dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · dynamic...
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![Page 1: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/1.jpg)
Dynamic pricing and inventory control withlarge replenishment lead times
Xin Chen
University of Illinois at Urbana-Champaign
Joint work with Sasha Stolyar and Linwei Xin
IMA Workshop on Data-Driven Supply Chain ManagementOctober 4, 2018
Funding support: NSF, JD.com, UIUC-ZJU Institute
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Model description Prior work Main result Proof sketch Conclusion
Outline
1 Model description
2 Prior work
3 Main resultAsymptotic optimality of constant-order list-price policies
4 Proof sketchThree steps in the proof
5 Conclusion
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Model description Prior work Main result Proof sketch Conclusion
Model and notation
Single item, periodic-review, backorder model, long-run averageprofit
Unit ordering, holding and backorder costs: c, h and b
Demand Dt , γtD(pt) + βt , D(pt) strictly decreasing
{γt} i.i.d. with mean one
{βt} i.i.d. with zero mean
pt ∈ [pmin, pmax ], where pmin < pmax
Deterministic lead time L > 0
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Model description Prior work Main result Proof sketch Conclusion
Model and notation
Single item, periodic-review, backorder model, long-run averageprofit
Unit ordering, holding and backorder costs: c, h and b
Demand Dt , γtD(pt) + βt , D(pt) strictly decreasing
{γt} i.i.d. with mean one
{βt} i.i.d. with zero mean
pt ∈ [pmin, pmax ], where pmin < pmax
Deterministic lead time L > 0
![Page 5: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/5.jpg)
Model description Prior work Main result Proof sketch Conclusion
Model and notation
Single item, periodic-review, backorder model, long-run averageprofit
Unit ordering, holding and backorder costs: c, h and b
Demand Dt , γtD(pt) + βt , D(pt) strictly decreasing
{γt} i.i.d. with mean one
{βt} i.i.d. with zero mean
pt ∈ [pmin, pmax ], where pmin < pmax
Deterministic lead time L > 0
![Page 6: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/6.jpg)
Model description Prior work Main result Proof sketch Conclusion
Model and notation
Single item, periodic-review, backorder model, long-run averageprofit
Unit ordering, holding and backorder costs: c, h and b
Demand Dt , γtD(pt) + βt , D(pt) strictly decreasing
{γt} i.i.d. with mean one
{βt} i.i.d. with zero mean
pt ∈ [pmin, pmax ], where pmin < pmax
Deterministic lead time L > 0
![Page 7: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/7.jpg)
Model description Prior work Main result Proof sketch Conclusion
Model and notation
Single item, periodic-review, backorder model, long-run averageprofit
Unit ordering, holding and backorder costs: c, h and b
Demand Dt , γtD(pt) + βt , D(pt) strictly decreasing
{γt} i.i.d. with mean one
{βt} i.i.d. with zero mean
pt ∈ [pmin, pmax ], where pmin < pmax
Deterministic lead time L > 0
![Page 8: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/8.jpg)
Model description Prior work Main result Proof sketch Conclusion
Model and notation
Single item, periodic-review, backorder model, long-run averageprofit
Unit ordering, holding and backorder costs: c, h and b
Demand Dt , γtD(pt) + βt , D(pt) strictly decreasing
{γt} i.i.d. with mean one
{βt} i.i.d. with zero mean
pt ∈ [pmin, pmax ], where pmin < pmax
Deterministic lead time L > 0
![Page 9: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/9.jpg)
Model description Prior work Main result Proof sketch Conclusion
Model and notation
Single item, periodic-review, backorder model, long-run averageprofit
Unit ordering, holding and backorder costs: c, h and b
Demand Dt , γtD(pt) + βt , D(pt) strictly decreasing
{γt} i.i.d. with mean one
{βt} i.i.d. with zero mean
pt ∈ [pmin, pmax ], where pmin < pmax
