Order-based backorders in multi-item inventory systems

Operations Research Letters 38 (2010) 27–32

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Operations Research Letters journal homepage: www.elsevier.com/locate/orl

Order-based backorders in multi-item inventory systems Jiejian Feng a , Liming Liu a , Yat-wah Wan b,∗ a

Faculty of Business, Hong Kong Polytechnic University, Hong Kong

b

Graduate Institute of Global Operations Strategy and Logistics Management, National Dong Hwa University, Hualien, Taiwan

article

info

Article history: Received 15 September 2008 Accepted 28 August 2009 Available online 16 September 2009 Keywords: Multi-item inventory systems Order-based backorders Assemble-to-order

abstract A new exact method is provided for evaluating the average order-based backorders of a multi-item inventory system under the (r , nQ ) inventory policy for items. The key to the method is as follows: the joint inventory positions for the same epoch and for different epochs share the same conditions of being independent and uniformly distributed. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In a multi-item inventory system, such as the assemble-toorder system, a customer order (or a production order) for an end product usually requires a number of different (kinds of) items (components). The order is satisfied only if all the required items are available; otherwise, it is lost or becomes a backorder. The order-based backorders (OBBs) in terms of numbers of customer orders backordered can be very different from the item-based backorders in terms of numbers of individual items backordered. Item-based backorders are useful for replenishment decisions and have been studied thoroughly. OBBs on the other hand provide a better measure of the customer fulfillment performance, the average order delay time, etc. However, the literature on this important subject is very recent and limited due to its difficulty. Song [1] recognizes that the starting point for evaluating the OBBs is the steady-state distributions of the inventory positions. Under the (r , nQ ) policy, studies [2–4,1] give conditions such that the joint steady-state inventory positions are independent and uniformly distributed. Knowing the distributions of the inventory positions, one can then focus on computing the average backorders of each inventory position. Song [5] finds closed-form expressions for OBBs under the base-stock policy and provides an approximation. A different approach in [6] is to evaluate the average backorders in the context of machine repair. There, failed machines (i.e., unfilled orders in [5]) can switch components instantaneously, while in [5] the First-Come–First-Served (FCFS) allocation discipline is adopted

∗

Corresponding author. E-mail address: [email protected] (Y.-w. Wan).

0167-6377/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2009.09.001

and available items are reserved for orders waiting for unavailable items. The order-based expected delay is also studied. By an itembased index, Seshadri and Swaminathan [7] minimizes the orderbased expected delay under the budget constraint. In this paper, we contribute to the literature and, potentially, to the practice by providing an implementable and fairly complete approach to the analysis of OBB problems. First, the key in Song [1] for establishing the result is for the stationary joint inventory positions at an order arriving epoch to be independent and uniformly distributed. We establish the result by requiring the stationary joint inventory positions at an arbitrary sequence of time epochs to be independent and uniformly distributed. We actually show that the two facts are indeed equivalent for independent Poisson arrivals. All these results are interesting in their own right. Second, both Song [5] and the present authors establish a recursive relation for evaluating the average OBBs. A successor in our recursion has at least one type of item less than its precedent. Moreover, orders after items deleted in our recursion may be combined. These reductions reduce our computational effort. Third, we develop a new approximation for average OBBs. In its calculation, the linkage of items by orders is decoupled, though the effect of orders on OBBs is still accounted for. 2. Model description Consider an inventory system with I different items where a customer demand may ask for multiple items, and possibly for multiple units for some items requested. (For clarity, from here onwards, the term ‘‘demand’’ refers to a customer order.) Assume that there are N demand types which form the demand family Θ for the system. Let row vector dj = (d1j , d2j , . . . , dIj ) be the jth member of Θ , where dij is the number of item i required by demand

