Asymptotically-optimal component allocation for Assemble-to-Order ...
Operations Research Letters 43 (2015) 304–310
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Asymptotically-optimal component allocation for Assemble-to-Order production–inventory systems Haohua Wan, Qiong Wang ∗ Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, United States
article
abstract
info
Article history: Received 23 September 2014 Received in revised form 2 March 2015 Accepted 27 March 2015 Available online 4 April 2015
We consider component allocation in Assemble-to-Order production–inventory systems. We prove that asymptotic optimality on the diffusion scale can be achieved under a continuous-review policy. We also show that in many systems, meeting this optimality criterion requires component reservation. © 2015 Elsevier B.V. All rights reserved.
Keywords: Asymptotic optimality Assemble-to-Order Production–inventory Allocation
1. Introduction
respectively. Assume B(0) = I(0) = 0, so the allocation decision is subject to
We consider an Assemble-to-Order (ATO) production/inventory system that builds m products from n components. The Bill of Material (BOM) is given by a nonnegative integer matrix A, where aji is the amount of component j (1 ≤ j ≤ n) needed to produce a unit of product i (1 ≤ i ≤ m). Inventories are kept at the component level and the planning horizon starts from time 0. Let Di (t ) be the amount of demand for product i (1 ≤ i ≤ m) arrived and Rj (t ) be the amount of component j (1 ≤ j ≤ n) produced during [0, t ]. Denote
Z(t ) ≤ D (t ) and AZ(t ) ≤ R(t ),
D (t ) = (D1 (t ), . . . , Dm (t )) and R(t ) = (R1 (t ), . . . , Rn (t )),
Z(t ) = (Z1 (t ), . . . , Zm (t )),
t ≥ 0,
where Zi (t ) is the amount of demand i (1 ≤ i ≤ m) served during
[0, t ].
We consider a backlog model and denote backlog and inventory levels by B(t ) = (B1 (t ), . . . , Bm (t )) and I(t ) = (I1 (t ), . . . , In (t )),
∗
t ≥0
Corresponding author. E-mail address: [email protected] (Q. Wang).
http://dx.doi.org/10.1016/j.orl.2015.03.010 0167-6377/© 2015 Elsevier B.V. All rights reserved.
(1)
and backlog and inventory levels are determined by B(t ) = D (t ) − Z(t ),
and
I(t ) = R(t ) − AZ(t ),
t ≥ 0.
(2)
Let δ > 0 be the discount rate, pi be the price of product i (1 ≤ i ≤ m), bi be the per-unit backlog cost of product i (1 ≤ i ≤ m), and hj be the per-unit inventory holding cost of component j (1 ≤ j ≤ n). Our objective is to maximize the following Net Present Value (NPV) of the expected profit
t ≥ 0.
The problem is to define an allocation policy that decides which demands to serve at time t based on information available by that time, i.e., to determine
t ≥ 0,
E
m i =1
0
∞
−
∞
e−δ t pi dZi (t )
e
0
−δ t
m i=1
bi Bi (t ) +
n
hj Ij (t ) dt .
(3)
j =1
The problem is known to be difficult and optimal policies have only been developed for special systems with memoryless demand and production processes [1,5]. For systems in the ‘‘highvolume’’ asymptotic regime, Plambeck and Ward develop pricing, capacity, and allocation policies that are asymptotically optimal on the diffusion scale [6]. Under their approach, the percentage difference of the NPV from its optimum converges to zero as demand arrival rates increase. Most relevant to our work is their allocation policy, which is a periodic-review policy with the review interval inversely related to a norm of demand arrival rates. By
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
this design, sufficient safety stocks of demands and components are accumulated in each period to offset impacts of randomness. This policy has been applied to ATO inventory systems with N and W structures [4]. The latter work makes component ordering and allocation decisions dynamically over time for asymptotically optimal inventory management. While asymptotically optimal, the above allocation policy does not allow managers to act immediately on continuously-arriving demands. Outside the high-volume regime, demands arrive at slow rates, and thus can lead to an excessively long review interval and high inventory costs. We address this limitation by proving that the same level of asymptotic optimality can be attained under continuous-review allocations. Our candidate policies are those defined by the Allocation Principle in [7], which have been proved to be asymptotically optimal for ATO inventory systems (components are ordered from uncapacitated suppliers instead of produced under capacity constraints) with identical lead times, and shown to outperform alternative approaches outside the asymptotic regime [2,3,7]. While applying these policies to our production/inventory systems is straightforward, proving they remain asymptotically optimal requires a new analysis. Unlike [7], which features a stationary system and a long-run average cost objective function, our systems are non-recurrent and our objective is a discounted value function. Developing a proof of asymptotic optimality, under a relaxed moment condition from [6], is a major contribution of this paper. Following the Allocation Principle in [7] often requires reserving components for future high-value demands [3,7]. Reservation, which holds back components from some existing demands, is an insurance against hindsight regrets about current allocation decisions. However, in the high-volume asymptotic regime, these regrets can be addressed rapidly by fast production of component replenishments. Thus one may ask whether reservation is necessary for asymptotic optimality. To answer this question, we prove that in many systems, any policy that does not reserve components cannot be asymptotically optimal. We will present problem background in Section 2, prove asymptotic optimality in Section 3, and demonstrate the need for reservation in Section 4.
By applying the equality ∞
e−δ t pi dZi (t ) = 0
e−δ t δ pi Zi (t )dt ,
−δ t
E
e 0
−
m
δ pi +
i =1 m
ci Bi (t ) −
n
hj aji Di (t )
hj Rj (t ) dt ,
n
aji hj ,
1≤i≤m
j =1
is the total value of serving a unit of demand for product i (1 ≤ i ≤ m). Given D (t ) and R(t ), maximizing (3) is equivalent to choosing B(t ) (t ≥ 0) to minimize −δ t
C=E
e 0
AB(t ) ≥ Q(t ),
t ≥ 0.
min
m
B ≥0
ci Bi |AB ≥ Q(t ) .
(6)
i=1
Following the discussion in Section 3.2 of [7], we can let B∗ (t ) to be Lipschitz-continuous in Q(t ). Observe that the objective value C in (4) has a lower bound
∞
C=E
e 0
−δ t
m
ci B (t )dt . ∗
(7)
i=1
We refer to B∗ (t ) (t ≥ 0) as backlog targets. While reaching these targets at all times is generally impossible, the LP (6) that defines them is closely related to the aforementioned feasible policies. Component allocation in [6] relies on solving an LP that is equivalent to (6) with an additional constraint:
min B ≥0
m
ci Bi |AB ≥ Q(t ), B ≤ B (t ) , −
(8)
i=1
where B− (t ) denotes backlog levels at t (t ≥ 0) before serving demands at that time. Demands are served to reduce their backlog levels to B0 (t ), the optimal solution of (8), which are feasible to reach because B0 (t ) ≤ B− (t ). However, reaching B0 (t ) (t ≥ 0) does not optimize (4). Moreover, to be asymptotically optimal, components can only be allocated periodically with a carefullychosen period length [6]. The Allocation Principle in [7] uses B∗ (t ) directly and requires that
m i=1
ci Bi (t )dt .
1 ≤ i ≤ m, t ≥ 0.
(9)
2. Backlog Bi (t ) (1 ≤ i ≤ m) should not be reduced when it is below or at the target B∗i (t ).
3. Asymptotic analysis
where
∞
and
Let B∗ (t ) (t ≥ 0) be the optimal solution of the Linear Program (LP)
These two requirements specify to different allocation policies in different systems [2,3]. Demands are served continuously over time, but components can be held back from some low-value demands [3,7].
j =1
B(t ) ≥ 0
= 0,
j =1 n
i=1
ci = δ pi + bi +
(5)
as component shortage (for all j such that Qj (t ) > 0) or surplus (for all j such that Qj ≤ 0) at time t (t ≥ 0). We then rewrite constraints in (1) as
j:aji >0
1 ≤ i ≤ m,
and using (2) to replace Z(t ) and I(t ), we rewrite the objective function (3) as ∞
Q(t ) ≡ AD (t ) − R(t ) = AB(t ) − I(t ),
[Bi (t ) − B∗i (t )]+ ∧ [ min {(Ij (t ) − aji + 1)+ }]
∞
0
Using (2), we define
1. Backlog Bi (t ) (1 ≤ i ≤ m) should not exceed its target B∗i (t ) when all components needed to reduce the excess are available, i.e.,
2. Preliminaries
305
(4)
The same as in [6], we assume demand arrival and component production follow independent renewal processes. Define m + n sequences of i.i.d. random variables, where xli (l = 1, . . .) are interarrival times of demand i (1 ≤ i ≤ m) and ylj (l = 1, . . . , ) are
production times of component j (1 ≤ j ≤ n). Assume x1i (1 ≤ i ≤ m) and y1j (1 ≤ j ≤ n) have unit means and finite second moments, which relaxes the assumption in [6] that requires finite (2 + ϵ) moments (ϵ > 0). Consider a series of systems k = 1, . . . , with kλi as the arrival rate of demand i (1 ≤ i ≤ m) and kµj as the production rate
306
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
of component j (1 ≤ j ≤ n). Define demand and production processes respectively by
(k)
Di (t ) ≡ max L :
xli ≤ kλi t
,
k→∞
1 ≤ i ≤ m, t ≥ 0,
(k)
L
and Rj (t ) ≡ max L :
ylj ≤ kµj t
,
1 ≤ j ≤ n, t ≥ 0.
l =1
Denote the corresponding amounts occurred during a period [t1 , t2 ] by D(k) (t1 , t2 ) ≡ D (k) (t2 ) − D (k) (t1 ) and R(k) (t1 , t2 ) ≡ R(k) (t2 ) − R(k) (t1 ),
respectively (0 ≤ t1 < t2 ). In [6], demand and production rates are (k) allowed to have a small imbalance, θj (1 ≤ j ≤ n), that satisfies
√
(k)
kθj
lim
k→∞
√ ≡ lim
k
k→∞
m
(k)
aji λi
(k)
− µj
= constants.