Deterministic lead time L > 0
![Page 10: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/10.jpg)
Model description Prior work Main result Proof sketch Conclusion
Inventory dynamics (in period t)
Inventoryreview
Itemsdelivered
Pricingdecision
Neworder
placed
Demandrealized
(It , xt ) x1,t pt qt Dt
On-hand inventory ItPipeline vector xt = (x1,t , x2,t , . . . , xL,t)
Orders already placed but not yet received
Decision variables pt ,qt
pt : pricing deicsionqt : new order placed
Inventory update:It+1 = It + x1,t − Dt , xt+1 = (x2,t , . . . , xL,t , qt)
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Model description Prior work Main result Proof sketch Conclusion
Inventory dynamics (in period t)
Inventoryreview
Itemsdelivered
Pricingdecision
Neworder
placed
Demandrealized
(It , xt ) x1,t pt qt Dt
On-hand inventory ItPipeline vector xt = (x1,t , x2,t , . . . , xL,t)
Orders already placed but not yet received
Decision variables pt ,qt
pt : pricing deicsionqt : new order placed
Inventory update:It+1 = It + x1,t − Dt , xt+1 = (x2,t , . . . , xL,t , qt)
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Model description Prior work Main result Proof sketch Conclusion
Inventory dynamics (in period t)
Inventoryreview
Itemsdelivered
Pricingdecision
Neworder
placed
Demandrealized
(It , xt ) x1,t pt qt Dt
On-hand inventory ItPipeline vector xt = (x1,t , x2,t , . . . , xL,t)
Orders already placed but not yet received
Decision variables pt ,qt
pt : pricing deicsionqt : new order placed
Inventory update:It+1 = It + x1,t − Dt , xt+1 = (x2,t , . . . , xL,t , qt)
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Model description Prior work Main result Proof sketch Conclusion
Inventory dynamics (in period t)
Inventoryreview
Itemsdelivered
Pricingdecision
Neworder
placed
Demandrealized
(It , xt ) x1,t pt qt Dt
On-hand inventory ItPipeline vector xt = (x1,t , x2,t , . . . , xL,t)
Orders already placed but not yet received
Decision variables pt ,qt
pt : pricing deicsionqt : new order placed
Inventory update:It+1 = It + x1,t − Dt , xt+1 = (x2,t , . . . , xL,t , qt)
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Model description Prior work Main result Proof sketch Conclusion
Inventory dynamics (in period t)
Inventoryreview
Itemsdelivered
Pricingdecision
Neworder
placed
Demandrealized
(It , xt ) x1,t pt qt Dt
On-hand inventory ItPipeline vector xt = (x1,t , x2,t , . . . , xL,t)
Orders already placed but not yet received
Decision variables pt ,qt
pt : pricing deicsionqt : new order placed
Inventory update:It+1 = It + x1,t − Dt , xt+1 = (x2,t , . . . , xL,t , qt)
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Model description Prior work Main result Proof sketch Conclusion
Inventory dynamics (in period t)
Inventoryreview
Itemsdelivered
Pricingdecision
Neworder
placed
Demandrealized
(It , xt ) x1,t pt qt Dt
On-hand inventory ItPipeline vector xt = (x1,t , x2,t , . . . , xL,t)
Orders already placed but not yet received
Decision variables pt ,qt
pt : pricing deicsionqt : new order placed
Inventory update:It+1 = It + x1,t − Dt , xt+1 = (x2,t , . . . , xL,t , qt)
![Page 16: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/16.jpg)
Model description Prior work Main result Proof sketch Conclusion
Inventory dynamics (in period t)
Inventoryreview
Itemsdelivered
Pricingdecision
Neworder
placed
Demandrealized
(It , xt ) x1,t pt qt Dt
On-hand inventory ItPipeline vector xt = (x1,t , x2,t , . . . , xL,t)
Orders already placed but not yet received
Decision variables pt ,qt
pt : pricing deicsionqt : new order placed
Inventory update:It+1 = It + x1,t − Dt , xt+1 = (x2,t , . . . , xL,t , qt)
![Page 17: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/17.jpg)
Model description Prior work Main result Proof sketch Conclusion
Inventory dynamics (in period t)
Inventoryreview
Itemsdelivered
Pricingdecision