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J. Feng et al. / Operations Research Letters 38 (2010) 27–32

dj , such that dij ≥ 0 and i=1 dij > 0 for i = 1, 2, . . . , I, and j = 1, 2, . . . , N. As defined, the demand family Θ = {d1 , . . . , dN }, and a batch demand on item i is requested whenever dij ≥ 2. For convenience of exposition, the notation d without a subscript is used to represent a generic demand type in Θ . Assume that the arrivals of type-d demands follow a Poisson process of rate λd , single itemP or not, independent of the arrivals of d other demand types. Let λ = d∈Θ λ be the rate of the combined Poisson arrival stream of all demand types. Obviously, an arriving demand is of type d with probability qd = λd /λ. For this system the inventory of each of the I kinds of items is controlled independently by an (ri , nQi ) policy, that is, whenever the inventory position of item i is reduced to ri or below, a replenishment order for a minimum quantity in multiples of Qi (>0) is placed for the item to bring its inventory position above ri (≥0), i = 1, 2, . . . , I. (From here onwards, the term ‘‘order’’ refers to a system replenishment order of an item.) Thus, the inventory position of item i always falls within ri + 1, . . . , ri + Qi . A replenishment order for item i comes after a constant lead time Li . Without the loss of generality, assume that L1 ≤ L2 ≤ · · · ≤ LI . An arriving demand is satisfied and leaves immediately if all of its requested items are available. If any of the requested items is out of stock, the demand is backordered. The FCFS rule is applied in the system: when a demand is backordered due to the shortage of one or more items, the available units of all items requested by this demand are reserved for this demand and are not available for other future demands. Let IPi (t ) be the inventory position at time t, and Bd (t ) be the number of (order-based) type-d backorders at time t. Under very mild conditions to be specified later, the steady-state versions of these quantities IPi and Bd exist; and the following long-run averages are then well-defined: the average backorders of type-

PI

d

d demands B

P

= E [Bd ]; the average backorders of all demands

d

B= d B ; and the average time in system of a (generic) demand W = B/λ. For ease of exposition, let g (·|µ) and G(·|µ) denote the probability mass function and the cumulative distribution function of the Poisson distribution with parameter (mean) µ, respectively, and G0 = 1 − G. Let η ∧ ν = min{η, ν} for any real numbers η and ν . Normally, the superscript ‘‘d’’ indicates the demand type and the subscript ‘‘i’’ identifies the item type.

For the idea of the proof, consider a two-item system in the steady state, and t1 ≥ t2 . IP1 (t1 ) depends only on IP1 (t2 ) and on terms independent of IP2 (t2 ), which gives the independence of IP1 (t1 ) and IP2 (t2 ). The uniform distribution of the joint inventory positions then follows readily from the marginal distributions. When the conclusion of Theorem 1 holds, the PASTA result shows that for the steady state, (IP1 (t1 ), . . . , IPI (tI )) are independent and uniformly distributed at arrival epochs t1 ≥ t2 ≥ · · · ≥ tI . This is how we evaluate the average OBBs in subsequent sections. In our derivation, the cases for unit and batch demands follow exactly the same procedure, only with heavier notation for batch demands. (The OBB for a single-item demand is a well-known standard result ([5]).) For clarity, the OBBs of a two-item, threedemand system where a single unit is requested for each item in demand is discussed in Section 3.1. The generalization to batch demands is outlined at the end of Section 3.1 and discussed in detail in Section 3.2. The numerical work in Section 4 certainly considers the general case of batch demands. 3.1. Two-item systems Consider an arbitrary type-(1, 1) demand arriving in the stationary state for a product family Θ = {(1, 1), (1, 0), (0, 1)}. Let W (1,1) be the time that this type-(1, 1) demand spends in the system. Let us consider whether W (1,1) is less than or equal to w for a particular w . From the arrival epoch, we look back L1 − w time units for the inventory position of item 1 and L2 − w time units for the inventory position of item 2. Let Ψ (s1 , s2 ) denote the event where the observed inventory positions are s1 and s2 for items 1 and 2, respectively. For convenience, let the arrival be at time 0; take times −(L1 − w) and −(L2 − w) to be L1 − w and L2 − w time units before the arrival epoch, respectively, as shown in Fig. 1. Similarly, time w is actually w time units after the arrival epoch. Let W (1,1) (s1 , s2 ) be the waiting time of the type(1,1)

(1, 1) demand conditioning on the event Ψ (s1 , s2 ). Let B (s1 , s2 ) be the conditional expected type-(1, 1) backorders when Ψ (s1 , s2 ) occurs. By Theorem 1 and Little’s law, the expected type-(1, 1) backorders is given by (1,1)