(10)
i=1
Our results apply to this assumption, which is evident from their (k) proofs: under (10), θj (1 ≤ j ≤ n) would have been dominated by other terms. Nevertheless, for brevity of presentation, we impose a slightly stronger assumption that m
aji λi − µj = 0,
1 ≤ j ≤ n.
i =1
For systems k = 1, . . . , we center and scale demand and production processes by (k)
(k)
√
Dˆ i (t ) ≡ Di (t ) − kλi t / k,
1 ≤ i ≤ m, t ≥ 0 ,
√ (k) (k) Rˆ j (t ) ≡ Rj (t ) − kµj t / k,
1 ≤ j ≤ n, t ≥ 0.
√ ˆ (i k) (t1 , t2 ) ≡ D(i k) (t1 , t2 ) − kλi (t2 − t1 ) / k, 1 ≤ i ≤ m, D √ (k) (k) Rˆ j (t1 , t2 ) ≡ Rj (t1 , t2 ) − kµj (t2 − t1 ) / k, 1 ≤ j ≤ n. (11) For t ≥ 0, we scale backlogs, inventories, and component shortage/surplus by
√ ˜I(k) (t ) = I(k) (t )/ k,
d
ˆ (k)
d
and
Di (t ) → σi B (t ) (1 ≤ i ≤ m), t ≥ 0,
(k)
√
lim C (k) − C (k) / k = 0.
(1 ≤ j ≤ n, t ≥ 0),
ˆ i (s)|2 ] ≤ κ1 t + κ2 and E [ sup |D
(1 ≤ i ≤ m, t ≥ 0).
(15)
k→∞
√
FCLT implies that C (k) is on the order of k. Therefore, when (15) holds, the percentage difference of the objective value from its optimum diminishes to zero. Condition (14) is satisfied by the periodic-review policy in [6], which sets the review interval on the order of k−2/3 to make the impact of the additional constraint in (8), B ≤ B− (t ), invisible on the diffusion scale. To prove (14) can be attained under any continuous-review policy that satisfies the Allocation Principle in [7], we first extract two properties of the principle from the latter paper and present them as Lemmas 1 and 2. Lemma 1 implies that when the backlog level of a product significantly exceeds its target, another product must have its backlog level below the target. Lemma 2 bounds the shortfall of a product’s backlog from its target. Our main √ theorem, Theorem 1, shows that this bound is on the order of o( k), so any difference between a product’s backlog level and its target disappears on the diffusion scale. Lemma 1. Let a be the smallest non-zero element and a¯ be the largest element of matrix A. Define φ = 1 − 1/¯a. Then under any policy that satisfies the aforementioned Allocation Principle, m a¯
a l =1
(B∗l (t ) − Bl (t ))+ ,
1 ≤ i ≤ m, t ≥ 0.
(16)
Proof. From (9), if Bi (t ) > B∗i (t ), then there exists component j′ such that aj′ i > 0 and Ij′ (t ) ≤ aj′ i − 1. Apply the constraint in (6) to component j′ , m
aj′ l B∗l (t ) ≥ Qj′ (t ) =
m
aj′ l Bl (t ) − Ij′ (t )
l =1 m
aj′ l Bl (t ) − (aj′ i − 1).
l =1
Since aj′ i − 1 ≤ aj′ i φ , the above implies that
(12)
=
m aj ′ l
aj′ i l=1 m aj ′ l
aj′ i l =1
(Bl (t ) − B∗l (t )) [(Bl (t ) − B∗l (t ))+ − (B∗l (t ) − Bl (t ))+ ]
≥ (Bi (t ) − B∗i (t ))+ −
m a¯
a l =1
(B∗l (t ) − Bl (t ))+ ,
and (16) follows by rearranging terms.
0≤s≤t
(k)
then the objective value converges to its lower bound on the diffusion scale, i.e.,
φ ≥
where σi (1 ≤ i ≤ m) and γj (1 ≤ j ≤ n) are constants. Second, following the proof of Lemma 2 in [6], which does not require finite 2 + ϵ (ϵ > 0) moments, there exist constants κ1 and κ2 such that
ˆ j (s)|2 ] ≤ κ1 t + κ2 E [ sup |R
√ (t ) dt / k = 0, (14)
≥
Rj (t ) → γj B (t ) (1 ≤ j ≤ n), t ≥ 0,
(k)∗
0
l =1
Our analysis will use two properties of these processes. First, let B (t ) (t ≥ 0) be a standard Brownian Motion. By FCLT,
ˆ (k)
(k)
(Bi (t ) − B∗i (t ))+ ≤ φ +
For a period [t1 , t2 ] where 0 ≤ t1 < t2 , define
√ ˜ (k) (t ) = B(k) (t )/ k, B √ ˜ (k) (t ) = Q(k) (t )/ k. Q
e−δ t Bi (t ) − Bi
1 ≤ i ≤ m,
l=1
∞
lim E
L
under an allocation policy,
(13)
0≤s≤t
Let B(k)∗ (t ), C (k) , and C (k) be corresponding variables of B∗ (t ), C , and C in system k (k = 1, . . .) respectively. By (4) and (7), if
Lemma 2. For any t > 0 and demand i such that B∗i (t ) > Bi (t ), define ti = sup{τ : 0 ≤ τ ≤ t and Bi (τ ) > B∗i (τ )}.
(17)
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
Then there exist constants h1 > 0 and h2 > 0 such that
for all j = 1, . . . , n. Apply the above and (5) to substitute Q s in (18),
n
B∗i (t ) − Bi (t ) ≤ h1
|Qj (t ) − Qj (ti )|
(k)∗
Bi
j =1
+ h2
n
307
(t ) − B(i k) (t ) ≤ G(k) (t ),
t ≥ 0,
(22)
where
|Qj (ti ) − Qj (ti− )| − Di (ti , t ).
(18) G(k) (t ) ≡ h1
j =1
n (k) (k) (k) (k) Aj D (ti , t ) − Rj (ti , t ) j =1
Proof. By the definition of ti , Bi (t ) ≤ B∗i (t ) during [ti , t ]. Under the Allocation Principle, no demand i is served during that period, and thus Bi (t ) = Bi (ti ) + Di (ti , t ).
(19)
∗ Demand i is served at ti only if B− i (ti ) > Bi (ti ), so (17) implies that − ∗ ∗ ∗ + Bi (ti ) = B− i (ti ) ∧ Bi (ti ) = Bi (ti ) − (Bi (ti ) − Bi (ti )) .
Bi (ti ) ≥ B∗i (ti ) − |B∗i (ti ) − B∗i (ti− )|.
(20)
Eqs. (19) and (20) imply that
and (18) follows because Bi (t ) is Lipschitz continuous in Q(t ). ∗
e−δ t
lim E
k→∞
0
(k)
ci B˜ i (t )dt
∞
e−δ t
= lim E k→∞
m
0
(k)∗
ci B˜ i
m
Y˜ (k) (t ) ≡ h1
(k)
|ˆal Dˆ (l k) (ti(k) , t )| + h1
n
|Rˆ (j k) (ti(k) , t )|
j =1
√ − (t − ti(k) ) kλi − Dˆ (i k) (ti(k) , t ),
t ≥ 0.
Then G(k) (t ) − h2 Y˜ (k) (t ) ≥
n
ηj
j =1
√
,
k
∞
e−δ t (Y˜ (k) (t ))+ dt
lim E
k→∞
t ≥ 0.
= 0,
(23)
0
which we prove next. Define
ϕ ≡ λi
i =1
(k)
Use (11) to center and scale D(k) (ti , t ) and R(k) (ti , t ). Define
Theorem 1. Under any allocation policy that satisfies the Allocation Principle, m
ηj − D(i k) (ti(k) , t ).
j =1
Because ηj (1 ≤ j ≤ n) are constants, (22) implies that (21) holds if
B∗i (t ) − Bi (t ) ≤ B∗i (t ) − B∗i (ti ) + |B∗i (ti ) − B∗i (ti− )| − Di (ti , t ),
∞
n
l =1
By the definition of ti , Bi (ti− ) > B∗i (ti− ). Because B− i (ti ) precedes − ∗ − allocation at ti , B− i (ti ) ≥ Bi (ti ) > Bi (ti ). Thus the above inequalities imply that
+ h2
h1
m
aˆ l + nh1 + 1
> 0.
l=1
(t )dt .
Then
i=1
Hence the policy is asymptotically optimal on the diffusion scale.
