto improving. However, the shorter lead time usually requires larger investment by the supplier

into inventory and capacity. [14] explores the impact of manufacturing flexibility on inventory investments

in a two-echelon periodic review inventory system. It argues that the lead time of outstanding orders in

the pipeline can be shortened by the use of the manufacturing flexibility. On the other hand, expediting

deliveries when there is shortage is also common in practice. [23] analyzes a periodic review inventory sys-

tem with emergency orders. Regular orders are placed periodically with a deterministic lead time according

to a base stock policy. Emergency orders are placed with a shorter lead time and higher acquisition cost

in case of stockouts. It is established that the proposed inventory system offers substantial cost savings

relative to an inventory system without the emergency replenishment option. [11] considers a two-stage SC

under centralized control. The upstream always meets the supply requests from the downstream, and the

shortage will be filled by either overtime production or premium freight shipments if the lead time is short.

It develops the optimal inventory policy with expedited order deliveries.

Research in dual sourcing and multiple delivery modes is similar to regular and emergency orders. But

the motivation for the latter is mostly limited to reducing shortages. [10] analyzes a SC with an option

of utilizing a reliable and an unreliable supplier. It considers both inventory and transportation costs of

supplier. [15] studies two supply options with different lead times in a continuous review inventory sys-

tem. When the inventory level drops below R1 an order Q1 is placed with the long lead time supplier. If

the inventory level further decreases to R2 another order with size Q2 is placed with the faster supplier.

The (R1, Q1, R2, Q2) model is an extension of (R, Q) but it is analyzed for only the special case of one

outstanding order. For periodic review inventory system, [6] discusses the optimal ordering policies with

two or three delivery modes, and with different variable ordering costs. [19] analyzes an inventory model

with two supply modes: one instantaneous and the other with a one period lead time. They obtain the

generalized optimal (s, S) policy with the fixed and variable ordering cost. [20] extends their work to a

periodic inventory model with fast, medium and slow delivery modes and demand forecast updates. It

shows the optimal ordering policy is a modified state-dependent base-stock policy. Instead of using different

supply modes, our model uses one supply with fixed quantity, but the supplier has the lead time option.

Using different lead times can effectively hedge the demand variability. Shorter lead times can be used

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when the demand surges unexpectedly, and longer lead times can be used when the demand drops. In

current competitive business environment, we believe that optimal choice of the lead time can improve the

profitability. Motivated by this, we develop a lead time option model in continuous review inventory system.

In the SC flexibility research, the main underlying model is the newsvendor model. Since the newsvendor

model is a single period model, it cannot be used to study lead time options. In our paper, we base our

model on (R, Q) system to address the important issue of lead time management. In (R, Q) systems, when

the inventory position is below the reorder point R, a fixed order quantity Q is placed immediately. Thus,

these systems are easy to operate, and are widely used in practice. In the SC literature, the (R, Q) model,

however, received less attention than the newsvendor model with the following notable exceptions. [26] works

on measuring the backorders for the (R, Q) system and shows that the expected cost of the (R,Q) system

is jointly convex in R and Q. [25] analyzes the properties of the (R,Q) system and compares to an EOQ

model. [5] develops an efficient algorithm to determine the optimal decision variables for the (R,Q) sys-

tem. [7] explores the cost bounds for (R,Q) system and a heuristic method to calculate the optimal R and Q.

3 Properties of (R, Q) Systems with Variable Lead Times

This section lays out important properties of the (R, Q) inventory systems. These properties will be used

in studying lead time flexibility in the next section. First, some standard notation is needed:

R, Q: Retailer’s reorder point and order quantity. Terms lot size and order quantity are synonymously used.

IP (t): Inventory position at time t.

T : Regular lead time (LT for short) for delivery to the retailer.

µ: Expected demand rate per time.

h, b: Inventory holding and backorder costs, each is in units of per unit and per time.

K: Fixed ordering cost.

∇if(x1, . . . , xi, . . . , xn): The derivative of f w.r.t ith argument.

∇ijf(x1, ..., xi, ...xn): The derivative of f w.r.t ith and jth arguments.

For t1 > 0, D(t1, t2) denotes the random demand over [t1, t2]. When t1 = 0, we suppress t1 in the

notation and write D(t2) := D(t1 = 0, t2). Demand rates over disjoint but equal time intervals are assumed

to be identically and independently distributed. We also assume that D(T ) stochastically increases in T .

This weak assumption is a natural outcome of nonnegative demands so a reasonable demand process should

satisfy it.

The adjective unimodal is used several times in the paper, its meaning slightly varies as follows. If the

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adjective unimodal applies to a density function, the density function is first increasing and then decreasing.

If it applies to a cost function, the cost is first nonincreasing and then nondecreasing.

3.1 The cost function

We now briefly present the components of the (R, Q) costs: ordering costs, inventory holding cost and the

backordering cost. Let G(y, T ) denote the expected inventory holding and backorder cost accumulation rate

at time t given the inventory position was y at time t− T :

G(y, T ) := E[h(y −D(T ))+ + b(D(T )− y)+] = (h + b)E(D(T )− y)+ − h(Tµ− y) (1)

The last equality follows from (y −D(T ))+ = (D(T )− y)+ − (D(T )− y).

When the cumulative demand is a nondecreasing stochastic process with stationary increments and

continuous sample paths ([18] and [25]), the expected per time cost C(R, Q, T ) of the (R, Q) system is

C(R,Q, T ) =Kµ +

∫ R+QR G(y, T )dy

Q= Kµ/Q + h(Q/2 + R− Tµ) + (h + b)B(R, Q, T ) (2)

where the B(R,Q,T) is average backorders per time

B(R, Q, T ) =

∫ R+QR E(D(T )− y)+dy

Q. (3)

From the discussion in [25] and [26], B(R, Q, T ), C(R, Q, T ) and G(y, T ), are all convex in their arguments

(R, Q) and y. For fixed lead time T , this is summarized in the next lemma. We also show that G(y, T ) has

nonpositive cross partial derivatives.

Lemma 1. For fixed lead time T , B(R, Q, T ) and C(R, Q, T ) are both convex in (R, Q), and G(y, T ) is

convex in y. In addition, ∇13B(R, Q, T ) ≤ 0, ∇13C(R,Q, T ) ≤ 0 and ∇23B(R, Q, T ) ≤ 0, ∇23C(R,Q, T ) ≤0, also ∇12G(y, T ) ≤ 0.

Nonpositive cross partials imply that longer lead times are more preferable with higher inventory po-

sitions. That is why the flexible lead times improve the cost. The assumption of stochastically increasing

D(T ) suffices for the proof of lemma.

In our lead time flexibility discussion, we would need to compute the per time costs from C(R,Q, T )

given IP (t = τ) at time τ > 0. Starting from the per time cost, we can rewrite it as follows

C(R, Q, T ) = Kµ/Q +∫ R+Q

RE[h(y −D(0, T ))+ + b(D(0, T )− y)+]dy/Q

= Kµ/Q + ED(0,τ)

∫ R+Q

RED(τ,T )[h(y −D(0, τ)−D(τ, T ))+ + b(D(0, τ) + D(τ, T )− y)+]dy/Q

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where the equality follows from interchanging expected value and integration which is justified by the

nonnegativity of the integrand. Now make a change of variable u = y−D(τ) and use D(τ, T ) = D(0, T − τ)

in distribution.

C(R, Q, T ) = Kµ/Q + ED(τ)

∫ R−D(τ)+Q

R−D(τ)ED(T−τ)[h(u−D(T − τ))+ + b(D(T − τ)− u)+]du/Q

= ED(τ) (C(R−D(τ), Q, T − τ)) (4)

Namely the per time cost is the average of the per time costs if it is known that IP (t = τ) = R + Q− d(τ)

and the average is taken over all possible values of d(τ).

3.2 Sensitivity of R and Q to lead time

Naturally the reorder points and lot sizes that minimize C(R, Q, T ) both change with T . However, the

variable that is more sensitive to T must be treated more carefully while optimizing costs under flexible

lead times. To measure the sensitivities, we respectively use the optimality equations for R and Q given in

[25] as G(R, T ) = G(R + Q,T ) and C(R,Q, T ) = G(R + Q,T ). The next lemma presents the sensitivity

of optimal R and Q with respect to lead time T . We also use numerical results of Table 1 to illustrate the

magnitude of these sensitivites. The computation of the derivatives in the table are based on the functions

defined in the proof of lemma.

Lemma 2. Optimal R increases but optimal Q decreases with T when the demand is normally distributed.

– Table 1 –

Observe that R is more sensitive to T than Q. Because of this and the reasons below, we will be primarily

concerned with finding optimal reorder points:

• There could be physical reasons limiting the lot size: truck/container capacities in transportation or

machine capabilities in production.

• We are not studying transportation or production costs in detail. These costs might have economies

of scale or quantity based discounts which may suggest a different Q than optimizing C(R,Q, T ) alone

does.

In Lemma 2, we study a normally distributed demand. This is a special case of scalable demands (used

for example in [26]) and defined by

Scalable demand: P (D(T ) ≤ y) = P

(D(T )− µT

σ√

T≤ y − µT

σ√

T

)= F

(y − µT

σ√

T

)(5)

where σ is the standard deviation of the demand rate. Scalability stipulates that (D(T )−µT )/(σ√

T ) has the

same distribution for different T values. Thus we can let f and F denote probability and cumulative density

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functions for (D(T ) − µT )/(σ√

T ). An important feature of scalability is that it leads to stochastically

increasing demand D(T ) over a sufficiently large part of the support of D(T ). This easily follows from the

equivalence of i) and ii)

i)∂F ((y − µT )/(σ

√T ))

∂T≤ 0 ii) y ≥ −µT = −E(D(T )). (6)

As it is clear in (2), we deal with inventory position y for y ≥ R. Since it is safe to assume that

R ≥ −E(D(T )), y ≥ R ≥ −E(D(T )) so (6) holds.

Before closing this subsection, we present another lemma on sensitivity. However, this time one variable

is fixed while the sensitivity of the other variable is discussed. This lemma will be useful while finding the

reorder points to minimize the costs.

