We first cite the following lemma by Rust (1997) and Andrews (1992) without proof, then we present a proof of Proposition 4.6.
Lemma 5.1 (Uniform Strong Law of Large Numbers) Let x1, x2,· · · be i.i.d. random variables choosing value from domain X (X is a Borel space). Let g(x, θ) =
h(x, θ)− Ex(h(x, θ)) be a measurable function of x for all θ ∈ Θ, and a continuous
function of θ for almost all x ∈ X, where Ex is the expectation with respect to x. Assume
1. Θ is compact.
2. 1 N
N
n=1g(xn, θ)→ 0 almost surely for all θ ∈ Θ.
3. |g(x, θ)| ≤ d(x) for some function d satisfying Ex(d(x)) <∞.
Then, we have as N → ∞, supθ∈Θ| 1 N
N
n=1g(xn, θ)| → 0, almost surely.
Proof of Proposition 4.6:
Let’s consider the case of no information sharing and when the manufacturer uses linear least square estimation based on Qn, Qn−1,· · ·. Proposition 4.5 gives
mse∞ = eP
P = π1 0πln(i∈Iωi(x))dx
= π1 0πln[|I|( 1|I|i∈Iωi(x))]dx
= ln|I| + 1π 0πln[ 1|I|i∈Iωi(x)]dx.
Then, consider the case of information sharing. From Proposition 4.1 we have
mse = eP
P = ln[i∈I(bi+ 1)2δi2]
= ln|I| + ln[ 1
|I|i∈I(bi+ 1)2δi2].
Thus, the forecast error ratio equals
eP−P = exp(ln[ 1 |I| i∈I (bi+ 1)2δ2i]− 1 π π 0 ln[ 1 |I| i∈I ωi(x)]dx).
Following the Strong Law of Large Numbers (Karr 1993 Theorem 5.33), lim|I|→∞|I|1 i∈I(bi+ 1)2δ2i = ρ,δ(b + 1)2δ2f (ρ)g(δ)dρdδ, a.s.
lim|I|→∞ln[ 1|I|i∈I(bi+ 1)2δi2] = ln[ρ,δ(b + 1)2δ2f (ρ)g(δ)dρdδ], a.s.
The second equality comes from Karr 1993 Theorem 5.23 (continuous mappings). Similarly, for all x∈ [0, π],
lim |I|→∞ 1 |I| i∈I ωi(x) = ρ,δ ω(x)f (ρ)g(δ)dρdδ, a.s.
Notice that ω(x) = σ2+2λ cos(x)− ρ 1− 2ρcos(x) + ρ2 =
δ2(1 + 2b(b + 1)− 2b(b + 1)cos(x)) 1− 2ρcos(x) + ρ2 . It’s easy to see
0 < δ
2
(1 + ρ)2 < ω(x) <
δ2(1 + 4b(b + 1)) (1− ρ)2 <∞,
for ρ∈ [0, 1 − ] and δ > 0.
We now verify the conditions of lemma 5.1 for ω(x): [0, π] is compact. ω(x)−
Eρ,δ(ω(x)) is continuous in x for all ρ, δ, and it satisfies the third condition of lemma 5.1 if we choose d(ρ, δ) = δ2(1 + 4b(b + 1))
(1− ρ)2 . From the Strong Law of Large Numbers, it is easily seen that| 1|I|i∈I(ωi(x)−Eρ,δ(ω(x)))| → 0 almost surely given
∀x ∈ [0, π]. Thus, the lemma 5.1 implies supx∈[0,π]| 1|I|
i∈I(ωi(x)−Eρ,δ(ω(x)))| → 0,
a.s.
By definition, the Uniform Strong Law of Large Numbers implies
P{(ρ, δ) : | lim |I|→∞ 1 |I| i∈I ωi(x) = Eρ,δ(ω(x)),∀x ∈ [0, π]} = 1,
where (ρ, δ) is an element in the sample space. Define
W ={(ρ, δ) : | lim |I|→∞ 1 |I| i∈I ωi(x) = Eρ,δ(ω(x)),∀x ∈ [0, π]}.
For any (ρ, δ)∈ W , we have
lim|I|→∞|I|1 i∈Iωi(x) = Eρ,δ(ω(x)),∀x ∈ [0, π] lim|I|→∞ln[ 1|I|i∈Iωi(x)] = ln[Eρ,δ(ω(x))],∀x ∈ [0, π].
The second equation comes from the fact that ln(·) is continuous and finite. Because
ln[ 1
|I|i∈Iωi(x)] is uniformly bounded for all x ∈ [0, π] and ρ, δ, by Dominated
convergence theory, we have
π 0 ln[ 1 |I| i∈I ωi(x)]dx→ π 0 ln[Eρ,δ(ω(x))]dx,∀(ρ, δ) ∈ W. Which implies lim |I|→∞ 1 π π 0 ln[ 1 |I| i∈I ωi(x)]dx = 1 π π 0 ln[ ρ,δ ω(x)f (ρ)g(δ)dρdδ]dx, a.s.