(Y˜ (k) (t ))+ ≤ h1
m
(k)
(k)
(k)
√
ˆ l (ti , t )| − (t − ti ) kϕ aˆ l |D
+
l =1
√ + + |Dˆ (i k) (ti(k) , t )| − (t − ti(k) ) kϕ
Proof. Apply Lemma 1 to system k (k = 1, . . .), ∞
−δ t
E
e
φ (k)∗ (k) ˜ ˜ (Bi (t ) − Bi (t ))dt ≤ √ k
0
∞ −δ t
×E
e 0
∞
e−δ t dt + 0
m ( k )∗ ( k ) + (B˜ l (t ) − B˜ l (t )) dt ,
a¯ a
l =1
∞
lim E
k→∞
e
−δ t
˜ (k)∗
(Bi
˜ (k)
(t ) − Bi (t )) dt = 0, +
1 ≤ i ≤ m. (21)
0
Let Aj be the jth row of matrix A (1 ≤ j ≤ n). Define
ηj =
m
aji + 1 (1 ≤ j ≤ n) and
i=1
aˆ l =
n
ajl
n
√ + |Rˆ (k) (ti(k) , t )| − (t − ti(k) ) kϕ ,
j=1
1 ≤ i ≤ m.
Since δ and φ are constants, the above implies that (14) holds if
+ h1
(1 ≤ l ≤ m).
j=1
(k)
Let ti be the time as defined in (17) for system k (k = 1, 2 . . . , ). Since D (k) (t ) and R(k) (t ) are renewal processes with step size 1,
|Qj(k) (ti(k) ) − Qj(k) (ti(k)− )| = |Aj D(k) (ti(k)− , ti(k) ) − R(j k) (ti(k)− , ti(k) )| ≤ ηj ,
which allows us to prove (23) by showing that ∞
e−δ t sup
lim E
k→∞
0≤s≤t
0
= 0,
1 ≤ l ≤ m, ∞
e−δ t sup
and lim E k→∞
= 0,
√ + |Dˆ (l k) (s, t )| − (t − s) kϕ dt
0≤s≤t
0
√ + |Rˆ (j k) (s, t )| − (t − s) kϕ dt
1 ≤ j ≤ n.
Below we prove the first equation. A similar proof applies to the second one. For any given l, define (k)
Ψ1
∞
−δ t
≡E
e
0≤s≤t
0
(k)
Ψ2
ˆ (l k) (s, t )|1(t − s ≤ k−1/4 )dt sup |D
∞
≡ E
√
ˆ (l k) (s, t )| − (t − s) kϕ)+ e−δ t ( sup |D 0≤s≤t
0
−1/4
1(t − s > k
)dt .
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H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
ˆ (l k) (s, t ) = Dˆ l(k) (t ) − Dˆ l(k) (s) and because of (13), Since D
Then ∞
√ + (k) ˆ sup |Dl (s, t )| − (t − s) kϕ dt
e−δ t
E
ˆ (l k) (s, t )| − k1/4 ϕ)+ ] E [( sup |D 0≤s≤t
0≤s≤t
0
(k)
≤ E [( sup |2Dˆ l(k) (s)| − k1/4 ϕ)+ ] 0≤s≤t ∞ (k) P sup |Dˆ l (s)| ≥ x dx =2
(k)
≤ Ψ1 + Ψ2 , (k)
→ 0 as k → ∞ (i = 1, 2).
so we only need to prove that Ψi By Tonelli’s Theorem, (k)
Ψ1
∞
e−δ t E
=
sup (t −k−1/4 )+ ≤s≤t
0
k1/4 ϕ/2
(k)
|Dˆ (l k) (s, t )| dt . ≤2
(t −k−1/4 )+ ≤s≤t
≤2
0≤s≤t
≤ 1 + sup |Dˆ l(k) (s)|2 ,
(24)
0≤s≤t
κ1 t + κ2
∞
dx
x2
k1/4 ϕ/2
|Dˆ (l k) (s, t )| ≤ 2 sup |Dˆ l(k) (s)|
ˆ l (s)|2 ] E [ sup |D
∞
For k = 1, . . . , and t ≥ 0, sup
0≤s≤t
x2
k1/4 ϕ/2
dx.
= 2(k1/4 ϕ/2)−1 (κ1 t + κ2 ),
t ≥ 0.
Applying the above to (28) proves the result.
0≤s≤t
and by (13),
(k)
ˆ l (s)|2 )e−δt E (1 + sup |D
4. Need for component reservation
≤ e−δt (1 + κ1 t + κ2 ),
0≤s≤t
We consider systems with the BOM
where the right-hand side is integrable over [0, ∞). Thus if for each t > 0,
lim E e
−δ t
k→∞
ˆ (k)
sup (t −k−1/4 )+ ≤s≤t
|Dl (s, t )| = 0,
lim Ψ1
∞
−δ t
=
e
k→∞
0
lim E
k→∞
ˆ (k)
sup (t −k−1/4 )+ ≤s≤t
1 A = 1
1 0
(25)
then we can apply the Dominated Convergence Theorem to exchange the limit and integral signs to arrive at (k)
|Dl (s, t )| dt = 0. −1/4
To prove (25), for any ϵ > 0, choose k0 ≥ 1/t 4 (so t − k0 such that
≥ 0)
0....0 0....0 .
0 1 A˜
(29)
Slightly deviating from the general notation, we index products by 0, 1, . . . , m − 1. The first two rows of A show that component i (i = 1, 2) is used by products 0 and i by one unit each. The usage of other components is given by the rest of the matrix. For simplicity, we assume that all entries in A are binary. So any policy that does not reserve component satisfies the condition that
Bi (t ) ∧
Ij (t ) = 0,
0 ≤ i ≤ m − 1, t ≥ 0,
(30)
j:aji =1
(26)
i.e., no demand can have backlog if components to serve it are all available.
(B (s) is a Standard Brownian Motion). By (12) and the Continuous Mapping Theorem,
Theorem 2. For systems in which the BOM is given by (29); values of serving products 0, 1, and 2 satisfy c0 > c1 + c2 ; and demand arrival and component production follow m + n independent Poisson processes, any policy that satisfies (30) is not asymptotically optimal, i.e., under such a policy,
e−δ t E [
sup −1/4 0≤s≤k0
|σl B (s)|] < ϵ/2,
|Dˆ (l k) (s, t )| −→ d
sup −1/4 t −k0 ≤s ≤t
sup −1/4 0≤s≤k0
|σl B (s)| as k → ∞,
for any given k0 . For each t, (13) and (24) imply that
{e−δt
sup −1/4 t −k0 ≤s≤t
|Dˆ (l k) (s, t )|},
sup −1/4 t −k0 ≤s≤t
|Dˆ (l k) (s, t )|] ≤ e−δt E [ sup
−1/4 0≤s≤k0
|σl B (s)|]
+ ϵ/2,
(27)
and (25) follows from (26) and (27) because for all k ≥ max(k0 , k1 ), e
−δ t
E[
sup (t −k−1/4 )+ ≤s≤t
|Dˆ (l k) (s, t )|] ≤ e−δt E [ < ϵ. (k)
To prove limk→∞ Ψ2 definition, (k)
Ψ2
∞
−δ t
≤
e 0
sup −1/4 t −k0 ≤s≤t
|Dˆ (l k) (s, t )|]
= 0, replace (t − s) with k−1/4 in its
ˆ (l k) (s, t )| − k1/4 ϕ)+ ]dt . E [( sup |D 0≤s≤t
(31)
k→∞
k = 1, 2, . . . ,
is uniformly integrable. So for given ϵ and k0 , ∃k1 such that for all k ≥ k1 , e−δ t E [
√
lim C (k) − C (k) / k > 0.
(28)
We prove the theorem by showing that during a period [1/2, 1], there is a set of sample paths (Ω (k) ) associated with a strictly positive probability (Eq. (35)). On these paths, the difference between√ the discounted inventory cost and its minimum is on the order of k (Eq. (36)). Proof. Let ∆ > 0 be a constant. Define the following sets of sample paths (k)
(k)
(k)
(k)
(k)
(k)
Ω (k) = E1 ∧ E2 ∧ E3 ∧ E4 ∧ E5 ∧ E6 ,
k = 1, . . . ,
where (k)
(k)
(k)
(k)
(k)
ˆ (i k) (1/2, s)| ≤ ∆/8, i = 0, 1, 2 sup |D
ˆ i (1/2) ≤ 2∆, i = 0, 1, 2 E1 = ∆ ≤ D
ˆ j (1/2) ≤ ∆/2, j = 1, 2 E2 = R E3 =
1/2≤s≤1
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
(k)
E4 =
(k)
sup Rˆ j (1/2, s) ≤ ∆/4, j = 1, 2
Since c0 > c1 + c2 , (31) follows immediately from the above inequality and (34). To prove (35), since D (k) (t ) and R(k) (t ) (t ≥ 0) are independent (k) Poisson Processes, Eq (q = 1, 2, 3, 4, 5, 6) are independent of each other, so if
1/2≤s≤1
(k)
(k)
E5 =
ˆ sup Di (s) ≤ 0, i = 3, . . . , m − 1
1/2≤s≤1
E6 =
ˆ inf Rj (s) ≥ 7∆, j = 3, . . . , n .