B

=

3. Computation of the average type-d backorders The distribution of the joint inventory positions of items is not straightforward, irrespective to their uniform marginal distributions. In general, an item can be requested by different kinds of demands, whose interactions make the average OBBs hard to evaluate. Let IP(t ) = (IP1 (t ), . . . , IPI (t )) be the joint inventory positions at t, and P = {(IP1 , . . . , IPI ) : IPi ∈ {ri + 1, . . . , ri + Qi }, i = 1, . . . , I } be the set of all possible joint inventory positions. Theorem 1 of [1] shows that under the (r , nQ ) policy, the steadystate joint inventory positions IP of the discrete-time Markov chain {IP(t )} are independent and uniformly distributed if {IP(t )} forms an irreducible and aperiodic Markov chain. This condition can be weakened by dropping the aperiodicity in the condition ([4]). The subsequent development in this paper assumes that such a condition holds, e.g., the embedded chain {IP− } of {IP(t )} being irreducible, leading to independent and uniformly distributed joint inventory positions (IP1 (t ), . . . , IPI (t )) for all t. The derivation of the OBBs relies on a result that is simple, but interesting in its own right: Theorem 1. Suppose that the steady-state joint inventory positions (IP1 (t ), . . . , IPI (t )) are independent and uniformly distributed for any t. Then the joint inventory positions (IP1 (t1 ), . . . , IPI (tI )) are independent and uniformly distributed for a sequence of arbitrary epochs t1 ≥ t2 ≥ · · · ≥ tI .

=

1 Q1 Q2

r1 +Q1

r2 +Q2

X

B

( s1 , s2 )

s1 =r1 +1 s2 =r2 +1

r +Q λ(1,1) 1X1

Q1 Q2

(1,1)

X

r2 +Q2

X

E [W (1,1) (s1 , s2 )],

s1 =r1 +1 s2 =r2 +1

where E [W

(1,1)

(s1 , s2 )] =

Z

L2

P (W (1,1) (s1 , s2 ) > w)dw

0

Z

L2

[1 − P (W (1,1) (s1 , s2 ) ≤ w)]dw

= 0

L1

Z = L2 −

P (W (1,1) (s1 , s2 ) ≤ w)dw

0

Z

L2

−

P (W (1,1) (s1 , s2 ) ≤ w)dw.

(1)

L1

What remains to be computed are the two integrals in Eq. (1). RL Consider first L 2 P (W (1,1) (s1 , s2 ) ≤ w)dw . For w ∈ (L1 , L2 ), the 1

event {W (1,1) (s1 , s2 ) ≤ w} depends only on item 2. Note that all outstanding replenishment orders at time −(L2 − w) for item 2 will have arrived by time w . Thus, given the inventory position s2 at −(L2 − w), if the total demands for item 2 between −(L2 − w) and the arrival epoch are fewer than s2 , the time that this type-(1, 1) demand spends waiting in the system will be w or

J. Feng et al. / Operations Research Letters 38 (2010) 27–32

29

3.2. Multi-item systems The analysis in this section is for general type-d demand, posd

Fig. 1. Waiting time of a type-(1, 1) demand.

shorter. Otherwise, this type-(1, 1) demand will be satisfied after time order after −(L2 − w). Let λi = P w by a new replenishment dj be the arrival rate of demands for item i (by λ dj ∈{dj :dij >0, 1≤j≤N } summing up all related orders). Thus, L2

Z

P (W (1,1) (s1 , s2 ) ≤ w)dw =

L2

Z

G(s2 − 1|λ2 (L2 − w))dw

L1

L1

=

s2 −1 1 X

λ2

G0 (x1 |λ2 (L2 − L1 )).

x 1 =0

For w ∈ (0, L1 ), any delay of a type-(1, 1) demand may be caused by the shortage of item 1, item 2, or both items. Given s1 at −(L1 − w) and s2 at −(L2 − w), the time that this type-(1, 1) demand spends in the system is not over w if the demand for item 1 between −(L1 − w) and its arrival is less than s1 and the demand for item 2 between −(L2 − w) and its arrival is less than s2 , i.e., L1

Z

P (W (1,1) (s1 , s2 ) ≤ w)dw

0 L1 s1 ∧s2 −1

Z

X

= 0

g (x1 |λ(1,1) (L1 − w))

x 1 =0

× G(s1 − x1 − 1|λ(1,0) (L1 − w))

s2 −x1 −1

X

g (x3 |λ(0,1) (L1 − w))

x 3 =0

×

2

−

x 1 =0

x 2 =0

(2)