Neworder
placed
Demandrealized
(It , xt ) x1,t pt qt Dt
On-hand inventory ItPipeline vector xt = (x1,t , x2,t , . . . , xL,t)
Orders already placed but not yet received
Decision variables pt ,qt
pt : pricing deicsionqt : new order placed
Inventory update:It+1 = It + x1,t − Dt , xt+1 = (x2,t , . . . , xL,t , qt)
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Model description Prior work Main result Proof sketch Conclusion
Performance measure and optimal policy
G(x) ∆= hx+ + bx−
Profit in period t :
Ct∆= ptDt − [cx1,t + G (It + x1,t − Dt)]
Long run average profit of policy π:
C(π) , lim infT→∞
1T
T∑t=1
E [Cπt ]
Optimal long run average profit:
OPT(L) , supπ
C(π)
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Model description Prior work Main result Proof sketch Conclusion
Performance measure and optimal policy
G(x) ∆= hx+ + bx−
Profit in period t :
Ct∆= ptDt − [cx1,t + G (It + x1,t − Dt)]
Long run average profit of policy π:
C(π) , lim infT→∞
1T
T∑t=1
E [Cπt ]
Optimal long run average profit:
OPT(L) , supπ
C(π)
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Model description Prior work Main result Proof sketch Conclusion
Performance measure and optimal policy
G(x) ∆= hx+ + bx−
Profit in period t :
Ct∆= ptDt − [cx1,t + G (It + x1,t − Dt)]
Long run average profit of policy π:
C(π) , lim infT→∞
1T
T∑t=1
E [Cπt ]
Optimal long run average profit:
OPT(L) , supπ
C(π)
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Model description Prior work Main result Proof sketch Conclusion
Performance measure and optimal policy
G(x) ∆= hx+ + bx−
Profit in period t :
Ct∆= ptDt − [cx1,t + G (It + x1,t − Dt)]
Long run average profit of policy π:
C(π) , lim infT→∞
1T
T∑t=1
E [Cπt ]
Optimal long run average profit:
OPT(L) , supπ
C(π)
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Model description Prior work Main result Proof sketch Conclusion
Literature review
First studied in Whitin (1955)
Optimality of a base-stock list price policy in the zero lead-timesetting (Federgruen and Heching (1999))
Extension to the setting with setup costs (e.g. Chen and Simchi-Levi (2004a, 2004b), Yao et al. (2007), Huh and Janakarimian(2008)
Setting with lead-timesbase-stock list price policy is no longer optimal in generalCurse of dimensionality
“. . . it remains a significant challenge to incorporate lead time intostochastic models. Indeed, the zero lead time assumption is re-quired for all the multi-period models reviewed here. . . " (Chen andSimchi-Levi 2012)
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Model description Prior work Main result Proof sketch Conclusion
Literature review
First studied in Whitin (1955)
Optimality of a base-stock list price policy in the zero lead-timesetting (Federgruen and Heching (1999))
Extension to the setting with setup costs (e.g. Chen and Simchi-Levi (2004a, 2004b), Yao et al. (2007), Huh and Janakarimian(2008)
Setting with lead-timesbase-stock list price policy is no longer optimal in generalCurse of dimensionality
“. . . it remains a significant challenge to incorporate lead time intostochastic models. Indeed, the zero lead time assumption is re-quired for all the multi-period models reviewed here. . . " (Chen andSimchi-Levi 2012)
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Model description Prior work Main result Proof sketch Conclusion
Literature review cont.
Selected literature: Thomas (1974), Petruzzi and Dada (1999),Agrawal and Seshadri (2000), Elmaghraby and Keskinocak (2003),Chen, Xu and Zhang (2009), Li, Lim and Rodrigues (2009), Chen,Pang and Pan (2014), Chen, Chao and Ahn (2015), Chen, Chaoand Shi (2016)
Bernstein, Li and Shang (2015): positive lead time, focusing ondesigning effective heuristics
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Model description Prior work Main result Proof sketch Conclusion
Literature review cont.