lim P[E(qk) ] > 0,
1/2≤s≤1
(k)
(k)
(k)
lim P[E1 ] =
+ Dˆ (j k) (1/2, t )] − [Rˆ j(k) (1/2) + Rˆ (j k) (1/2, t )]
k→∞
≥ ∆ (> 0),
(32)
and for components j = 3, . . . , n (recall that all entries in A˜ are binary), Qj (t ) =
2
(k)
k→∞
(k)
lim P[E3 ] =
k→∞
ˆ i(k) (t ) − Rˆ j(k) (t ) ≤ 0. aji D
(k)∗
(33)
(k)∗
B˜ 2
(k)
lim P[E4 ] ≥
(t ) = Q˜ 2(k)+ (t ),
0≤s≤1/2
P[ sup |B (s)| ≤ ∆/(4γj )|B (0) = 0] 0≤s≤1/2 m−1
(k)
lim P[E5 ] ≥ lim
and
k→∞
k→∞
1/2≤s≤1
(t ) = B˜ (i k) (t ) − Q˜ i(k)+ (t ) = B˜ i(k) (t ) − [B˜ (0k) (t ) + B˜ (i k) (t ) − ˜Ii(k) (t )]+
=
P[B (1/2) ≤ −∆/σi ]
i=3
× P[ sup |B (s)| ≤ ∆/σi |B (0) = 0] 0≤s≤1/2
≥ −B0 (t ) (k)∗
k→∞
k→∞
(t ) = B˜ (i k) (t ) (≥ 0) (k)
m−1
C
−C √ k
j=3 1/2≤s≤1
(k)∗
=
n
P[B (1/2) ≥ 8∆/γj ]
j =3
(k)
× P[ sup |B (s)| ≤ ∆/γj |B (0) = 0]. 0≤s≤1/2
≥ E 1(Ω (k) )
≥ E 1(Ω (k) )
m−1
1
e−δ t 1/2
1/2
(k)∗
(t ) dt
The above implies that (35) holds because for any given constant ξ > 0,
i=0
1
(k)
ci E B˜ i (t ) − B˜ i
(k)
e−δ t (c0 − c1 − c2 )B˜ 0 (t )dt ,
(34)
P[B (1/2) ≥ ξ ] > 0,
0≤s≤1/2
where 1(Ω ) is the indicator function that a sample path is in Ω (k) . Hence we can prove Theorem 2 by showing that lim P[Ω (k) ] > 0,
(35)
k→∞
and when k is sufficiently large,
∆ (k) e−δ t B˜ 0 (t )dt ≥ e−δ , 4 1/2
in Ω
lim E 1(Ω
k→∞
a.s.,
(36)
(k)
)
1
e 1/2
−δ t ˜ (k)
B0 (t )dt
(k)
t0
= inf {t : B˜ (1k) (t ) × B˜ (2k) (t ) = 0}. t ≥1/2
≥ e−δ > 0.
(k)
(k)
If t0 > 3/4, then for t ∈ [1/2, 3/4], B˜ 1 (t ) > 0, B˜ 2 (t ) > 0, and thus by (30),
˜I1(k) (t ) = ˜I2(k) (t ) = 0.
. These two conditions imply that
To prove (36), define
(k)
1
(k)
P[B (1/2) ≤ −ξ ] > 0,
P[ξ ≤ B (1/2) ≤ 2ξ ] > 0, and P[ sup |B (s)| ≤ ξ |B (0) = 0] > 0.
(k)
(k)
P[Rˆ j (1/2) ≥ 8∆]
× P[ sup |Rˆ (j k) (1/2, s)| ≤ ∆]
for i = 0 and i = 3, . . . , m−1. Since i=0 ci [B˜ i (t )−B˜ i (t )] ≥ 0 for each t ≥ 0 and on every sample path, the above indicates (k)
n
(k)
lim P[E6 ] ≥ lim
for i = 1, 2, and (k)
i=3
m−1
˜ (k)
B˜ i (t ) − B˜ i
(k)
P[Dˆ i (1/2) ≤ −∆]
× P[ sup |Dˆ (i k) (1/2, s)| ≤ ∆]
Hence at any time t ∈ [1/2, 1] and on any sample path in Ω (k) , (k)∗
P[ sup |B (s)| ≤ ∆/(8σi )|B (0) = 0]
j =1
(t ) = 0 (i = 0, 3, . . . , m − 1).
(k)
2
2
k→∞
(t ) = Q˜ 1(k)+ (t ),
B˜ i (t ) − B˜ i
P[B (1/2) ≤ ∆/(2γj )].
i=0
˜ (k) (t ) in the above, A in (29), and c0 > c1 + c2 , (6) yields Given Q B˜ i
2 j =1
ˆ (k)
i =3
(k)∗
P[∆/σi ≤ B (1/2) ≤ 2∆/σi ]
i=0
lim P[E2 ] =
aji [Di (1/2) + Di (1/2, t )]
m−1
B˜ 1
2
Applying FCLT and Continuous Mapping Theorem,
ˆ (k)
i=0
+
then (35) holds. Following (12),
(k)
ˆ 0 (1/2) + Dˆ 0 (1/2, t )] + [Dˆ j (1/2) Q˜ j (t ) = [D
˜ (k)
q = 1, 2, 3, 4, 5, 6,
k→∞
The definition implies that at any time t ∈ [1/2, 1] and on any sample path in Ω (k) , for components j = 1, 2, (k)
309
∆
lim P[Ω (k) ]
4 k→∞
The above implies that during [1/2, 3/4], demand 0 can be served only when components 1 and 2 arrive at the same time, which (k) (k) does not happen (a.s.) because R1 (t ) and R2 (t ) are independent Poisson processes. Therefore,
310
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
(k)
B˜ 0 (t ) ≥
(k)
D0 (1/2, t )
√
k
√
kλ0 (t − 1/2) −
≥
(k)
∆ 8
,
1 1/2
(k)
e−δ t B˜ 0 (t )dt ≥
3/4
≥
a.s.
(k)
For cases where t0 (32) applies, (k)
(k)
√
1/2
> e−δ
(k)
e−δ t
∆ 4
kλ0 (t − 1/2) −
∆ 8
(k)
(k)
(k)
(37)
a.s., l = 1, 2, . . . .
a.s., l = 1, 2, . . .
which, with no arrival of demand 1 or 2 at υl (a.s.), implies that (k)
(k)
B˜ 1 (υl ) ∧ B˜ 2 (υl ) = 0,
(k)
a.s., l = 1, 2, . . . .
Since υl ∈ [1/2, 1], (32) holds, and thus at every υl (l = 1, 2, . . . , ),
(38)
Since demand 0 is not served at t ∈ [1/2, 1] when t ̸= υl (l = (k) 1, . . .), B˜ 0 (t ) does not decrease at these times. Hence (37) and (38) imply that a.s.,
and (36) follows immediately.
To satisfy (30), the above and (33) imply that (k)
(k)
Q˜ 1 (s) ∧ Q˜ 2 (s) ≥ ∆.
(k)
Under (30), demand 0 is served only when it has a new arrival or one of components 1 or 2 is received. Let {υ1 , υ2 , . . .} be the set of these times during [1/2, 1]. Demand arrival and component production follow independent Poisson processes, so demand 1 or 2 does not arrive and components 1 and 2 are not simultaneously received at υl (l = 1, . . . , ) (a.s.). Thus for demand 0 to be served,
B˜ 1 (υl− ) ∧ B˜ 2 (υl− ) = 0,
1/2≤s≤1
.
B˜ 0 (t0 ) ≥ Q˜ 1 (t0 ) ∧ Q˜ 2 (t0 ) ≥ ∆.
˜I1(k) (υl− ) ∨ ˜I2(k) (υl− ) > 0,
inf
B˜ 0 (t ) ≥ ∆, 3/4 ≤ t ≤ 1,
dt
≤ 3/4, since B˜ (1k) (t0(k) ) ∧ B˜ 2(k) (t0(k) ) = 0 and
(k)
(k)
i=1,2
for all t ∈ [1/2, 3/4] (the last inequality holds because the sample (k) path is in E3 ). When k is sufficiently large, (36) follows from
(k)
B˜ 0 (υl ) ≥ max{Q˜ i (υl ) − B˜ i (υl )}
(k)
ˆ 0 (1/2, t ) kλ0 (t − 1/2) + D
=
√
Acknowledgments This paper is based upon work supported by the National Science Foundation Grant 1363314 (CMMI). We thank Marty Reiman and reviewers of this paper for their helpful comments and suggestions. References [1] Saif Benjaafar, Mohsen ElHafsi, Production and inventory control of a single product Assemble-to-Order system with multiple customer classes, Manag. Sci. 52 (12) (2006) 1896–1912. [2] M.K. Doğru, M.I. Reiman, Qiong Wang, A stochastic programming based inventory policy for Assemble-to-Order systems with application to the W model, Oper. Res. 58 (4-part-1) (2010) 849–864. [3] M. Doğru, M.I. Reiman, Q. Wang, Stochastic programming based inventory control for the M system. Working Paper, 2014. [4] L. Lu, J.S. Song, H. Zhang, Optimal and asymptotically optimal policies for Assemble-to-Order N- and W-systems. Working Paper, 2014. [5] E. Nadar, M. Akan, A. Scheller-Wolf, Technical note—optimal structural results for Assemble-to-Order generalized M-systems, Oper. Res. 62 (6) (2014) 571–579. [6] E.L. Plambeck, A.R. Ward, Optimal control of a high-volume Assemble-to-Order system, Math. Oper. Res. 31 (3) (2006) 453–477. [7] M.I. Reiman, Q. Wang, Asymptotically optimal inventory control of Assembleto-Order systems with identical lead times, Oper. Res. (2015) forthcoming.
