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1-1-2009

An Inventory Allocation System with Backorders and Sales Rejection Yanyi Xu Shanghai University

Arnab Bisi Purdue University

Maqbool Dada John Hopkins University

Follow this and additional works at: http://docs.lib.purdue.edu/ciberwp Xu, Yanyi; Bisi, Arnab; and Dada, Maqbool, "An Inventory Allocation System with Backorders and Sales Rejection" (2009). Purdue CIBER Working Papers. Paper 59. http://docs.lib.purdue.edu/ciberwp/59

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I

An Inventory Allocation System with Backorders and Sales Rejection ∗

†

Yanyi Xu • Arnab Bisi • Maqbool Dada* ∗ †

School of Management, Shanghai University, Shanghai 200444, China

Krannert School of Management, Purdue University, West Lafayette, IN 47907, USA *Carey Business School, Johns Hopkins University, Baltimore, MD 21201, USA ∗

[email protected] •

†

[email protected] • *[email protected]

November 11, 2009

Abstract: We consider a variant of the periodic review system with a virtual warehouse that is due to Eppen and Schrage (1981). Depending on the realized pattern of demand during the delivery leadtime, this inventory is dynamically allocated or rationed to each of the retailers. While Eppen and Schrage (1981) assume that all unfilled demand is backordered, we allow some demand to be rejected to keep inventories in balance. This results in an exact analysis of the system. We develop conditions for the unique solution in a two retailer model and discuss the implications for the multi-retailer systems. Our analysis works for both discrete and continuous demands. Illustrative numerical examples suggest that, with our policy, only a very small fraction of demand would be rejected. The scope of our policy for general distribution and transshipment models has been explained. Key words: Stochastic Inventory Models, Base Stock Systems, Backorders, Lost Sales

1. Introduction Motivated by an industrial application, Eppen and Schrage (1981) proposed and analyzed an innovative periodic review system to manage inventories in a distribution system. In its simplest setting of their system, there are N distribution centers or retailers, whose inventory is managed from a central warehouse by a central planner who periodically places a joint order quantity. Demand unmet from stock is backordered to be filled by subsequent shipments. Depending on the realized pattern of demand during the delivery leadtime, this inventory is dynamically allocated or rationed to each of the retailers; thus the warehouse may be modeled as a virtual or a cross-docking facility. Their allocation is called the Fair Share Rule (FS) since it attempts to balance inventories in such a way that the probability of stockout is equalized at each retailer. Eppen and Schrage assume that 1) a balanced allocation is always possible, 2)

1

there is sufficient incoming inventory to clear all backorders, and 3) that demand is independently normally distributed at each retail location. Given these assumptions, Eppen and Schrage (1981) perform an approximate analysis that strives for the optimal target inventory level for this important distribution system. This seminal paper of Eppen and Schrage (1981) has spawned a large number of scholarly studies that generalize the applicability of their modeling framework. An insightful review is presented in Section 4 of Diks et al. (1996). One stream has been operational in the sense that it finds modifications to the fair share rationing rule that have better performance characteristics. Another, more strategic stream seeks to widen the scope of the model. For example, Jonsson and Silver (1987) allow for the possibility of retaining some stock at the central warehouse for subsequent redistribution. And, less directly, Tagaras and Cohen (1992) allow for transshipments at the end of each period to clear some backorders. In this paper we contribute to the modeling framework of Eppen and Schrage (1981) in such a way that it guarantees an exact analysis of the system. In the basic variant of our system developed in the next section, we only accept a demand if there is enough pipeline stock to clear backorders while maintaining the fair share assumption; hence in our system some lost sales may occur.

However, we are able to

calculate exact system performance for a wide range of demand processes including discrete distributions. The scope of the applications is illustrated in Section 4 where a rich generalization of the basic system is considered. That section also discusses the relationship of work to recent more dynamic transhshipment systems like those of Archibald et al. (1997) and Comez et al. (2006). The paper is organized as follows. In Section 2, we revisit the model of Eppen & Schrage (1981) and generalize their allocation equalization result. In Section 3, we focus on the analysis of the two retailer case and establish uniqueness of the solutions for both continuous and discrete demands. Illustrative numerical examples are provided and implications for the multi-retailer systems are discussed. Finally, we conclude and discuss the scope of our model in Section 4. All proofs except Lemma 1 are presented in the Appendix.

2

2. The Multiple Retailer Model We study a multi-retailer system that is a variant of the first model of Eppen and Schrage (1981). As in that model, each retailer i ( 1 ≤ i ≤ N ), faces demand Dti each of which is independently and identically distributed and is independent of that of other retailers. The system is managed by a central decision maker who decides at the beginning of each period, the amount to order for the system that brings the system inventory position to R. The supplier delivery lead-time is Lw periods. This order is received at a virtual warehouse that may also be interpreted as a cross-docking facility; it is then instantaneously allocated to the retailers who receive it after a delay of Lr periods. Hence, at the beginning of each period, the warehouse receives the order that was placed Lw periods ago, and, each retailer receives its allocation of the order that was placed Lw + Lr periods ago and allocated to the retailer Lr periods ago. The objective is to choose the order up to level R that minimizes the long-run average cost. To proceed with the development we will use the following notation: Demand and Supply Information

Dti = the demand of retailer i in period t .

ξti = a random observation of Dti . Fi (.) = cumulative distribution function of Dti . fi (.) = probability density function of Dti .

(ui , σ i ) = the mean and standard deviation of Dti . Lw = the lead time between warehouse and outside supplier (nonnegative integer). Lr = the lead time between warehouse and the retailers (nonnegative integer). Performance-Related Variables Sti = the actual sales of retailer i in period t. St = the total sales in period t. G Sti⋅n = ( Sti , Sti+1 ,...Sti+ n −1 ) = the sales vector of retailer i starting with Sti of mode n.

Sti⋅n = ∑ j =t S ij = the total sales of retailer i for n consecutive periods starting with period t. G Stn = ( St , St +1 ,...St + n −1 ) = the total sales vector starting with St of mode n. t + n −1

Stn = ∑ j =t S j = the total sales of n consecutive periods starting with period t. t + n −1

Qt = the order quantity in period t (with our base stock policy, Qt = St −1 ). G Qtn = (Qt , Qt +1 ,...Qt + n −1 ) = the order vector starting with Qt of mode n.

Qtn = ∑ j =t Q j = the total order quantity of n consecutive periods starting with period t. t + n −1

3

IPt i = the starting inventory position of retailer i in period t. IPt = the starting inventory position at the retailer level in period t. IOti = the on-hand inventory of retailer i at the beginning of period t

Φ it (.) = the cumulative distribution function of Sti .

φti (.) = the probability density function of Sti . Ψ t (.) = the cumulative distribution function of St . ψ t (.) = the probability density function of St . Cost Functions

TC ( R, Lw , Lr ) = the expected total cost per period after the system becomes stable. PC ( R, Lw , Lr ) = the expected order quantity per period after the system becomes stable. HC ( R, Lw , Lr ) = the expected leftovers per period after the system becomes stable. BO( R, Lw , Lr ) = the expected backorders per period after the system becomes stable. LS ( R, Lw , Lr ) = the expected lost sales per period after the system becomes stable.

The key to the successful analysis of this model is to focus on the performance of the single cycle that has the duration of Lw + Lr + 1 periods. In particular, if the system inventory position is invariant at the beginning of each period, then stationarity is assured, and the single cycle analysis yields the optimal control. In their analysis of this model, Eppen and Schrage (1981) invoke this property, by making three critical assumptions that result in an amenable, but approximate, single-cycle analysis: 1) Sufficiency: The incoming order is sufficient to clear all backorders; hence, there are no residual backorders.

