This document provides an overview of dynamic pricing and revenue management for a single resource with independent demands. It introduces static and dynamic models for allocating capacity among different fare classes to maximize expected revenues. Static models assume demands arrive in a fixed low-to-high order, while dynamic models explicitly consider time and allow more general demand patterns. Key concepts discussed include Littlewood's rule for optimal allocation in the two-class case, spill rates, callable products, and dynamic programming formulations. Both static and dynamic approaches are compared, finding dynamic policies better handle non-ordered demand arrivals.
Dynamic pricing: Lessons from Airline Revenue ManagementTom Bacon
Dynamic pricing is practiced across industries to personalize prices based on customer data analytics. Pioneered by airlines in the 1980s as revenue management, it involves segmenting customers, adjusting prices based on demand forecasts, and unbundling/rebundling services. Tom Bacon has applied these techniques at several airlines, leading to increased ancillary revenue and competitive fare positioning through integrated pricing, merchandising, and operations.
Elevate 2017- Pricing evolution: The Road to Dynamic Offer GenerationATPCO
Tom Gregorson is the Vice President of Products & Solutions at ATPCO. The document outlines ATPCO's research into next-generation pricing mechanisms in the airline industry. It discusses six potential new mechanisms including more frequent fare updates, dynamic availability of fare products, advanced RBD capabilities, dynamic pricing engines, continuous pricing, and dynamic offer generation. The implications of these new approaches are also assessed, such as their impact on airline revenues, competition, customers, and internal processes. While next-gen pricing could provide revenue gains, it also poses significant risks and challenges to existing airline systems and practices.
- Yield optimization strategies aim to maximize rental revenues through balancing occupancy and rental rates using revenue management techniques.
- Statistical concepts like elasticity of demand are used to understand how price changes impact demand and total revenue.
- AMLI Residential implemented revenue management software and key performance indicators to centrally manage pricing, monitor performance, and optimize yields across its portfolio.
The document discusses methods for determining how costs behave and estimating cost functions. It explains that total costs can often be explained by changes in a single activity level and that cost behavior is commonly approximated by linear functions within a relevant range. It then outlines different types of linear cost functions and approaches to estimating cost functions, including the industrial engineering method, conference method, account analysis, and quantitative regression analysis. Key steps in estimation include selecting a dependent cost variable and driver, collecting data, plotting relationships, and evaluating estimated functions.
Airline pricing strategies and revenue managementAMALDASKH
Revenue management is a technique airlines use to optimize revenue from a fixed resource (seats on flights) by selling to the right customers at the right price. It implements supply and demand principles tactically. Key capabilities include forecasting demand and customer price sensitivity to set optimal prices, allocating inventory efficiently, and responding quickly to changes. Cargo revenue management similarly aims to enhance profits across a network. Airlines employ various pricing strategies considering costs, sales goals, competition, and customer value over time. Popular strategies include premium, penetration, economy, price skimming, competition-based, and cost-plus pricing. Demand is inversely related to price - as price falls, quantity demanded rises, and vice versa. Non-price factors
Travel Management Company (TMC) Transformation Solutions | WNS TRAVOGUERNayak3
Explore WNS Travogue's solutions for corporate travel management companies across travel risk management, revenue management, shared services, and recovery to drive efficiency across the value chain.
The document discusses the history and concepts of algorithmic marketing. It begins by explaining how the rise of digital marketing created an environment requiring millions of micro-decisions that cannot be made efficiently without algorithms. It then discusses how algorithmic marketing automates marketing processes in areas like product, promotion, price, and placement. The document also provides historical examples that helped establish algorithmic marketing, such as how online advertising exchanges developed and how airlines pioneered revenue management systems. Finally, it proposes a framework for programmatic marketing services involving data collection, feature transformation, objective-driven modeling and decision-making, and feedback loops.