Lemma 3. i) When Q is fixed, the optimal R is nondecreasing with lead time T .

ii) When R is fixed, the optimal Q is nondecreasing with lead time T .

iii) When Q is fixed, the optimal total per time cost minR C(R,Q, T ) for each T is nondecreasing with lead

time T .

3.3 Convexity of the cost in lead time

Intuitively speaking for every fixed Q > 0 and R > 0, the cost C(R,Q, T ) will decrease when T changes

from 0 to a positive number. On the other hand, the cost also decreases when T decreases from a very large

value. When demand variability is small enough to be ignored, the cost increases as T deviates from R/µ

in either direction. Consequently, it is reasonable to search for conditions when C(R, Q, T ) is convex in T .

One such condition dealing with mode Mode(D(T )) of a unimodal demand is given by the next lemma.

Lemma 4. C(R,Q, T ) is convex in T for R > Mode(D(T )).

Several remarks regarding R > Mode(D(T )) are in order. If f is unimodal and the demand is scalable,

then the density function f((y − µT )/(σ√

T ))/(σ√

T ) for D(T ) is unimodal as well. If f is also symmetric,

then Mode(D(T )) = µT . In that case, R > Mode(D(T )) merely says safety stock is nonnegative. Oth-

erwise, with negative safety stock, there will be shortages in every cycle with at least probability of 1/2.

When the cycle service level is high, then R > Mode(D(T )) will be satisfied easily so we believe that this

condition will hold for reasonable R values. Lemma 4 can be specialized for normally distributed demands

to weaken its assumption; see Lemma 11 in the appendix.

When demands have Poisson distribution, R and Q must be integers. Then the cost computation and

optimization become tedious although managerial insights are unlikely to change. It is possible to develop

the concepts in this paper with Poisson demand, in that case the following lemma is useful. Note that this

lemma does not require a service type constraint unlike Lemma 4.

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Lemma 5. C(R,Q, T ) is convex in T when demands have Poisson distribution.

In this section, we have introduced the cost function and studied its convexity in lead time. Moreover, we

have established that R is more sensitive to lead time changes than Q so we will primarily study optimizing

R in the remainder.

4 Lead Time Flexibility

In this section, the traditional (R,Q) inventory system is extended by introducing the lead time flexibility.

The lead time time flexibility is defined by two nonnegative parameters τ and α for τ + α < T . The former

(latter) determines when (by how much) the lead time can be modified. The classical (R, Q) systems corre-

spond to (τ ≥ 0, α = 0). On the other hand, (τ = 0, α > 0) indicates choosing a lead time for orders when

they are placed. However, a system with (τ = 0, α > 0) does not have the capability to observe a portion of

the demand and react accordingly. Moreover, such a system always trivially chooses the shortest lead time;

see Lemma 3.iii. Conversely, our intention with (τ > 0, α > 0) is to make the lead time a function of the

observed demand so that it can be used to fine tune inventory levels. In a sense, our lead time choice is a

recourse decision. It provides a chance to adapt the inventory decisions to the observed demand. Lead time

flexibility works as a hedge against extreme demand observations: If the inventory position or level is too

low (high), it may be wise to expedite (postpone) an order to avoid excessive backorders (inventory).

We now discuss the lead time flexibility mechanism. Suppose that an order is placed at time t with lead

time T . A lead time modification opportunity happens at exactly t + τ . In general, the retailer may like

to have a window of opportunity but such a broad flexibility may be hard to provide via transportation

or manufacturing. Thus, we stipulate that there is only one instantaneous opportunity for each order. By

time t + τ , the retailer observes the demand d(t, t + τ) and accordingly makes one of the three choices: (i)

Keep the lead time constant at T ; (ii) Increase the lead time to T +α; (iii) Decrease the lead time to T −α.

The retailer is guaranteed to receive the orders within the lead times it chooses. Depending on d(t, t + τ),

the retailer may like to choose a lead time in between T and T − α or T + α. However, such continuous

lead time choices are hard to provide. For example, transportation companies provide discrete lead time

options such as one day delivery, two day delivery, etc. Clearly, this lead time mechanism corresponds to

a real option on the lead time. Thus, τ and α can be called the exercise time and the magnitude of the option.

Next we develop the per time cost expression for (R, Q) systems with lead time flexibility. First recall

that the inventory position distribution has the uniform limiting distribution over (R, R + Q). Also note

that the inventory position is not affected by the choice of lead times. In particular, whether an arbitrary

infinite sequence of {LTi} are used or all LTi are the same, the limiting inventory position distribution is the

same. Let IP := IP (t) as time t goes to infinity. Then the per time expected inventory cost (not including

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fixed ordering cost) is computed by

EIP [G(y, T )|y = IP ] =∫ R+Q

RG(y, T )dy/Q.

From the arguments used to obtain (4), this cost becomes

EIP [G(y, T )|y = IP ] = ED(τ)

∫ R+Q−D(τ)

R−D(τ)G(y, T − τ)dy/Q.

Although the last expression is obtained algebraically, it admits an intuitive explanation. When τ time

passes after placing an order, the demand D(τ) during this time is observed. Supposing no other order

placements during τ time, the inventory position τ time after an order placement has the same distribution

as IP −D(τ). However, the remaining lead time is T − τ not T . As a result, the last expression is simply

the traditional cost formula with new inventory position random variable IP−D(τ) and new lead time T−τ .

When lead times T and T + α are available but one of them must be chosen at the time of ordering, the

per time expected inventory cost becomes

min

{ED(τ)

∫ R+Q−D(τ)

R−D(τ)G(y, T − τ)dy , ED(τ)

∫ R+Q−D(τ)

R−D(τ)G(y, T − τ + α)dy

}/Q.

This cost function is not very interesting to study because it only compares short and long lead time costs.

The more interesting case is delaying the choice of the lead time by τ times after an order placement. In

that case, the lead time responds to the initial demand D(τ) or the lead time is said to be adaptable to

the demand. To account for adaptability, we need to exchange min and ED(τ) in our per time expected

inventory cost expression

ED(τ) min

{∫ R+Q−D(τ)

R−D(τ)G(y, T − τ)dy ,

∫ R+Q−D(τ)

R−D(τ)G(y, T − τ + α)dy

}/Q.

Finally, we can incorporate the lead time T − α and fixed cost K to obtain the per time expected cost for

our lead time flexibility model

CF (R, Q, T ; α, τ) = ED(τ) min

C(R−D(τ), Q, T − τ − α),C(R−D(τ), Q, T − τ),C(R−D(τ), Q, T − τ + α)

. (7)

With the expected cost expression in hand, we turn to the optimal lead time policy. An intuitive policy

is as follows. When the inventory position is high (low) after observing d(τ) the retailer opts for the longer

(shorter) lead time. The policy is formally presented below.

Theorem 1. When C(R, Q, T ) is unimodal in T , there exist two unique critical levels I and I such that

the optimal lead time LT chosen by the retailer to minimize CF (R,Q, T ; α, τ) is

LT =

T − α, R− d(τ) ≤ I,T, I ≤ R− d(τ) ≤ I ,T + α, I ≤ R− d(τ)

. (8)

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Proof: Let I = R− d(τ) and define ∆(I) representing the per time cost difference when the lead time T +α

is used instead of T . It follows from (7) that

∆(I) = C(I, Q, T − τ + α)− C(I, Q, T − τ)

which can be rewritten by using (2)

∆(I) =∫ I+Q

I(G(y, T − τ + α)−G(y, T − τ))dy/Q

= −hµα + (h + b)∫ I+Q

I[E(D(T − τ + α)− y)+ − E(D(T − τ)− y)+]dy/Q.

Then the first derivative is

∇∆(I) = (h + b)∫ I+Q

I[P (D(T − τ + α) ≤ x)− P (D(T − τ) ≤ x)]dx/Q (9)

Since the demand during the lead time is stochastically increasing with lead time, the integrand is negative

so ∇∆(I) ≤ 0. Note that ∆(I = −Q) = bµα > 0. When inventory level I → ∞, ∆(I) = −hµα < 0.

By monotonicity and continuity of ∆(I), there is an I ′ such that ∆(I ′) = 0. Let I = min{I ′, R}. By

construction for R− d(τ) ≥ I, C(R− d(τ), Q, T − τ) ≥ C(R− d(τ), Q, T − τ + α) and the retailer exercises

the lead time option by using T + α as opposed to T . Similarly we can argue that there exists I, such that

when R− d(τ) ≤ I the retailer should use T − α instead of T .

We now show that I ≤ I. Suppose to the contrary that I > I. For I < I < I,

C(I, Q, T − τ) ≥ C(I,Q, T − τ + α) and C(I, Q, T − τ) ≥ C(I,Q, T − τ − α)

These inequalities contradict the hypothesis of the theorem; the cost is unimodal in T . ¤

In the last section, we have presented some sufficient conditions for the convexity (hence unimodality)

of costs. Thus the conclusions of Theorem 1 apply to fairly general cases. Using the theorem, (7) can be

rewritten as

CF (R, Q, T ;α, τ) = ED(τ){1(D(τ) ≥ R− I) C(R−D(τ), Q, T − τ − α)

+1(R− I ≤ D(τ) ≤ R− I) C(R−D(τ), Q, T − τ)

+1(D(τ) ≤ R− I) C(R−D(τ), Q, T − τ + α)} (10)

where 1(.) is the indicator function. Clearly, the expected cost with flexibility is always less than that

without flexibility; see Figure 1.

In view of (9), the critical levels I and I are only related to the order quantity Q and they are independent

of R (unless they are equal to R). The sensitivity of these levels to Q can be examined after letting

Θ(y) := E(D(T − τ + α)− y)+ − E(D(T − τ)− y)+.

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Cost

Critical levels

T-α

T+αT

I I

Figure 1: The cost of optimal lead time policy is the expectation of lower envelope of the costs under leadtimes T − α, T and T + α.

Lemma 6. The critical levels I and I are nonincreasing in order quantity Q:

−1 ≤ dI

dQ≤ 0 and − 1 ≤ dI

dQ≤ 0.