2)

Allocation Equalization: The demand, and thus sales, occurs in such a way that after each incoming order is allocated, the stocking factor of the inventory position is identical across all retailers; that is, the inventory position at each retailer is such that IPt i = Lr ui + kt Lr σ i , where kt is the same across retailers but may differ from period to period. And, 3) Normality: The demand process at each retailer is described by a normally distributed random variable. While the normality assumption requires admitting negative demand, it enables them to interpret their allocation rule as yielding approximately an equal probability of stockout at each retailer in each period. As we know, the sufficiency assumption cannot be supported under demand variability since it is not possible to assure that all backorders clear. Moreover, the allocation equalization assumption puts additional pressure on the variability of the demand process.

4

We show below that it is possible to provide an exact treatment by allowing partial lost sales. Initially, we circumvent the normality assumption by representing demand as a continuous non-negative random variable; the case of discrete random variable is discussed later. Then, in our proposed control rule the sufficiency assumption is satisfied by ensuring that total sales in any period do not exceed R. And, to satisfy the allocation equalization assumption, we guarantee balance between retailers by requiring that the sales at any retailer, relative to the average sales at other retailers, do not differ too much after adjustments for inventory-on-order and variability. In particular, when there are two retailers who have identical demand distributions, this condition implies that the absolute gap St1 − St2 between the sales levels at the two retailers in period t does not exceed the total sales t − Lw periods ago, which is precisely the order quantity that is about to be allocated. This is formally stated as: Lemma 1 (After, Eppen and Schrage 1981)

If the system was in an equal stocking-factor position at the beginning of period t, then it will remain in an equal stocking-factor position at the beginning of period t + 1 if and only if ⎧∑ N Sti ≤ R , ⎪ i=1 ⎨ j i ⎪⎩ −σ j St − Lw ≤ St ∑ i ≠ j σ i − σ j ∑ i ≠ j St ≤ St − Lw ∑ i ≠ j σ i (1 ≤ i ≠ j ≤ N ).

(1)

Proof. (Necessity) Since the system was in an equal stocking-factor position at the beginning of period t, we can assume that IPt i = Lr ui + kt Lr σ i ( 1 ≤ i ≤ N ). Then demand Dti occurs and we decide to accept

sales Sti . At the beginning of period t + 1, Qt − Lw+1 (or St − Lw ) arrives for allocation. Then to return to equal stocking-factor position, we must have IPt i+1 = Lr ui + kt +1 Lr σ i = Lr ui + kt Lr σ i − Sti + α i (where

α i = amount allocated to retailer i) ( ∑ i=1α i =St − Lw ; 0 ≤ α i ≤ St − Lw ). Summing over N equations above N

gives

(

)( )(

)

N N ⎧( k − k ) = ∑ i =1 Sti − St − Lw / Lr ∑ i =1α i , t +1 ⎪ t ⎨ N N ⎪α j = St j − σ j ∑ i =1 Sti − St − Lw / ∑ i =1α i (1 ≤ j ≤ N ). ⎩

(

)

5

Since 0 ≤ α i ≤ St − Lw , we have ⎧S j − σ j ⎪ t ⎨ ⎪ St j − σ j ⎩

(∑ (∑

N i =1

Sti − St − Lw

N

S i − St − Lw i =1 t

) / ( ∑ α ) ≥ 0, ) / (∑ α ) ≤ S N

i =1

i

N

i =1

i

(1 ≤ j ≤ N )

(2)

t − Lw .

After some algebra, (2) gives the second inequality of (1), whereas the first inequality of (1) guarantees that the system inventory position is nonnegative. (Sufficiency) Notice that each step above is reversible, such that proof of sufficiency can be done in the

,

reverse direction.

Lemma 1 is a significant generalization of Lemma 1 of Eppen and Schrage (1981) who presented a condition for the result to hold. In contrast, we give precisely the set of conditions that must be met to guarantee equal stocking factors. In particular, when demand at each retailer is from a scalable distribution (Porteus 2002), which includes Normal, Gamma, Weibull, Power and Pareto distributions, then our model could guarantee that the equalization of stocking factors yields the same probability of stockout at each retailer in each period. In particular, our results also hold if retailers had demand distributions which had been shifted only by a constant reflecting a different mean demand. And, it would also yield equivalence if retailers had identical distributions. While Lemma 1 identifies the conditions that sales must satisfy, it does not specify how to choose from the infinitely many choices for Sti . For example, the trivial solution Sti = 0, satisfies the conditions in (1). Therefore, from the multitude of solutions, we choose the one that maximizes total immediate sales, which is the optimal solution to an easily solvable linear program. This is now described in detail for the case of two retailers.

3. The Two Retailer Case To show how to operationalize Lemma 1, we consider the case of two retailers. When N = 2, the two sets of inequalities in (1) give three inequalities as follows: St1 + St2 ≤ R ,

(3a)

6

σ 2 St1 − σ 1St2 ≤ σ 2 St − Lw ,

(3b)

σ 1St2 − σ 2 St1 ≤ σ 1 St − Lw .

(3c)

Depending on the realization of {Dt1 , Dt2 } , with the optimal choice for ( St1 , St2 ) , each of the constraints can be tight or loose. If we use the indicator variables {at , as } , {bt , bs } and {ct , cs } to denote whether each of these three constraints is tight or loose, it is easily seen that up to 23 = 8 combinations are candidate solutions. However, constraints (3b) and (3c) cannot be tight concurrently, so two solutions can be eliminated leaving the following six solutions, each of which can be optimal depending on the realizations of {Dt1 , Dt2 } (for expositional convenience, define St − Lw = z ): (as , bs , cs ) : {St1 = Dt1 ; St2 = Dt2 } if Dt1 + Dt2 ≤ R;σ 1 z ≤ σ 2 Dt1 − σ 1 Dt2 ≤ σ 2 z ,

(4a)

(as , bs , ct ) : {St1 = Dt1 ; St2 = σ 2 Dt1 / σ 1 + z} if 0 ≤ Dt1 < σ 1 ( R − z ) /(σ 1 + σ 2 ); Dt2 ≥ σ 2 Dt1 / σ 1 + z ;

(4b)

(as , bt , cs ) : {St1 = σ 1 Dt2 / σ 2 + z; St2 = Dt2 } if 0 ≤ Dt2 < σ 2 ( R − z ) /(σ 1 + σ 2 ); Dt1 ≥ σ 1 Dt2 / σ 2 + z ;

(4c)

(at , bs , cs ) : This corresponds to three different cases as follows: {St1 = Dt1 ; St2 = R − Dt1} if σ 1 ( R − z ) /(σ 1 + σ 2 ) ≤ Dt1 < σ 1 R /(σ 1 + σ 2 ); Dt2 ≥ R − Dt1 ,

(4d)

{St1 = R − Dt2 ; St2 = Dt2 } if σ 2 ( R − z ) /(σ 1 + σ 2 ) ≤ Dt2 < σ 2 R /(σ 1 + σ 2 ); Dt1 ≥ R − Dt2 ,

(4e)

{St1 = σ 1 R /(σ 1 + σ 2 ); St2 = σ 2 R /(σ 1 + σ 2 )} if Dt1 ≥ σ 1 R /(σ 1 + σ 2 ); Dt2 ≥ σ 2 R /(σ 1 + σ 2 ) .