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Asymptotically-optimal component allocation for Assemble-to-Order production–inventory systems Haohua Wan, Qiong Wang ∗ Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, United States
article
abstract
info
Article history: Received 23 September 2014 Received in revised form 2 March 2015 Accepted 27 March 2015 Available online 4 April 2015
We consider component allocation in Assemble-to-Order production–inventory systems. We prove that asymptotic optimality on the diffusion scale can be achieved under a continuous-review policy. We also show that in many systems, meeting this optimality criterion requires component reservation. © 2015 Elsevier B.V. All rights reserved.
Keywords: Asymptotic optimality Assemble-to-Order Production–inventory Allocation
1. Introduction
respectively. Assume B(0) = I(0) = 0, so the allocation decision is subject to
We consider an Assemble-to-Order (ATO) production/inventory system that builds m products from n components. The Bill of Material (BOM) is given by a nonnegative integer matrix A, where aji is the amount of component j (1 ≤ j ≤ n) needed to produce a unit of product i (1 ≤ i ≤ m). Inventories are kept at the component level and the planning horizon starts from time 0. Let Di (t ) be the amount of demand for product i (1 ≤ i ≤ m) arrived and Rj (t ) be the amount of component j (1 ≤ j ≤ n) produced during [0, t ]. Denote
Z(t ) ≤ D (t ) and AZ(t ) ≤ R(t ),
D (t ) = (D1 (t ), . . . , Dm (t )) and R(t ) = (R1 (t ), . . . , Rn (t )),
Z(t ) = (Z1 (t ), . . . , Zm (t )),
t ≥ 0,
where Zi (t ) is the amount of demand i (1 ≤ i ≤ m) served during
[0, t ].
We consider a backlog model and denote backlog and inventory levels by B(t ) = (B1 (t ), . . . , Bm (t )) and I(t ) = (I1 (t ), . . . , In (t )),
∗
t ≥0
Corresponding author. E-mail address: [email protected] (Q. Wang).
http://dx.doi.org/10.1016/j.orl.2015.03.010 0167-6377/© 2015 Elsevier B.V. All rights reserved.
(1)
and backlog and inventory levels are determined by B(t ) = D (t ) − Z(t ),
and
I(t ) = R(t ) − AZ(t ),
t ≥ 0.
(2)
Let δ > 0 be the discount rate, pi be the price of product i (1 ≤ i ≤ m), bi be the per-unit backlog cost of product i (1 ≤ i ≤ m), and hj be the per-unit inventory holding cost of component j (1 ≤ j ≤ n). Our objective is to maximize the following Net Present Value (NPV) of the expected profit
t ≥ 0.
The problem is to define an allocation policy that decides which demands to serve at time t based on information available by that time, i.e., to determine
t ≥ 0,
E
m i =1
0
∞
−
∞
e−δ t pi dZi (t )
e
0
−δ t
m i=1
bi Bi (t ) +
n
hj Ij (t ) dt .
(3)
j =1
The problem is known to be difficult and optimal policies have only been developed for special systems with memoryless demand and production processes [1,5]. For systems in the ‘‘highvolume’’ asymptotic regime, Plambeck and Ward develop pricing, capacity, and allocation policies that are asymptotically optimal on the diffusion scale [6]. Under their approach, the percentage difference of the NPV from its optimum converges to zero as demand arrival rates increase. Most relevant to our work is their allocation policy, which is a periodic-review policy with the review interval inversely related to a norm of demand arrival rates. By
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
this design, sufficient safety stocks of demands and components are accumulated in each period to offset impacts of randomness. This policy has been applied to ATO inventory systems with N and W structures [4]. The latter work makes component ordering and allocation decisions dynamically over time for asymptotically optimal inventory management. While asymptotically optimal, the above allocation policy does not allow managers to act immediately on continuously-arriving demands. Outside the high-volume regime, demands arrive at slow rates, and thus can lead to an excessively long review interval and high inventory costs. We address this limitation by proving that the same level of asymptotic optimality can be attained under continuous-review allocations. Our candidate policies are those defined by the Allocation Principle in [7], which have been proved to be asymptotically optimal for ATO inventory systems (components are ordered from uncapacitated suppliers instead of produced under capacity constraints) with identical lead times, and shown to outperform alternative approaches outside the asymptotic regime [2,3,7]. While applying these policies to our production/inventory systems is straightforward, proving they remain asymptotically optimal requires a new analysis. Unlike [7], which features a stationary system and a long-run average cost objective function, our systems are non-recurrent and our objective is a discounted value function. Developing a proof of asymptotic optimality, under a relaxed moment condition from [6], is a major contribution of this paper. Following the Allocation Principle in [7] often requires reserving components for future high-value demands [3,7]. Reservation, which holds back components from some existing demands, is an insurance against hindsight regrets about current allocation decisions. However, in the high-volume asymptotic regime, these regrets can be addressed rapidly by fast production of component replenishments. Thus one may ask whether reservation is necessary for asymptotic optimality. To answer this question, we prove that in many systems, any policy that does not reserve components cannot be asymptotically optimal. We will present problem background in Section 2, prove asymptotic optimality in Section 3, and demonstrate the need for reservation in Section 4.
By applying the equality ∞
e−δ t pi dZi (t ) = 0
e−δ t δ pi Zi (t )dt ,
−δ t
E
e 0
−
m
δ pi +
i =1 m
ci Bi (t ) −
n
hj aji Di (t )
hj Rj (t ) dt ,
n
aji hj ,
1≤i≤m
j =1
is the total value of serving a unit of demand for product i (1 ≤ i ≤ m). Given D (t ) and R(t ), maximizing (3) is equivalent to choosing B(t ) (t ≥ 0) to minimize −δ t
C=E
e 0
AB(t ) ≥ Q(t ),
t ≥ 0.
min
m
B ≥0
ci Bi |AB ≥ Q(t ) .
(6)
i=1
Following the discussion in Section 3.2 of [7], we can let B∗ (t ) to be Lipschitz-continuous in Q(t ). Observe that the objective value C in (4) has a lower bound
∞
C=E
e 0
−δ t
m
ci B (t )dt . ∗
(7)
i=1
We refer to B∗ (t ) (t ≥ 0) as backlog targets. While reaching these targets at all times is generally impossible, the LP (6) that defines them is closely related to the aforementioned feasible policies. Component allocation in [6] relies on solving an LP that is equivalent to (6) with an additional constraint:
min B ≥0
m
ci Bi |AB ≥ Q(t ), B ≤ B (t ) , −
(8)
i=1
where B− (t ) denotes backlog levels at t (t ≥ 0) before serving demands at that time. Demands are served to reduce their backlog levels to B0 (t ), the optimal solution of (8), which are feasible to reach because B0 (t ) ≤ B− (t ). However, reaching B0 (t ) (t ≥ 0) does not optimize (4). Moreover, to be asymptotically optimal, components can only be allocated periodically with a carefullychosen period length [6]. The Allocation Principle in [7] uses B∗ (t ) directly and requires that
m i=1
ci Bi (t )dt .
1 ≤ i ≤ m, t ≥ 0.
(9)
2. Backlog Bi (t ) (1 ≤ i ≤ m) should not be reduced when it is below or at the target B∗i (t ).
3. Asymptotic analysis
where
∞
and
Let B∗ (t ) (t ≥ 0) be the optimal solution of the Linear Program (LP)
These two requirements specify to different allocation policies in different systems [2,3]. Demands are served continuously over time, but components can be held back from some low-value demands [3,7].
j =1
B(t ) ≥ 0
= 0,
j =1 n
i=1
ci = δ pi + bi +
(5)
as component shortage (for all j such that Qj (t ) > 0) or surplus (for all j such that Qj ≤ 0) at time t (t ≥ 0). We then rewrite constraints in (1) as
j:aji >0
1 ≤ i ≤ m,
and using (2) to replace Z(t ) and I(t ), we rewrite the objective function (3) as ∞
Q(t ) ≡ AD (t ) − R(t ) = AB(t ) − I(t ),
[Bi (t ) − B∗i (t )]+ ∧ [ min {(Ij (t ) − aji + 1)+ }]
∞
0
Using (2), we define
1. Backlog Bi (t ) (1 ≤ i ≤ m) should not exceed its target B∗i (t ) when all components needed to reduce the excess are available, i.e.,
2. Preliminaries
305
(4)
The same as in [6], we assume demand arrival and component production follow independent renewal processes. Define m + n sequences of i.i.d. random variables, where xli (l = 1, . . .) are interarrival times of demand i (1 ≤ i ≤ m) and ylj (l = 1, . . . , ) are
production times of component j (1 ≤ j ≤ n). Assume x1i (1 ≤ i ≤ m) and y1j (1 ≤ j ≤ n) have unit means and finite second moments, which relaxes the assumption in [6] that requires finite (2 + ϵ) moments (ϵ > 0). Consider a series of systems k = 1, . . . , with kλi as the arrival rate of demand i (1 ≤ i ≤ m) and kµj as the production rate
306
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
of component j (1 ≤ j ≤ n). Define demand and production processes respectively by
(k)
Di (t ) ≡ max L :
xli ≤ kλi t
,
k→∞
1 ≤ i ≤ m, t ≥ 0,
(k)
L
and Rj (t ) ≡ max L :
ylj ≤ kµj t
,
1 ≤ j ≤ n, t ≥ 0.
l =1
Denote the corresponding amounts occurred during a period [t1 , t2 ] by D(k) (t1 , t2 ) ≡ D (k) (t2 ) − D (k) (t1 ) and R(k) (t1 , t2 ) ≡ R(k) (t2 ) − R(k) (t1 ),
respectively (0 ≤ t1 < t2 ). In [6], demand and production rates are (k) allowed to have a small imbalance, θj (1 ≤ j ≤ n), that satisfies
√
(k)
kθj
lim
k→∞
√ ≡ lim
k
k→∞
m
(k)
aji λi
(k)
− µj
= constants.
