Algorithms for Product Pricing and Energy Allocation in Energy - PowerPoint PPT Presentation

Algorithms for Product Pricing and Energy Allocation in Energy Harvesting Sensor Networks MSc(Engg.) Thesis Defense Sindhu P R Advisor: Prof. Shalabh Bhatnagar Department of Computer Science and Automation Indian Institute of Science, Bangalore

Stochastic System ✘ System ❊ ❬ f ✭ ✒❀ ✘ ✮❪ ✒ The output f ✭ ✒❀ ✘ ✮ depends on the control (or input) ✒ and a noise element ✘ Objective: Find input ✒ , which minimizes (or maximizes) ❊ ❬ f ✭ ✒❀ ✘ ✮❪

Overview 1 Product Pricing Problem Formulation Algorithm Results

Overview 1 Product Pricing Problem Formulation Algorithm Results 2 Energy Sharing in Sensor Networks Problem Formulation Algorithms Approximation Algorithms Results

Product Demand and Price New product introduced into market faces demand uncertainty Size of target population not known Response to promotional campaigns cannot be ascertained Product price, quality and technolgy influence demand Dynamically varying price gives better profit compared to fixed pricing

Product Pricing Problem Product sold over the time interval ❬ 0 ❀ T ❪ X ✭ t ✮ : Cumulative sales (or demand) upto time t u ✭ X ✭ t ✮✮ : Manufacturing cost per unit of the product P r ✭ t ✮ : Price of product at time t Objective Find P r ✭ t ✮ which maximizes expected cumulative profit over ❬ 0 ❀ T ❪ ✔❩ T ✕ J ✄ ❂ max ☞ P r ✭ t ✮ ❊ ✭ P r ✭ t ✮ � u ✭ X ✭ t ✮✮✮ X ✭ t ✮ dt ☞ X ✭ 0 ✮ ❂ X 0 ☞ 0

Bass Diffusion Model Used to quantify how demand grows with time Driving forces for product diffusion - mass-media communication and word-of-mouth communication Change in cumulative sales X ✭ t ✮ of a product is stated as dX ✭ t ✮ ❂ p ✭ M � X ✭ t ✮✮ ✰ qX ✭ t ✮ ✭ M � X ✭ t ✮✮ dt M - M : Market potential - p , q : Coefficients of innovation and imitation

Growth of Cumulative Demand Change in X ✭ t ✮ is modelled as a Stochastic Differential Equation (SDE)- ✒ p ✰ qX ✭ t ✮ ✓ dX ✭ t ✮ ❂ ✭ M � X ✭ t ✮✮ ✭ 1 � ✌ P r ✭ t ✮✮ dt ✰ ✛ ✭ X ✭ t ✮✮ dW ✭ t ✮ M - ✌ determines the extent of influence of P r ✭ t ✮ on the product diffusion - ✛ ✭ ✿ ✮ - diffusion term, ❢ W ✭ t ✮ ❀ t ✕ 0 ❣ - standard Brownian motion.

Growth of Cumulative Demand Change in X ✭ t ✮ is modelled as a Stochastic Differential Equation (SDE)- ✒ p ✰ qX ✭ t ✮ ✓ dX ✭ t ✮ ❂ ✭ M � X ✭ t ✮✮ ✭ 1 � ✌ P r ✭ t ✮✮ dt ✰ ✛ ✭ X ✭ t ✮✮ dW ✭ t ✮ M - ✌ determines the extent of influence of P r ✭ t ✮ on the product diffusion - ✛ ✭ ✿ ✮ - diffusion term, ❢ W ✭ t ✮ ❀ t ✕ 0 ❣ - standard Brownian motion. Approach Discretize SDE and solve in discrete time setting Formulate the problem in the setting of simulation optimization Tune the (discrete) price trajectory to obtain optimum performance

✑ ✭ ✮ ✭ ✮ ✑ ✭ ✮ � ✁ � ✁ ❀ ❂ ✭ � ✮✭ ✰ ✮ � ✌ ✭ ✮ ✭ ✮ � ✁ � ✁ ❀ ❂ ✭ ✮ � ✭ ✮ ✭ ✮ � ❂ ✛ ✭ ✮ ✑ ✛ ✭ ✮ ✛ ✓ ✭ ✁ ✮ ✛ ✭ ✁ ✮ SDE Discretization T ❂ Nh for some N ❃ 1, 0 ❁ h ❁ 1

✭ ✮ ✑ ✭ ✮ � ✁ � ✁ ❀ ❂ ✭ � ✮✭ ✰ ✮ � ✌ ✭ ✮ ✭ ✮ � ✁ � ✁ ❀ ❂ ✭ ✮ � ✭ ✮ ✭ ✮ � ❂ ✛ ✭ ✮ ✑ ✛ ✭ ✮ ✛ ✓ ✭ ✁ ✮ ✛ ✭ ✁ ✮ SDE Discretization T ❂ Nh for some N ❃ 1, 0 ❁ h ❁ 1 Cumulative demand at decision time j is X j and X j ✑ X ✭ jh ✮