IP2 −1 λ(1,1) X G0 (x1 |λ2 (L2 − L1 )) λ2 x =0

Z

Lh

The analysis leading to and including Eq. (1) holds also for batch demands, with more demand combinations considered in computing the integral in (1). Consider, for example, the demand family Θ = {(2, 5), (3, 8)}. The evaluation of the type-(3, 8) order-based RL backorder involves calculating 0 1 P (W (3,8) (s1 , s2 ) ≤ w)dw and P (W (3,8) (s1 , s2 ) ≤ w)dw . For the former, we need to list all the possible combinations of demands (2, 5), (3, 8) such that these demands require no more than s1 − 3 units of item 1 and no more than s2 − 8 units of item 2 in the interval (−(L1 − w), 0). For the latter, we only need to list all the possible combinations that require no more than s2 − 8 units of item 2 in the interval (−(L2 − w), 0). L1

P (W k (s|L) ≤ w)dw.

P (W k (s|L) ≤ w)dw =

(3)

Z

Lh

P W k−1 (s−1 |L−1 ) ≤ w dw,




L1

L1

in which W (s−1 |L−1 ) is the corresponding stationary waiting time for a system with one fewer item. Substituting this back to (3), k−1

E [W (s|L)] = k

Z

L1

P (W k−1 (s−1 |L−1 ) ≤ w)dw

0 L1

Z

x 3 =0

(x1 + x2 + x3 )! G(IP2 − x1 − x3 − 1|λ2 (L2 − L1 )) x1 !x2 !x3 ! × (q(1,1) )x1 (q(1,0) )x2 (q(0,1) )x3 G0 (x1 + x2 + x3 |λL1 ).

R L2

P (W k (s|L) ≤ w)dw

Note that item 1 does not affect the third term on the right-hand side of (3), i.e.

∧IP2 −1 IP1X −x1 −1 IP2X −x1 −1 (1,1) IP1 X

λ

L1

L1

1

λ

Lh

Z −

(x1 + x2 + x3 )! (1,1) x1 (1,0) x2 (0,1) x3 (q ) (q ) (q ) x1 !x2 !x3 !

(IP1 , IP2 ) = λ(1,1) L2 −

Z 0

Theorem 2. The expected type-(1, 1) OBBs is given by (1,1)

In general, to compute the average backorders of the general type-d demand that requires h different items, drop all non-d items from consideration to obtain a new system with these h items forming Nd kinds of demands for the demand family Θ d = {k1 , k2 , . . . , kNd }. Consolidate arrival rates of the new demands, if necessary. Let the original type d be changed in the new system to type k. For notational simplicity, take k = k1 = (k11 , . . . , kh1 )1×h . Clearly, ki1 ≥ 1 for i = 1, . . . , h. Denote the observed joint inventory position by s = (s1 , s2 , . . . , sh ) and the lead times by L = (L1 , L2 , . . . , Lh ). For any vector v = (v1 , . . . , vh ), let v−1 = (v2 , . . . , vh ) be the vector reduced from it by taking the first element out. Given lead times L, let W k (s|L) be the stationary waiting time of type-k demands at inventory positions s. Similar to (1), we can show that

3

× G0 (x1 + x2 + x3 |λL1 ).

B

Example 1. Consider a four-item system with Θ = {(3, 0, 0, 8), (2, 9, 0, 7), (0, 0, 6, 8), (0, 0, 0, 8)}. For d = (2, 9, 0, 7), item 3 is a non-d item. Dropping item 3 leads to a simplified system: The drelated new demand family is Θ d = {(3, 0, 8), (2, 9, 7), (0, 0, 8)}. The original demand type (2, 9, 0, 7) is reduced to k = (2, 9, 7). The relevant demand rates are consolidated: λ(3,0,8) = λ(3,0,0,8) , λ(2,9,7) = λ(2,9,0,7) , and λ(0,0,8) = λ(0,0,6,8) + λ(0,0,0,8) .

E [W k (s|L)] = Lh −

× G(s2 − x1 − x3 − 1|λ2 (L2 − L1 ))dw s ∧s2 −1 s1 − x 1 −1 s 2 − x 1 −1 X X 1 1X G(s2 − x1 − x3 − 1|λ2 (L2 − L1 )) = λ x =0 x =0 x =0 1

sibly with batch demands of items. To compute B , it suffices to consider only items in the type-d demand. Items not in d, i.e., the non-d items, can be treated as if they are absent; after all, our calculation is based on the demands for individual items. The following example illustrates this procedure.