Proof: Argument for I is presented. Set (9) equal to zero and rearrange to obtain

∫ I+Q

IΘ(y)dy =

hµαQ

h + b(11)

Taking the implicit derivative of (11) with respect to Q

(∂I

∂Q+ 1

)Θ(I + Q)− ∂I

∂QΘ(I) =

hµα

h + b

Since Θ(y) is decreasing in y,

Θ(I + Q) ≤ hµα/(h + b) ≤ Θ(I),

which in combination with the implicit derivative yields

−1 ≤ ∂I

∂Q=

hµα/(h + b)−Θ(I + Q)Θ(I + Q)−Θ(I)

≤ 0. ¤

When the lot size is high, the critical levels are lower. Then it is more likely for retailer to choose the

longer lead time as opposed to the shorter lead time. That is, the retailer should not expedite large lot

sizes as often as it expedites small lot sizes because, prematurely expedited large lot sizes increase inventory

11

unnecessarily to a high level.

We can obtain only the equations (such as (11)) defining the critical levels, no closed form for the critical

levels can be found without further assumptions, say on demand distributions. Later on, we will take the

opposite direction and look for bounds on the critical levels when there is no information about the demand

distribution.

5 Computing Lot Sizes and Reorder Points

The expected total cost CF (R, Q, T ; α, τ) under lead time flexibility is given by (7). Lot size Q and reorder

point R are determined solely by the retailer. A glance at (7) reveals that CF is a minimum of three convex

functions (in R and Q) evaluated at three lead times T , T + α and T − α. Unfortunately, the convexity

of these three functions is not inherited by CF . For example as the uncertainty in D(τ) is dropped, CF

looks like a concatenation of three convex functions along the R axis. Consequently, we will look for weaker

results in R and Q directions.

We first focus on the reorder point and take the lot size as given, and propose an approximation to the

cost function CF . Recall I = R− d(τ), and let

Cτ (I,Q, T ) = min{C(I,Q, T − τ − α), C(I,Q, T − τ), C(I,Q, T − τ + α)}which is the cost with flexibility if the demand during [0, τ ] is known. Sometimes this function is unimodal

and other times it will be approximated with a unimodal function. Formally, let I(T ) := arg minI{C(I, Q, T};since Q is constant throughout our discussion of R, it is not included in I(T ) notation. I(T ) is the optimal

reorder level when the lead time is T .

We can show that

I ≥ I(T − τ) and I ≥ I(T − τ − α). (12)

First, Lemma 3 says C(I(T−τ), Q, T−τ) ≤ C(I(T−τ+α), Q, T−τ+α) which by the definition of I(T−τ+α)

implies C(I(T − τ), Q, T − τ) ≤ C(I(T − τ), Q, T − τ + α). By the same lemma I(T − τ) ≤ I(T − τ + α).

Then for I − τ ≤ I(T − τ) ≤ I(T − τ + α), 0 ≥ ∇1C(I, Q, T − τ) ≥ ∇1C(I,Q, T − τ + α) due to Lemma

1. Namely C(I, Q, T − τ) is smaller than C(I,Q, T − τ + α) at I = I(T − τ) and decreases slower on the

left side of I(T − τ). Then C(I, Q, T − τ) ≤ C(I, Q, T − τ + α) for I ≤ I(T − τ), these functions cross after

I(T − τ), i.e. I ≥ I(T − τ). By replacing each τ with τ − α starting after (12) in this paragraph, we can

also argue for I ≥ I(T − τ − α).

Because of Lemma 3, I(T − τ − α) ≤ I(T − τ) ≤ I(T − τ + α). Combining this with I ≤ I and (12),

only the following four cases are possible:

12

• Case 1: I ≤ I(T − τ) ≤ I ≤ I(T − τ + α)

• Case 2: I ≤ I(T − τ) ≤ I(T − τ + α) ≤ I

• Case 3: I(T − τ) ≤ I ≤ I ≤ I(T − τ + α)

• Case 4: I(T − τ) ≤ I and I(T − τ + α) ≤ I

In Case 4, it does not matter whether I ≤ I(T − τ + α) or not. The cost Cτ (I, Q, T ) is unimodal in Case

4. However, it is not unimodal in the remaining three cases, see Figure 2. We will approximate the cost

Cτ (I, Q, T ) using unimodal functions C1, C2 and C3.

Case 3

Case 1 Case 2

Case 4

I I

I

I

I

I

I I

Figure 2: Unimodal Approximation for the cost at time τ .

C1τ (I, Q, T ) =

max{C(I,Q, T − τ − α), C(I, Q, T − τ − α)} I < I,C(I,Q, T − τ) I < I < I,max{C(I,Q, T − τ + α), C(I , Q, T − τ + α)} I > I.

. (13)

C2τ (I, Q, T ) =

max{C(I,Q, T − τ − α), C(I, Q, T − τ − α)} I < I,C(I,Q, T − τ) I < I < I,C(I,Q, T − τ + α) I > I.

. (14)

C3τ (I, Q, T ) =

C(I,Q, T − τ − α) I < I,C(I,Q, T − τ) I < I < I,max{C(I,Q, T − τ + α), C(I , Q, T − τ + α)} I > I.

. (15)

13

C4τ (I,Q, T ) = Cτ (I). (16)

Finally we set Cτ (I, Q, T ) = Ciτ (I, Q, T ) under Case i for 1 ≤ i ≤ 4. The approximated cost function has

the desirable unimodal property.

Lemma 7. The approximated cost Cτ (I, Q, T ) is between Cτ (I,Q, T ) and C(I, Q, T ), and it is unimodal in

critical level I.

Now we provide an explanation of how the four cases may arise as the standard deviation σ of demand

increases. First fix all the inventory and cost parameters. Suppose that σ = 0, i.e. demand is deterministic,

then I(T − τ) = (T − τ)µ − Qh/(h + b) and C(I(T − τ − α), Q, T − τ − α) = C(I(T − τ), Q, T − τ) =

C(I(T − τ + α), Q, T − τ + α) so that Case 1 happens. As the standard deviation increases, all the costs

C(I,Q, T − τ −α), C(I(T − τ), Q, T − τ) and C(I,Q, T − τ + α) go up. The effect of standard deviation on

the costs is larger when the lead time is longer. Thus, the cost under lead time T − τ + α increases faster

than those under T − τ , which in return increases faster than those under (T − τ −α). In other words, Case

1 molds first into Case 2 or Case 3 and finally into Case 4. Since we use actual costs under Case 4, our

approximation improves as Case 4 becomes more likely, i.e. for larger standard deviations.

Finally, we obtain the approximate cost by taking the expected value over the demand during [0, τ ]:

CF (R, Q, T ;α, τ) := ED(τ)Cτ (R−D(τ), Q, T ). The next lemma establishes the unimodality of CF (R,Q, T ; α, τ).

Lemma 8. The retailer’s approximate expected cost CF (R, Q, T ; α, τ) is between CF (R, Q, T ;α, τ) and

C(R, Q, T ), and it is unimodal in R if the demand has PF2 density.

Note that both Normal and Poisson demand densities are PF2 so approximate costs with these densities

will be unimodal. Let R∗ be the reorder level minimizing approximate costs,

R∗ = argminRCF (R, Q, T ;α, τ)

Lemma 8 guarantees that R∗ is the unique critical value of R for the approximate cost function.

We now discuss the order quantity. Since reorder points are modified to absorb the lead time changes

(recall Lemma 2), the order quantities are not as sensitive to lead times as reorder points. Thus, with or

without lead time flexibility, optimal order quantities should be close. Besides, the costs with flexibility

are not necessarily convex in order quantities. However, once R is fixed we can restrict the optimal order

quantity to a line segment and search over that segment. Let Q(T ) = argminQC(R,Q, T ). By Lemma 3.ii),

Q(T −α) ≤ Q(T + α). For any given R, the line segment [Q(T −α), Q(T + α)] contains the optimal lot size

as formalized below.

Lemma 9. Given the reorder point R, the cost under lead time flexibility is nonincreasing in Q when

Q ≤ Q(T − α) and it is nondecreasing in Q when Q ≥ Q(T + α).

14

Although the optimal order quantity is restricted to [Q(T −α), Q(T +α)], there still can be local optima

within this segment. Let Q∗ be one of them. Then a procedure of iterating between the following steps i)

and ii) can be used to find a solution, where i) Find R∗ for a given Q∗, possibly starting with Q∗ = EOQ;

ii) For fixed R∗, find a local optimal Q∗ ∈ [Q(T − α), Q(T + α)] for CF (R∗, Q, T, ;α, τ).

We now compare costs under various lead time option parameters. But, we first establish that τ can

influence costs monotonically; higher τ values yield smaller costs.

Lemma 10. Costs under flexibility is nonincreasing in exercise time: CF (R, Q, T ;α, τ1) ≥ CF (R, Q, T ; α, τ2)

for τ1 ≤ τ2.

If we recall that lead time flexibility is indeed a real option on lead times, we can refer to (financial)

options literature for similar results where delaying exercise time of a call option is always beneficial (e.g.

p.110 of [12]).

To make further cost comparisons let CD(R, Q, T ) be the cost when the demand is deterministic. Then

we can order costs with the following theorem.

Theorem 2. For any fixed T ,α and τ , costs can be compared as follows

i) For any fixed R,Q,

CD(R, Q, T ) ≤ CF (R, Q, T ; α, T − α) ≤ CF (R,Q, T ; α, τ) ≤ CF (R, Q, T ;α, τ) ≤ C(R, Q, T )

ii) For any fixed Q,

minR

CD(R,Q, T ) ≤ minR

CF (R, Q, T ;α, T − α) ≤ minR

CF (R, Q, T ;α, τ)

≤ minR

CF (R,Q, T ; α, τ) ≤ minR

C(R, Q, T )

iii)

minR,Q

CD(R,Q, T ) ≤ minR,Q

CF (R, Q, T ;α, T − α) ≤ minR,Q

CF (R, Q, T ;α, τ)

≤ min{CF (R∗, Q∗, T ;α, τ), minR,Q

C(R, Q, T )}

Proof: In case of i) and ii), the first inequality is due to switching from deterministic to stochastic demand.

The second, and the third and fourth inequalities are respectively by Lemmas 10 and 8. The argument for

iii) is the same except that CF (R∗, Q∗, T ;α, τ) > minR,Q C(R, Q, T ) is very unlikely but possible. Therefore,

we have only three inequalities as opposed to four in i) and ii). ¤

For fixed Q, these differences among costs can be named as

15

• UG: Uncertainty Gap = minR C(R,Q, T )−minR CD(R, Q, T ).

