Dynamic Pricing and Inventory Management under ... - Semantic Scholar
Dynamic Pricing and Inventory Management under Inventory-Dependent-Demand Nan Yang and Philip (Renyu) Zhang Olin School of Business Washington University in St. Louis
October 15, 2012
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Outline
Motivation and Introduction
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Outline
Motivation and Introduction
Literature Review
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Outline
Motivation and Introduction
Literature Review
Model Formulation
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Outline
Motivation and Introduction
Literature Review
Model Formulation
Main Results and Managerial Implications
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Outline
Motivation and Introduction
Literature Review
Model Formulation
Main Results and Managerial Implications
Q&A
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Motivation
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Motivation I
Campus Caf´es, dispose leftover coffee and bread every day.
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Motivation I
Campus Caf´es, dispose leftover coffee and bread every day.
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BMW Mini Cooper, deliberately limits its annual sales to 25, 000, and the average wait time is 2.5 months (New York Times, 2006).
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Motivation I
Campus Caf´es, dispose leftover coffee and bread every day.
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BMW Mini Cooper, deliberately limits its annual sales to 25, 000, and the average wait time is 2.5 months (New York Times, 2006).
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Apple iPad, limits customers to two iPads per order in online sales (CBS News, 2010).
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Motivation I
Campus Caf´es, dispose leftover coffee and bread every day.
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BMW Mini Cooper, deliberately limits its annual sales to 25, 000, and the average wait time is 2.5 months (New York Times, 2006).
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Apple iPad, limits customers to two iPads per order in online sales (CBS News, 2010).
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Impact of Inventory on Demand
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Drawbacks of high inventory levels: I I I
Poor turnover (sales/inventory ). Significant purchasing holding and managing costs. Investment opportunity lost.
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Drawbacks of high inventory levels: I I I I
Poor turnover (sales/inventory ). Significant purchasing holding and managing costs. Investment opportunity lost. Potential demand depressed.
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Drawbacks of high inventory levels: I I I I
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Poor turnover (sales/inventory ). Significant purchasing holding and managing costs. Investment opportunity lost. Potential demand depressed.
Excessive inventory is an implicit negative indicator of the product’s quality, popularity and freshness.
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Drawbacks of high inventory levels: I I I I
Poor turnover (sales/inventory ). Significant purchasing holding and managing costs. Investment opportunity lost. Potential demand depressed.
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Excessive inventory is an implicit negative indicator of the product’s quality, popularity and freshness.
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Future demand is negatively correlated with excessive inventory levels.
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Motivating Research Questions
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Motivating Research Questions
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What is the structure of the optimal price and inventory policy under inventory-dependent-demand?
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Motivating Research Questions
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What is the structure of the optimal price and inventory policy under inventory-dependent-demand?
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How inventory-dependent-demand will influence the optimal policy?
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Motivating Research Questions
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What is the structure of the optimal price and inventory policy under inventory-dependent-demand?
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How inventory-dependent-demand will influence the optimal policy?
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How will the flexibility in pricing and inventory disposal impact the optimal policy and the performance of the system?
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Benefit of Dynamic Pricing
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Benefit of Dynamic Pricing I
Dynamic pricing is most effective for perishable products and products facing high demand and/or supply variability.
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Benefit of Dynamic Pricing I
Dynamic pricing is most effective for perishable products and products facing high demand and/or supply variability.
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Inventory-Dependent-Demand amplifies the demand volatility.
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Benefit of Dynamic Pricing I
Dynamic pricing is most effective for perishable products and products facing high demand and/or supply variability.
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Inventory-Dependent-Demand amplifies the demand volatility.
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Effective price adjustments can help stabilize the inventory levels, thus reducing the demand variability. .
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Benefit of Inventory Disposal
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Benefit of Inventory Disposal
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Inventory disposal is an efficient way to liquidate surplus asset and reduce inventory holding and managing costs.