−

P (W k (s|L) ≤ w)dw + E [W k−1 (s−1 |L−1 )].

0

(4) Thus, (4) provides a recursive formula for E [W k (s|L)] as long as the integrals can be computed, which leads to a reduced system of (at least) one fewer item. The process continues until a single item is left. RL To evaluate 0 1 P (W k (s|L) ≤ w)dw, let Jk (s) be the collection of demand arrivals in the reduced system such that the quantity required for each item does not exceed the observed inventory position of this item. Define Jk (s) = {x = (x1 , x2 , . . . , xNd ) :

PNd

j=1 xj kij < si , for i = 1, 2, . . . , h}, where xj is the number of type-kj demands. In the same spirit as for the two-item case, given any w ∈ [0, L1 ], the waiting time of a type-k demand is not longer than w if the quantity of item i required in the interval (−(Li − w), 0) is fewer than si for all i, as illustrated in Fig. 1.

30

J. Feng et al. / Operations Research Letters 38 (2010) 27–32

Let 1s = (s2 −

PNd

j =1

xj k2j , . . . , sh −

PNd

j=1

xj khj ), 1L = (L2 −

L1 , , . . . , Lh − L1 ), and P (k|1s ,1L ) be the probability that the number of item i units required in interval 1L is fewer than 1s for demand k for i = 1, 2, . . . , h. For the current system, possibly a simplified system from an earlier iteration of recursion, let e λ be the total kj

arrival rate of all demands and e qkj = λe be the proportion of typeλ kj demands among the total demand arrivals. Let m = x1 + · · · + xNd . Then, conditioning on the demand arrivals in the time interval from −(L1 − w) to the arrival epoch, we have L1

Z

P (W k (s|L) ≤ w)dw

0

=

L1

X Z x∈Jk (s)

m! x1 ! . . . xNd !

0

k x q Nd ) Nd (e qk1 )x1 . . . (e

[e λ(L1 − w)]m −eλ(L1 −w) (k−1 |1s ,1L ) e P dw m! X m! 1 k x q Nd ) Nd G0 (m|e = λL1 )P (k−1 |1s ,1L ) , (e qk1 )x1 . . . (e e x ! . . . x ! λ 1 N d x∈J (s) ×

k

where P (k−1 |1p ,1L ) can be evaluated similarly by conditioning sequentially on the time intervals. The above approach can be RL applied to evaluate 0 1 P (W k−1 (s−1 |L−1 ) ≤ w)dw in (4). With known integrals, (4) becomes a recursive equation of E [W k (p|L)]. Moreover, from (4), Little’s formula gives the following result.

Basically, we single out item i in considering the componentbased backorders of this type of item. To illustrate the idea, consider the demand (3, 4) in the three-demand system {(2, 0), (0, 3), 3e

3e

spectively. BΘ1 is the OBB of demand 3e in the system of demand 4e

family Θ1 = {2e, 3e}, and BΘ2 is the OBB of demand 4e in the system of demand family Θ2 = {3e, 4e}. It is straightforward to 3e

k

B = λ

k

L1

Z

P (W

k−1

(s−1 |L−1 ) ≤ w)dw

0

Z − 0

L1

λk k−1 P (W k (s|L) ≤ w)dw + k B . λ −1 

When the replenishment lead times of two or more items are the same, the above procedure remains the same except that the number of iterations will be fewer. Basically, in an iteration, if two or more items share the current shortest lead time, we can remove all of these items at the same iteration. This obviously speeds up the reduction process. Remark 1. In addition to reducing one dimension of L in an iteration, our evaluation procedure also reduces the number of demand types in the demand family since some of the previously different demand types become identical and can be consolidated. In Example 1, after removing item 1, the demand (3, 0, 8) is no longer different from the original demand (0, 0, 8); and the arriving rate of the demand (0, 8) in the reduced system becomes λ(3,0,8) + λ(0,0,8) . The advantage from these reductions over Song’s approach will be more significant when the numbers of items and demand types are larger since it is more likely to remove more demand types in an iteration. 4. An approximation for average OBBs Our approximation requires the new concept of componentbased backorders to avoid confusion with the item-based counterparts for batch demands. Let e = (1) be a 1 × 1 unit vector. Definition 1. The average component-based backorders of the ith component of demand dj = (d1j , . . . , dIj ) in a system with demand family Θ = {d1 , . . . , dN } is the average of OBBs of demand dij e in the system consisting only of item i for demand family Θi = {di1 e, . . . , diN e} − {0e}.