(4f)

In (4d), if Dt1 = σ 1 ( R − z ) /(σ 1 + σ 2 ); Dt2 ≥ R − Dt1 , then (at , bs , ct ) holds; similarly in (4e), if Dt2 = σ 2 ( R − z ) /(σ 1 + σ 2 ); Dt1 ≥ R − Dt2 , then (at , bt , cs ) holds. Since we assume demand is continuous, the probability of Dt1 = σ 1 ( R − z ) /(σ 1 + σ 2 ) or Dt2 = σ 2 ( R − z ) /(σ 1 + σ 2 ) equals zero, that is why we do not specify these two cases in detail. From (4a) to (4f), we can conclude that: when both Dt1 and Dt2 are low (as in case (4a)), such that (4a), (4b) and (4c) hold naturally, both demands will be fully accepted as actual sales; when Dt1 (or Dt2 ) is much higher than the other one (as in case (4b), (4c), (4d) and (4e)) , it will be fully accepted, while Dt2 (or Dt1 ) will be partially accepted to the extent such that constraint (3a) or (3b) (or both) become

7

tight; when both Dt1 and Dt2 are too high (as in case (4f)), both demands will be rationed to ( St1 , St2 ) = (σ 1 R /(σ 1 + σ 2 ), σ 2 R /(σ 1 + σ 2 )) , which is a balanced allocation itself.

To summarize, the solution can be succinctly presented as: ⎧⎪ Sti = min{Dti ,[(σ i ⋅ Dt j ) / σ j ] + St − Lw , R − Dt j } if ⎨ i i if ⎪⎩ St = min{Dt ,(σ i ⋅ R) /(σ i + σ j )}

Dt j < σ j R /(σ i + σ j ),

(5)

Dt j ≥ σ j R /(σ i + σ j ).

In the remainder of the section we will use this explicit solution for the two retailer case to 1) build the connection between R, St − Lw , St and Sti ; and, 2) use these relationships to provide structural properties of the optimal solution.

3.1 Operating Characteristics Before proceeding with the formulation of the problem, it is necessary to identify how the operating characteristics of this system evolve over time. Then, taking the appropriate limits yields the stationary distributions of the key random variables. We begin by focusing on St , the total sales in period t, and Ψ t (.) , its cumulative distribution function. We show in Appendix A1 that Ψ t (.) depends only on the sales at time t − Lw. And, in particular, Lemma 2 x

x

0

0

Ψ t ( x) = ∫ f1 (ξ )F2 ( x − ξ )d ξ + ∫ +∫

x

∫

0

σ 2 ( x − z ) /(σ1 +σ 2 )

0

∫

σ 1 ( x − z ) /(σ 1 +σ 2 )

0

[ f1 (ξ )F2 ( x − ξ )]d ξ d Ψ t − Lw ( z )

[ f 2 (ξ )F1 ( x − ξ )]d ξ d Ψ t − Lw ( z )

(0 ≤ x < R)

(6)

Since (6) expresses a particular instance of the transition law, Ψ t = T ( Ψ t − Lw ) , St possesses the ergodic property, namely that the random variable St converges in the sense of distributions to a limiting random variable Z ( 0 ≤ Z ≤ R ). This limiting or stationary distribution (denote it by Ψ ) is a function of the policy chosen and the demand distribution. For a formal proof of the existence of limiting distributions, we refer the reader to Karlin (1958). Thus, it can be shown that (6) yields x

x

0

0

Ψ ( x ) = ∫ f1 (ξ )F2 ( x − ξ )d ξ + ∫ +∫

x

0

∫

σ 2 ( x − z ) /(σ1 +σ 2 )

0

∫

σ1 ( x − z ) /(σ1 +σ 2 )

0

[ f1 (ξ )F2 ( x − ξ )ψ ( z )]d ξ dz

[ f 2 (ξ )F1 ( x − ξ )ψ ( z )]d ξ dz

8

( 0 ≤ x < R ).

Since sales cannot exceed R, P{St = R} = 1 − Ψ ( R ) = Ψ ( R ) . Not only can we find the stationary distribution of St , but also we can find the stationary distribution of Sti ( St1 or St2 ) as (the proof is given in Appendix A2): Lemma 3 x

Φ i ( x) = P{Sti ≤ x} = Fi ( x) + Fi ( x) ∫ Fj [σ j ⋅ ( x − z ) / σ i ]ψ ( z ) dz 0

( 0 < x < (σ i ⋅ R ) /(σ i + σ j ) ),

P{Sti = (σ i ⋅ R) /(σ i + σ j )} = Fi [(σ i ⋅ R ) /(σ i + σ j )] ⋅ Fj [(σ j ⋅ R) /(σ i + σ j )] ( i, j = 1, 2; i ≠ j );

Φ i ( x) = Fj ( R − x) + Ψ ( x) F j ( R − x) Fi ( x) x

{

}

+ ∫ Fj [σ j ( x − z ) / σ i ] + Fi ( x) ⎣⎡ Fj ( R − x) − Fj [σ j ( x − z ) / σ i ]⎦⎤ ψ ( z )dz 0

(7a)

(7b)

( σ i R /(σ i + σ j ) < x < R ). (7c)

Clearly, Φ i ( x) is continuous and differentiable except at Sti = (σ i ⋅ R) /(σ i + σ j ) . We have now established the stationary distribution of St and Sti (i = 1, 2), and shown the clear dependence between St − Lw and St . To effectively use the single-cycle analysis, it remains to establish that St − Lw , St − Lw+1 ,…,and St −1 are mutually independent as are Sti− Lw , Sti− Lw+1 ,…, and Sti−1 (see Appendix A3). This leads to: Theorem 1

If Ψ (.) (resp. Φ i (.) ) is the stationary distribution function of sales per period St (resp. Sti ), then Ψ n (.) (resp. Φ in (.) ) is the stationary distribution of total sales Stn (resp. Sti⋅n ) for n consecutive periods ( n ≤ Lw ), where Ψ n (.) (resp. Φ in (.) ) is the n -fold convolution of Ψ (.) (resp. Φ i (.) ). Because we are employing an order-up-to policy, we have:

G G Lw Lw Lw QtLw − Lw − Lr +1 = St − Lw − Lr ; and Qt − Lw − Lr +1 = St − Lw − Lr . G Since QtLw − Lw − Lr +1 is the outstanding order vector at the beginning of period t − Lr, it is also the total pipeline inventory (of the warehouse) at the beginning of period t − Lr, so that Lw 1 2 IPt − Lr = R − QtLw − Lw − Lr +1 = R − St − Lw − Lr = IPt − Lr + IPt − Lr . Given allocation equalization, we have

9

Lw ⎧ 1 ⎪ IPt − Lr = ⎣⎡ Lr (u1σ 2 − u2σ 1 ) + σ 1 ( R − St − Lw− Lr ) ⎦⎤ /(σ 1 + σ 2 ), ⎨ 2 Lw ⎪⎩ IPt − Lr = ⎡⎣ Lr (u2σ 1 − u1σ 2 ) + σ 2 ( R − St − Lw− Lr ) ⎤⎦ /(σ 1 + σ 2 ).

(8)

Finally, note that the inventory-on-hand of each retailer at the beginning of period t is: i ⋅ Lr i ⎤ IOti = IPt i− Lr − Sti−. LrLr = ⎡⎣ Lr (uiσ j − u jσ i ) + σ i R ⎤⎦ /(σ i + σ j ) − ⎡⎣σ i StLw − Lw − Lr /(σ i + σ j ) + St − Lr ⎦ = Ri − Vt ,

(9)

i ⋅ Lr ⎤ where Ri = ⎡⎣ Lr (uiσ j − u jσ i ) + σ i R ⎤⎦ /(σ i + σ j ) and Vt i = ⎡⎣σ i StLw − Lw − Lr /(σ i + σ j ) + St − Lr ⎦ .

Now that we have identified how to characterize the key stationary distributions that describe system performance, these are used in the next sub-section to first derive the long-run average cost function, which is then used to find the optimal base-stock.

3.2 The Average Cost Formulation The long-run average cost function may be written as:

TC ( R, Lw , Lr ) = c ⋅ PC ( R, Lw , Lr ) + l ⋅ LS ( R, Lw , Lr ) + h ⋅ HC ( R, Lw , Lr ) + b ⋅ BO( R, Lw , Lr ) .