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Benefit of Inventory Disposal
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Inventory disposal is an efficient way to liquidate surplus asset and reduce inventory holding and managing costs.
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Potential demand loss caused by high excessive inventory levels is also saved.
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Benefit of Inventory Disposal
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Inventory disposal is an efficient way to liquidate surplus asset and reduce inventory holding and managing costs.
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Potential demand loss caused by high excessive inventory levels is also saved.
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Inventory disposal controls both the supply (service level) and demand (inventory-dependent-demand) sides of the story.
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Literature Review
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Literature Review I
Inventory dependent demand: I I I
Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
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Literature Review I
Inventory dependent demand: I I I
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Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
Dynamic pricing under stochastic demand: I I
Federgruen and Heching (1999), Chen and Simchi-Levi (2004 a,b, 2006).
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Literature Review I
Inventory dependent demand: I I I
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Dynamic pricing under stochastic demand: I I
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Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
Federgruen and Heching (1999), Chen and Simchi-Levi (2004 a,b, 2006).
Joint price & inventory control under inventory-dependent demand: I I
Dana and Petruzzi (2001), Balakrishnan et al. (2008).
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Literature Review I
Inventory dependent demand: I I I
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Dynamic pricing under stochastic demand: I I
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Federgruen and Heching (1999), Chen and Simchi-Levi (2004 a,b, 2006).
Joint price & inventory control under inventory-dependent demand: I I
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Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
Dana and Petruzzi (2001), Balakrishnan et al. (2008).
Our paper: Dynamic pricing and inventory control & stochastic demand negatively correlated with inventory.
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Literature Review I
Inventory dependent demand: I I I
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Dynamic pricing under stochastic demand: I I
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Federgruen and Heching (1999), Chen and Simchi-Levi (2004 a,b, 2006).
Joint price & inventory control under inventory-dependent demand: I I
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Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
Dana and Petruzzi (2001), Balakrishnan et al. (2008).
Our paper: Dynamic pricing and inventory control & stochastic demand negatively correlated with inventory.
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Models
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand.
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand. I
Model (1): perishable products.
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand. I
Model (1): perishable products.
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Model (2): nonperishable products, without inventory disposal.
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand. I
Model (1): perishable products.
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Model (2): nonperishable products, without inventory disposal.
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Model (3): nonperishable products, with inventory disposal.
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand. I
Model (1): perishable products.
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Model (2): nonperishable products, without inventory disposal.
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Model (3): nonperishable products, with inventory disposal.
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Notations and Assumptions
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Notations and Assumptions I
T periods in total, labeled backwards, full backorder.
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pt =the price set in period t. pt∗ is the optimal price. c =the unit procurement cost.
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It =the inventory level at the beginning of period t before replenishment.
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xt =the inventory level in period t after replenishment before demand realization. xt∗ is the optimal service level. Zero leadtime.
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Dt = δ(pt , It , ϵt ) =the random demand in period t, strictly decreasing in pt , decreasing in It . ϵt is a random vector.
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α =discount factor. .
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Notations and Assumptions I
T periods in total, labeled backwards, full backorder.
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pt =the price set in period t. pt∗ is the optimal price. c =the unit procurement cost.
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It =the inventory level at the beginning of period t before replenishment.
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xt =the inventory level in period t after replenishment before demand realization. xt∗ is the optimal service level. Zero leadtime.
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Dt = δ(pt , It , ϵt ) =the random demand in period t, strictly decreasing in pt , decreasing in It . ϵt is a random vector.
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α =discount factor. .
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Model (1): Sequence of Events
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
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Period t − 1 starts
Decides (xt , pt )
Excessive demand costs β(> c) per unit.
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Excessive demand costs β(> c) per unit.
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It−1 = xt − Dt perishes but has impact on Dt−1 .
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Excessive demand costs β(> c) per unit.
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It−1 = xt − Dt perishes but has impact on Dt−1 .