Selected literature: Thomas (1974), Petruzzi and Dada (1999),Agrawal and Seshadri (2000), Elmaghraby and Keskinocak (2003),Chen, Xu and Zhang (2009), Li, Lim and Rodrigues (2009), Chen,Pang and Pan (2014), Chen, Chao and Ahn (2015), Chen, Chaoand Shi (2016)
Bernstein, Li and Shang (2015): positive lead time, focusing ondesigning effective heuristics
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Model description Prior work Main result Proof sketch Conclusion
Constant-order policies
First studied in a lost-sales inventory model [Reiman (2004)]
Always order the same amount of inventory regardless of what onhands and in-transit
Example:Constant-order quantity: 100
If oh-hand=0, order 100
If oh-hand=1000, order 100
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Model description Prior work Main result Proof sketch Conclusion
Performance in a lost-sales model
Can beat base-stock as the lead time grows [Reiman (2004)]
Surprising computational results of [Zipkin (2008)]Compare to several heuristicsConstant-order policy did surprisingly well even when L = 4
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Model description Prior work Main result Proof sketch Conclusion
Asymptotic optimality
Lost-sales modelconstant-order is asymptotically optimal as the lead time grows[Goldberg, Katz-Rogozhnikov, Lu, Sharma, Squillante (2016)]
exponential convergence [Xin and Goldberg (2016)]
Dual-sourcing modelTailored Base-Surge policy (constant-order + base-stock) is asymp-totically optimal as the lead time difference grows [Xin and Gold-berg (2017)]
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Model description Prior work Main result Proof sketch Conclusion
Asymptotic optimality
Lost-sales modelconstant-order is asymptotically optimal as the lead time grows[Goldberg, Katz-Rogozhnikov, Lu, Sharma, Squillante (2016)]
exponential convergence [Xin and Goldberg (2016)]
Dual-sourcing modelTailored Base-Surge policy (constant-order + base-stock) is asymp-totically optimal as the lead time difference grows [Xin and Gold-berg (2017)]
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Model description Prior work Main result Proof sketch Conclusion
Assumptions
Assumption
The inverse function D−1 of D is continuous and strictly decreas-ing.The revenue dD−1(d) is a concave function of the expected de-mand d .dD−1(d) is Lipschitz continuous with a constant κ > 0.
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Model description Prior work Main result Proof sketch Conclusion
Constant-order list-price policy
dmin , D(pmax), dmax , D(pmin)
Compute the best constant-order policy:
maxx∈[dmin,dmax ]
maxπp
C(πx,πp)︸ ︷︷ ︸concave in x
The best constant x∗ ∈ (dmin,dmax)
Theorem
limL→∞
OPT(L) = maxx∈[dmin,dmax ]
maxπp
C(πx,πp)︸ ︷︷ ︸concave in x
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Model description Prior work Main result Proof sketch Conclusion
Constant-order list-price policy
dmin , D(pmax), dmax , D(pmin)
Compute the best constant-order policy:
maxx∈[dmin,dmax ]
maxπp
C(πx,πp)︸ ︷︷ ︸concave in x
The best constant x∗ ∈ (dmin,dmax)
Theorem
limL→∞
OPT(L) = maxx∈[dmin,dmax ]
maxπp
C(πx,πp)︸ ︷︷ ︸concave in x
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Model description Prior work Main result Proof sketch Conclusion
Constant-order list-price policy
dmin , D(pmax), dmax , D(pmin)
Compute the best constant-order policy:
maxx∈[dmin,dmax ]
maxπp
C(πx,πp)︸ ︷︷ ︸concave in x
The best constant x∗ ∈ (dmin,dmax)
Theorem
limL→∞
OPT(L) = maxx∈[dmin,dmax ]
maxπp
C(πx,πp)︸ ︷︷ ︸concave in x
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Model description Prior work Main result Proof sketch Conclusion
Constant-order list-price policy
dmin , D(pmax), dmax , D(pmin)
Compute the best constant-order policy:
maxx∈[dmin,dmax ]
maxπp
C(πx,πp)︸ ︷︷ ︸concave in x
The best constant x∗ ∈ (dmin,dmax)
Theorem
limL→∞
OPT(L) = maxx∈[dmin,dmax ]
maxπp
C(πx,πp)︸ ︷︷ ︸concave in x
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Model description Prior work Main result Proof sketch Conclusion
Proof overview
Step I: existence of a steady-stateperturbative approaches
Step II: an upper bound of the optimal valueconcavity argument
Step III: match constant-order to the upper boundvanishing discount approach