(10)

⎧ PC ( R, L , L ) = R zψ ( z )dz , w r ∫0 ⎪ +∞ ⎪ In (10), ⎨ LS ( R, Lw , Lr ) = ∫ ( z − R )ψ ( z )dz , R ⎪ + + ⎪ HC ( R, Lw , Lr ) = ∑ 2 HCi ( R, Lw , Lr ) = ∑ 2 E[ IOti − Sti ] = ∑ 2 E[ Ri − Vt i − Sti ] , i = i i 1 = = 1 1 ⎩ where HC1 ( R, Lw , Lr ) and HC2 ( R, Lw , Lr ) are the expected leftovers of retailers 1 and 2. Since Sti and Vt i are not independent, it is convenient to define ρi ( wi , vi ) as the joint density of ( Sti ,Vt i ) . Then,

HC ( R, Lw , Lr ) = ∫

R1

0

∫

R1 − w1

0

( R1 − w1 − v1 )ρ1 ( w1 , v1 )dv1dw1 + ∫

R2

0

∫

R2 − w2

0

( R2 − w2 − v2 )ρ 2 ( w2 , v2 )dv2 dw2 .

(11)

And we derive the expected backorders as BO( R, Lw , Lr ) = {Expected sales in Lw + Lr + 1 periods} − {R − Expected Leftovers} = ( Lw + Lr + 1) PC ( R, Lw , Lr ) − [ R − HC ( R, Lw , Lr )] .

Now we have the following

theorem whose proof is presented in Appendix A4. Theorem 2

2.1) If b ≤ (l − c) /( Lw + Lr + 1) , then TC ( R, Lw , Lr ) is convex in R, and there exists a unique R* which minimizes TC ( R, Lw , Lr ) ;

10

2.2) If (l − c) /( Lw + Lr + 1) < b ≤ (l − c) /( Lw + Lr ) , and the demand distributions of the retailers are independent and identical exponential or uniform, then TC ( R, Lw , Lr ) is unimodal and concave-convex, and there exists a unique R* which minimizes TC ( R, Lw , Lr ) ;

2.3) If b > (l − c) /( Lw + Lr ) , and the demand distributions of the retailers are independent and identical exponential or uniform, then TC ( R, Lw , Lr ) is concave-convex..

Theorem 2.1) implies that the cost function is convex when the unit backorder cost is small relative to l. Theorem 2.2) and 2.3) state that if the two retailers are independent and identical, unimodality or concavity-convexity is assured for exponential and uniform distributions. Since we have established that under appropriate technical conditions determining the optimal value of R requires a simple search, we complete the analysis of the model by considering the case of discrete demand.

3.3 The Case of Discrete Demand In Section 3.1 we established the stationary distribution of St , and subsequently found the stationary distribution of Sti , Stn and Sti ⋅n (for n ≤ Lw ). Since the stationary distribution of St involves difficult functions, it is not easy to compute even for relatively simple demand distributions. However, considerable simplification occurs when demand is discrete since the nested structure of this distribution is discernible. However, when demand is discrete, the target inventory position R and sales are also discrete creating a technical problem regarding the allocation assumption. To see this, consider that IPt1− Lr and IPt 2− Lr in (8) may not be integer with our allocation equalization. One way that this can be redressed is by slightly modifying the balance allocation to: ⎧ IP1 = ⎢ ⎡ L (u σ − u σ ) + σ ( R − S Lw ⎥ ⎤ t − Lw − Lr ) ⎦ /(σ 1 + σ 2 ) , 2 1 1 ⎪ t − Lr ⎣ ⎣ r 1 2 ⎦ ⎨ ⎤ ⎤ ⎪ IP 2 = ⎡ ⎡ L (u σ − u σ ) + σ 2 ( R − StLw − Lw − Lr ) ⎦ /(σ 1 + σ 2 ) , ⎥ ⎩ t − Lr ⎢ ⎣ r 2 1 1 2 where ⎢⎣ x ⎥⎦ denotes the largest integer less than or equal to x and ⎡⎢ x ⎤⎥ denotes the smallest integer greater than or equal to x. While we can execute the analysis using the above construct, it is simpler to assume

11

that demand at each retailer in each period is at least one unit ensuring that it is appropriate to assume that R ≥ 2 so that total sales in each period takes values in ΩR = {2,3,..., R} . Then, as for the case of

continuously distributed random variables, we can build the connection among R , St − Lw and St , using a transition matrix to characterize the transition law as: Pij = P{St = j | St − Lw = i} , for i, j ∈ ΩR .

(12)

Based on Equation (5), Ρij is established as follows: 3.1) If j = R ,

Pij = P{Dt1 + Dt2 ≥ R; Dt1 ≥ σ 1 ( R − i ) /(σ 1 + σ 2 ); Dt2 ≥ σ 2 ( R − i) /(σ 1 + σ 2 )} .

(13a)

3.2) If j < R , j ∈{2,3,...R − 1} , then

Pij = P{Dt1 + Dt2 = j; −σ 1i ≤ σ 2 Dt1 − σ 1 Dt2 ≤ σ 2i} + P{(1 + σ 2 / σ 1 ) Dt1 = j − i;σ 1 Dt2 − σ 2 Dt1 ≥ σ 1i} + P{(1 + σ 1 / σ 2 ) Dt2 = j − i;σ 2 Dt1 − σ 1 Dt2 ≥ σ 2i}.

(13b)

While the terms given by (13a) depend on j = R, the terms given by (13b) are independent of R; a characteristic which we will exploit. First notice that when i < j , (13b) is naturally independent of R. Then notice that when i ≥ j and j < R , Dt1 ≥ 1 ( Dt2 ≥ 1 ), {Dt1 + Dt2 = j} ⊂ {−σ 1i ≤ σ 2 Dt1 − σ 1 Dt2 ≤ σ 2i} ; and {(1 + σ 2 / σ 1 ) Dt1 = j − i} = {(1 + σ 1 / σ 2 ) Dt2 = j − i} = ∅ . Hence, Pij = P{Dt1 + Dt2 = j} + 0 + 0 = P{Dt1 + Dt2 = j} ( j ≤ i ≤ R ; j < R ).

(13c)

For example, if both retailers are identical, the transition matrices AR = {aijR } ( i, j ∈ Ω R ) for R = 4, 5, are: ⎡ ⎢2 A4 = ⎢ ⎢3 ⎢ ⎣4

2 P1 2 P1 2 P1 2

3 2 P1 P2 2 P1 P2 2 P1 P2

4 ⎤ 2 P2 + 2 P1 P3 ⎥⎥ , P2 2 + 2 P1 P3 ⎥ ⎥ P2 2 + 2 P1 P3 ⎦

⎡ ⎢2 ⎢ 5 A = ⎢3 ⎢ ⎢4 ⎢⎣ 5

2 P1 2 P1 2

3 2 P1 P2 2 P1 P2

4 P2 + 2 P1 P3 P2 2 + 2 P1 P3

P1 2 P1 2

2 P1 P2 2 P1 P2

P2 2 + 2 P1 P3 P2 2 + 2 P1 P3

2

5 ⎤ ⎥ 2 2 P2 P3 − P3 ⎥ 2 2 P2 P3 − P3 + 2 P1 P4 ⎥ . ⎥ 2 P2 P3 − P3 2 + 2 P1 P4 ⎥ 2 P2 P3 − P3 2 + 2 P1 P4 ⎥⎦

In A4 and A5 , the first column (row) represents the value of i (j). From these two illustrative matrixes, we can easily see that only the last column depends on the value of R. And, we can also see that the entries in the square sub-matrix comprised of the four north-west elements of A4 are replicated in A5 . More

12

formally, comparing the entries of AR and AR +1 as above, and using (13a), (13b) and (13c), we can easily find the following properties: Lemma 4

⎧∑ R aijR = 1 ⎪ j =1 ⎪a R = a R ij ⎪⎪ i +1 j R R +1 ⎨aij = aij ⎪ R R +1 R +1 ⎪aiR = aiR + aiR +1 ⎪a R +1 = a R +1 Rj ⎪⎩ R +1 j

for i ∈ Ω R , for i ≥ j; i, j ∈ Ω R , for i ∈ Ω R ; j ∈ Ω R −1 , for i ∈ Ω R , for

j ∈ Ω R +1 .