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VtP (It ) =the optimal expected profit to go with inventory level It .
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Excessive demand costs β(> c) per unit.
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It−1 = xt − Dt perishes but has impact on Dt−1 .
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VtP (It ) =the optimal expected profit to go with inventory level It .
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Model (1): Optimality Equation
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Model (1): Optimality Equation P VtP (It ) = maxxt ,pt E{pt Dt − cxt − β(xt − Dt )− + αVt−1 (xt − Dt )}.
V0P (I0 ) = 0.
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Model (1): Optimality Equation P VtP (It ) = maxxt ,pt E{pt Dt − cxt − β(xt − Dt )− + αVt−1 (xt − Dt )}.
V0P (I0 ) = 0. I
pt Dt =revenue in period t.
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cxt =procurement cost in period t.
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β(xt − Dt )− =penality incurred by excessive demand in period t.
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P (xt − Dt ) =the profit in later periods. αVt−1
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Model (1): Optimality Equation P VtP (It ) = maxxt ,pt E{pt Dt − cxt − β(xt − Dt )− + αVt−1 (xt − Dt )}.
V0P (I0 ) = 0. I
pt Dt =revenue in period t.
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cxt =procurement cost in period t.
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β(xt − Dt )− =penality incurred by excessive demand in period t.
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P (xt − Dt ) =the profit in later periods. αVt−1
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Model (2): Sequence of Events
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
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Period t − 1 starts
Decides (xt , pt )
Procurement cost: c(xt − It ).
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Procurement cost: c(xt − It ).
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Procurement cost: c(xt − It ).
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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VtND (It ) =the optimal expected profit to go with inventory level It .
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Procurement cost: c(xt − It ).
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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VtND (It ) =the optimal expected profit to go with inventory level It .
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Model (2): Optimality Equation
VtND (It ) = max E{pt Dt − c(xt − It ) − h(xt − Dt )+ − b(xt − Dt )− xt ≥It ,pt
ND + αVt−1 (xt − Dt )}.
V0ND (I0 ) = −cI0− + sI0+ .
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Model (2): Optimality Equation
VtND (It ) = max E{pt Dt − c(xt − It ) − h(xt − Dt )+ − b(xt − Dt )− xt ≥It ,pt
ND + αVt−1 (xt − Dt )}.
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V0ND (I0 ) = −cI0− + sI0+ . pt Dt =revenue in period t.
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c(xt − It ) =procurement cost in period t.
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h(xt − Dt )+ + b(xt − Dt )− =operational cost in period t.
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ND (xt − Dt ) =the profit in later periods. αVt−1
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Model (2): Optimality Equation
VtND (It ) = max E{pt Dt − c(xt − It ) − h(xt − Dt )+ − b(xt − Dt )− xt ≥It ,pt
ND + αVt−1 (xt − Dt )}.
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V0ND (I0 ) = −cI0− + sI0+ . pt Dt =revenue in period t.
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c(xt − It ) =procurement cost in period t.
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h(xt − Dt )+ + b(xt − Dt )− =operational cost in period t.
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ND (xt − Dt ) =the profit in later periods. αVt−1
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Model (3): Sequence of Events
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
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Period t − 1 starts
Decides (xt , pt )
Salvage value less procurement cost: s(xt − It )− − c(xt − It )+ .
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Salvage value less procurement cost: s(xt − It )− − c(xt − It )+ .
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Salvage value less procurement cost: s(xt − It )− − c(xt − It )+ .
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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Vt (It ) =the optimal expected profit to go with inventory level It .
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Salvage value less procurement cost: s(xt − It )− − c(xt − It )+ .
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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Vt (It ) =the optimal expected profit to go with inventory level It .
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Model (3): Optimality Equation
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Model (3): Optimality Equation
Vt (It ) =
max
xt ≥min{0,It },pt
E{pt Dt − c(xt − It )+ + s(xt − It )− − h(xt − Dt )+
ND − b(xt − Dt )− + αVt−1 (xt − Dt )}.