Thus, the forecast error ratio converges to exp(ln[ ρ,δ (b + 1)2δ2f (ρ)g(δ)dρdδ]− 1 π π 0 ln[ ρ,δ ω(x)f (ρ)g(δ)dρdδ]dx) almost surely. Observe that ρ,δ (b + 1)2δ2f (ρ)g(δ)dρdδ = δ(2) δ(1) δ2g(δ)dδ× S, ρ,δω(x)f (ρ)g(δ)dρdδ = δ(2) δ(1) δ2g(δ)dδ× 1− 0 (σ2+ 2λ1− 2ρcos(x) + ρcos(x)− ρ 2)f (ρ)dρ = δ(1)δ(2)δ2g(δ)dδ× [r + u(x)], where S = 01−(b + 1)2f (ρ)dρ r = 01− 1 + ρ− 2ρL+2− 2ρL+3+ 2ρ2L+4 (1− ρ)(1 − ρ2) f (ρ)dρ u(x) = 01− 2ρ L+2(1 + ρ− ρL+2) 1− ρ2 cos(x)− ρ 1− 2ρcos(x) + ρ2f (ρ)dρ. Canceling the terms associated with δ, the forecast error ratio converges to
S/ exp(1 π
π
0 ln[r + u(x)]dx), a.s.
Second, we prove (3). As L→ ∞, b + 1 → 11− ρ thus
S → 01− 1 (1− ρ)2f (ρ)dρ r → 01− 1 + ρ (1− ρ)(1 − ρ2)f (ρ)dρ = 01− 1 (1− ρ)2f (ρ)dρ u(x) → 0. Finally, exp( 1π0πln(r)dx) = r =01− 1 (1− ρ)2f (ρ)dρ. As desired.
Andrews, D. (1992). Generic Uniform Convergence. Economic Theory, 8, 241-247. Aviv, Y. (2001). The effect of collaborative forecasting on supply chain performance.
Management Science, 47, 1326-1343.
Aviv, Y. (2002). Gaining benefits from joint forecasting and replenishment pro- cesses: the case of auto-correlated demand. Manufacturing and Service Opera-
tions Management, 4, 55-74.
Aviv, Y. and A. Federgruen (1997). Stochastic inventory models with limited pro- duction capacities and periodically varying parameters. Probability in the En-
gineering and Informational Science, 11, 107-135.
Aviv, Y. and A. Federgruen (1998). The Operational Benefits of Information Shar- ing and Vendor Managed Inventory (VMI) Programs. Washington University, St. Louis.
Bertsekas, D. P. (1995). Dynamic Programming and Optimal Control, vol. 1. Athena Scientific, Belmont, Massachusetts.
Bertsekas, D. P. (1987). Dynamic Programming: Deterministic and Stochastic Mod- els. Prentice-Hall, Englewood Cliffs, New York.
Billingsley, P. (1979). Probability and measure. John Wiley and Sons, Inc., New York.
Billingsley, P. (1999). Convergence of probability measures. John Wiley and Sons Inc., New York.
Cachon, G. P. and M. Fisher (2000). Supply chain inventory management and the value of shared information. Management Science, 46, 1032-1050.
Chen, F. (1998). Echelon reorder points, installation reorder points, and the value of centralized demand information. Management Science, 44, 0221-0234.
Chen, F. (2001). Market segmentation, advanced demand information, and supply chain performance. Manufacturing and Service Operations Management, 3, 53- 67.
Chen, F. and Y. S. Zheng (1997). One warehouse multi-retailer systems with cen- tralized stock information. Operations Research, 45, 275-287.
Chen F., I. Drezner, J. K. Ryan and D. Simchi-Levi (2000). The bullwhip effect: managerial insight on the impact of forecasting and information on variability in a supply chain. Management Science 46, 436–443.
Cheung, K. L. and H. L. Lee (2002). The inventory benefits of shipment coordina- tion and stock rebalancing in a supply chain. Management Science, 48, 300-306. Federgruen, A., P. Schweitzer and H. Tijms (1983). Denumerable Undiscounted Decision Processes with Unbounded Rewards. Mathematics of Operations Re-
search, 8, 298-314.
Federgruen, A. and P. Zipkin (1986a). An inventory model with limited production capacity and uncertain demands 1: The average-cost criterion. Mathematics of
Operations Research, 11, 193-207.
Federgruen, A. and P. Zipkin (1986b). An inventory model with limited production capacity and uncertain demands 2: The discounted-cost criterion. Mathematics
of Operations Research, 11, 208-215.
Fu, M. C. (1994). Optimization via simulation: A review. Annals of Operations
Research, 53, 199-247.
Fahrenwald, B., D. Wise and D. Glynn (2001) Supply chain collaboration: close encounters of the best kind. Business week, 26.
Gallego, G., Y. Huang, K. Katircioglu and Y. T. Leung (2000). When to share demand information in a simple supply chain? Dept. of Industrial Engineer- ing and Operations Research, Columbia University, and T. J. Watson Research Center, Yorktown Heights, NY.
Gallego, G. and O. Ozer (2001). Integrating replenishment decisions with advanced demand information. Management Science, 47, 1344-1360.