(10)
i=1
Our results apply to this assumption, which is evident from their (k) proofs: under (10), θj (1 ≤ j ≤ n) would have been dominated by other terms. Nevertheless, for brevity of presentation, we impose a slightly stronger assumption that m
aji λi − µj = 0,
1 ≤ j ≤ n.
i =1
For systems k = 1, . . . , we center and scale demand and production processes by (k)
(k)
√
Dˆ i (t ) ≡ Di (t ) − kλi t / k,
1 ≤ i ≤ m, t ≥ 0 ,
√ (k) (k) Rˆ j (t ) ≡ Rj (t ) − kµj t / k,
1 ≤ j ≤ n, t ≥ 0.
√ ˆ (i k) (t1 , t2 ) ≡ D(i k) (t1 , t2 ) − kλi (t2 − t1 ) / k, 1 ≤ i ≤ m, D √ (k) (k) Rˆ j (t1 , t2 ) ≡ Rj (t1 , t2 ) − kµj (t2 − t1 ) / k, 1 ≤ j ≤ n. (11) For t ≥ 0, we scale backlogs, inventories, and component shortage/surplus by
√ ˜I(k) (t ) = I(k) (t )/ k,
d
ˆ (k)
d
and
Di (t ) → σi B (t ) (1 ≤ i ≤ m), t ≥ 0,
(k)
√
lim C (k) − C (k) / k = 0.
(1 ≤ j ≤ n, t ≥ 0),
ˆ i (s)|2 ] ≤ κ1 t + κ2 and E [ sup |D
(1 ≤ i ≤ m, t ≥ 0).
(15)
k→∞
√
FCLT implies that C (k) is on the order of k. Therefore, when (15) holds, the percentage difference of the objective value from its optimum diminishes to zero. Condition (14) is satisfied by the periodic-review policy in [6], which sets the review interval on the order of k−2/3 to make the impact of the additional constraint in (8), B ≤ B− (t ), invisible on the diffusion scale. To prove (14) can be attained under any continuous-review policy that satisfies the Allocation Principle in [7], we first extract two properties of the principle from the latter paper and present them as Lemmas 1 and 2. Lemma 1 implies that when the backlog level of a product significantly exceeds its target, another product must have its backlog level below the target. Lemma 2 bounds the shortfall of a product’s backlog from its target. Our main √ theorem, Theorem 1, shows that this bound is on the order of o( k), so any difference between a product’s backlog level and its target disappears on the diffusion scale. Lemma 1. Let a be the smallest non-zero element and a¯ be the largest element of matrix A. Define φ = 1 − 1/¯a. Then under any policy that satisfies the aforementioned Allocation Principle, m a¯
a l =1
(B∗l (t ) − Bl (t ))+ ,
1 ≤ i ≤ m, t ≥ 0.
(16)
Proof. From (9), if Bi (t ) > B∗i (t ), then there exists component j′ such that aj′ i > 0 and Ij′ (t ) ≤ aj′ i − 1. Apply the constraint in (6) to component j′ , m
aj′ l B∗l (t ) ≥ Qj′ (t ) =
m
aj′ l Bl (t ) − Ij′ (t )
l =1 m
aj′ l Bl (t ) − (aj′ i − 1).
l =1
Since aj′ i − 1 ≤ aj′ i φ , the above implies that
(12)
=
m aj ′ l
aj′ i l=1 m aj ′ l
aj′ i l =1
(Bl (t ) − B∗l (t )) [(Bl (t ) − B∗l (t ))+ − (B∗l (t ) − Bl (t ))+ ]
≥ (Bi (t ) − B∗i (t ))+ −
m a¯
a l =1
(B∗l (t ) − Bl (t ))+ ,
and (16) follows by rearranging terms.
0≤s≤t
(k)
then the objective value converges to its lower bound on the diffusion scale, i.e.,
φ ≥
where σi (1 ≤ i ≤ m) and γj (1 ≤ j ≤ n) are constants. Second, following the proof of Lemma 2 in [6], which does not require finite 2 + ϵ (ϵ > 0) moments, there exist constants κ1 and κ2 such that
ˆ j (s)|2 ] ≤ κ1 t + κ2 E [ sup |R
√ (t ) dt / k = 0, (14)
≥
Rj (t ) → γj B (t ) (1 ≤ j ≤ n), t ≥ 0,
(k)∗
0
l =1
Our analysis will use two properties of these processes. First, let B (t ) (t ≥ 0) be a standard Brownian Motion. By FCLT,
ˆ (k)
(k)
(Bi (t ) − B∗i (t ))+ ≤ φ +
For a period [t1 , t2 ] where 0 ≤ t1 < t2 , define
√ ˜ (k) (t ) = B(k) (t )/ k, B √ ˜ (k) (t ) = Q(k) (t )/ k. Q
e−δ t Bi (t ) − Bi
1 ≤ i ≤ m,
l=1
∞
lim E
L
under an allocation policy,
(13)
0≤s≤t
Let B(k)∗ (t ), C (k) , and C (k) be corresponding variables of B∗ (t ), C , and C in system k (k = 1, . . .) respectively. By (4) and (7), if
Lemma 2. For any t > 0 and demand i such that B∗i (t ) > Bi (t ), define ti = sup{τ : 0 ≤ τ ≤ t and Bi (τ ) > B∗i (τ )}.
(17)
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
Then there exist constants h1 > 0 and h2 > 0 such that
for all j = 1, . . . , n. Apply the above and (5) to substitute Q s in (18),
n
B∗i (t ) − Bi (t ) ≤ h1
|Qj (t ) − Qj (ti )|
(k)∗
Bi
j =1
+ h2
n
307
(t ) − B(i k) (t ) ≤ G(k) (t ),
t ≥ 0,
(22)
where
|Qj (ti ) − Qj (ti− )| − Di (ti , t ).
(18) G(k) (t ) ≡ h1
j =1
n (k) (k) (k) (k) Aj D (ti , t ) − Rj (ti , t ) j =1
Proof. By the definition of ti , Bi (t ) ≤ B∗i (t ) during [ti , t ]. Under the Allocation Principle, no demand i is served during that period, and thus Bi (t ) = Bi (ti ) + Di (ti , t ).
(19)
∗ Demand i is served at ti only if B− i (ti ) > Bi (ti ), so (17) implies that − ∗ ∗ ∗ + Bi (ti ) = B− i (ti ) ∧ Bi (ti ) = Bi (ti ) − (Bi (ti ) − Bi (ti )) .
Bi (ti ) ≥ B∗i (ti ) − |B∗i (ti ) − B∗i (ti− )|.
(20)
Eqs. (19) and (20) imply that
and (18) follows because Bi (t ) is Lipschitz continuous in Q(t ). ∗
e−δ t
lim E
k→∞
0
(k)
ci B˜ i (t )dt
∞
e−δ t
= lim E k→∞
m
0
(k)∗
ci B˜ i
m
Y˜ (k) (t ) ≡ h1
(k)
|ˆal Dˆ (l k) (ti(k) , t )| + h1
n
|Rˆ (j k) (ti(k) , t )|
j =1
√ − (t − ti(k) ) kλi − Dˆ (i k) (ti(k) , t ),
t ≥ 0.
Then G(k) (t ) − h2 Y˜ (k) (t ) ≥
n
ηj
j =1
√
,
k
∞
e−δ t (Y˜ (k) (t ))+ dt
lim E
k→∞
t ≥ 0.
= 0,
(23)
0
which we prove next. Define
ϕ ≡ λi
i =1
(k)
Use (11) to center and scale D(k) (ti , t ) and R(k) (ti , t ). Define
Theorem 1. Under any allocation policy that satisfies the Allocation Principle, m
ηj − D(i k) (ti(k) , t ).
j =1
Because ηj (1 ≤ j ≤ n) are constants, (22) implies that (21) holds if
B∗i (t ) − Bi (t ) ≤ B∗i (t ) − B∗i (ti ) + |B∗i (ti ) − B∗i (ti− )| − Di (ti , t ),
∞
n
l =1
By the definition of ti , Bi (ti− ) > B∗i (ti− ). Because B− i (ti ) precedes − ∗ − allocation at ti , B− i (ti ) ≥ Bi (ti ) > Bi (ti ). Thus the above inequalities imply that
+ h2
h1
m
aˆ l + nh1 + 1
> 0.
l=1
(t )dt .
Then
i=1
Hence the policy is asymptotically optimal on the diffusion scale.