V0 (I0 ) = −cI0− + sI0+ .
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Model (3): Optimality Equation
Vt (It ) =
max
xt ≥min{0,It },pt
E{pt Dt − c(xt − It )+ + s(xt − It )− − h(xt − Dt )+
ND − b(xt − Dt )− + αVt−1 (xt − Dt )}.
V0 (I0 ) = −cI0− + sI0+ . I
pt Dt =revenue in period t.
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c(xt − It )+ /s(xt − It )− =procurement cost/salvage value.
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h(xt − Dt )+ + b(xt − Dt )− =operational cost in period t.
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αVt−1 (xt − Dt ) =the profit in later periods.
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Model (3): Optimality Equation
Vt (It ) =
max
xt ≥min{0,It },pt
E{pt Dt − c(xt − It )+ + s(xt − It )− − h(xt − Dt )+
ND − b(xt − Dt )− + αVt−1 (xt − Dt )}.
V0 (I0 ) = −cI0− + sI0+ . I
pt Dt =revenue in period t.
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c(xt − It )+ /s(xt − It )− =procurement cost/salvage value.
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h(xt − Dt )+ + b(xt − Dt )− =operational cost in period t.
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αVt−1 (xt − Dt ) =the profit in later periods.
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Main Results (Model 3)
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order I
Keep
It Dispose
pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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pt∗ , xt∗ , ItL and ItH are higher with inventory-disposal opportunity.
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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pt∗ , xt∗ , ItL and ItH are higher with inventory-disposal opportunity.
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The benefits of dynamic pricing and inventory disposal (in profit) are as high as 1/3 (numerical).
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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pt∗ , xt∗ , ItL and ItH are higher with inventory-disposal opportunity.
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The benefits of dynamic pricing and inventory disposal (in profit) are as high as 1/3 (numerical).
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When the wait-list effect is strong enough, ItL < ItH = 0.
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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pt∗ , xt∗ , ItL and ItH are higher with inventory-disposal opportunity.
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The benefits of dynamic pricing and inventory disposal (in profit) are as high as 1/3 (numerical).
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When the wait-list effect is strong enough, ItL < ItH = 0.
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Managerial Insights and Implications I
Inventory-dependent-demand strengthens overstocking risk by depressing potential demand.
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Managerial Insights and Implications I
Inventory-dependent-demand strengthens overstocking risk by depressing potential demand.
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Price and operational flexibility helps mitigate demand loss driven by high inventory levels.
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Managerial Insights and Implications I
Inventory-dependent-demand strengthens overstocking risk by depressing potential demand.
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Price and operational flexibility helps mitigate demand loss driven by high inventory levels.
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Suggestions: I I I I I
Never ignore inventory-dependent-demand. Limit order quantity. Decrease sales price. Dispose unnecessary inventory (campus caf´e). Take advantage of the wait-list effect (BMW and Apple).
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Managerial Insights and Implications I
Inventory-dependent-demand strengthens overstocking risk by depressing potential demand.
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Price and operational flexibility helps mitigate demand loss driven by high inventory levels.
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Suggestions: I I I I I
Never ignore inventory-dependent-demand. Limit order quantity. Decrease sales price. Dispose unnecessary inventory (campus caf´e). Take advantage of the wait-list effect (BMW and Apple).
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Q&A
Thank you! Questions?
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October 15, 2012
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Outline
Motivation and Introduction
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Outline
Motivation and Introduction
Literature Review
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Outline
Motivation and Introduction
Literature Review
Model Formulation
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Outline
Motivation and Introduction
Literature Review
Model Formulation
Main Results and Managerial Implications
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Outline
Motivation and Introduction
Literature Review
Model Formulation
Main Results and Managerial Implications
Q&A
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Motivation
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Motivation I
Campus Caf´es, dispose leftover coffee and bread every day.