(3,4)

4e

3e

4e

prove that max{BΘ1 , BΘ2 } ≤ B ≤ BΘ1 + BΘ2 : An average component-based backorder underestimates the average OBBs because a demand may wait for an item not considered by the component; on the other hand, the sum of the average component-based backorders overestimates the OBBs because two component-based backorders are generated for one backorder when both items are unavailable (cf. [5]). The desirable approximation is from compensating this overestimation. Again consider demand 3e and 4e generated from demand (3, 4). Let p3e Θ1 be the probability that demand 3e does not need to wait in the demand family Θ1 , and p4e Θ2 be the corresponding probability for demand 4e in the demand family Θ2 . 4e Suppose p3e Θ1 > pΘ2 . Intuitively, when demand (3, 4) waits for both items, item 1 is more likely to be available earlier, and we use a form of conditional average backorders of item 2 to compensate (3,4)

3e

BΘ1 to get B

4e . A similar argument applies when p3e Θ1 ≥ pΘ2 . Thus,

an approximation of B

(

Theorem 3. The expected OBB is given by

4e

(3, 4)}. Let BΘ1 and BΘ2 be the average component-based backorders of the first and the second component of demand (3, 4), re-

3e

(3,4)

4e

BΘ1 + p3e Θ1 BΘ2 , e B(ne3,w4) = 4e 3e BΘ2 + p4e Θ2 BΘ1 ,

is 4e if p3e Θ1 > p Θ2 ,

otherwise.

To generalize to more components, consider demand (3, 4, 6) as an (3,4,6)

3e

4e

6e

3e 4e 3e 4e 6e example. e Bnew = BΘ1 + p3e Θ1 BΘ2 + pΘ1 pΘ2 BΘ3 if pΘ1 > pΘ2 > pΘ3 ; other cases are found similarly. Consider a two-item system underqthe (r , nQ ) policies such that

ri = max{1, b0.5 + Li

P λdj dij + zi Li λdj dij c}, where b(·)c is the largest integer smaller than (·), and zi is the parameter used P

to control the fill rate (i.e., service level). In total, there are 60 groups of experiments under different parameter combinations: λ ∈ {2, 5}, L1 ≡ 1, L2 ∈ {1.5, 2, 3}, Q ∈ {3, 7} and zi ∈ {1.04, 1.28, 1.64, 2.33, 2.58}. The values zi correspond roughly to fill rates 85%, 90%, 95%, 99% and 99.5%, respectively. In each group of experiments, the two items have the same zi value, and 500 replications are run for each group. For each replication, let qd1 , qd2 and qd3 be uniform (0, 1) random variables P independent normalized to qdj = 1. Let the demand rate λdj = λqdj + 0.02 for j = 1, 2, 3 where 0.02P for avoiding a demand is very close to zero. Then, qdj = λdj / λdj . Finally, generate dij randomly such as d1 = (d11 , 0), d2 = (0, d22 ) and d3 = (d13 , d23 ) where 1 ≤ d11 , d13 < Q and 1 ≤ d22 , d23 < Q . Certainly, batch demands and arbitrary demand vectors are included in our numerical work. To get the benchmark, take the average of the lower and upper bounds of the average OBB to be the approximation, which is the only approximation available in the literature ([5]). This approximation is denoted as the ‘‘Average-Bound’’ method in Table 1 whereas the proposed approximation in this  is  paper

B(·) −B × B

denoted as ‘‘Our-Method’’. In Table 1, MAE = mean

100%, where B is the exact value and B(·) is the corresponding

  B(·) −B × 100% . B

approximation method (·); and VAE = v ariance

Table 1 shows that our approximation is more accurate than the ‘‘Average-Bound’’ method, especially for systems whose lead times are very different. We have the same observations for other settings and for a three-item system, results that we omit for brevity. Although the lead times appear to be the key for the

J. Feng et al. / Operations Research Letters 38 (2010) 27–32

31

Table 1 Approximation of the demand requiring two items in the systems with two items.

