Single Period Stochastic Inventory Problem with CVaR Risk Constraint

1
Single Period Stochastic Inventory Problem with
CVaR Risk Constraint

• Semra AGRALI
• Ahu SOYLU

2
OUTLINE

• Introduction and Motivation
• Literature Review
• Newsvendor Problem with CVaR
• CVaR Optimization via an LP Approach
• CVaR-Constrained Newsvendor Problem via an LP
  Approach
• Analysis of the Model
• CVaR Optimization
• Profit Maximization with CVaR constraint
• Conclusion

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Introduction and Motivation

• Single period stochastic inventory model
  (newsvendor)
• Goal Maximize profit
• Demand D FD
• r retail price
• c unit cost
• s salvage value
• Expected profit function
• Classical newsvendor problem does not take risk
  into account

cu r - c co c - s
4
Literature Review

• Risk Measures
• Satisficing Probability Maximization
• Lau and Lau, 1988 Li et al., 1991
• Scalarization Method
• Parlar and Weng, 2003
• Utility Functions
• Anvari, 1987
• VaR and CVaR
• Gan et al., 2003 Chen et al., 2004

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The Goal of the Project

• Product portfolio selection in multi-product
  newsvendor model with CVaR risk constraint

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Newsvendor Problem with CVaR

• CVaR Minimization
• Profit Maximization with CVaR Constraint

subject to
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CVaR Minimization via an LP Approach

• Minimization of CVaR for m products for a single
  period

VaR
Loss-VaR
Qj - Dij
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CVaR-Constrained Newsvendor Problem

• Maximization of expected profit, subject to a
  CVaR Constraint

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Analysis of the Model

• Two products
• Multi-variate normally distributed demand
• Discretization 100 x 10010000 scenarios
• Analysis of the effects of system parameters

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CVaR Minimization

• The Effect of Average Demand

(µ2 10, s1 1.5, s 21.5, ? 0.2, cu5, c02 )
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CVaR Minimization

• The Effect of Average Demand

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CVaR Minimization

• The Effect of the Standard Deviation of Demand

(µ2 10, µ2 10, s 21.5, ? 0.2, cu5, c02 )
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CVaR Minimization

• The Effect of the Standard Deviation of Demand

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CVaR Minimization

• The Effect of the Correlation between Products

(µ1 10, µ2 10, s 11.5, s21.5, cu5, c02 )
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CVaR Minimization

• The Effect of the Threshold Probability

(µ2 10, µ2 10, s 11.5, s21.5, cu5, c02 )
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CVaR Minimization

• The Effect of Underage Cost

(µ1 10, µ2 10, s1 1.5, s 21.5, ? 0.2, c02 )
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CVaR Minimization

• The Effect of Underage Cost

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CVaR Minimization

• The Effect of Overage Cost

(µ1 10, µ2 10, s1 1.5, s 21.5, ? 0.2, cu5 )
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CVaR Minimization

• The Effect of Overage Cost

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Product Maximization with CVaR Constraint
(µ1 5, µ2 10, s1 3, s 21.5, ? 0, cu5, c03 )
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Product Maximization with CVaR Constraint
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Conclusion

• Multi-product single period newsvendor problem
  with CVaR constraint
• Two models presented
• Minimization of CVaR
• Profit maximization with CVaR constraint
• Effect of system parameters on the product
  portfolio selection are analyzed

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• QUESTIONS
• COMMENTS

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Solution Statistics