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Motivation I
Campus Caf´es, dispose leftover coffee and bread every day.
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BMW Mini Cooper, deliberately limits its annual sales to 25, 000, and the average wait time is 2.5 months (New York Times, 2006).
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Motivation I
Campus Caf´es, dispose leftover coffee and bread every day.
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BMW Mini Cooper, deliberately limits its annual sales to 25, 000, and the average wait time is 2.5 months (New York Times, 2006).
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Apple iPad, limits customers to two iPads per order in online sales (CBS News, 2010).
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Motivation I
Campus Caf´es, dispose leftover coffee and bread every day.
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BMW Mini Cooper, deliberately limits its annual sales to 25, 000, and the average wait time is 2.5 months (New York Times, 2006).
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Apple iPad, limits customers to two iPads per order in online sales (CBS News, 2010).
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Impact of Inventory on Demand
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Drawbacks of high inventory levels: I I I
Poor turnover (sales/inventory ). Significant purchasing holding and managing costs. Investment opportunity lost.
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Drawbacks of high inventory levels: I I I I
Poor turnover (sales/inventory ). Significant purchasing holding and managing costs. Investment opportunity lost. Potential demand depressed.
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Drawbacks of high inventory levels: I I I I
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Poor turnover (sales/inventory ). Significant purchasing holding and managing costs. Investment opportunity lost. Potential demand depressed.
Excessive inventory is an implicit negative indicator of the product’s quality, popularity and freshness.
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Impact of Inventory on Demand I
Operations objective: avoid excessive inventory.
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Drawbacks of high inventory levels: I I I I
Poor turnover (sales/inventory ). Significant purchasing holding and managing costs. Investment opportunity lost. Potential demand depressed.
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Excessive inventory is an implicit negative indicator of the product’s quality, popularity and freshness.
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Future demand is negatively correlated with excessive inventory levels.
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Motivating Research Questions
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Motivating Research Questions
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What is the structure of the optimal price and inventory policy under inventory-dependent-demand?
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Motivating Research Questions
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What is the structure of the optimal price and inventory policy under inventory-dependent-demand?
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How inventory-dependent-demand will influence the optimal policy?
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Motivating Research Questions
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What is the structure of the optimal price and inventory policy under inventory-dependent-demand?
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How inventory-dependent-demand will influence the optimal policy?
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How will the flexibility in pricing and inventory disposal impact the optimal policy and the performance of the system?
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Benefit of Dynamic Pricing
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Benefit of Dynamic Pricing I
Dynamic pricing is most effective for perishable products and products facing high demand and/or supply variability.
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Benefit of Dynamic Pricing I
Dynamic pricing is most effective for perishable products and products facing high demand and/or supply variability.
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Inventory-Dependent-Demand amplifies the demand volatility.
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Benefit of Dynamic Pricing I
Dynamic pricing is most effective for perishable products and products facing high demand and/or supply variability.
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Inventory-Dependent-Demand amplifies the demand volatility.
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Effective price adjustments can help stabilize the inventory levels, thus reducing the demand variability. .
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Benefit of Inventory Disposal
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Benefit of Inventory Disposal
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Inventory disposal is an efficient way to liquidate surplus asset and reduce inventory holding and managing costs.
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Benefit of Inventory Disposal
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Inventory disposal is an efficient way to liquidate surplus asset and reduce inventory holding and managing costs.
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Potential demand loss caused by high excessive inventory levels is also saved.
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Benefit of Inventory Disposal
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Inventory disposal is an efficient way to liquidate surplus asset and reduce inventory holding and managing costs.
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Potential demand loss caused by high excessive inventory levels is also saved.
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Inventory disposal controls both the supply (service level) and demand (inventory-dependent-demand) sides of the story.