(Y˜ (k) (t ))+ ≤ h1
m
(k)
(k)
(k)
√
ˆ l (ti , t )| − (t − ti ) kϕ aˆ l |D
+
l =1
√ + + |Dˆ (i k) (ti(k) , t )| − (t − ti(k) ) kϕ
Proof. Apply Lemma 1 to system k (k = 1, . . .), ∞
−δ t
E
e
φ (k)∗ (k) ˜ ˜ (Bi (t ) − Bi (t ))dt ≤ √ k
0
∞ −δ t
×E
e 0
∞
e−δ t dt + 0
m ( k )∗ ( k ) + (B˜ l (t ) − B˜ l (t )) dt ,
a¯ a
l =1
∞
lim E
k→∞
e
−δ t
˜ (k)∗
(Bi
˜ (k)
(t ) − Bi (t )) dt = 0, +
1 ≤ i ≤ m. (21)
0
Let Aj be the jth row of matrix A (1 ≤ j ≤ n). Define
ηj =
m
aji + 1 (1 ≤ j ≤ n) and
i=1
aˆ l =
n
ajl
n
√ + |Rˆ (k) (ti(k) , t )| − (t − ti(k) ) kϕ ,
j=1
1 ≤ i ≤ m.
Since δ and φ are constants, the above implies that (14) holds if
+ h1
(1 ≤ l ≤ m).
j=1
(k)
Let ti be the time as defined in (17) for system k (k = 1, 2 . . . , ). Since D (k) (t ) and R(k) (t ) are renewal processes with step size 1,
|Qj(k) (ti(k) ) − Qj(k) (ti(k)− )| = |Aj D(k) (ti(k)− , ti(k) ) − R(j k) (ti(k)− , ti(k) )| ≤ ηj ,
which allows us to prove (23) by showing that ∞
e−δ t sup
lim E
k→∞
0≤s≤t
0
= 0,
1 ≤ l ≤ m, ∞
e−δ t sup
and lim E k→∞
= 0,
√ + |Dˆ (l k) (s, t )| − (t − s) kϕ dt
0≤s≤t
0
√ + |Rˆ (j k) (s, t )| − (t − s) kϕ dt
1 ≤ j ≤ n.
Below we prove the first equation. A similar proof applies to the second one. For any given l, define (k)
Ψ1
∞
−δ t
≡E
e
0≤s≤t
0
(k)
Ψ2
ˆ (l k) (s, t )|1(t − s ≤ k−1/4 )dt sup |D
∞
≡ E
√
ˆ (l k) (s, t )| − (t − s) kϕ)+ e−δ t ( sup |D 0≤s≤t
0
−1/4
1(t − s > k
)dt .
308
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
ˆ (l k) (s, t ) = Dˆ l(k) (t ) − Dˆ l(k) (s) and because of (13), Since D
Then ∞
√ + (k) ˆ sup |Dl (s, t )| − (t − s) kϕ dt
e−δ t
E
ˆ (l k) (s, t )| − k1/4 ϕ)+ ] E [( sup |D 0≤s≤t
0≤s≤t
0
(k)
≤ E [( sup |2Dˆ l(k) (s)| − k1/4 ϕ)+ ] 0≤s≤t ∞ (k) P sup |Dˆ l (s)| ≥ x dx =2
(k)
≤ Ψ1 + Ψ2 , (k)
→ 0 as k → ∞ (i = 1, 2).
so we only need to prove that Ψi By Tonelli’s Theorem, (k)
Ψ1
∞
e−δ t E
=
sup (t −k−1/4 )+ ≤s≤t
0
k1/4 ϕ/2
(k)
|Dˆ (l k) (s, t )| dt . ≤2
(t −k−1/4 )+ ≤s≤t
≤2
0≤s≤t
≤ 1 + sup |Dˆ l(k) (s)|2 ,
(24)
0≤s≤t
κ1 t + κ2
∞
dx
x2
k1/4 ϕ/2
|Dˆ (l k) (s, t )| ≤ 2 sup |Dˆ l(k) (s)|
ˆ l (s)|2 ] E [ sup |D
∞
For k = 1, . . . , and t ≥ 0, sup
0≤s≤t
x2
k1/4 ϕ/2
dx.
= 2(k1/4 ϕ/2)−1 (κ1 t + κ2 ),
t ≥ 0.
Applying the above to (28) proves the result.
0≤s≤t
and by (13),
(k)
ˆ l (s)|2 )e−δt E (1 + sup |D
4. Need for component reservation
≤ e−δt (1 + κ1 t + κ2 ),
0≤s≤t
We consider systems with the BOM
where the right-hand side is integrable over [0, ∞). Thus if for each t > 0,
lim E e
−δ t
k→∞
ˆ (k)
sup (t −k−1/4 )+ ≤s≤t
|Dl (s, t )| = 0,
lim Ψ1
∞
−δ t
=
e
k→∞
0
lim E
k→∞
ˆ (k)
sup (t −k−1/4 )+ ≤s≤t
1 A = 1
1 0
(25)
then we can apply the Dominated Convergence Theorem to exchange the limit and integral signs to arrive at (k)
|Dl (s, t )| dt = 0. −1/4
To prove (25), for any ϵ > 0, choose k0 ≥ 1/t 4 (so t − k0 such that
≥ 0)
0....0 0....0 .
0 1 A˜
(29)
Slightly deviating from the general notation, we index products by 0, 1, . . . , m − 1. The first two rows of A show that component i (i = 1, 2) is used by products 0 and i by one unit each. The usage of other components is given by the rest of the matrix. For simplicity, we assume that all entries in A are binary. So any policy that does not reserve component satisfies the condition that
Bi (t ) ∧
Ij (t ) = 0,
0 ≤ i ≤ m − 1, t ≥ 0,
(30)
j:aji =1
(26)
i.e., no demand can have backlog if components to serve it are all available.
(B (s) is a Standard Brownian Motion). By (12) and the Continuous Mapping Theorem,
Theorem 2. For systems in which the BOM is given by (29); values of serving products 0, 1, and 2 satisfy c0 > c1 + c2 ; and demand arrival and component production follow m + n independent Poisson processes, any policy that satisfies (30) is not asymptotically optimal, i.e., under such a policy,
e−δ t E [
sup −1/4 0≤s≤k0
|σl B (s)|] < ϵ/2,
|Dˆ (l k) (s, t )| −→ d
sup −1/4 t −k0 ≤s ≤t
sup −1/4 0≤s≤k0
|σl B (s)| as k → ∞,
for any given k0 . For each t, (13) and (24) imply that
{e−δt
sup −1/4 t −k0 ≤s≤t
|Dˆ (l k) (s, t )|},
sup −1/4 t −k0 ≤s≤t
|Dˆ (l k) (s, t )|] ≤ e−δt E [ sup
−1/4 0≤s≤k0
|σl B (s)|]
+ ϵ/2,
(27)
and (25) follows from (26) and (27) because for all k ≥ max(k0 , k1 ), e
−δ t
E[
sup (t −k−1/4 )+ ≤s≤t
|Dˆ (l k) (s, t )|] ≤ e−δt E [ < ϵ. (k)
To prove limk→∞ Ψ2 definition, (k)
Ψ2
∞
−δ t
≤
e 0
sup −1/4 t −k0 ≤s≤t
|Dˆ (l k) (s, t )|]
= 0, replace (t − s) with k−1/4 in its
ˆ (l k) (s, t )| − k1/4 ϕ)+ ]dt . E [( sup |D 0≤s≤t
(31)
k→∞
k = 1, 2, . . . ,
is uniformly integrable. So for given ϵ and k0 , ∃k1 such that for all k ≥ k1 , e−δ t E [
√
lim C (k) − C (k) / k > 0.
(28)
We prove the theorem by showing that during a period [1/2, 1], there is a set of sample paths (Ω (k) ) associated with a strictly positive probability (Eq. (35)). On these paths, the difference between√ the discounted inventory cost and its minimum is on the order of k (Eq. (36)). Proof. Let ∆ > 0 be a constant. Define the following sets of sample paths (k)
(k)
(k)
(k)
(k)
(k)
Ω (k) = E1 ∧ E2 ∧ E3 ∧ E4 ∧ E5 ∧ E6 ,
k = 1, . . . ,
where (k)
(k)
(k)
(k)
(k)
ˆ (i k) (1/2, s)| ≤ ∆/8, i = 0, 1, 2 sup |D
ˆ i (1/2) ≤ 2∆, i = 0, 1, 2 E1 = ∆ ≤ D
ˆ j (1/2) ≤ ∆/2, j = 1, 2 E2 = R E3 =
1/2≤s≤1
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
(k)
E4 =
(k)
sup Rˆ j (1/2, s) ≤ ∆/4, j = 1, 2
Since c0 > c1 + c2 , (31) follows immediately from the above inequality and (34). To prove (35), since D (k) (t ) and R(k) (t ) (t ≥ 0) are independent (k) Poisson Processes, Eq (q = 1, 2, 3, 4, 5, 6) are independent of each other, so if
1/2≤s≤1
(k)
(k)
E5 =
ˆ sup Di (s) ≤ 0, i = 3, . . . , m − 1
1/2≤s≤1
E6 =
ˆ inf Rj (s) ≥ 7∆, j = 3, . . . , n .