• FG: Flexibility Gap = minR C(R,Q, T )− CF (R∗, Q, T ; α, τ).

• AFG: Approximate Flexibility Gap = minR C(R,Q, T )− CF (R∗, Q, T ; α, τ).

In our numerical experiments, we will specifically investigate AFG and FG, and their relative magnitude

against the optimal costs for the classical (R,Q) system.

6 Distribution Free Upper Bounds on Critical Levels

We now develop closed form upper bounds for two critical levels. These bounds are distribution free so they

are applicable in a wide range of cases, even when we lack the distribution information on demand but know

the mean and variance.

The first step of finding bounds is bounding Θ(y) from above. We present derivations only for I, first

note that G(y, T ) = (h + b)E(D(T )− y)+ − h(Tµ− y). Thus,∫ R+Q

RE(D(T )− y)+dy = [

∫ R+Q

RG(y, T )dy +

∫ R+Q

Rh(Tµ− y)dy]/(h + b)

≤ [(h + b)σ2/2 + h(R + Q− µ)2/2 + b(µ−R)2/2 + h

∫ R+Q

R(Tµ− y)dy]/(h + b)

=Tσ2 + (R− Tµ)2

2(17)

where the inequality is due to Lemma 1 of [7]. Note that Tσ2 is the demand variance during the lead time

T and R− µT is the safety stock.

On the other hand, using the Jensen’s inequality, we have

E(D(T − τ)− y)+ ≥ (E(D(T − τ))− y)+ = (µ(T − τ)− y)+; (18)

Then, we obtain∫ I+Q

IΘ(y)dy =

∫ I+Q

I[E(D(T − τ + α)− y)+ − E(D(T − τ)− y)+]dy

≤ (T − τ + α)σ2 + ((T − τ + α)µ− I)2

2−

∫ I+Q

I(µ(T − τ)− y)+dy (19)

where the inequality is due to the (17) and (18).

Let I0 be the solution for the following equation

(T − τ + α)σ2 + ((T − τ + α)µ− I)2

2−

∫ I+Q

I(µ(T − τ)− y)+dy =

hµαQ

h + b(20)

16

Since Θ(y) is decreasing in y, I0 is an upper bound for I. We compute the integral above under three cases

∫ I+Q

I(µ(T − τ)− y)+dy =

Q(µ(T − τ)− I0 −Q/2) Case 0 : I0 ≤ µ(T − τ)−Q,1/2(µ(T − τ)− I2

0 ) Case 1 : µ(T − τ)−Q ≤ I0 ≤ µ(T − τ),0 Case 2 : µ(T − τ) ≤ I0.

. (21)

Case 0: I0 ≤ µ(T − τ)−Q, (20) becomes

(T − τ + α)σ2 + ((T − τ + α)µ− I0)2

2−Q(µ(T − τ)− I0 −Q/2) =

hµαQ

h + b

Observing that the discriminant of this quadratic equation is negative, we cannot find a bound for I. The

reason for this is I0 + Q < µ(T − τ); When the order is received, the inventory level will be below zero.

Clearly, the order must have been expedited which can be achieved with a larger I0.

Case 1: µ(T − τ)−Q ≤ I0 ≤ µ(T − τ), equation (20) becomes

(T − τ + α)σ2

2+

(µ(T − τ + α)− I0)2

2− (µ(T − τ)− I0)2

2=

hµαQ

h + b(22)

Solving for I0, we find that

I0 = µ(T − τ) +µα

2+

(T − τ + α)σ2

2µα− hQ

h + b(23)

The solution exists only if the minimum value of the left hand side of (22) is smaller than hµαQ/(h + b).

Since the left hand side is increasing in I0, it takes the minimum value at I0 = µ(T − τ)−Q and becomes

(T − τ + α)σ2/2 + (µα)2/2. Then the condition for existence is

(T − τ + α)σ2/2 + (µα)2/2 ≤ hµαQ/(h + b). (24)

Note also that under (24), I0 ≤ µ(T − τ). To guarantee that I0 ≥ µ(T − τ)−Q, we need µα/2 + (T − τ +

α)σ2/(2µα) ≥ −Qh/(h+ b) which is trivially satisfied. As a result under (24), we use (23) to obtain I0, and

I0 ∈ [µ(T − τ)−Q,µ(T − τ)] as desired.

Case 2: I0 ≥ µ(T − τ), equation (20) becomes

(T − τ + α)σ2 + ((T − τ + α)µ− I0)2

2=

hµαQ

h + b

this has the solution

I0 = µ(T − τ + α)−√

2hµαQ

h + b− (T − τ + α)σ2 (25)

Clearly, the solution exist only if

(T − τ + α)σ2/2 ≤ hµαQ/(h + b) (26)

17

It is also straight forward that I0 ≥ µ(T − τ) under this condition.

To find the upper bound for I, we first check if condition (24) and/or (26) hold. Note that (24) implies

(26), so both may hold simultaneously. Since upper bounds under Case 1 are smaller, we prefer Case 1.

Hence, we use (23) to compute I0 if both conditions hold. If (24) fails but (26) holds, (25) yields the

correct upper bound for I0. If (26) fails, an upper bound cannot be found from (20). This is because when

the demand variance is large the approximation for∫ I+QI

Θ(y)dy is crude in that the actual function is

nonincreasing in I while the approximation is not. We will later numerically investigate the quality of upper

bounds and when they exist especially as the variance increases.

7 Flexible Lead Time Providers

Up to now, the benefits of the lead time flexibility for the retailer is examined. Now we discuss how delivery

lead times can be (made) flexible by adjusting transportation times or manufacturing operations in the

upstream SC. With this discussion, the fundamental trade offs in a SC can be analyzed.

Transportation Time Flexibility: Delivery lead times to a retailer include transportation times from

an upstream facility to the retailer. There often are several transportation options available, these options

can be used to build lead time flexibility. For a simple example, consider cargo airlines which charge different

transportation rates for different transportation times, such as overnight, two-day and there-day deliveries.

See Figure 3 where two delivery options are shown with α = 1 day.

Manufacturing Time Transportation Time

0 Tτ

Two day delivery

Manufacturing Time Transportation Time

0 Tτ

Three day delivery

T+α

Figure 3: Transportation lead time flexibility.

Different transportation options have different prices. Since we want to study the effect of the lead time

18

on a SC, we assume that all costs are incurred by the SC and the benefits are reaped by the SC. In practice,

the SC pays higher costs for shorter lead times than the the longer lead times. To keep the analysis simple,

we assume that when the delivery lead time is decreased or increased by α, the SC pays cT — the fixed cost

of transportation flexibility. The relevant per time cost of the lead time flexibility to SC is

cT

Q/µP (LT = T − α or LT = T + α). (27)

This cost will allow us to study the trade off between the inventory cost savings and extra transportation

costs due to lead time flexibility. We will numerically evaluate the contribution of the lead time flexibility

to a SC by comparing these costs.

Manufacturing Time Flexibility: In general, how (if) a manufacturer can respond to lead time mod-

ifications depends on to what extent the manufacturer can reschedule its operations. Most manufacturers

use the concept of frozen and flexible time zones for rescheduling. Inside the frozen zone, a job can be

cancelled or terminated prematurely but these modifications cannot affect the other jobs. We believe that

lead times are often short enough to fall within the frozen zone. Consequently a manufacturer working juts-

in-time (JIT) has only one option to deal with emergency lead time reductions: If the lead time is pulled

down to T − α by the retailer, the manufacturer produces at its usual rate Rm until the new delivery time.

At the new delivery time, the cumulative production will be less than the lot size Q so the manufacturer

completes the lot by subcontracting, overtime or buying from a spot market. In all of these cases, the

production/purchase cost per item will be larger than planned in order to provide lead time flexibility, let

cm denote the difference. When the manufacturer uses overtime or subcontracting, cm may increase slightly

with τ . When the spot market is used, cm is independent of τ . We adapt the latter point of view. On the

other hand, if the retailer changes the lead time to T +α, the lot size Q is finished at the previously planned

time. However, then the manufacturer carries an inventory of Q units over α time and deliver them to the

retailer α time later.

If Q/Rm ≥ α, the manufacturer must stop producing prematurely and lacks αRm many items to complete

the lot to Q. The cost of obtaining these αRm items using the subcontracting/overtime or spot market is

αRmcm. If Q/Rm ≤ α, the production, according to the frozen schedule, does not start before the desired

delivery time. Consequently, it must be cancelled and all of Q units must be obtained using alternative

sources, hence leading to the extra cost of Qcm. There is a bright side of shorter lead times for the

manufacturer; the finished items are sent to the retailer faster. This saves the manufacturer from paying an

inventory holding cost at a rate of hm. If Q/Rm ≥ α, a part of the lot Q is shipped early which gives an

inventory holding cost saving of (Q − αRm/2)αhm. Otherwise, the inventory cost savings is for the entire

lot, i.e. (Q/2)αhm. When the retailer wants a longer lead time, the inventory holding cost works against

the manufacturer; The entire lot of Q is held in the inventory for α time more which lead to extra inventory

costs of αQhm. The manufacturer’s flexibility costs can be converted from per order to per time and be

19

expressed as

CM =[cm min{αRm, Q} − αhm(Q− (min{αRm, Q})/2]P (LT = T − α) + [αhmQ]P (LT = T + α)

Q/µ(28)

To compute this cost, the manufacturer must know how probable the retailer is to change the lead times.

These probabilities can be computed / updated from historical data. Thus, the retailer cannot cheat by

underestimating the probabilities, at least over a long run. Note that CM is increasing in cm and Rm. Rm

is the production rate the manufacturer allocates to the retailer. Smaller Rm allows the manufacturer to

extend the production over a long time so that when the retailer wants LT = T − α, most of the lot size

Q is already finished. This observation applies only to our manufacturer which uses JIT within a relatively

long frozen zone for scheduling.

The lead time option will improve the performance of SC only if the cost saving at the retailer is bigger

than the cost increases at the manufacturer. This condition can be written as

SCFG := FG− CM ≥ 0

where SCFG is SC’s flexibility gap. The condition depends on many cost and demand parameters. We use

numerical experiments to discuss when the SC can improve its performance with lead time options.