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Literature Review
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Literature Review I
Inventory dependent demand: I I I
Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
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Literature Review I
Inventory dependent demand: I I I
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Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
Dynamic pricing under stochastic demand: I I
Federgruen and Heching (1999), Chen and Simchi-Levi (2004 a,b, 2006).
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Literature Review I
Inventory dependent demand: I I I
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Dynamic pricing under stochastic demand: I I
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Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
Federgruen and Heching (1999), Chen and Simchi-Levi (2004 a,b, 2006).
Joint price & inventory control under inventory-dependent demand: I I
Dana and Petruzzi (2001), Balakrishnan et al. (2008).
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Literature Review I
Inventory dependent demand: I I I
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Dynamic pricing under stochastic demand: I I
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Federgruen and Heching (1999), Chen and Simchi-Levi (2004 a,b, 2006).
Joint price & inventory control under inventory-dependent demand: I I
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Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
Dana and Petruzzi (2001), Balakrishnan et al. (2008).
Our paper: Dynamic pricing and inventory control & stochastic demand negatively correlated with inventory.
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Literature Review I
Inventory dependent demand: I I I
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Dynamic pricing under stochastic demand: I I
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Federgruen and Heching (1999), Chen and Simchi-Levi (2004 a,b, 2006).
Joint price & inventory control under inventory-dependent demand: I I
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Gerchak and Wang (1994), Urban (2005), Sapra et al. (2010).
Dana and Petruzzi (2001), Balakrishnan et al. (2008).
Our paper: Dynamic pricing and inventory control & stochastic demand negatively correlated with inventory.
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Models
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand.
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand. I
Model (1): perishable products.
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand. I
Model (1): perishable products.
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Model (2): nonperishable products, without inventory disposal.
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand. I
Model (1): perishable products.
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Model (2): nonperishable products, without inventory disposal.
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Model (3): nonperishable products, with inventory disposal.
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Models
We develop the following dynamic programming models to investigate the inventory-dependent-demand. I
Model (1): perishable products.
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Model (2): nonperishable products, without inventory disposal.
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Model (3): nonperishable products, with inventory disposal.
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Notations and Assumptions
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Notations and Assumptions I
T periods in total, labeled backwards, full backorder.
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pt =the price set in period t. pt∗ is the optimal price. c =the unit procurement cost.
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It =the inventory level at the beginning of period t before replenishment.
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xt =the inventory level in period t after replenishment before demand realization. xt∗ is the optimal service level. Zero leadtime.
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Dt = δ(pt , It , ϵt ) =the random demand in period t, strictly decreasing in pt , decreasing in It . ϵt is a random vector.
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α =discount factor. .
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Notations and Assumptions I
T periods in total, labeled backwards, full backorder.
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pt =the price set in period t. pt∗ is the optimal price. c =the unit procurement cost.
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It =the inventory level at the beginning of period t before replenishment.
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xt =the inventory level in period t after replenishment before demand realization. xt∗ is the optimal service level. Zero leadtime.
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Dt = δ(pt , It , ϵt ) =the random demand in period t, strictly decreasing in pt , decreasing in It . ϵt is a random vector.
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α =discount factor. .
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Model (1): Sequence of Events
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
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Period t − 1 starts
Decides (xt , pt )
Excessive demand costs β(> c) per unit.
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Excessive demand costs β(> c) per unit.
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It−1 = xt − Dt perishes but has impact on Dt−1 .
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Excessive demand costs β(> c) per unit.
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It−1 = xt − Dt perishes but has impact on Dt−1 .
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VtP (It ) =the optimal expected profit to go with inventory level It .
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Model (1): Sequence of Events I
In model (1), It is the inventory level at the end of period t + 1, which is not usable but has impact on Dt . xt ≥ 0.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Excessive demand costs β(> c) per unit.
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It−1 = xt − Dt perishes but has impact on Dt−1 .
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VtP (It ) =the optimal expected profit to go with inventory level It .