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Model description Prior work Main result Proof sketch Conclusion
Proof overview
Step I: existence of a steady-stateperturbative approaches
Step II: an upper bound of the optimal valueconcavity argument
Step III: match constant-order to the upper boundvanishing discount approach
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Model description Prior work Main result Proof sketch Conclusion
Proof overview
Step I: existence of a steady-stateperturbative approaches
Step II: an upper bound of the optimal valueconcavity argument
Step III: match constant-order to the upper boundvanishing discount approach
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Model description Prior work Main result Proof sketch Conclusion
Proof overview
Step I: existence of a steady-stateperturbative approaches
Step II: an upper bound of the optimal valueconcavity argument
Step III: match constant-order to the upper boundvanishing discount approach
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Model description Prior work Main result Proof sketch Conclusion
Proof overview
Step I: existence of a steady-stateperturbative approaches
Step II: an upper bound of the optimal valueconcavity argument
Step III: match constant-order to the upper boundvanishing discount approach
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Model description Prior work Main result Proof sketch Conclusion
Proof overview
Step I: existence of a steady-stateperturbative approaches
Step II: an upper bound of the optimal valueconcavity argument
Step III: match constant-order to the upper boundvanishing discount approach
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Model description Prior work Main result Proof sketch Conclusion
Step I: existence of a steady-state
LemmaWithout loss of generality, there exists a stationary measure(IL,∗, χL,∗
1 , . . . , χL,∗L
)of the Markov chain under an optimal stationary
policy, and it satisfies
OPT(L) = E[dL,∗
1 D−1(
dL,∗1
)]− cE[χL,∗
1 ]− E[G(IL,∗)] .
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Model description Prior work Main result Proof sketch Conclusion
Step II: upper bound of OPT (L)
Use concavity to obtain an upper bound:
IL,∗ χL,∗1
. . . χL,∗L
E[IL,∗] E
[χL,∗
1
]. . . E
[χL,∗
L
]
constant-order!
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Model description Prior work Main result Proof sketch Conclusion
Step II: upper bound of OPT (L)
Use concavity to obtain an upper bound:
IL,∗ χL,∗1
. . . χL,∗L
E[IL,∗] E
[χL,∗
1
]. . . E
[χL,∗
L
]
constant-order!
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Model description Prior work Main result Proof sketch Conclusion
Step II: upper bound of OPT (L)
Use concavity to obtain an upper bound:
IL,∗ χL,∗1
. . . χL,∗L
E[IL,∗] E
[χL,∗
1
]. . . E
[χL,∗
L
]
constant-order!
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Model description Prior work Main result Proof sketch Conclusion
OPT(L) is at most
E[dL,∗
1 D−1(
dL,∗1
)]− cE[χL,∗
1 ]− E[G(IL,∗)]
=1− α1− αL
L∑k=1
αk−1E
[dL,∗
k D−1(dL,∗k )− cχL,∗
1
−G
(IL,∗ +
k∑t=1
(χL,∗
t − γtdL,∗t − βt
))].
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Model description Prior work Main result Proof sketch Conclusion
Applying Jensen’s inequalty, OPT(L) is at most
1− α1− αL
L∑k=1
αk−1E
[E[dL,∗
k |ε[k−1]]D−1(E[dL,∗
k |ε[k−1]])− cE
[χL,∗
1
]
−G
(E[IL,∗] +
k∑t=1
(E[χL,∗
t ]− γtE[dL,∗t |ε[t−1]]− βt
))],
Thus,
OPT(L) ≤ 1− α1− αL max
S∈[−S,S]Vα
L (xL,S) for each α ∈ (0,1),
andlim infL→∞
OPT(L) ≤ (1− α) lim supL→∞
maxS∈[−S,S]
VαL (xL,S)
≤ (1− α) maxS∈[−S,S]
Vα∞ (x∞,S) .
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Model description Prior work Main result Proof sketch Conclusion
Step III: match constant-order to the upper bound
Upper bound
Total discounted profit over an infinite horizon with initial on-hand in-ventory S and constant-order x∞.
Constant-order policy
Long-run average profit under the best constant-order policy
convergence of a discounted problem to its long-run average counter-part
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Model description Prior work Main result Proof sketch Conclusion
Step III: match constant-order to the upper bound
Upper bound
Total discounted profit over an infinite horizon with initial on-hand in-ventory S and constant-order x∞.