As in the continuous demand case, the transition matrix here constitutes a Markov chain, which obviously has finite state space with all states being positive recurrent. Hence, we can easily compute the stationary distribution of St , where we use Π R = (π 2R , π 3R , ..., π RR ) to represent the stationary distribution of St . The following properties of Π R which can be easily derived from Lemma 4 significantly accelerate the computation

of

the

stationary

distribution

(the

proof

is

given

in

Appendix

A5):

Theorem 3

If we define Π R = (π 2R , π 3R ,..., π RR ) and Π R +1 = (π 2R +1 , π 3R +1 ,..., π RR +1 , π RR++11 ) , then ⎧π R = π R +1 for i ∈ Ω , i R −1 ⎪⎪ i R R +1 R +1 ⎨π R = π R + π R +1 , ⎪ R +1 R −1 R +1 2 R +1 R 1 ⎪⎩π R = ∑ i = 2 π i ⋅ aiR + π R ⋅ P{Dt + Dt = R}.

The most appealing consequence of Theorem 3 is that computing the stationary distribution of sales per period and then using it to compute the expected long-run average cost can be executed very efficiently with computation time that is a function of R − 1. This is illustrated next with an example.

3.4 Illustrative Examples In the model developed in this paper, a key innovation is the strategic use of lost sales to keep inventories in balance. To this end we will focus on showing how effectively the discrete approach works in demonstrating that the fraction of demand lost is likely to be small. To proceed, we consider a system with two retailers, each of which faces independently and identically distributed demand from a uniform

13

distribution. We consider three cases with mean demand of 10.5 per period per retailer or 21 per period per system. Specifically demand is drawn from U(1,20), U(5,16) and U(9,12), representing decreasing variability. Since they are independent of the leadtimes and economic parameters, we found it convenient to only calculate the stationary distributions of sales for different values of R. From these stationary distributions, for each value of R, we computed the expected sales, the corresponding fill rate, the probability that all demand is accepted, the probability that the demand of one retailer is fully accepted while part of the demand of the other retailer is lost, and the probability that both retailers have lost sales. For expositional convenience, we define: R⎫ ⎧ R⎫ ⎧ Pr1 = P{S1 < D1 , S 2 < D2 } = P ⎨ D1 > ⎬ ⋅ P ⎨ D2 > ⎬ ; 2⎭ ⎩ 2⎭ ⎩

Pr2 = P{S1 = D1 , S 2 = D2 } = Pr3 = P{S1 = D1 , S2 < D2 } =

R

∑ π ⋅ P {D + D i

i= 2 R

∑π

i

R⎫ ⎧R −i ⋅P⎨ ≤ D1 ≤ ⎬ ⋅ P {D2 > R − D1} ; 2⎭ ⎩ 2

i

R−i⎫ ⎧ ⋅ P ⎨ D2 < ⎬ ⋅ P {D1 > D2 + i} 2 ⎭ ⎩

i

R⎫ ⎧R −i ⋅P⎨ ≤ D2 ≤ ⎬ ⋅ P {D1 > R − D2 }. 2⎭ ⎩ 2

i=2

Pr4 = P{S1 < D1 , S 2 = D2 } =

R

∑π i =2

+

R

∑π i=2

≤ R, −i ≤ D1 − D2 ≤ i};

R−i⎫ ⎧ ⋅ P ⎨ D1 < ⎬ ⋅ P {D2 > D1 + i} 2 ⎭ ⎩

R

∑π

2

i

i=2

+

1

Before discussing the numerical results it is insightful to notice that the maximum sales in each period cannot exceed the maximum demand which is 40, 32 or 24. Hence, we only have to compute the stationary distribution of sales for values of R for that level. The stationary distribution for other cases can be computed from these probabilities which are presented in Tables 1a, 2a and 3a. For example, when demand is U(1,20) and R = 39, from Theorem 3 it follows that π 2 to π 38 will remain unchanged and π 39 would become 1/200+1/400=3/400. In contrast, for R = 41, π 2 to π 40 would remain unchanged and π 41 would be identically equal to zero.

14

The lower panels of these three tables report the service measures Pr1 to Pr4 . As R increases, the probability that there would be lost sales at both retailers decreases and ultimately reaches zero. A similar pattern is also observed for Pr3 and Pr4 which represent the probability that only one retailer has lost sales. Also observe that Pr2 , the probability of accepting all demand increases, as does expected sales as R increases. Also notice that the fill rate becomes very high as R approaches the upper bound on demand. It varies from .987 in Table 1b to .998 in Table 2b to 1.00 in Table 3b, suggesting that the fill rate improves with decreasing variability. The same is also true for Pr1 to Pr4 and E(Sales). Table 1a: Stationary Distribution of Sales with R = 41 and U(1,20) Demand π 2 = 1/ 400

π 22 = 9073359220912783199 /(2048 ×1017 )

π 3 = 1/ 200

π 23 = 1731473981118598381/(4096 ×1016 )

π 4 = 617 / 80000

π 24 = 82451034053314310 /(2048 ×1015 )

π 5 = 831/ 80000

π 25 = 39043441348753021/(1024 ×1015 )

π 6 = 42371/ 3200000

π 26 = 1843682896202930 /(512 ×1014 )

π 7 = 256217 /16000000

π 27 = 172993906895803 /(512 ×1013 )

π 8 = 12110423/ 640000000

π 28 = 16165735268100 /(512 ×1012 )

π 9 = 69450349 / 3200000000

π 29 = 749377328627 /(256 × 1011 )

π 10 = 3145182173 /(128 ×109 )

π 30 = 44605610000 /(200 × 1010 )

π 11 = 139584027000 /(512 ×1010 )

π 31 = 3149733177 /(128 × 109 ) π 32 = 71171383 / 3200000000

π 12 = 767697474997 /(256 ×1011 )

π 33 = 12684269 / 640000000

π 13 = 8316729337070 /(256 ×1012 )

π 34 = 278369 /16000000

π 14 = 179028264488179 /(512 × 10 ) 13

π 15 = 3806565705200900 /(1024 ×10 )

π 35 = 238983 /16000000

π 16 = 8050071615219299 /(2048 ×10 )

π 36 = 499 / 40000

π 17 = 42116004938358801/(1024 ×10 )

π 37 = 799 / 80000

14

14

15

π 38 = 3 / 400

π 18 = 1754211699560853577 /(4096 ×1016 ) π 19 = 9048858274430287507 /(2048 ×10 )

π 39 = 1/ 200

π 20 = 371701315102376673337 /(8192 ×10 )

π 40 = 1/ 400

π 21 = 378487849957854330823 /(8192 × 10 )

π 41 = 0

17

18

18

15

Table 1b: Performance Measures with Different Values of R for U(1,20) Demand R

Pr1

Pr2

Pr3

Pr4

E(Sales)

Fill rate

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0.2500 0.2025 0.2025 0.1600 0.1600 0.1225 0.1225 0.0900 0.0900 0.0625 0.0625 0.0400 0.0400 0.0225 0.0225 0.0100 0.0100 0.0025 0.0025 0.0000 0.0000

0.446291 0.489296 0.530872 0.570904 0.609291 0.645929 0.680726 0.713588 0.744434 0.773182 0.796791 0.824114 0.846174 0.865891 0.883226 0.898137 0.910599 0.920587 0.928087 0.933087 0.935587