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Model (1): Optimality Equation
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Model (1): Optimality Equation P VtP (It ) = maxxt ,pt E{pt Dt − cxt − β(xt − Dt )− + αVt−1 (xt − Dt )}.
V0P (I0 ) = 0.
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Model (1): Optimality Equation P VtP (It ) = maxxt ,pt E{pt Dt − cxt − β(xt − Dt )− + αVt−1 (xt − Dt )}.
V0P (I0 ) = 0. I
pt Dt =revenue in period t.
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cxt =procurement cost in period t.
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β(xt − Dt )− =penality incurred by excessive demand in period t.
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P (xt − Dt ) =the profit in later periods. αVt−1
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Model (1): Optimality Equation P VtP (It ) = maxxt ,pt E{pt Dt − cxt − β(xt − Dt )− + αVt−1 (xt − Dt )}.
V0P (I0 ) = 0. I
pt Dt =revenue in period t.
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cxt =procurement cost in period t.
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β(xt − Dt )− =penality incurred by excessive demand in period t.
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P (xt − Dt ) =the profit in later periods. αVt−1
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Model (2): Sequence of Events
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
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Period t − 1 starts
Decides (xt , pt )
Procurement cost: c(xt − It ).
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Procurement cost: c(xt − It ).
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Procurement cost: c(xt − It ).
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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VtND (It ) =the optimal expected profit to go with inventory level It .
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Model (2): Sequence of Events I
In model (2), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ It
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Procurement cost: c(xt − It ).
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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VtND (It ) =the optimal expected profit to go with inventory level It .
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Model (2): Optimality Equation
VtND (It ) = max E{pt Dt − c(xt − It ) − h(xt − Dt )+ − b(xt − Dt )− xt ≥It ,pt
ND + αVt−1 (xt − Dt )}.
V0ND (I0 ) = −cI0− + sI0+ .
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Model (2): Optimality Equation
VtND (It ) = max E{pt Dt − c(xt − It ) − h(xt − Dt )+ − b(xt − Dt )− xt ≥It ,pt
ND + αVt−1 (xt − Dt )}.
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V0ND (I0 ) = −cI0− + sI0+ . pt Dt =revenue in period t.
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c(xt − It ) =procurement cost in period t.
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h(xt − Dt )+ + b(xt − Dt )− =operational cost in period t.
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ND (xt − Dt ) =the profit in later periods. αVt−1
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Model (2): Optimality Equation
VtND (It ) = max E{pt Dt − c(xt − It ) − h(xt − Dt )+ − b(xt − Dt )− xt ≥It ,pt
ND + αVt−1 (xt − Dt )}.
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V0ND (I0 ) = −cI0− + sI0+ . pt Dt =revenue in period t.
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c(xt − It ) =procurement cost in period t.
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h(xt − Dt )+ + b(xt − Dt )− =operational cost in period t.
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ND (xt − Dt ) =the profit in later periods. αVt−1
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Model (3): Sequence of Events
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
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Period t − 1 starts
Decides (xt , pt )
Salvage value less procurement cost: s(xt − It )− − c(xt − It )+ .
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Salvage value less procurement cost: s(xt − It )− − c(xt − It )+ .
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Salvage value less procurement cost: s(xt − It )− − c(xt − It )+ .
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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Vt (It ) =the optimal expected profit to go with inventory level It .
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Model (3): Sequence of Events I
In model (3), It is the inventory level at the beginning of period t, which is usable and has impact on Dt . xt ≥ min{0, It }.
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The sequence of events in period t: Dt realized
Observes It Period t starts
Period t − 1 starts
Decides (xt , pt )
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Salvage value less procurement cost: s(xt − It )− − c(xt − It )+ .
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− + It−1 = xt − Dt , which incurs cost: hIt−1 + bIt−1 .
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Vt (It ) =the optimal expected profit to go with inventory level It .