A word of caution is in order here. It appears as if the transportation and manufacturing flexibility costs

do not vary with the exercise time τ . However, τ affects these costs via the probabilities P (LT = T − α) =

P (D(τ) ≥ R− I) and P (LT = T − α) = P (D(τ) ≤ R− I).

8 Numerical Experiments

To deepen our managerial insights in the absence of closed-form expressions, we numerically manipulate

these expressions to study the performance measures of interest such as AFG, FG, SC savings with lead

time options. We vary the following aspects: i) Exercise time τ ; ii) Magnitude α of the option; iii) Demand

uncertainty σ; iv) Costs K,h, b. We first fix a base case defined by K = 50, h = 15, b = 35, µ = 50, σ = 12,

for the retailer, T = 1, τ = 0.5 and α = 0.25 for the lead time option and hm = 5, cm = 5 and Rm = 60

for the manufacturer. We do not set a parameter value for cT of the transporter. Instead we solve for the

maximum transportation flexibility cost cT that makes flexibility profitable for the SC when the retailer

dictates the reorder point and the quantity to the SC. The parameters are usually varied one by one around

the base case to measure the performance of supply chain.

Numerical results are presented in Tables 2 - 6. In all tables, we put the varying parameter(s) into

the first (and second) column. The next three columns include optimal decision variables (R∗, Q∗) and

costs C := C(R∗, Q∗, T ) for the classical (R, Q) system. The next column has the optimal reorder point

20

R∗ for the approximate cost function CF (R, Q∗, T ; α, τ). The next four columns include approximate

CF (R∗, Q∗, T ; α, τ) and actual CF (R∗, Q∗, T ; α, τ) costs , and how they compare against the classical (R, Q)

system cost, i.e. AFG/C(R∗, Q∗, T ) and FG/C(R∗, Q∗, T ) in percentages. After the performance measures,

four columns are dedicated to values of critical levels I and I and their upper bounds denoted by I0 and

I0. The next column includes the maximum transportation flexibility cost cT . The next two columns have

the manufacturer’s flexibility cost CM and the corresponding SC flexibility gap SCFG. The last column of

each table is an upper bound Pc on the order crossing probabilities.

In all of our experiments, we first solve the retailer’s problem. Then the retailer’s decisions are imposed

onto the flexible lead time providers to compute the SC costs. It is possible to jointly optimize retailer’s and

the flexible lead time provider’s decisions with an objective of a total SC costs. Since our aim is to study

lead time flexibility rather than the virtues of the joint optimization, we use sequential optimization. The

SC costs savings with joint optimization would even be higher than we report below.

– Tables 2 - 6 –

Exercise time τ : Examining Table 2, we can easily verify Lemma 10 that the optimal exercise time

for the retailer is as late as possible, i.e. T − α. As τ increases, the retailer reaps more savings and can

share those savings with the transporter so cT increases. On the other hand, the cost of manufacturing

time flexibility is increasing with exercise time. We calculate the probability of modifying lead time for

τ = 0.1 and τ = 0.7, the corresponding values are 0.13 to 0.54: The later the retailer exercises the lead

time option, the higher the probability is to modify the lead time. However, SCFG still increase with ex-

ercise time. Therefore, under our parameter values, the optimal exercise time for the SC appears to be T−α.

The magnitude of lead time option α: In Table 3, the retailer’s savings with lead time flexibility

as indicated by FG is first increasing until α reaches 25% of the lead time and decreasing afterwards. This

is easy to understand for small α as it indicates limited flexibility. But, the large values of α does not really

help fine-tuning the lead time either. Since then flexibility is larger but seldom used, its benefits drop. How-

ever high transportation costs can still be afforded when flexibility is used, so cT increases. Manufacturing

flexibility costs also first increase and then decrease in α. This is because, first rising α is the dominating

factor in how the manufacturing flexibility cost moves, but then the probability of the lead time modification

becomes the dominating factor.

Demand uncertainty σ: From Table 4, the benefits of flexibility to the retailer increase with de-

mand uncertainty. These higher benefits justify higher transportation flexibility costs as cT increase. The

manufacturing time flexibility costs have a mixed behavior — first going up then coming down. The drop is

due to growing lot size Q with σ. If we examine manufacturing flexibility costs per cycle as opposed to per

21

time, those costs go up. Namely, manufacturer finds it harder to deal with an uncertain demand in each

cycle but each cycle becomes longer with more demand uncertainty. As SCFG indicates, higher σ makes

lead time flexibility attractive for the SC. When the uncertainty is too high, upper bounds I0 or I0 do not

exist as there is no solution to (26). Then we write N/A into I0 or I0 columns.

Ratio of holding cost to backordering cost h/b: Instead of varying h and b independently, we

vary h and b together while keeping h + b constant in Table 5. Both classical and flexible lead time (R, Q)

system costs increase with h/(h + b) and both attempt to keep costs reasonable by reducing inventory —

with smaller reorder points. However, the flexible lead time system can also regulate lead time so its cost

is increasing at a smaller rate than the classical system. With larger savings due to flexibility, the retailer

can afford to pay higher transportation costs so cT increases. The manufacturing costs slightly increase in

h/(h + b) mostly because the probability of lead time modification goes up.

Fixed cost K: Table 6 illustrates that the significance of the lead time flexibility disappears (AFG

and FG drop) as K increases because the fixed cost constitutes most of the per time costs. However, cT

still increases as it is paid per cycle which become longer with K. Because of the same reason, the per time

cost CM also drops so the lead time flexibility remains as a viable option for the SC. In this case, however,

it may be wiser to reduce high fixed costs before adjusting demand uncertainty via flexible lead times.

In order to appreciate the lead time flexibility better, we can measure its effectiveness in a way that does

not depend on K. Consider a stochastic demand (R, Q) system and the corresponding deterministic model

with backorders. Clearly, their cost difference UG is an upper bound for cost savings due to flexibility. We

can compute what percentage of UG can be saved by using lead time flexibility. To compute the UG for

the base case, we first minimize the deterministic cost Cd(R, Q, T ), which yields Rd = µT − (hQd)/(h + b),

Qd = (2µK(h + b)/(hb))0.5, and the optimal cost Cd(Rd, Qd, T ) = hbQd/(h + b) (see e.g. p.94 of [25]).

Using the parameter values of the base case, Rd = 43.45, Q = 21.82, Cd(Rd, Qd, T ) = 229.13 and

UG = 341.68 − 229.13 = 112.55. In the base case, FG = 41.42. Therefore, flexible lead times decrease

36.8% (=41.42/112.55) of the cost that can be decreased.

Upper bounds I0 and I0: In all of our numerical studies, upper bounds approximate actual I and

I fairly well. The bounds become worse as σ increases because we used Jensen’s inequality in their derivation.

Upper bound for order crossing probability Pc: When there is only one outstanding order, i.e.

Q/µ is large enough with regard to α and σ is small, there is no order crossing. Otherwise, orders can cross

although we build our models on no-order crossing assumption. We now compute simple upper bounds for

order crossing probabilities. Consider two arbitrary orders placed one after another with lead times LT1 and

LT2. These orders can cross in the following three cases: i) LT1 = T + α, LT2 = T , D(α) ≥ Q; ii) LT1 = T ,

22

LT2 = T − α, D(α) ≥ Q; iii) LT1 = T + α, LT2 = T − α, D(2α) ≥ Q. Recall that Pc is an upper bound for

Pc — the order crossing probability.

Pc = P (LT1 = T + α, LT2 = T, D(0, α) ≥ Q) + P (LT1 = T, LT2 = T − α, D(0, α) ≥ Q)

+ P (LT1 = T + α, LT2 = T − α, D(0, 2α) ≥ Q)

≤ P (D(0, τ) ≤ R− I , D(0, α) ≥ Q) + P (R− I ≤ D(0, τ) ≤ R− I,D(0, α) ≥ Q)

+ P (D(0, τ) ≤ R− I , D(0, 2α) ≥ Q)

where the inequality follows by ignoring the probability for the second order. Furthermore, depending on

the values of α and τ , the P c can be written as follows:

If 0 ≤ τ ≤ α,

Pc ≤ P (D(0, τ) ≤ R− I , D(τ, α) ≥ Q−R + I) + P (D(0, τ) ≤ R− I,D(τ, α) ≥ Q−R + I)

+ P (D(0, τ) ≤ R− I)P (D(τ, 2α) ≥ Q−R + I)

= P (D(τ) ≤ R− I)P (D(α− τ) ≥ Q−R + I) + P (D(τ) ≤ R− I)P (D(α− τ) ≥ Q−R + I)

+ P (D(τ) ≤ R− I)P (D(2α− τ) ≥ Q−R + I) (29)

where the inequality follows from the implication relationships among the following events

(D(0, τ) ≤ R− I , D(0, α) ≥ Q

) ⇒ (D(0, τ) ≤ R− I , D(τ, α) ≥ Q−R + I

)(R− I ≤ D(0, τ) ≤ R− I,D(0, α) ≥ Q

) ⇒ (D(0, τ) ≤ R− I,D(τ, α) ≥ Q−R + I)(D(0, τ) ≤ R− I , D(0, 2α) ≥ Q

) ⇒ (D(0, τ) ≤ R− I , D(τ, 2α) ≥ Q−R + I

)

and the equality follows from the independence of demands over disjoint time intervals. Similarly we have

two other cases: 0 ≤ α ≤ τ ≤ 2α and 0 ≤ 2α ≤ τ . The upper bounds for these cases can similarly be

developed.

Depending on values of α and τ , we calculated the order crossing probability upper bound Pc, the values

of Pc are reported in Tables 2 - 6. The largest bound is 6.54% (when α = 0.4), the average is less than

1% and in most cases it is almost 0. Such small values of the bound imply even smaller values for actual

crossing probabilities and support our assumption of no-order crossing.