� ✁ � ✁ ❀ ❂ ✭ � ✮✭ ✰ ✮ � ✌ ✭ ✮ ✭ ✮ � ✁ � ✁ ❀ ❂ ✭ ✮ � ✭ ✮ ✭ ✮ � ❂ ✛ ✭ ✮ ✑ ✛ ✭ ✮ ✛ ✓ ✭ ✁ ✮ ✛ ✭ ✁ ✮ SDE Discretization T ❂ Nh for some N ❃ 1, 0 ❁ h ❁ 1 Cumulative demand at decision time j is X j and X j ✑ X ✭ jh ✮ Price set at j is P r ✭ j ✮ ✑ P r ✭ jh ✮

� ✁ � ✁ ❀ ❂ ✭ ✮ � ✭ ✮ ✭ ✮ � ❂ ✛ ✭ ✮ ✑ ✛ ✭ ✮ ✛ ✓ ✭ ✁ ✮ ✛ ✭ ✁ ✮ SDE Discretization T ❂ Nh for some N ❃ 1, 0 ❁ h ❁ 1 Cumulative demand at decision time j is X j and X j ✑ X ✭ jh ✮ Price set at j is P r ✭ j ✮ ✑ P r ✭ jh ✮ ❂ ✭ M � X j ✮✭ p ✰ q � ✁ � ✁ Drift term, b X j ❀ P r ✭ j ✮ M X j ✮ 1 � ✌ P r ✭ j ✮

� ❂ ✛ ✭ ✮ ✑ ✛ ✭ ✮ ✛ ✓ ✭ ✁ ✮ ✛ ✭ ✁ ✮ SDE Discretization T ❂ Nh for some N ❃ 1, 0 ❁ h ❁ 1 Cumulative demand at decision time j is X j and X j ✑ X ✭ jh ✮ Price set at j is P r ✭ j ✮ ✑ P r ✭ jh ✮ ❂ ✭ M � X j ✮✭ p ✰ q � ✁ � ✁ Drift term, b X j ❀ P r ✭ j ✮ M X j ✮ 1 � ✌ P r ✭ j ✮ � ✁ � ✁ g X j ❀ P r ✭ j ✮ ❂ u ✭ X j ✮ � P r ✭ j ✮

✛ ✭ ✮ ✑ ✛ ✭ ✮ ✛ ✓ ✭ ✁ ✮ ✛ ✭ ✁ ✮ SDE Discretization T ❂ Nh for some N ❃ 1, 0 ❁ h ❁ 1 Cumulative demand at decision time j is X j and X j ✑ X ✭ jh ✮ Price set at j is P r ✭ j ✮ ✑ P r ✭ jh ✮ ❂ ✭ M � X j ✮✭ p ✰ q � ✁ � ✁ Drift term, b X j ❀ P r ✭ j ✮ M X j ✮ 1 � ✌ P r ✭ j ✮ � ✁ � ✁ g X j ❀ P r ✭ j ✮ ❂ u ✭ X j ✮ � P r ✭ j ✮ X � 1 ❂ 0 and X 0 is assumed to be known

✛ ✓ ✭ ✁ ✮ ✛ ✭ ✁ ✮ SDE Discretization T ❂ Nh for some N ❃ 1, 0 ❁ h ❁ 1 Cumulative demand at decision time j is X j and X j ✑ X ✭ jh ✮ Price set at j is P r ✭ j ✮ ✑ P r ✭ jh ✮ ❂ ✭ M � X j ✮✭ p ✰ q � ✁ � ✁ Drift term, b X j ❀ P r ✭ j ✮ M X j ✮ 1 � ✌ P r ✭ j ✮ � ✁ � ✁ g X j ❀ P r ✭ j ✮ ❂ u ✭ X j ✮ � P r ✭ j ✮ X � 1 ❂ 0 and X 0 is assumed to be known ✛ ✭ X j ✮ ✑ ✛ ✭ jh ✮

SDE Discretization T ❂ Nh for some N ❃ 1, 0 ❁ h ❁ 1 Cumulative demand at decision time j is X j and X j ✑ X ✭ jh ✮ Price set at j is P r ✭ j ✮ ✑ P r ✭ jh ✮ ❂ ✭ M � X j ✮✭ p ✰ q � ✁ � ✁ Drift term, b X j ❀ P r ✭ j ✮ M X j ✮ 1 � ✌ P r ✭ j ✮ � ✁ � ✁ g X j ❀ P r ✭ j ✮ ❂ u ✭ X j ✮ � P r ✭ j ✮ X � 1 ❂ 0 and X 0 is assumed to be known ✛ ✭ X j ✮ ✑ ✛ ✭ jh ✮ ✛ ✓ ✭ ✁ ✮ is the derivative of ✛ ✭ ✁ ✮