λ=2 Q =7

Our-Method

L2 = 1.5

z

MAE

2.58 2.33 1.64 1.28 1.04

3.80 3.90 3.77 3.63 3.72

L2 = 2 Average-Bound

Our-Method

VAE

MAE

VAE

MAE

17.58 17.51 13.75 10.49 8.90

5.06 4.93 4.54 4.61 4.20

12.68 12.51 10.11 8.70 9.49

2.84 2.92 2.88 2.87 3.01

difference in accuracy between two methods, we believe that the underlying principle is in fact the complexity of interaction among demands. For complex systems, it is hard to predict the true value relative to the bounds. It seems that the true value may spread fairly uniformly in this range. Thus, the Average-Bound method may work well in such situations. However, when lead times have a large difference, the interaction will be weak. In Fig. 1, the OBBs from time −(L2 − w) to −(L1 − w) depend only on item 2, a period without interaction of the two items. Hence, the complexity of interaction will be relatively weak if L2 is much greater than L1 .

L2 = 3 Average-Bound

Our-Method

VAE

MAE

VAE

MAE

VAE

MAE

VAE

8.49 8.09 6.21 5.49 5.47

5.82 5.40 4.69 4.07 4.17

15.00 13.56 11.24 9.47 8.39

1.78 1.88 2.18 2.60 2.90

2.92 2.46 2.52 3.58 4.89

6.94 6.58 5.47 4.73 4.29

13.71 13.13 11.55 10.42 9.36

To evaluate L 2 P (W (1,1) (s1 , s2 ) ≤ w)dw , check that item 1 has no 1 effect on the backorders when s1 ≥ 1; in this case, the evaluation is equivalent to a single-item system consisting of item 2. When s1 ≤ 0, the time of waiting for item 1 will not be over w if there is ever any replenishment order for the item issued in the time interval [0, w − L1 ). Thus, when s1 ≤ 0,

RL

Z

= λ(1,1)

L1

Z ∞

+

P (W (1,1) > w)dw +

Z

L2

P (W (1,1) > w)dw

L1



P (W (1,1) > w)dw .

L2

With W (1,1) (s1 , s2 ) defined in Section 3.1, L1

Z

P (W (1,1) > w)dw = L1 −

0

1 Q1 Q2 L1

Z ×

G(s2 − 1|λ2 (L2 − w))G0 (s1 − r1 − 2|λ1 (w − L1 ))dw

P (W (1,1) > w)dw = L2 − L1 −

L1

Z

L2

× L1

1 Q1 Q2

0

Z

∞

P (W (1,1) > w)dw =

L2

1

X

Q1 Q2

s1 ≤0 or s2 ≤0

∞

Z ×

P (W (1,1) (s1 , s2 ) > w)dw.

L2

Check that P (W (1,1) (s1 , s2 ) > w) = 0 if for each item a replenishment order is triggered in the time interval [0, w − Li ), i = 1, 2. There are three subcases to consider, depending on whether one or both of si ≤ 0. If s1 > 0 and s2 ≤ 0, ∞

P (W (1,1) (s1 , s2 ) > w)dw

L2

∞

Z

G(s2 − r2 − 2|λ2 (w − L2 ))dw =

=

(s1 , s2 ) ≤ w)dw.

r1 +Q1 r2 +Q2

X

j

where the closed form of 0 wx2 +x1 −j e(λ2 −λ1 )w dw can easily be found by noting the Rthree cases λ2 > λ1 , λ2 = λ1 , and λ2 < λ1 . ∞ For the last term L P (W (1,1) > w)dw , for both items we con2 dition on the inventory positions at epoch 0. Now, for w ≥ L2 , (1,1) P (W (s1 , s2 ) > w) = 0 if s1 ≥ 1 and s2 ≥ 1; i.e.,

s1 =1 s2 =1

P (W

s1 −r1 −2 s2 −1

G0 (x2 |λ2 (L2 − L1 )) −

R L2 −L1

X X

(1,1)