Short or flexible lead times: Finally, we compare SC costs under short lead time LT = T − α and

flexible lead time LT ∈ {T, T ± α}. SC costs with lead time flexibility is computed by summing up the

retailer’s (R, Q) system costs and the manufacturer’s costs CM defined in (28). When only the short lead

time is available to the retailer, its cost is C(R, Q, T −α). In that case, the manufacturer’s cost is computed

by using (28) after setting P (LT = T −α) = 1−P (LT = T )−P (LT = T +α) = 1 and using the optimal Q

corresponding to LT = T − α. The computation of the retailer’s and manufacturer’s costs under lead time

23

flexibility is the same as before in Tables 2 - 6. Table 7 presents the SC costs and their difference in the last

column. Since the last column is always positive, the SC with lead time flexibility performs better than the

SC with short lead time in our experimental settings. This appears to contradict the established notion that

short lead times are always better than longer lead times in inventory systems. Our results indicate that

by using a careful mixture of lead times from {T, T ± α} instead of the minimum lead time in the mixture

(T − α), SC performance can be improved.

– Table 7 –

9 Conclusion

This paper explores the continuous review (R, Q) inventory system with lead time flexibility. It illustrates

that the reorder point is more sensitive to the lead time than the order quantity, hence it mostly focuses on

analyzing the optimal reorder point. The paper discusses the convexity of the costs in lead time, and con-

firms it when the demand has Poisson or Normal (under some conditions) distributions. Although convexity

is of interest on its own, it is used to derive the optimal lead time policy for operating the flexible lead time

(R, Q) system. Since the parameters of this policy is not in closed form, the paper provides closed form and

distribution free upper bounds for the parameters. For the reorder point under the lead time flexibility, the

paper first approximates the cost with a unimodal function and then minimizes its expectation which is also

shown to be unimodal. With this approximate solution, a considerable portion of inventory costs can be

saved. A major contribution of the paper is in tackling lead time flexibility, especially because the inventory

theoretic study of lead times does not seem to be very popular among researchers.

The paper aims to draw managerial insights for lead time flexibility and provides the following observa-

tions. The retailer should always delay the choice of lead time as late as possible, i.e. should exercise its lead

time option later rather than earlier. In our settings, delaying the exercise time, in spite of increasing man-

ufacturer’s costs, provides higher profits for the SC. On the other hand, flexibility savings are not monotone

in the magnitude of the lead time option. A moderate value of α, 25% of the lead time in our experiments,

provides the most savings to the retailer. We illustrate that the lead time flexibility helps the retailer most

when the demand uncertainty is high. Also our experiments on inventory cost parameters indicate that

retailer’s savings due to flexibility grows when the holding cost rises. In addition, when the fixed cost is

large, flexibility cost savings are small; then the retailer must first deal with the large fixed cost instead

of demand uncertainty. To provide a broader perspective, flexible lead time providers, a transporter and a

manufacturer in a SC, are discussed and their costs are modelled. The costs of the retailer and the flexible

lead time providers determine if the SC should implement a flexible lead time strategy. As our experiments

illustrate, flexible lead times do not necessarily improve the SC performance. We are not advocating for

their common use but for their analysis.

24

In order to formulate a flexible lead time model, various assumptions are made: scalability assumption

plays a role in some of our results. Our model assumes full backordering as opposed to lost sales. We as-

sume no-order crossing and support this with numerical experiments. Lead time flexibility magnitude α is a

number, it could be given as an interval requiring the lead time to be modified to a number in [T −α, T +α].

Optimization of lot size can also be studied in more details. We assume that the manufacturer uses frozen

production schedule with JIT. Moreover, we sequentially optimize SC costs after noting that joint optimiza-

tion can yield better SC cost savings. Dealing with each of these issues and assumptions will yield a venue

for future research.

Acknowledgement: This research is supported by the University of Texas at Dallas summer research

grants.

25

Appendix: Proofs of Lemmas

Proof of Lemma 1: We first prove ∇13B(R, Q, T ) ≤ 0 which implies ∇13C(R, Q, T ) ≤ 0 by (2). Note that

∇1B(R, Q, T ) = −∫ R+Q

RP (D(T ) ≥ x)dx/Q.

Since P (D(T ) ≥ x) grows as T increases, ∇1B(R,Q, T ) is nonincreasing in T so ∇13B(R, Q, T ) ≤ 0. For

∇23B(R,Q, T ), first consider

∇2B(R, Q, T ) = E(D(T )−R−Q)+/Q−∫ R+Q

RE(D(T )−y)+dy/Q2 = −

∫ R+Q

R

∫ R+Q

yP (D(T ) ≥ x)dxdy/Q2.

Once more P (D(T ) ≥ x) increases in T so ∇23B(R,Q, T ) ≤ 0 and by (2) ∇23C(R, Q, T ) ≤ 0. Argument

for ∇12G(y, T ) ≤ 0 is similar. ¤

Proof of Lemma 2: First write the optimality equation for Q as

Kµ +∫ R+Q

RG(y, T )dy = QG(R + Q,T ).

By taking the implicit derivative of the optimality equations for R and Q with respect to T , we obtain two

simultaneous equations. After solving these equations

dR

dT= −∇2G(R, T )

∇1G(R, T )and

dQ

dT=∇2G(R, T )∇1G(R, T )

− ∇2G(R + Q,T )∇1G(R + Q,T )

.

Let φ and Φ be pdf and cdf for a standard normal random variable. Using the properties of these functions,∫∞t Φ(u)du = −t(1− Φ(t)) + φ(t)

G(y, T ) = (h + b)(

σ√

Tφ

(y − Tµ

σ√

T

)− (y − Tµ)

(1− Φ

(y − Tµ

σ√

T

)))+ h(y − Tµ).

Taking derivatives

∇1G(y, T ) = −(h + b)(

1− Φ(

y − Tµ

σ√

T

)− h

h + b

)

∇2G(y, T ) = (h+b)µ(

ρT

2φ

(y − Tµ

σ√

T

)+ 1− Φ

(y − Tµ

σ√

T

)− h

h + b

)= (h+b)

µρT

2φ

(y − Tµ

σ√

T

)−µ∇1G(y, T ).

where ρT = σ/(µ√

T ). Since G(y, T ) is convex in y and G(R, T ) = G(R + Q,T ), ∇1G(R, T ) ≤ 0. This

inequality implies ∇2G(R, T ) ≥ 0. Last two inequalities yield dR/dT ≥ 0.

For convenience, we define the sensitivity of R as H(R), that is

H(R) := −∇2G(R, T )∇1G(R, T )

= µ +µρT

2

φ(

R−Tµ

σ√

T

)

1− Φ(

R−Tµ

σ√

T

)− h

h+b

.

26

Now using ∇1G(R, T ) ≤ 0 and ∇1G(R + Q,T ) ≥ 0, clearly H(R) ≥ µ and H(R + Q) ≤ µ. Going back to

the amount of change in QdQ

dT= −H(R) + H(R + Q) ≤ 0.

It is more interesting to study the relative magnitude of sensitivities as T varies

|dQ/dT ||dR/dT | =

| −H(R) + H(R + Q)||H(R)| = 1− H(R + Q)

H(R).

Thus R changes more than Q if and only if H(R + Q) ≥ 0. Using the definition of H, substituting z =

(R+Q−Tµ)/(σ√

T ) and some algebra, the last condition can be written as ρT φ(z)/2 ≤ −1+Φ(z)+h/(h+b).

The inequality will generally be satisfied; see Table 1. ¤

Proof of Lemma 3: i) and ii) follow respectively from ∇13C(R,Q, T ) ≤ 0 and ∇23C(R, Q, T ) ≤ 0, and

the convexity of C(R, Q, T ) in R and Q, see Lemma 1. For iii) we provide a constructive proof. Let R(T )

to denote the optimal reorder point corresponding to lead time T . Consider three (R, Q) inventory systems,

see Figure 4, parameterized for an arbitrary τ ≥ 0 as follows.

System 1. Lead time T + τ and reorder point R(T + τ).

System 2. Lead time T and reorder point R(T + τ), but release orders τ time after IP reaches R(T + τ).

System 3. Lead time T , use reorder point R(T ).

Clearly the costs of system 1 and 2 are the same: C(R(T + τ), Q, T + τ). In system 2, we delay an order

by τ time and release at IP ′ = R(T + τ) −D(τ). C(IP ′, Q, T ) ≥ C(R(T ), Q, T ), because R(T ) is optimal

with respect to T . Then using (4)

C(R(T + τ), Q, T + τ) = ED(τ)C(R(T + τ)−D(τ), Q, T ) ≥ C(R(T ), Q, T )

which implies minR C(R, Q, T + τ) ≥ minR C(R, Q, T ). ¤

Proof of Lemma 4: We show that B(R, Q, T ) is convex in T . Since B(R,Q, T ) is computed by integrating

E(D(T ) − y)+ from y = R to y = R + Q, R > Mode(D(T )) implies y > Mode(D(T )). For small a1, we

first argue that E(D(T )− (y − a1))+ has a first derivative in a1 that is convex in a1:

∂E(D(T )− (y − a1))+

∂a1= P (D(T ) ≥ y − a1)

Note that P (D(T ) ≥ y−a1) is convex for y−a1 ≥ Mode(D(T )) because the density of D(T ) is nonincreasing

in this region.

Then for any sufficiently small number a2 ≥ 0, E(D(T ) − (y − a1 − a2))+ − E(D(T ) − (y − a1))+ is

convex in a1. Moreover, this difference is nondecreasing in a1 because E(D(T )− (y − a1))+ is convex in a1

27

R(T)

R(T+τ)

TimeT

T+τ

τ

InventoryPosition

Figure 4: Operation of system 2 and 3 with different demand realizations.

by Lemma 1. For some small but positive ε and δ,

∂E(D(T + ε)− y)+

∂T= lim

δ→0E

((D(T )− (y −D(ε)−D(δ)))+ − (D(T )− (y −D(ε)))+

δ

)

≥ limδ→0

E

((D(T )− (y −D(δ)))+ − (D(T )− y)+

δ

)

=∂E(D(T )− y)+

∂T(30)

The inequality follows from the increasing and convex properties of the difference E(D(T )−(y−a1−a2))+−E(D(T ) − (y − a1))+ where a1 and a2 are taken as D(ε) and D(δ) respectively. It is also due to the fact

that D(ε) is more variable than 0 in the sense of p.433 of [22]. Finally, (30) directly implies the convexity

of E(D(T )− y)+ in T . On the other hand,

B(R, Q, T ) =∫ R+

RE(D(T )− y)+dy/Q +

∫ R+Q

R+

E(D(T )− y)+dy/Q

= µT (R+ −R)/Q +∫ R+Q

R+

E(D(T )− y)+dy/Q

Since the first term is linear in T and the second term is just shown to be convex, the sum B(R, Q, T ) is

convex in T . ¤

Lemma 11. C(R,Q, T ) is convex in T if demand is normally distributed and σ2 ≤ 4µR.