Gallego, G. and A. S. Wolf (2000). Capacitated inventory problems with fixed order costs: some optimal policy structure. European Journal of Operational
Research, 126, 603-613.
Gavirneni, S., R. Kapuscinski and S. Tayur (1999). Value of information in capaci- tated supply chains. Management Science, 45, 16-24.
Glasserman P., S. Tayur (1995). Sensitivity analysis for base-stock levels in multi echelon production-inventory systems. Management Science, 41, 263-281. Hariharan, R. and P. Zipkin (1995). Customer-order information, leadtimes, and
inventories. Management Science, 41, 1599-1607.
Heyman, D. and M. Sobel (1984). Stochastic models in operations research. Vol. 2, McGrow-Hill, New York.
Iyer, A. V. and J. Ye (2000). Assessing the value of information sharing in a promo- tional retail environment. Manufacturing and Service Operations Management, 2, 128-143.
Kapuscinski, R. and S. Tayur (1998). A capacitated production-inventory model with periodic demand. Operations Research, 46, 899-911.
Karlin, S. (1960a). Dynamic inventory policy with varying stochastic demands.
Management Science, 6, 231-258.
Karlin, S. (1960b). Optimal policy for dynamic inventory process with stochastic demands subject to seasonal variations. J. SIAM. 8, 611-629.
Karr, A. F. (1993). Probability. Springer-Verlag New York, Inc.
Kemeny, J. G., J. L. Snell and A. W. Knapp (1966). Denumerable Markov Chains. D. Van Nostrand Company, Inc.
Kolmogorov, A. N. (1941) Interpolation and extrapolation of stationary random se- quences. In Linear Least-Squares Estimation, Thomas Kailath (eds.), Dowden, Hutchingon and Ross, Inc. Stroudsburg. Pennsylvania. 52-65.
Kulkarni, V. G. (1995). Modeling and analysis of stochastic systems. Chapman and Hall, New York.
Lee, H. L. and S. Nahmias (1993). Single-product, single-location models. Hand-
books in Operations Research and Management Science. Vol. 4, North-Holland,
Amsterdam.
Lee, H., P. Padmanabhan and S. Whang, (1997a). Information distortion in a sup- ply chain: the bullwhip effect. Management Science, 43, 546-558.
Lee, H., P. Padmanabhan and S. Whang, (1997b). Bullwhip effect in a supply chain.
Sloan Management Review, 38, 93-102.
Lee, H., K. C. So and C. Tang (2000). The value of information sharing in a two- level supply chain. Management Science, 46, 626-643.
Lee, H. and S. Whang (1998). Information sharing in a supply chain. Working Pa- per, Dept. of Industrial Engineering and Engineering Management, and Grad- uate School of Business, Stanford University, CA.
Lippman, S. (1975). On dynamic programming with unbounded rewards. Manage-
ment Science, 21, 1225-1233.
Meyn, S. P. and R. L. Tweedie (1993). Markov chains and stochastic stability. Springer-Verlag, London.
Meyn, S. P. (1997). The policy iteration algorithm for average reward Markov De- cision Processes with general state space. IEEE Transactions on Automatic
Control, 42, 1663-1679.
Nelson, C. R. (1973). Applied Time Series Analysis for Managerial Forecasting. Holden-Day, Inc., San Francisco.
Ozer, O. and W. Wei (2001). Inventory control with limited capacity and advance demand information. Working Paper, Dept. of Management Science and Engi- neering, Stanford University. CA.
Puterman, M. L. (1994). Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley & Sons, Inc, New York.
Raghunathan, S. (2001). Information sharing in a supply chain: A note on its value when demand is nonstationary. Management Science, 47, 605-610.
Rust, J. (1997). Lecture notes for Economics. Un-published manuscripts.
Sethi, S. P. and F. Cheng (1997). Optimality of (s,S) policies in inventory models with Markovian demand. Operations Research, 45, 931-939.
Sennott, L. I. (1989). Average cost optimal stationary policies in infinite state Markov decision processes with unbounded costs. Operations Research, 37,
261-272.
Simchi-Levi, D., F. Kaminsky and E. Simchi-Levi (1999). Designing and managing the Supply chain. Irwin/McGraw-Hill, IL.
Simchi-Levi, D. and Y. Zhao (2001a). The Value of Information Sharing in a Two- stage Supply Chain with Production Capacity Constraints. Working paper, Northwestern University.
Simchi-Levi, D. and Y. Zhao (2001b). The Value of Information Sharing in a Two- stage Supply Chain with Production Capacity Constraints: the infinite horizon case. Working paper, Northwestern University.
Simchi-Levi, D. and Y. Zhao (2002). The impact of information sharing on forecast- ing accuracy in a multi-stage supply chain with stationary demand. Working paper, Northwestern University.
Song, J. S. and P. Zipkin (1993). Inventory control in a fluctuating environment.
Operations Research, 41, 351-370.
Stein T. and J. Sweat (1998). Killer supply chains. Information Week.
Yu, B. and F. Chen (2001). Quantifying the benefits of supply chain visibility. Graduate School of Business, Columbia University, NY.
Zipkin, P. (1989). Critical number policies for inventory models with periodic data.