0.151855 0.154102 0.266628 0.269096 0.115354 0.115785 0.098387 0.098206 0.082783 0.082159 0.070354 0.067943 0.056913 0.055804 0.047137 0.045932 0.039700 0.038457 0.034707 0.033457 0.032207

0.151855 0.154102 0.266628 0.269096 0.115354 0.115785 0.098387 0.098206 0.082783 0.082159 0.070354 0.067943 0.056913 0.055804 0.047137 0.045932 0.039700 0.038457 0.034707 0.033457 0.032207

16.966920 17.470000 17.931213 18.345956 18.718428 19.050640 19.344723 19.602797 19.827084 20.019796 20.183236 20.319701 20.431558 20.521174 20.590971 20.643370 20.680833 20.705820 20.720820 20.728320 20.730820

0.807948 0.830000 0.850000 0.873617 0.891354 0.907173 0.921177 0.933467 0.944147 0.953324 0.961106 0.967605 0.972931 0.977199 0.980522 0.983018 0.984802 0.985991 0.986706 0.987063 0.987182

Table 2a: Stationary Distribution of Sales with R = 33 and U(5,16) Demand

π 10 =1/400 π 11 =1/72 π 12 =1/48 π 13 =1/36 π 14 =5/144 π 15 =1/24

π 16 =7/144 π 17 =1/18 π 18 =1/16 π 19 =5/72 π 20 =793/10368 π 21 =863/10368

16

π 22 =11/144 π 23 =5/72 π 24 =1/16 π 25 =1/18 π 26 =7/144 π 27 =1/24

π 28 =5/144 π 29 =1/36 π 30 =1/48 π 31 =1/72 π 32 =1/144 π 33 = 0

Table 2b: Performance Measures with Different Values of R for U(5,16) Demand R

Pr1

Pr2

Pr3

Pr4

E(Sales)

Fill rate

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0.4400 0.3400 0.3403 0.2500 0.2500 0.1736 0.1736 0.1111 0.1111 0.0625 0.0625 0.0278 0.0278 0.0069 0.0069 0.0000 0.0000

0.1900 0.2500 0.3100 0.3800 0.4600 0.5400 0.6200 0.6900 0.7500 0.8100 0.8500 0.8900 0.9300 0.9500 0.9700 0.9900 0.9955

0.180556 0.204861 0.173611 0.184028 0.145000 0.142378 0.104184 0.100712 0.069462 0.065990 0.041684 0.040203 0.022919 0.019508 0.009138 0.005696 0.002240

0.180556 0.204861 0.173611 0.184028 0.145000 0.142378 0.104184 0.100712 0.069462 0.065990 0.041684 0.040203 0.022919 0.019508 0.009138 0.005696 0.002240

15.63778 16.44778 17.20222 17.89417 18.51667 19.06268 19.52546 19.91185 20.22879 20.48324 20.68213 20.71240 20.81657 20.88601 20.92768 20.94851 20.95546

0.744656 0.783228 0.819153 0.852103 0.881746 0.907747 0.929784 0.948183 0.963276 0.975392 0.984863 0.986305 0.991265 0.994572 0.996556 0.997548 0.997879

Table 3a: Stationary Distribution of Sales with R = 25 and U(9,12) Demand

π 18 =1/16 π 19 =1/8 π 20 =3/16 π 21 =1/4

π 22 =3/16 π 23 =1/8 π 24 =1/16 π 25 = 0

Table 3b: Performance Measures with Different Values of R for U(9,12) Demand R

Pr1

Pr2

Pr3

Pr4

E(Sales)

Fill rate

20 21 22 23 24

0.25 0.06 0.06 0.00 0.00

0.38 0.50 0.69 0.81 1.00

0.18500 0.21875 0.12500 0.09375 0.00000

0.18500 0.21875 0.12500 0.09375 0.00000

19.7500 20.3750 20.7500 20.9375 21.0000

0.940476 0.970238 0.988095 0.997024 1.000000

From a practical perspective, we would set inventory level R = ( Lw + Lr + 1) μ + k Lw + Lr + 1σ , which would tend to be on the order of at least three times the mean demand, which would be higher than

17

the maximum demand per period. Thus, in an optimal system, we would expect that the fill rate could be high and that it would be very unlikely that both retailers would have lost sales in the same period. This observation can show the path for how to handle the case with more than two retailers. Implications for Multi-Retailer Systems While it is the case that constraints of the form in equations (3) may be written for the case when N > 2, the combinatorial nature of them would make their manipulation difficult. Thus, one practical approach would be to relax some of them by invoking insight gleaned from the above examples. A practical approach would be to assume that the probability that more than one of N retailers has lost sales is zero; under this relaxation only O( N ) constraints would be needed. If such a relaxation is too coarse,, it could be replaced by the assumption that the probability that more than two of N retailers have lost sales is zero; under this relaxation only O( N 2 ) constraints would be needed, as in the two-retailer case. Thus, our approach appears to be pragmatic enough to use for multiple retailer systems.

4. Summary and Scope In this paper we propose a novel control to manage demand in the virtual warehouse system of Eppen and Schrage (1981). For an arbitrary sized network, we are able to show that under scalable demand, the fair share interpretation prevails. Subsequently, we focused on the two-retailer case to develop insight into the structure of the underlying optimal policy.

These analytical results are complemented by an

illustrative numerical study which suggests that there would be relatively limited lost sales in our system. Interestingly, these results suggest that the likelihood is small that two retailers will have lost sales in the same period. This suggests that the combinatorial problem that would arise as the number of retailers increases could be circumvented by assuming that no more than two retailers would have lost sales in the same period. The analysis of this relaxed problem dynamics would mimic that of the two retailer model. As the above development of a variant of a model due to Eppen and Schrage (1981) shows, the sufficiency and allocation assumptions yield an exact analysis when used with our proposed control. In an analogous way, our approach can be adapted to the other model studied by them in which an order is

18

placed every T (T > 1) periods. In that model, Eppen and Schrage allow the leadtime to be shorter than the review period. In fact, the key ideas from this model have been generalized directly by Jonsson and Silver (1987) to allow for redistribution of stocks among retailers before demand is realized; and, less directly by Tagaras and Cohen (1992) to allow for transshipments to clear backlogs after demand is realized. In this closing section, we illustrate the richness and scope of our modeling approach by presenting an overview of how our control can be used to present a unified treatment of such models. To simplify the exposition, we consider an inventory system with one warehouse and two retailers. Demand for each retailer is modeled by a positive random variable that is independently and identically distributed. The system employs a (R, T) policy to control the inventory position, i.e., the warehouse places an order every T periods to bring the system-wide inventory position back to R. To present the resulting dynamics we will use the following Figure 1. Redistribution made