✷ ✸ � ☞ ❳ � ✁ ✭ ✮ ❀ ✭ ✮ ❀ ✁ ✁ ✁ ❀ ❂ ❊ ✭ ❀ ✭ ✮ ✮✭ � � ✮ ☞ ✭ � ✮ ✹ ✺ ☞ ❂ SDE Discretization (contd.) Discretized SDE ♣ hZ j ✰ 1 ✰ 1 ✭ X j ✮ ✛ ✭ X j ✮ h ✭ Z 2 X j ✰ 1 ❂ X j ✰ b ✭ X j ❀ P r ✭ j ✮ ✮ h ✰ ✛ ✭ X j ✮ 2 ✛ ✓ j ✰ 1 � 1 ✮ - For 0 ✔ j ✔ N � 1, Z j ✰ 1 are independent N ✭ 0 ❀ 1 ✮ distributed samples

SDE Discretization (contd.) Discretized SDE ♣ hZ j ✰ 1 ✰ 1 ✭ X j ✮ ✛ ✭ X j ✮ h ✭ Z 2 X j ✰ 1 ❂ X j ✰ b ✭ X j ❀ P r ✭ j ✮ ✮ h ✰ ✛ ✭ X j ✮ 2 ✛ ✓ j ✰ 1 � 1 ✮ - For 0 ✔ j ✔ N � 1, Z j ✰ 1 are independent N ✭ 0 ❀ 1 ✮ distributed samples Discretized Objective Function ✷ ✸ N � 1 ☞ ❳ � ✁ J P r ✭ 0 ✮ ❀ P r ✭ 1 ✮ ❀ ✁ ✁ ✁ ❀ P r ✭ N � 1 ✮ ❂ ❊ g ✭ X j ❀ P r ✭ j ✮ ✮✭ X j � X j � 1 ✮ ☞ X 0 ☞ ✹ ✺ j ❂ 0

Optimal Price Vector r ✭ N � 1 ✮ ✮ ❃ where, Objective: Find P ✄ r ❂ ✭ P ✄ r ✭ 0 ✮ ❀ P ✄ r ✭ 1 ✮ ❀ ✁ ✁ ✁ ❀ P ✄ ❢ P ✄ r ✭ 0 ✮ ❀ P ✄ r ✭ 1 ✮ ❀ ✁ ✁ ✁ ❀ P ✄ r ✭ N � 1 ✮ ❣ ❂ arg min J X 0 ✭ P r ✭ 0 ✮ ❀ P r ✭ 1 ✮ ❀ ✁ ✁ ✁ ❀ P r ✭ N � 1 ✮ ✮ ❢ P r ✭ 0 ✮ ❀ ✁✁✁ ❀ P r ✭ N � 1 ✮ ❣ Price set at stage k influences the demand in that stage as well as in future stages The initial price influences the single-stage costs of all stages

Gradient Search Using Simulations f ✭ ✒ ✮ Need to minimize f ✭ ✒ ✮ Find r ✒ f Update ✒ in the negative direction of gradient f ✭ ✒ 1 ✮ In pricing problem, f ✭ ✒ ✮ ❂ J ✭ P r ✭ 0 ✮ ❀ P r ✭ 1 ✮ ❀ ✁ ✁ ✁ ❀ P r ✭ N � 1 ✮ ✮ ✒ � ☞ r ✒ f Figure: Gradient Search

❤ ✑ ✭ ✐ ✭ ✮ ❂ ☞ ❊ ✭ ✰ ☞ ✑ ✮ � ✭ � ☞ ✑ ✮✮ ❥ ☞ Smoothed Functional Technique Smooth r J ✭ P r ✭ 0 ✮ ❀ P r ✭ 1 ✮ ❀ ✁ ✁ ✁ ❀ P r ✭ N � 1 ✮ ✮ by convolution with N � dimensional Gaussian density function, G ✭ ✑ ✮ , ✑ ✷ ❘ N ☞ ❃ 0 is a small constant x f ✭ x ✮ f ✭ x ✮ ✍ G ✭ ✑ ✮

Smoothed Functional Technique Smooth r J ✭ P r ✭ 0 ✮ ❀ P r ✭ 1 ✮ ❀ ✁ ✁ ✁ ❀ P r ✭ N � 1 ✮ ✮ by convolution with N � dimensional Gaussian density function, G ✭ ✑ ✮ , ✑ ✷ ❘ N ☞ ❃ 0 is a small constant x f ✭ x ✮ f ✭ x ✮ ✍ G ✭ ✑ ✮ The gradient estimate is D ☞ J X 0 ✭ P r ✮ ❂ 1 ❤ ✑ ✐ 2 ✭ J X 0 ✭ P r ✰ ☞ ✑ ✮ � J X 0 ✭ P r � ☞ ✑ ✮✮ ❥ P r ☞ ❊

Gradient Estimate In our simulation-based algorithm, expectation is replaced by samples obtained from simulation. For large L , small ☞ , ✧ L ★ r J X 0 ✭ P r ✮ ❂ 1 1 ✑ ✭ n ✮ ❳ � ✁ J X 0 ✭ P r ✰ ☞ ✑ ✭ n ✮✮ � J X 0 ✭ P r � ☞ ✑ ✭ n ✮✮ L 2 ☞ n ❂ 1 ❦ D ☞ J X 0 ✭ P r ✮ � r J X 0 ✭ P r ✮ ❦✦ 0 as ☞ ✦ 0 and L ✦ ✶