Constant-order policy
Long-run average profit under the best constant-order policy
convergence of a discounted problem to its long-run average counter-part
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Model description Prior work Main result Proof sketch Conclusion
Step III: match constant-order to the upper bound
Upper bound
Total discounted profit over an infinite horizon with initial on-hand in-ventory S and constant-order x∞.
Constant-order policy
Long-run average profit under the best constant-order policy
convergence of a discounted problem to its long-run average counter-part
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Model description Prior work Main result Proof sketch Conclusion
Schäl’s conditions
Consider the following MDP withstate space S,action spaces A(s) for each s ∈ S,probability transition function q(.|s,a),deterministic and nonnegative single-period cost function c(s,a).
Given a feasible policy π, a discount factor α ∈ (0,1), and an initialstate s, the expected long-run average cost and total discounted costare denoted as Jπ(s) and Jπα(s) respectively.
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Model description Prior work Main result Proof sketch Conclusion
Schäl’s conditions
Consider the following MDP withstate space S,action spaces A(s) for each s ∈ S,probability transition function q(.|s,a),deterministic and nonnegative single-period cost function c(s,a).
Given a feasible policy π, a discount factor α ∈ (0,1), and an initialstate s, the expected long-run average cost and total discounted costare denoted as Jπ(s) and Jπα(s) respectively.
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Model description Prior work Main result Proof sketch Conclusion
Schäl’s conditions
1 S is a locally compact space with a countable base, i.e., thereexists a countable collection B of open sets in a locally compactspace S such that any open set containing x ∈ S contains at leastone of the open sets in B.
2 For each s ∈ S, A(s) is nonempty and compact. Furthermore,A(.) is upper semicontinuous, i.e., for every open set B ⊆ R, theset {s : A(s) ⊆ B} is open in S.
3 The probability transition function q : {(s,a) : a ∈ A(s)} → P(S)is continuous with respect to weak convergence on P(S), whereP(S) denotes the set of all probability measures on S.
4 The single-period cost function c is lower semicontinuous, i.e.,{(s,a) : c(s,a) > γ} is an open set for all γ ∈ R.
5 There exists a policy π and an initial state s ∈ S such that Jπ(s) <∞.
6 supα<1
(infπ Jπα(s)− infs′∈S infπ Jπα(s′)
)<∞ for all s ∈ S.
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Model description Prior work Main result Proof sketch Conclusion
Schäl’s conditions
1 S is a locally compact space with a countable base, i.e., thereexists a countable collection B of open sets in a locally compactspace S such that any open set containing x ∈ S contains at leastone of the open sets in B.
2 For each s ∈ S, A(s) is nonempty and compact. Furthermore,A(.) is upper semicontinuous, i.e., for every open set B ⊆ R, theset {s : A(s) ⊆ B} is open in S.
3 The probability transition function q : {(s,a) : a ∈ A(s)} → P(S)is continuous with respect to weak convergence on P(S), whereP(S) denotes the set of all probability measures on S.
4 The single-period cost function c is lower semicontinuous, i.e.,{(s,a) : c(s,a) > γ} is an open set for all γ ∈ R.
5 There exists a policy π and an initial state s ∈ S such that Jπ(s) <∞.
6 supα<1
(infπ Jπα(s)− infs′∈S infπ Jπα(s′)
)<∞ for all s ∈ S.
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Model description Prior work Main result Proof sketch Conclusion
Vanishing discount approach
Theorem (Schäl 1993)
Under the above conditions, there exists an optimal stationary policyπ∗ such that for all s ∈ S,
Jπ∗(s) = inf
s′∈Sinfπ
Jπ(s′) = limα↑1
[(1− α) inf
s′∈Sinfπ
Jπα(s′)
].
In our setting, we need to prove
lim infα↑1
[(1− α) max
S∈[−S,S]Vα∞(x∞,S)
]= C(πx∞).
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Model description Prior work Main result Proof sketch Conclusion
Verifying Conditions
It suffices to verify condition
supα∈(0,1)
[maxS′∈R
Vα∞(x∞,S′)− Vα
∞(x∞,S)
]<∞.
Assume S∗α solves maxS′∈R Vα∞(x∞,S′) with an optimal policy π∗. In
the inventory system starting with S∗α following policy π∗,
I∗n = I∗n−1 + x∞ − γd∗n − β.