Allocation made

Redistribution received

Transshipment made

Allocation made

R

Redistribution received

R − Lwu R − Lu

R − ( L + T )u

−1 − Lr

−T Current Order placed

−L Allocation made

− Lr

-1 0 Allocation received

T −L Next order placed

Figure 1: Two Typical Cycles of the Hybrid Policy

19

T − 1 − Lr

Redistribution made

T −1 Transshipm ent made

As shown in Figure 1 (where u = u1 + u2 ), an order is placed at the beginning of period −L (L = Lw + Lr, T > L), this order reaches the warehouse at the beginning of period − Lr. At this point the order is allocated to equalize the stocking-factor of the two retailers. This allocation reaches the retailers at time 0. As in Jonsson and Silver (1987), inventory on-hand is redistributed at the beginning of period T − 1 − Lr. In this redistribution the retailer who has the higher inventory on-hand will contribute a portion of his extra inventory to the other one. Moreover, it is assumed, consistently with Jonsson and Silver (1987), that the redistribution lead time is the same as Lr, i.e., the other retailer will receive it at the beginning of period T − 1. And, adapting the protocol of Tagaras and Cohen (1992), after demand is realized in each period, an expedited delivery is allowed to transship the leftovers at one retailer to the other retailer when it has unsatisfied demand. The next order will be placed at the beginning of period T − L initiating the next cycle. As we show in Appendix A6, accommodating these features add additional constraints to (1). Since all these constraints only determine how we exchange inventory between the two retailers, but have no impact on the target level R, it should be possible to show that stationary distribution of sales in each period is also nested in R, which would also facilitate as in Section 3 the computation of the stationary distribution of sales and the expected cost accordingly as we proceed. One complication that could arise is that it may not be possible to establish unimodality in R of the cost function, necessitating a line search for the optimal value of R. In the models discussed above, transshipments occur at one or two points during the review cycle. In contrast there have been recent developments in which transshipments occur more dynamically. For example, in Archibald et al. (1997), Poisson demand is reassigned dynamically from a stocked-out retailer to another retailer if it has sufficient stock on-hand. Otherwise the demand is lost. In this model, overflow demand is accepted as long as the inventory-on-hand is above a threshold that gradually decreases during the period. In contrast, Comez et al. (2006) consider a two-retailer system similar to that of Archibald et al. (1997) but with a more general demand process. While Archibald et al. (1997) consider systems with zero leadtime, Comez et al. (2006) propose and test heuristics for the two-retailer

20

case when leadtimes are positive. In our model in contrast, the decision to accept or reject demand depends on the inventory position and not inventory on hand. Some insight into how our approach could accommodate such dynamic models may be found in Xu et al. (2008) who consider a single-retailer model that has somewhat more complex controls than those considered here.

Appendix A1. Proof of Lemma 2 Ψ t ( x) = P{St ≤ x} = P{St1 + St2 ≤ x} = P{Dt1 + Dt2 ≤ x;0 ≤ St − Lw < x} + P{Dt1 + Dt2 ≤ x; x ≤ St − Lw < R} + P{Dt1 + Dt2 > x;0 < Dt1 ≤ σ 1 ( x − St − Lw ) /(σ 1 + σ 2 );0 ≤ St − Lw < x} + P{Dt1 + Dt2 > x;0 < Dt2 ≤ σ 2 ( x − St − Lw ) /(σ 1 + σ 2 );0 ≤ St − Lw < x} x

x

0

0

=∫

∫ f (ξ )F ( x − ξ )dξ d Ψ ( z ) + Ψ σ σ σ +∫ ∫ f (ξ )F ( x − ξ ) d ξ d Ψ σ σ σ +∫ ∫ f (ξ )F ( x − ξ )d ξ d Ψ 1

t − Lw

2

0

0

2

t − Lw

( z)

2

1

t − Lw

( z)

x

x

0

0

= ∫ f1 (ξ )F2 ( x − ξ )d ξ + ∫ x

∫

σ 2 ( x − z ) /(σ1 +σ 2 )

0

0

1

2 ( x − z ) /( 1 + 2 )

x

0

0

( x) ∫ f1 (ξ ) F2 ( x − ξ )d ξ

1 ( x − z ) /( 1 + 2 )

x

0

+∫

x

t − Lw

∫

σ1 ( x − z ) /(σ1 +σ 2 )

0

Q.E.D.

f1 (ξ )F2 ( x − ξ ) d ξ d Ψ t − Lw ( z )

f 2 (ξ )F1 ( x − ξ ) d ξ d Ψ t − Lw ( z ).

A2. Proof of Lemma 3

First of all, by (5), if Dti ≥ σ i R /(σ i + σ j ) , Dt j ≥ σ j R /(σ i + σ j ) , then Sti = min{Dti ,σ i R /(σ i + σ j )} =

σ i R /(σ i + σ j ) . Therefore, P{Sti = (σ i ⋅ R) /(σ i + σ j )} = Fi [(σ i ⋅ R) /(σ i + σ j )] ⋅ Fj [(σ j ⋅ R) /(σ i + σ j )]

(i, j = 1, 2; i ≠ j).

Next, we calculate P{Sti ≤ x} for x < σ i R /(σ i + σ j ) and x > σ i R /(σ i + σ j ) separately. For the first case, by (5), we know R − Dt j > σ i R /(σ i + σ j ) , hence Sti ≤ x could occur in two cases: 1) Dti ≤ x ; 2) x

σ i Dt j / σ j + z ≤ x < Dti , then Φ i ( x) = Fi ( x) + Fi ( x) ∫0 Fj [σ j ⋅ ( x − z ) / σ i ]ψ ( z )dz .

21

For the second case, we know from (5) that if Dt j ≥ R − x , then R − Dt j ≤ x , hence Sti ≤ x holds. Moreover, if Dt j < R − x , min{Dti ,σ i Dt j / σ j + z , R − Dt j } ≤ x ⇒ min{Dti ,σ i Dt j / σ j + z} ≤ x , which happens in the following three cases: 2a) if z ≥ x , then σ i Dt j / σ j + z ≥ x , min{Dti ,σ i Dt j / σ j + z} ≤ x ⇒ Dti ≤ x ; 2b) if z < x , σ i Dt j / σ j + z ≤ x ⇒ Dt j ≤ σ j ( x − z ) / σ i then min{Dti ,σ i Dt j / σ j + z} ≤ x ; and 2c) if

σ j ( x − z ) / σ i ≤ Dt j ≤ R − x , then min{Dti ,σ i Dt j / σ j + z} ≤ x ⇒ Dti ≤ x . Therefore, we could conclude that Φ i ( x) = Fj ( R − x) + Ψ ( x) F j ( R − x) Fi ( x) x

{

}

+ ∫ F j [σ j ( x − z ) / σ i ] + Fi ( x) ⎡⎣ F j ( R − x) − Fj [σ j ( x − z ) / σ i ]⎤⎦ ψ ( z )dz 0

( x > σ i R /(σ i + σ j ) ).

Q.E.D. A3. Independence among St − Lw , St − Lw+1 , …, and St −1 ( Sti− Lw , Sti− Lw+1 , …, and Sti−1 )

----- St − Lw ___ St − Lw+1 ___ St − Lw+ 2 ___------ St −1 ___ St ___ St +1 ___ St + 2 ------ St + Lw−1 ___ St + Lw ---

Figure 2: Plot of Relationship among St Figure 2 can be read as: 3.1) St − Lw determines St ; St in turn determines St + Lw and so on; 3.2) St − Lw+1 determines St +1 ; St +1 in turn determines St + Lw+1 and so on; 3.3) St − Lw+ 2 determines St + 2 ; St + 2 in turn determines St + Lw+ 2 and so on;

# 3. Lw ) St −1 determines St − Lw+1 ; St − Lw+1 in turn determines St − 2 Lw+1 and so on. Namely, the sequence of sales consists of Lw independent subsequences of Markov chains, and then it becomes clear that within any Lw consecutive periods, the sales are independent. Similar arguments hold also for Sti− Lw , Sti− Lw+1 , …, and Sti−1 .

Q.E.D.

22

A4. Proof of Theorem 2

2.1) We write TC as

TC ( R, Lw , Lr ) = [c + ( Lw + Lr + 1)b]PC ( R, Lw , Lr ) − bR + l ⋅ LS ( R, Lw , Lr ) + ( h + b) HC ( R, Lw , Lr ) .