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Model (3): Optimality Equation
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Model (3): Optimality Equation
Vt (It ) =
max
xt ≥min{0,It },pt
E{pt Dt − c(xt − It )+ + s(xt − It )− − h(xt − Dt )+
ND − b(xt − Dt )− + αVt−1 (xt − Dt )}.
V0 (I0 ) = −cI0− + sI0+ .
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Model (3): Optimality Equation
Vt (It ) =
max
xt ≥min{0,It },pt
E{pt Dt − c(xt − It )+ + s(xt − It )− − h(xt − Dt )+
ND − b(xt − Dt )− + αVt−1 (xt − Dt )}.
V0 (I0 ) = −cI0− + sI0+ . I
pt Dt =revenue in period t.
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c(xt − It )+ /s(xt − It )− =procurement cost/salvage value.
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h(xt − Dt )+ + b(xt − Dt )− =operational cost in period t.
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αVt−1 (xt − Dt ) =the profit in later periods.
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Model (3): Optimality Equation
Vt (It ) =
max
xt ≥min{0,It },pt
E{pt Dt − c(xt − It )+ + s(xt − It )− − h(xt − Dt )+
ND − b(xt − Dt )− + αVt−1 (xt − Dt )}.
V0 (I0 ) = −cI0− + sI0+ . I
pt Dt =revenue in period t.
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c(xt − It )+ /s(xt − It )− =procurement cost/salvage value.
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h(xt − Dt )+ + b(xt − Dt )− =operational cost in period t.
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αVt−1 (xt − Dt ) =the profit in later periods.
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Main Results (Model 3)
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order I
Keep
It Dispose
pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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pt∗ , xt∗ , ItL and ItH are higher with inventory-disposal opportunity.
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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pt∗ , xt∗ , ItL and ItH are higher with inventory-disposal opportunity.
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The benefits of dynamic pricing and inventory disposal (in profit) are as high as 1/3 (numerical).
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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pt∗ , xt∗ , ItL and ItH are higher with inventory-disposal opportunity.
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The benefits of dynamic pricing and inventory disposal (in profit) are as high as 1/3 (numerical).
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When the wait-list effect is strong enough, ItL < ItH = 0.
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Main Results (Model 3) I
Inventory-dependent order-up-to/dispose-down-to list-price policy. ItH
ItL Order
Keep
It Dispose
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pt∗ , xt∗ , ItL and ItH are lower with inventory-dependent-demand.
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pt∗ , xt∗ , ItL and ItH are higher with inventory-disposal opportunity.
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The benefits of dynamic pricing and inventory disposal (in profit) are as high as 1/3 (numerical).
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When the wait-list effect is strong enough, ItL < ItH = 0.
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Managerial Insights and Implications I
Inventory-dependent-demand strengthens overstocking risk by depressing potential demand.
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Managerial Insights and Implications I
Inventory-dependent-demand strengthens overstocking risk by depressing potential demand.
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Price and operational flexibility helps mitigate demand loss driven by high inventory levels.
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Managerial Insights and Implications I
Inventory-dependent-demand strengthens overstocking risk by depressing potential demand.
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Price and operational flexibility helps mitigate demand loss driven by high inventory levels.
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Suggestions: I I I I I
Never ignore inventory-dependent-demand. Limit order quantity. Decrease sales price. Dispose unnecessary inventory (campus caf´e). Take advantage of the wait-list effect (BMW and Apple).
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Managerial Insights and Implications I
Inventory-dependent-demand strengthens overstocking risk by depressing potential demand.
I
Price and operational flexibility helps mitigate demand loss driven by high inventory levels.
I
Suggestions: I I I I I
Never ignore inventory-dependent-demand. Limit order quantity. Decrease sales price. Dispose unnecessary inventory (campus caf´e). Take advantage of the wait-list effect (BMW and Apple).
.
.
.
.
.
.
Q&A
Thank you! Questions?
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