For the system starting with S, we want to construct a policy π topursue I∗ so that the profit difference is bounded (independent of α),
In = In−1 + x∞ − γdn − β.
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Model description Prior work Main result Proof sketch Conclusion
Mapping to Random Yield Model with CapacityConstraint
By treating supply as “demand” and demand as “supply”, the pric-ing and inventory control problem becomes a random yield modelwith capacity constraint on orders
In = In−1 + γdn − (x∞ − β).
Federgruen and Yang (2014) address infinite horizon random yieldmodel without capacity constraint using the vanishing discount ap-proach
|In − I∗n | decreases geometricallythe idea cannot be extended to the case with capacity constraint
We show for the infinite horizon random yield model with capacityconstraint
order to capacity when inventory is too low (uniformly on α)carefully bound the cost difference
![Page 57: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/57.jpg)
Model description Prior work Main result Proof sketch Conclusion
Mapping to Random Yield Model with CapacityConstraint
By treating supply as “demand” and demand as “supply”, the pric-ing and inventory control problem becomes a random yield modelwith capacity constraint on orders
In = In−1 + γdn − (x∞ − β).
Federgruen and Yang (2014) address infinite horizon random yieldmodel without capacity constraint using the vanishing discount ap-proach
|In − I∗n | decreases geometricallythe idea cannot be extended to the case with capacity constraint
We show for the infinite horizon random yield model with capacityconstraint
order to capacity when inventory is too low (uniformly on α)carefully bound the cost difference
![Page 58: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/58.jpg)
Model description Prior work Main result Proof sketch Conclusion
Mapping to Random Yield Model with CapacityConstraint
By treating supply as “demand” and demand as “supply”, the pric-ing and inventory control problem becomes a random yield modelwith capacity constraint on orders
In = In−1 + γdn − (x∞ − β).
Federgruen and Yang (2014) address infinite horizon random yieldmodel without capacity constraint using the vanishing discount ap-proach
|In − I∗n | decreases geometricallythe idea cannot be extended to the case with capacity constraint
We show for the infinite horizon random yield model with capacityconstraint
order to capacity when inventory is too low (uniformly on α)carefully bound the cost difference
![Page 59: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/59.jpg)
Model description Prior work Main result Proof sketch Conclusion
Mapping to Random Yield Model with CapacityConstraint
By treating supply as “demand” and demand as “supply”, the pric-ing and inventory control problem becomes a random yield modelwith capacity constraint on orders
In = In−1 + γdn − (x∞ − β).
Federgruen and Yang (2014) address infinite horizon random yieldmodel without capacity constraint using the vanishing discount ap-proach
|In − I∗n | decreases geometricallythe idea cannot be extended to the case with capacity constraint
We show for the infinite horizon random yield model with capacityconstraint
order to capacity when inventory is too low (uniformly on α)carefully bound the cost difference
![Page 60: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/60.jpg)
Model description Prior work Main result Proof sketch Conclusion
Mapping to Random Yield Model with CapacityConstraint
By treating supply as “demand” and demand as “supply”, the pric-ing and inventory control problem becomes a random yield modelwith capacity constraint on orders
In = In−1 + γdn − (x∞ − β).
Federgruen and Yang (2014) address infinite horizon random yieldmodel without capacity constraint using the vanishing discount ap-proach
|In − I∗n | decreases geometricallythe idea cannot be extended to the case with capacity constraint
We show for the infinite horizon random yield model with capacityconstraint
order to capacity when inventory is too low (uniformly on α)carefully bound the cost difference
![Page 61: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/61.jpg)
Model description Prior work Main result Proof sketch Conclusion
Mapping to Random Yield Model with CapacityConstraint
By treating supply as “demand” and demand as “supply”, the pric-ing and inventory control problem becomes a random yield modelwith capacity constraint on orders
In = In−1 + γdn − (x∞ − β).