Proof: According to [26], backorders can be written as a function of order quantity Q, safety stock s

(= R− µT ) and lead time demand standard deviation σT = σ√

T :

B(Q,R, T ) = B(Q, s, σT ) :=σ2

T {β(s/σT )− β((s + Q)/σT )}Q

(31)

28

where β(z) =∫∞z (t− z)[1−Φ(t)]dt. Taking the second derivative of B(Q, s, σT ) with respect to T, we have:

∂2B(Q, s, σT )∂T 2

=∂2B(Q, s, σT )

∂σ2T

(∂σT

∂T)2 +

∂B(Q, s, σT )∂σT

∂2σT

∂T 2+

∂2B(Q, s, σT )∂σT ∂s

∂σT

∂T

∂s

∂T

+∂2B(Q, s, σT )

∂s2(∂s

∂T)2 +

∂B(Q, s, σT )∂s

∂2s

∂T 2+

∂2B(Q, s, σT )∂σT ∂s

∂σT

∂T

∂s

∂T

where ∂σT /∂T = σT−1/2/2, ∂2σT /∂T 2 = −σT−3/2/4, and ∂s/∂T = −µ,∂2s/∂T 2 = 0. We drop arguments

of B, then

∂2B

∂T 2=

σ2

4T

(∂2B

∂σ2T

− 1σT

∂B

∂σT

)+

∂2B

∂s2µ2 − ∂2B

∂σT ∂sµσT−1/2

so that after recalling ρT = σ√

T/(µT )

1µ2

∂2B

∂T 2=

ρ2T

4

(∂2B

∂σ2T

− 1σT

∂B

∂σT

)+

∂2B

∂s2− ρT

∂2B

∂σT ∂s(32)

We will show that the right hand side of (32) is nonnegative. Towards that we set z = s/σT and δ = Q/σT ,

we borrow the following equalities from [26]:

∂B

∂σT=

σT {2β(z)− 2β(z + δ)− zβ′(z) + (z + δ)β′(z + δ)}Q

∂2B

∂σ2T

=[2β(z)− 2zβ′(z) + z2β′′(z)]− [2β(z + δ)− 2(z + δ)β′(z + δ) + (z + δ)2β′′(z + δ)]

Q

∂2B

∂s∂σT=

[β′(z)− zβ′′(z)]− [β′(z + δ)− (z + δ)β′′(z + δ)]Q

∂2B

∂s2=

β′′(z)− β′′(z + δ)Q

Using these equations along with β′(z) = − ∫∞z [1 − Φ(t)]dt = −φ(z) + z[1 − Φ(z)], β′′(z) = 1 − Φ(z) and

β′′′(z) = −φ(z),

∂2B

∂σ2T

− 1σT

∂B

∂σT=

−zβ′(z) + (z + δ)β′(z + δ) + z2β′′(z)− (z + δ)2β′′(z + δ)Q

=zφ(z)− (z + δ)φ(z + δ)

Q

∂2B

∂s2− ρT

∂2B

∂σT ∂s=

β′′(z)− β′′(z + δ)− ρT [β′(z)− zβ′′(z)] + ρT [β′(z + δ)− (z + δ)β′′(z + δ))]Q

=−Φ(z) + ρT φ(z) + Φ(z + δ)− ρT φ(z + δ)

Q

29

We combine the last two equations to get

Q

µ2

∂2B

∂T 2= D(z)−D(z + δ) where D(z) = (ρT /2)2zφ(z)− Φ(z) + ρT φ(z).

To finish the proof, it suffices to argue that D(z) is decreasing. After recalling φ′(z) = −zφ(z),

D′(z) =(−(ρT /2)2z2 − ρT z + (ρT /2)2 − 1

)φ(z)

Choosing the largest root of the quadratic term inside the parentheses less than or equal to z, we find that

D′(z) ≤ 0 if z = s/σT ≥ 1− 2/ρT .

The condition on the safety stock is equivalent to (R − µT )/(σ√

T ) ≥ 1 − 2µT/(σ√

T ). Studying this

inequality further, we can show that it holds if

√T ≤ σ

2µ−

√σ2

4µ2− R

µor

√T ≥ σ

2µ+

√σ2

4µ2− R

µ(33)

The inequalities on T confirms our expectations put forward at the beginning of Section 3.3: Convexity is

very likely to hold for relatively small and relatively large T . Moreover, when σ2 − 4µR ≤ 0, the ranges in

(33) collapse and the cost is convex for every T . Note that in all of our numerical examples σ2 − 4µR ≤ 0

holds.

Due to (31), the backorders are a function of σT = σ√

T . To highlight the difference between Lemma

11 and convexity results in [26], we note that the former studies convexity in T while the latter studies the

same in√

T . Since the backorders increase and convex in σT , and√

T increases in T , convexity in T implies

convexity in√

T . Therefore Lemma 11 shoots for a stronger result by introducing an extra condition. The

condition can be interpreted as keeping at least (1− 2/ρT )σT units of safety stock. For ρT = ∞, 2 and 1,

this implies a service level of at least 84%, 50% and 16%. When T approaches infinity, ρT drops to zero and

required minimum service level for convexity to hold converges to 0%. Thus the convexity condition is not

very restrictive. ¤

Proof of Lemma 5: It suffices to show that B(R,Q, T ) is convex in T .

B(R, Q, T ) =1Q

R+Q∑

y=R+1

∞∑n=y

(n− y)(µT )ne−µT

n!.

Then

∂2B(R,Q, T )∂T 2

=1Q

R+Q∑

y=R+1

∞∑n=y

(n− y)µ2(µT )n−2e−µT ((n− µT )2 − n)

n!

=µ2

Q

R+Q∑

y=R+1

{∞∑

n=y

(n− y)(µT )n−2e−µT

(n− 2)!− 2

∞∑n=y

(n− y)(µT )n−1e−µT

(n− 1)!+

∞∑n=y

(n− y)(µT )ne−µT

n!}

30

=µ2

Q

R+Q∑

y=R+1

{∞∑

n=y−2

(n− (y − 2))(µT )ne−µT

n!− 2

∞∑

n=y−1

(n− (y − 1))(µT )ne−µT

n!

+∞∑

n=y

(n− y)(µT )ne−µT

n!}

=µ2

Q

R+Q∑

y=R+1

{E(D(T )− (y − 2))+ − 2E(D(T )− (y − 1))+ + E(D(T )− y)+} ≥ 0

Since E(D(T )−y)+ is convex in y, the term inside the sum is nonnegative which yields the last inequality. ¤

Proof of Lemma 7: The first part of the claim follows from the definitions of C1τ , C2

τ and C3τ . For the

second part, we need to argue that each of Ciτ for 1 ≤ i ≤ 4 is unimodal. Consider Case 1 only. For

I ≤ I, the approximated cost is max{C(I,Q, T − τ − α), C(I,Q, T − τ − α)}. Since C(I, Q, T − τ − α)

is convex in I, the approximated cost is nonincreasing over I ≤ I and is exactly C(I,Q, T − τ − α) at

I = I. For I ≤ I ≤ I, the approximated cost is convex in I. For I ≥ I, the approximated cost is

max{C(I,Q, T − τ + α), C(I , Q, T − τ + α)}, it is nondecreasing and takes the value C(I , Q, T − τ + α) at

I = I. We have established that C1τ (I, Q, T ) is constructed by patching a nonincreasing, a convex and a

nondecreasing function together and it is continuous. Then C1τ (I,Q, T ) must be unimodal.

We next consider Case 4 where Cτ,Q,T (I) = Cτ (I, Q, T ). For I ≤ I, the cost is C(I, Q, T − τ −α) and is

convex. For I ≤ I ≤ I, the cost is C(I, Q, T − τ) and it is nondecreasing because I(T − τ) ≤ I. Similarly,

for I ≥ I, the cost is C(I,Q, T − τ + α) and it is nondecreasing because I(T − τ + α) ≤ I. The cost is first

convex and then nondecreasing in I so it must be unimodal. The proofs for Case 2 and Case 3 invokes a

combination of the arguments for Case 1 and 4. ¤

Proof of Lemma 8: By the definition of the approximate cost

CF (R,Q, T ; α, τ) = ED(τ)Cτ (R−D(τ), Q, T ) ≥ ED(τ)Cτ (R−D(τ), Q, T ) = CF (R, Q, T ;α, τ)

CF (R,Q, T ; α, τ) = ED(τ)Cτ (R−D(τ), Q, T ) ≤ ED(τ)C(R−D(τ), Q, T ) = C(R, Q, T )

establishing the first claim. On the other hand, using unimodality of Cτ (I, Q, T ) in I, PF2 property for the

demand density and the result of Schoenberg (see Theorem 9.1 of [17] or Proposition 3.1 of [4]), we establish

the unimodality of CF (R,Q, T ; α, τ)). ¤

Proof of Lemma 9: Differentiating (10) and using C(I,Q, T − α) = C(I,Q, T ) and C(I , Q, T ) =

31

C(I , Q, T + α),

∇2CF (R,Q, T ) = ED(τ) {1(D(τ) ≥ R− I)∇2C(R−D(τ), Q, T − τ − α)

+1(R− I ≤ D(τ) ≤ R− I)∇2C(R−D(τ), Q, T − τ)

+1(D(τ) ≤ R− I)∇2C(R−D(τ), Q, T − τ + α)}

Then for Q ≤ Q(T − α),

0 ≥ ∇2C(R, Q, T − α) = ED(τ){1(D(τ) ≥ R− I)∇2C(R−D(τ), Q, T − τ − α)

+1(R− I ≤ D(τ) ≤ R− I)∇2C(R−D(τ), Q, T − τ − α)

+1(D(τ) ≤ R− I)∇2C(R−D(τ), Q, T − τ − α)}≥ ED(τ){1(D(τ) ≥ R− I)∇2C(R−D(τ), Q, T − τ − α)

+1(R− I ≤ D(τ) ≤ R− I)∇2C(R−D(τ), Q, T − τ)