X X λx2 λx1 2 1 e−λ2 (L2 −L1 ) λ 2 x =0 x2 !x1 ! x = 0 x = 0 2 1 2   Z L2 −L1 x2 X x 2 × (L2 − L1 )j (−1)x2 −j w x2 +x1 −j e(λ2 −λ1 )w dw, s 2 −1 1 X

j =0

Z

The equality holds because for a non-positive inventory position, any missing item will take at least an order lead time to be available, which makes P (W (1,1) (s1 , s2 ) ≤ w) = 0 for w ≤ Li . RL The actual evaluation of 0 1 P (W (1,1) (s1 , s2 ) ≤ w)dw is similar to that of (2). RL For L 2 P (W (1,1) > w)dw , we condition on the joint inventory 1 positions at time −(L2 − w) and time 0 for items 2 and 1, respectively. Thus, W (1,1) (s1 , s2 ) should be interpreted accordingly. Note that for s2 ≤ 0, P (W (1,1) (s1 , s2 ) ≤ w) = 0 for L1 ≤ w ≤ L2 , and L2

=

r1 +Q1 r2 +Q2

0

Z

L2

Z

L1

0

Z

P (W (1,1) (s1 , s2 ) ≤ w)dw

=

Up to now our analysis has been for ri ≥ 0, i.e., positive inventory positions. In systems with very low shortage cost and relatively high holding cost, we may deliberately let customer demands accumulate and wait before issuing a replenishment order, i.e., ri < 0. For notational simplicity, we use the two-item system to illustrate the analysis. Note the subtle difference between a positive and a nonnegative inventory position. If the inventory position of an item is positive, either the item is available, or a replenishment order of the item has been issued and the item will be available in a time period no longer than its lead time. For ri < 0, a customer demand that finds non-positive inventory position has to wait for future demand arrivals to trigger a new replenishment order; this waiting time can be longer than an order lead time. To evaluate OBBs, the key is still to compute E [W (1,1) ] as in (1). (1,1)

L2 L1

5. The analysis for ri < 0

B

Average-Bound

X

s1 =r1 +1 s2 =1

P (W (1,1) (s1 , s2 ) ≤ w)dw.

L2

1

λ2

max{0, s2 − r2 − 1}.

If s1 ≤ 0 and s2 > 0,

Z

∞

P (W (1,1) (s1 , s2 ) > w)dw

L2

Z

∞

G(s1 − r1 − 2|λ1 (w − L1 ))dw

= L2

s1 −r1 −2

=

X

1

j =0

λ1

G(j|λ1 (L2 − L1 )).

If s1 ≤ 0 and s2 ≤ 0, the event W 1,1 (s1 , s2 ) > w is equivalent to either the event D1 (w − L1 |s1 − r1 − 2) ≡ {demands of item 1 in (0, w− L1 ) ≤ s1 − r1 − 2}, or the event D2 (w− R L∞2 |s2 − r2 − 2) ≡ {demands of item 2 in (0, w− L2 ) ≤ s2 − r2 − 2}. L P (W (1,1) (s1 , s2 ) >

R∞ R2∞ w)dw = L2 P (D1 (w − L1 |s1 − r1 − 2))dw + L2 P (D2 (w − L2 |s2 − R∞ r2 −2))dw− L P (D1 (w−L1 |s1 −r1 −2)∩D2 (w−L2 |s2 −r2 −2))dw . 2

32

Z

J. Feng et al. / Operations Research Letters 38 (2010) 27–32

∞

∞

Z

L2 s1 −r1 −2

=

X

1

j =0

λ1

G(j|λ1 (L2 − L1 )).

∞

References

P (D2 (w − L2 |s2 − r2 − 2))dw L2

Z

∞

G(s2 − r2 − 2|λ2 (w − L2 ))dw =

= Z

We thank an anonymous referee and the area editor for their valuable comments. This work was partially supported by grant PolyUG-YX82 and RGC grant PolyU6145/04E of Hong Kong, and grant NSC 97-2410-H-259-020 of Taiwan.

G(s1 − r1 − 2|λ1 (w − L1 ))dw

=

Z

Acknowledgments

P (D1 (w − L1 |s1 − r1 − 2))dw L2

∞

max(0, s2 − r2 − 1)

L2

λ2

P (D2 (w − L2 |s2 − r2 − 2) ∩ D1 (w − L1 |s1 − r1 − 2))dw L2

=

V (s2 −r2 −2X ) (s1 −r1 −2) s1 −rX 1 −2 −x 1 x 1 =0

x 2 =0

s2 −r2 −2−x1

X

x 3 =0

×

G(s1 − r1 − 2 − x1 − x2 |λ1 (L2 − L1 ))

(x1 + x2 + x3 )! 1 (1,1)
x1 (1,0)
x2 (0,1)
x3 q q q . x1 !x2 !x3 ! λ

.

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