Federgruen and Yang (2014) address infinite horizon random yieldmodel without capacity constraint using the vanishing discount ap-proach
|In − I∗n | decreases geometricallythe idea cannot be extended to the case with capacity constraint
We show for the infinite horizon random yield model with capacityconstraint
order to capacity when inventory is too low (uniformly on α)carefully bound the cost difference
![Page 62: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/62.jpg)
Model description Prior work Main result Proof sketch Conclusion
Mapping to Random Yield Model with CapacityConstraint
By treating supply as “demand” and demand as “supply”, the pric-ing and inventory control problem becomes a random yield modelwith capacity constraint on orders
In = In−1 + γdn − (x∞ − β).
Federgruen and Yang (2014) address infinite horizon random yieldmodel without capacity constraint using the vanishing discount ap-proach
|In − I∗n | decreases geometricallythe idea cannot be extended to the case with capacity constraint
We show for the infinite horizon random yield model with capacityconstraint
order to capacity when inventory is too low (uniformly on α)carefully bound the cost difference
![Page 63: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/63.jpg)
Model description Prior work Main result Proof sketch Conclusion
Conclusion
Main contributionEstablish asymptotic optimality of constant-order policies for jointpricing and inventory control with large replenishment lead times
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Model description Prior work Main result Proof sketch Conclusion
Conclusion
Main contributionEstablish asymptotic optimality of constant-order policies for jointpricing and inventory control with large replenishment lead times
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Model description Prior work Main result Proof sketch Conclusion
Future research directions
Other inventory modelsnot universally heldcounter-example: single-sourcing backlogged model
Open problemsfixed ordering costsgeneral MDPs. . .
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Model description Prior work Main result Proof sketch Conclusion
Future research directions
Other inventory modelsnot universally heldcounter-example: single-sourcing backlogged model
Open problemsfixed ordering costsgeneral MDPs. . .
![Page 67: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/67.jpg)
Model description Prior work Main result Proof sketch Conclusion
Future research directions
Other inventory modelsnot universally heldcounter-example: single-sourcing backlogged model
Open problemsfixed ordering costsgeneral MDPs. . .
![Page 68: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/68.jpg)
Model description Prior work Main result Proof sketch Conclusion
Future research directions
Other inventory modelsnot universally heldcounter-example: single-sourcing backlogged model
Open problemsfixed ordering costsgeneral MDPs. . .
![Page 69: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/69.jpg)
Model description Prior work Main result Proof sketch Conclusion
Extension and Implications
Finite state, finite action MDPsFirst type decisions: takes effect right awaySecond type decisions: takes effect after a long lead time
Claim: Open-loop control is asymptotically optimal for the secondtype decisionsImplications for data-driven models
Ignore real time information when making the second type decisionsother examples?
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Model description Prior work Main result Proof sketch Conclusion
Extension and Implications
Finite state, finite action MDPsFirst type decisions: takes effect right awaySecond type decisions: takes effect after a long lead time
Claim: Open-loop control is asymptotically optimal for the secondtype decisionsImplications for data-driven models
Ignore real time information when making the second type decisionsother examples?
![Page 71: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/71.jpg)
Model description Prior work Main result Proof sketch Conclusion
Extension and Implications
Finite state, finite action MDPsFirst type decisions: takes effect right awaySecond type decisions: takes effect after a long lead time
Claim: Open-loop control is asymptotically optimal for the secondtype decisionsImplications for data-driven models
Ignore real time information when making the second type decisionsother examples?
![Page 72: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/72.jpg)
Model description Prior work Main result Proof sketch Conclusion
Extension and Implications
Finite state, finite action MDPsFirst type decisions: takes effect right awaySecond type decisions: takes effect after a long lead time
Claim: Open-loop control is asymptotically optimal for the secondtype decisionsImplications for data-driven models
Ignore real time information when making the second type decisionsother examples?
![Page 73: Dynamic pricing and inventory control with large replenishment lead … · 2018-10-04 · Dynamic pricing and inventory control with large replenishment lead times Xin Chen University](https://reader033.vdocuments.us/reader033/viewer/2022060402/5f0e55447e708231d43ebbf4/html5/thumbnails/73.jpg)
Model description Prior work Main result Proof sketch Conclusion
Extension and Implications
Finite state, finite action MDPsFirst type decisions: takes effect right awaySecond type decisions: takes effect after a long lead time
Claim: Open-loop control is asymptotically optimal for the secondtype decisionsImplications for data-driven models
Ignore real time information when making the second type decisionsother examples?