lim P[E(qk) ] > 0,
1/2≤s≤1
(k)
(k)
(k)
lim P[E1 ] =
+ Dˆ (j k) (1/2, t )] − [Rˆ j(k) (1/2) + Rˆ (j k) (1/2, t )]
k→∞
≥ ∆ (> 0),
(32)
and for components j = 3, . . . , n (recall that all entries in A˜ are binary), Qj (t ) =
2
(k)
k→∞
(k)
lim P[E3 ] =
k→∞
ˆ i(k) (t ) − Rˆ j(k) (t ) ≤ 0. aji D
(k)∗
(33)
(k)∗
B˜ 2
(k)
lim P[E4 ] ≥
(t ) = Q˜ 2(k)+ (t ),
0≤s≤1/2
P[ sup |B (s)| ≤ ∆/(4γj )|B (0) = 0] 0≤s≤1/2 m−1
(k)
lim P[E5 ] ≥ lim
and
k→∞
k→∞
1/2≤s≤1
(t ) = B˜ (i k) (t ) − Q˜ i(k)+ (t ) = B˜ i(k) (t ) − [B˜ (0k) (t ) + B˜ (i k) (t ) − ˜Ii(k) (t )]+
=
P[B (1/2) ≤ −∆/σi ]
i=3
× P[ sup |B (s)| ≤ ∆/σi |B (0) = 0] 0≤s≤1/2
≥ −B0 (t ) (k)∗
k→∞
k→∞
(t ) = B˜ (i k) (t ) (≥ 0) (k)
m−1
C
−C √ k
j=3 1/2≤s≤1
(k)∗
=
n
P[B (1/2) ≥ 8∆/γj ]
j =3
(k)
× P[ sup |B (s)| ≤ ∆/γj |B (0) = 0]. 0≤s≤1/2
≥ E 1(Ω (k) )
≥ E 1(Ω (k) )
m−1
1
e−δ t 1/2
1/2
(k)∗
(t ) dt
The above implies that (35) holds because for any given constant ξ > 0,
i=0
1
(k)
ci E B˜ i (t ) − B˜ i
(k)
e−δ t (c0 − c1 − c2 )B˜ 0 (t )dt ,
(34)
P[B (1/2) ≥ ξ ] > 0,
0≤s≤1/2
where 1(Ω ) is the indicator function that a sample path is in Ω (k) . Hence we can prove Theorem 2 by showing that lim P[Ω (k) ] > 0,
(35)
k→∞
and when k is sufficiently large,
∆ (k) e−δ t B˜ 0 (t )dt ≥ e−δ , 4 1/2
in Ω
lim E 1(Ω
k→∞
a.s.,
(36)
(k)
)
1
e 1/2
−δ t ˜ (k)
B0 (t )dt
(k)
t0
= inf {t : B˜ (1k) (t ) × B˜ (2k) (t ) = 0}. t ≥1/2
≥ e−δ > 0.
(k)
(k)
If t0 > 3/4, then for t ∈ [1/2, 3/4], B˜ 1 (t ) > 0, B˜ 2 (t ) > 0, and thus by (30),
˜I1(k) (t ) = ˜I2(k) (t ) = 0.
. These two conditions imply that
To prove (36), define
(k)
1
(k)
P[B (1/2) ≤ −ξ ] > 0,
P[ξ ≤ B (1/2) ≤ 2ξ ] > 0, and P[ sup |B (s)| ≤ ξ |B (0) = 0] > 0.
(k)
(k)
P[Rˆ j (1/2) ≥ 8∆]
× P[ sup |Rˆ (j k) (1/2, s)| ≤ ∆]
for i = 0 and i = 3, . . . , m−1. Since i=0 ci [B˜ i (t )−B˜ i (t )] ≥ 0 for each t ≥ 0 and on every sample path, the above indicates (k)
n
(k)
lim P[E6 ] ≥ lim
for i = 1, 2, and (k)
i=3
m−1
˜ (k)
B˜ i (t ) − B˜ i
(k)
P[Dˆ i (1/2) ≤ −∆]
× P[ sup |Dˆ (i k) (1/2, s)| ≤ ∆]
Hence at any time t ∈ [1/2, 1] and on any sample path in Ω (k) , (k)∗
P[ sup |B (s)| ≤ ∆/(8σi )|B (0) = 0]
j =1
(t ) = 0 (i = 0, 3, . . . , m − 1).
(k)
2
2
k→∞
(t ) = Q˜ 1(k)+ (t ),
B˜ i (t ) − B˜ i
P[B (1/2) ≤ ∆/(2γj )].
i=0
˜ (k) (t ) in the above, A in (29), and c0 > c1 + c2 , (6) yields Given Q B˜ i
2 j =1
ˆ (k)
i =3
(k)∗
P[∆/σi ≤ B (1/2) ≤ 2∆/σi ]
i=0
lim P[E2 ] =
aji [Di (1/2) + Di (1/2, t )]
m−1
B˜ 1
2
Applying FCLT and Continuous Mapping Theorem,
ˆ (k)
i=0
+
then (35) holds. Following (12),
(k)
ˆ 0 (1/2) + Dˆ 0 (1/2, t )] + [Dˆ j (1/2) Q˜ j (t ) = [D
˜ (k)
q = 1, 2, 3, 4, 5, 6,
k→∞
The definition implies that at any time t ∈ [1/2, 1] and on any sample path in Ω (k) , for components j = 1, 2, (k)
309
∆
lim P[Ω (k) ]
4 k→∞
The above implies that during [1/2, 3/4], demand 0 can be served only when components 1 and 2 arrive at the same time, which (k) (k) does not happen (a.s.) because R1 (t ) and R2 (t ) are independent Poisson processes. Therefore,
310
H. Wan, Q. Wang / Operations Research Letters 43 (2015) 304–310
(k)
B˜ 0 (t ) ≥
(k)
D0 (1/2, t )
√
k
√
kλ0 (t − 1/2) −
≥
(k)
∆ 8
,
1 1/2
(k)
e−δ t B˜ 0 (t )dt ≥
3/4
≥
a.s.
(k)
For cases where t0 (32) applies, (k)
(k)
√
1/2
> e−δ
(k)
e−δ t
∆ 4
kλ0 (t − 1/2) −
∆ 8
(k)
(k)
(k)
(37)
a.s., l = 1, 2, . . . .
a.s., l = 1, 2, . . .
which, with no arrival of demand 1 or 2 at υl (a.s.), implies that (k)
(k)
B˜ 1 (υl ) ∧ B˜ 2 (υl ) = 0,
(k)
a.s., l = 1, 2, . . . .
Since υl ∈ [1/2, 1], (32) holds, and thus at every υl (l = 1, 2, . . . , ),
(38)
Since demand 0 is not served at t ∈ [1/2, 1] when t ̸= υl (l = (k) 1, . . .), B˜ 0 (t ) does not decrease at these times. Hence (37) and (38) imply that a.s.,
and (36) follows immediately.
To satisfy (30), the above and (33) imply that (k)
(k)
Q˜ 1 (s) ∧ Q˜ 2 (s) ≥ ∆.
(k)
Under (30), demand 0 is served only when it has a new arrival or one of components 1 or 2 is received. Let {υ1 , υ2 , . . .} be the set of these times during [1/2, 1]. Demand arrival and component production follow independent Poisson processes, so demand 1 or 2 does not arrive and components 1 and 2 are not simultaneously received at υl (l = 1, . . . , ) (a.s.). Thus for demand 0 to be served,
B˜ 1 (υl− ) ∧ B˜ 2 (υl− ) = 0,
1/2≤s≤1
.
B˜ 0 (t0 ) ≥ Q˜ 1 (t0 ) ∧ Q˜ 2 (t0 ) ≥ ∆.
˜I1(k) (υl− ) ∨ ˜I2(k) (υl− ) > 0,
inf
B˜ 0 (t ) ≥ ∆, 3/4 ≤ t ≤ 1,
dt
≤ 3/4, since B˜ (1k) (t0(k) ) ∧ B˜ 2(k) (t0(k) ) = 0 and
(k)
(k)
i=1,2
for all t ∈ [1/2, 3/4] (the last inequality holds because the sample (k) path is in E3 ). When k is sufficiently large, (36) follows from
(k)
B˜ 0 (υl ) ≥ max{Q˜ i (υl ) − B˜ i (υl )}
(k)
ˆ 0 (1/2, t ) kλ0 (t − 1/2) + D
=
√
Acknowledgments This paper is based upon work supported by the National Science Foundation Grant 1363314 (CMMI). We thank Marty Reiman and reviewers of this paper for their helpful comments and suggestions. References [1] Saif Benjaafar, Mohsen ElHafsi, Production and inventory control of a single product Assemble-to-Order system with multiple customer classes, Manag. Sci. 52 (12) (2006) 1896–1912. [2] M.K. Doğru, M.I. Reiman, Qiong Wang, A stochastic programming based inventory policy for Assemble-to-Order systems with application to the W model, Oper. Res. 58 (4-part-1) (2010) 849–864. [3] M. Doğru, M.I. Reiman, Q. Wang, Stochastic programming based inventory control for the M system. Working Paper, 2014. [4] L. Lu, J.S. Song, H. Zhang, Optimal and asymptotically optimal policies for Assemble-to-Order N- and W-systems. Working Paper, 2014. [5] E. Nadar, M. Akan, A. Scheller-Wolf, Technical note—optimal structural results for Assemble-to-Order generalized M-systems, Oper. Res. 62 (6) (2014) 571–579. [6] E.L. Plambeck, A.R. Ward, Optimal control of a high-volume Assemble-to-Order system, Math. Oper. Res. 31 (3) (2006) 453–477. [7] M.I. Reiman, Q. Wang, Asymptotically optimal inventory control of Assembleto-Order systems with identical lead times, Oper. Res. (2015) forthcoming.