+1(D(τ) ≤ R− I)∇2C(R−D(τ), Q, T − τ + α)}= ∇2CF (R, Q, T )

where the first inequality follows from the convexity of cost in order quantity Q and the first equality is due

to (4). The second inequality is due to ∇23C(R, Q, T ) ≤ 0, see Lemma 1. We obtain 0 ≥ ∇2CF (R,Q, T )

which implies that the cost with flexibility is decreasing when Q ≤ Q(T − α). A similar argument can be

used to show that the cost will increase in Q if Q ≥ Q(T + α). ¤

Proof of Lemma 10: First recall from (7) that

CF (R, Q, T ; α, τ1) = ED(τ1) min

C(R−D(τ1), Q, T − τ1 − α),C(R−D(τ1), Q, T − τ1),C(R−D(τ1), Q, T − τ1 + α)

. (34)

Then using (4),

C(R−D(τ1), Q, T − τ1) = ED(τ2−τ1)C(R−D(τ1)−D(τ2 − τ1), Q, T − τ1 − (τ2 − τ1))

≥ ED(τ2−τ1) min

C(R−D(τ2), Q, T − τ2 − α),C(R−D(τ2), Q, T − τ2),C(R−D(τ2), Q, T − τ2 + α)

Repeating the last argument for every term in the minimization in (34) we obtain

min

C(R−D(τ1), Q, T − τ1 − α),C(R−D(τ1), Q, T − τ1),C(R−D(τ1), Q, T − τ1 + α)

≥ ED(τ2−τ1) min

C(R−D(τ2), Q, T − τ2 − α),C(R−D(τ2), Q, T − τ2),C(R−D(τ2), Q, T − τ2 + α)

Taking the expected value over D(τ1) of the both sides establishes the result. ¤

32

References

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34

T 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0dR/dT 53.71 54.67 54.93 54.99 55.00 54.98 54.95 54.90 54.86 54.82dQ/dT -3.81 -5.26 -5.96 -6.35 -6.59 -6.75 -6.85 -6.92 -6.96 -6.99|dQ/dR| 0.071 0.096 0.108 0.116 0.120 0.123 0.125 0.126 0.127 0.127

Table 1: Sensitivity analysis of R and Q as T varies with parameters µ=50, σ = 12, h = 15, b = 35, K = 50.

35

τR∗

Q∗

CR∗

CF

CF

AF

G%

FG

%I

II

0I 0

c TC

MS

CF

G10

0Pc

0.1

43.3

528

.93

341.

6842

.49

339.

9633

8.19

0.50

1.02

33.1

043

.35

35.6

543

.35

14.9

711

.00

-7.5

10.

460.

243

.35

28.9

334

1.68

41.0

133

4.55

328.

442.

083.

8727

.85

40.9

829

.91

43.3

524

.74

25.1

1-1

1.89

0.05

0.3

43.3

528

.93

341.

6839

.73

327.

6631

7.90

4.10

6.96

22.5

935

.73

24.2

238

.54

32.8

033

.98

-10.

200.

010.

443

.35

28.9

334

1.68

39.1

232

0.26

308.

646.

279.

6717

.33

30.4

818

.57

32.7

740

.34

38.0

9-5

.05

0.00

0.5

43.3

528

.93

341.

6839

.02

312.

7130

0.26

8.48

12.1

212

.07

25.2

212

.96

27.0

647

.69

39.9

21.

490.

000.

643

.35

28.9

334

1.68

39.1

530

5.06

292.

1110

.72

14.5

16.

8119

.96

7.37

21.3

954

.85

41.0

38.

550.

160.

743

.35

28.9

334

1.68

39.3

629

7.25

283.

8213

.00

16.9

31.

5814

.70

1.80

15.7

661

.69

42.0

515

.79

1.35

Tab

le2:

The

effec

tof

exer

cise

tim

eτ.

αR∗

Q∗

CR∗

CF

CF

AF

G%

FG

%I

II

0I 0

c TC

MS

CF

G10

0Pc

0.10

43.3

528

.93

341.

6841

.98

313.

9831

1.56

8.11

8.81

15.9

021

.16

N/A

N/A

22.8

921

.02

9.08

0.00

0.20

43.3

528

.93

341.

6839

.90

309.

8930

1.09

9.30

11.8

813

.34

23.8

514

.92

26.4

741

.01

34.6

45.

950.

000.

2543

.35

28.9

334

1.68

39.0

231

2.71

300.

268.

4812

.12

12.0

725

.22

12.9

627

.06

47.6

939

.92

1.49

0.00

0.30

43.3

528

.93

341.

6839

.25

317.

0930

3.14

7.19

11.2

810

.80

26.6

011

.27

27.9

553

.76

40.9

9-2

.45

0.05

0.40

43.3

528

.93

341.

6841

.01

326.

2031

5.54

4.53

7.65

8.30

29.4

28.

4130

.25

63.3

832

.73

-6.5

96.

54

Tab

le3:

The

effec

tof

the

mag

nitu

deof

lead

tim

eop

tion

α.

36

σR∗

Q∗

CR∗

CF

CF

AF

G%

FG

%I

II

0I 0

c TC

MS

CF

G10

0Pc

742

.91

25.5

127

6.71

42.0

527

0.11

266.

152.

383.

8211

.76

24.4

912

.08

25.0

711

.81

39.2

9-2

8.72

0.00

1243

.35

28.9

334

1.68

39.0

231

2.71

300.

268.

4812

.12

12.0

725

.22

12.9

627

.06

47.6

939

.92

1.49

0.00

1744

.29

31.8

741

5.16

38.9

336

4.74

349.

3012

.15

15.8

613

.18

26.8

515

.28

32.7

879

.33

36.5

029

.35

0.01

2245

.52

34.4

049

2.76

39.6

842

6.43

407.

7913

.46

17.2

414

.97

29.1

921

.00

N/A

104.

9433

.60

51.3

50.

0527

46.9

436

.62

572.

6940

.66

495.

1047

1.73

13.5

517

.63

17.2

932

.11

N/A

N/A

122.

8031

.99

68.9

70.

10

Tab

le4:

The

effec

tof

dem

and

unce

rtai

nty

(mea

sure

din

stan

dard

devi

atio

nσ).

hb

R∗

Q∗

CR∗

CF

CF

AF

G%

FG

%I

II

0I 0

c TC

MS

CF

G10

0Pc

545

51.7

639

.57

206.

6846

.59

193.

3518

7.61

6.45

9.23

18.7

632

.79

19.8

1N

/A32

.61

22.1

4-3

.06

0.00

1040

47.0

731

.81

290.

3842

.36

267.

9125

8.21

7.74

11.0

815

.03

28.5

615

.67

30.3

642

.20

33.3

0-1

.13

0.00

1535

43.3

528

.93

341.

6839

.02

312.

7130

0.26

8.48

12.1

212

.07

25.2

212

.96

27.0

647

.69

39.9

21.

490.

0020

3039

.89

27.6

737

0.27

35.9

033

7.27

323.

128.

9112

.73

9.30

22.1

110

.56

24.5

050

.33

44.0

73.

070.

0025

2536

.35

27.3

037

9.50

32.7

434

4.85

330.

069.

1313

.03

6.43

18.9

37.

9821

.92

50.7

446

.20

3.25

0.00

Tab

le5:

The

effec

tof

hold

ing

cost

hre

lati

veto

back

orde

ring

cost

bw

hen

h+

b=

50.

KR∗

Q∗

CR∗

CF

CF

AF

G%

FG

%I

II

0I 0

c TC

MS

CF

G10

0Pc

2545

.96

22.5

229

3.25

41.6

526

2.06

248.

2110

.64

15.3

614

.39

27.7

015

.16

29.6

941

.30

55.3

1-1

0.26

0.46

5043

.35

28.9

334

1.68

39.0

231

2.71

300.

268.

4812

.12

12.0

725

.22

12.9

627

.06

47.6

939

.92

1.49

0.00

7541

.52

33.6

638

1.58

37.1

735

4.33

342.

847.

1410

.15

10.4

523

.49

11.5

325

.48

51.0

532

.06

6.67

0.00

100

40.0

537

.58

416.

6635

.74

390.

8138

0.08

6.20

8.78

9.16

22.1

10.3

624

.30

53.2

326

.85

9.73

0.00

500

28.1

073

.54

774.

5324

.06

758.

3775

2.21

2.09

2.88

-1.8

710

.66

-0.4

313

.51

60.5

23.

0219

.29

0.00

Tab

le6:

The

effec

tof

fixed

cost

K.

37

CM with SC cost with SC cost with SC costC(R, Q, T − α) LT = T − α LT = T − α LT ∈ {T, T ± α} Difference

α0.10 333.07 30.19 363.26 332.58 30.680.20 324.12 67.39 391.51 335.73 55.790.25 319.50 88.80 408.30 340.18 68.120.30 314.77 112.18 426.95 344.13 82.820.40 304.97 165.31 470.28 348.27 122.01

τ0.1 319.50 88.80 408.30 349.19 59.110.2 319.50 88.80 408.30 353.55 54.750.3 319.50 88.80 408.30 351.87 56.420.4 319.50 88.80 408.30 346.73 61.560.5 319.50 88.80 408.30 340.18 68.120.6 319.50 88.80 408.30 333.13 75.160.7 319.50 88.80 408.30 325.87 82.42

h, b5,45 196.13 46.65 242.78 209.75 33.03

10,40 272.90 74.54 347.44 291.51 55.9215,35 319.50 88.80 408.30 340.18 68.1220,30 345.36 96.00 441.36 367.18 74.1725,25 353.70 98.22 451.92 376.27 75.65

K25 269.15 132.64 401.79 303.52 98.2750 319.50 88.80 408.30 340.18 68.1275 360.84 67.13 427.97 374.90 53.07

100 397.07 53.30 450.37 406.93 43.44500 762.53 -4.33 758.20 755.23 2.97

σ7 266.27 107.39 373.66 305.44 68.22

12 319.50 88.80 408.30 340.18 68.1217 381.01 75.42 456.43 385.80 70.6322 446.65 65.54 512.19 441.39 70.8027 514.67 57.90 572.57 503.72 68.85

Table 7: SC cost comparisons between short lead time LT = T −α and flexible lead time LT ∈ {T, T ±α}.

38