(A4.1)

Let us look at the first and second order derivatives of TC ( R, Lw , Lr ) : dTC ( R, Lw , Lr ) / dR = [c − l + ( Lw + Lr + 1)b]Ψ ( R ) − b + (h + b) ⎡ ∑ i =1σ i ∫ 0 ⎣⎢

ρi ( wi , vi )dvi dwi ⎤ /(σ 1 + σ 2 );

(A4.2)

Ri 2 dTC 2 ( R, Lw , Lr ) / dR 2 = [l − c − ( Lw + Lr + 1)b]ψ ( R ) + (h + b) ⎡ ∑ i =1σ i2 ∫ ρi ( wi , Ri − wi )dw⎤ /(σ 1 + σ 2 ) 2 . 0 ⎣⎢ ⎦⎥

(A4.3)

2

Ri

∫

Ri − wi

0

⎦⎥

Clearly, if (l − c) − ( Lw + Lr + 1)b ≥ 0 , dTC 2 ( R, Lw , Lr ) / dR 2 ≥ 0 and TC ( R, Lw , Lr ) is convex in R. 2.2) and 2.3) If Dt1 and Dt2 follow identical distribution, then f1 (ξ ) = f 2 (ξ ) , and dHC 2 ( R, Lw , Lr ) / dR 2 = ⎡ ∫ ⎣⎢ 0

R/2

ρ ( w, R / 2 − w)dw⎤ / 2 = ⎡ ∫ ⎦⎥

⎣⎢

R/2

0

ρ ( R / 2 − v, v)dv ⎤⎥ / 2 . ⎦

(A4.4)

And the conditional densities of ρ ( R / 2 − v, v) and ψ ( R ) given any realizations of v and z ( St − Lw and i ⋅ Lr ⎤ Vt i = ⎡⎣ StLw − Lw − Lr / 2 + St − Lr ⎦ are already known at the beginning of period t) are given as:

ρ ( R / 2 − v, v | v, z ) = φ ( R / 2 − v | v, z ) =

d Φ ( x) |x = R / 2 − v = f ( R / 2 − v) F ( R / 2 − v − z ) + f ( R / 2 − v − z ) F ( R / 2 − v); dx

(A4.5)

(taking derivative of (7a) at x = R / 2 − v < R / 2 ), where φ (⋅) ≡ φi (⋅) and Φ (⋅) = Φ i (⋅) as the retailers are identical. Also R

ψ ( R | z ) = ∫ f (ξ ) f ( R − ξ )d ξ − 2∫ 0

(R−z) / 2

0

f (ξ ) f ( R − ξ )d ξ + f [( R − z ) / 2]F [( R + z ) / 2] .

(A4.6)

It can be easily checked that φ ( R / 2 − v | v, z ) /ψ ( R | z ) is non-decreasing in R for any realization of z and v when demand distribution is uniform or exponential; this implies that dTC 2 ( R, Lw , Lr ) / dR 2 could change sign at most once as R increases, hence TC ( R, Lw , Lr ) is concave-convex. If (l − c) /( Lw + Lr + 1) < b ≤ (l − c) /( Lw + Lr ) , then ⎧dTC ( R, Lw , Lr ) / dR |R = 0 = c − l + ( Lw + Lr + 1)b − b = c − l + ( Lw + Lr )b ≤ 0, ⎨ ⎩dTC ( R, Lw , Lr ) / dR |R =+∞ = −b + (h + b) = h > 0,

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it follows that TC ( R, Lw , Lr ) is unimodal; if b > (l − c) /( Lw + Lr ) , then

dTC ( R, Lw , Lr ) / dR |R = 0 = c − l + ( Lw + Lr + 1)b − b = c − l + ( Lw + Lr )b > 0 , then unimodality is not guaranteed. This completes the proofs of Theorems 2.2) and 2.3).

Q.E.D.

A5. Proof of Theorem 3

We will prove by induction. If R = 2, π 22 = 1 = π 22 +1 + π 32 +1 , so Theorem 3 holds trivially. If R > 2, π 2R = ∑ i = 2 π iR ⋅ Ρi 2 = ∑ i = 2 π iR ⋅ P{Dt1 + Dt2 = 2} = P{Dt1 + Dt2 = 2} (this follows from (13c) since R

R

R 2 ≤ i, 2< R); and if R > 3, π 3R = ∑ i= 2 π iR ⋅ Ρi 3 = π 2R ⋅ a23 + (1 − π 2R ) P{Dt1 + Dt2 = 3} is independent of R too R

R (by (13b), a23 is independent of R since 2 < 3 < R). Similarly, we can show that π iR is independent of R

for i ∈ Ω R −1 , which means when the target is increased from R to R +1, π iR ( i ∈ Ω R−1 ) does not change, and π RR = π RR +1 + π RR++11 must hold given ∑ i = 2 π iR = ∑ i = 2 π iR +1 = 1 . If Π R = (π 2R ,..., π RR ) are computed, then R

R +1

for Π R+1 , we only need to compute π RR +1 and π RR++11 , where

π RR +1 = ∑ i = 2 π iR +1 ⋅ aiRR +1 + (π RR +1 + π RR++11 ) P{Dt1 + Dt2 = R} and π RR++11 = π RR − π RR +1 . R −1

Q.E.D.

A6. Constraints for Hybrid Policy

To make sure that sufficiency and allocation equalization is viable, we need the following constraints similar to (1): ⎧ Sti + St j ≤ IPt i + IPt j + Q− L , ⎪ j i ⎨−σ i Q− L ≤ σ i ( IPt +1 − Lr u j ) − σ j ( IPt +1 − Lr ui ) ≤ σ j Q− L , (−L≤ t ≤ −1 −Lr; i, j = 1, 2; i ≠ j) ⎪ i i i ⎩ IPt +1 = IPt − St ,

(A6.1)

where Q− L is outstanding at the warehouse. As the set of equations make clear, the inventory position is R in period −L and reduces by the sales in the following periods within the cycle. Specifically, from

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period− L to period −1 − Lr, Q− L is not allocated, hence the two retailers have no pipeline inventory, and IPt i = IOti ; while from −Lr to period −1, both retailers have pipeline inventory (since allocation is made at the start of period −Lr); from period 0 to T − L − 1, IPt i = IOti again since the orders have arrived at the retailers. To accommodate redistribution, we define two new variables X −ij1− Lr (i, j = 1, 2; i ≠ j) here, whose difference represents the net number of shipment from the other retailer. Then,

⎧ IO −i 1− Lr = IO−i 1− Lr − X −ij1− Lr , ⎪ ij i j ji ⎨ X −1− Lr − X −1− Lr ≤ σ i ( IO−1− Lr − IO−1− Lr ) /(σ i + σ j ), ⎪ ij i ⎩0 ≤ X −1− Lr ≤ IO−1− Lr .

(A6.2)

In (A6.2), IO −i 1− Lr ( IP−i1− Lr ) represents inventory-on-hand (inventory position) after redistribution of retailer i (i = 1, 2). In general, a unit redistribution cost c−ij1− Lr (i, j = 1, 2; i ≠ j) will be incurred associated with redistribution amount X −ij1− Lr , thereby assuring that cross flows will not arise (actually, redistribution could be made in any period within one cycle, and redistribution leadtime could be different from Lr). Finally, at the end of each period, transshipment is allowed for higher service level, which is formulated as the following set of inequalities (and equalities). For expositional convenience, we define two new variables Yt ij (i, j =1, 2; i ≠ j), which represents the partial demand of retailer j that is satisfied from inventory-on-hand of retailer i. ⎧ IOti + = IOti − + Yt ji − Yt ij − Sti , ⎪ i+ i− ji ij i ⎨ IPt = IPt + Yt − Yt − St , ⎪ ij i− i ⎩0 ≤ Yt ≤ IOt − St .

(A6.3)

In (34), IOti − and IPt i − ( IOti + and IPt i + ) represent the starting inventory-on-hand and inventory position (ending inventory-on-hand and inventory position) of retailer i (i = 1, 2). Since a unit transshipment cost ctij (i, j = 1, 2) will be incurred associated with transshipment amount Yt ij , it assures that cross flows will

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not arise (The above control occurs in the current cycle (from period −L to period T − L − 1), similar control occurs in other cycles also).

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