Conjoint analysis is a technique used to understand how consumers develop preferences for products and services. It involves presenting respondents with profiles consisting of different attribute levels and measuring their preferences. The preference data is then analyzed, usually via regression, to estimate the part-worth or utility that respondents associate with each attribute level. These utilities can then be used to simulate market shares of potential new products and identify key drivers of preferences. Conjoint analysis sees wide commercial use across various industries to guide new product design and selection.
Dynamic pricing: Lessons from Airline Revenue ManagementTom Bacon
Dynamic pricing is practiced across industries to personalize prices based on customer data analytics. Pioneered by airlines in the 1980s as revenue management, it involves segmenting customers, adjusting prices based on demand forecasts, and unbundling/rebundling services. Tom Bacon has applied these techniques at several airlines, leading to increased ancillary revenue and competitive fare positioning through integrated pricing, merchandising, and operations.
Elevate 2017- Pricing evolution: The Road to Dynamic Offer GenerationATPCO
Tom Gregorson is the Vice President of Products & Solutions at ATPCO. The document outlines ATPCO's research into next-generation pricing mechanisms in the airline industry. It discusses six potential new mechanisms including more frequent fare updates, dynamic availability of fare products, advanced RBD capabilities, dynamic pricing engines, continuous pricing, and dynamic offer generation. The implications of these new approaches are also assessed, such as their impact on airline revenues, competition, customers, and internal processes. While next-gen pricing could provide revenue gains, it also poses significant risks and challenges to existing airline systems and practices.
- Yield optimization strategies aim to maximize rental revenues through balancing occupancy and rental rates using revenue management techniques.
- Statistical concepts like elasticity of demand are used to understand how price changes impact demand and total revenue.
- AMLI Residential implemented revenue management software and key performance indicators to centrally manage pricing, monitor performance, and optimize yields across its portfolio.
The document discusses methods for determining how costs behave and estimating cost functions. It explains that total costs can often be explained by changes in a single activity level and that cost behavior is commonly approximated by linear functions within a relevant range. It then outlines different types of linear cost functions and approaches to estimating cost functions, including the industrial engineering method, conference method, account analysis, and quantitative regression analysis. Key steps in estimation include selecting a dependent cost variable and driver, collecting data, plotting relationships, and evaluating estimated functions.
Airline pricing strategies and revenue managementAMALDASKH
Revenue management is a technique airlines use to optimize revenue from a fixed resource (seats on flights) by selling to the right customers at the right price. It implements supply and demand principles tactically. Key capabilities include forecasting demand and customer price sensitivity to set optimal prices, allocating inventory efficiently, and responding quickly to changes. Cargo revenue management similarly aims to enhance profits across a network. Airlines employ various pricing strategies considering costs, sales goals, competition, and customer value over time. Popular strategies include premium, penetration, economy, price skimming, competition-based, and cost-plus pricing. Demand is inversely related to price - as price falls, quantity demanded rises, and vice versa. Non-price factors
Travel Management Company (TMC) Transformation Solutions | WNS TRAVOGUERNayak3
Explore WNS Travogue's solutions for corporate travel management companies across travel risk management, revenue management, shared services, and recovery to drive efficiency across the value chain.
The document discusses the history and concepts of algorithmic marketing. It begins by explaining how the rise of digital marketing created an environment requiring millions of micro-decisions that cannot be made efficiently without algorithms. It then discusses how algorithmic marketing automates marketing processes in areas like product, promotion, price, and placement. The document also provides historical examples that helped establish algorithmic marketing, such as how online advertising exchanges developed and how airlines pioneered revenue management systems. Finally, it proposes a framework for programmatic marketing services involving data collection, feature transformation, objective-driven modeling and decision-making, and feedback loops.
Conjoint analysis is a technique used to understand how consumers develop preferences for products and services. It involves presenting respondents with profiles consisting of different attribute levels and measuring their preferences. The preference data is then analyzed, usually via regression, to estimate the part-worth or utility that respondents associate with each attribute level. These utilities can then be used to simulate market shares of potential new products and identify key drivers of preferences. Conjoint analysis sees wide commercial use across various industries to guide new product design and selection.
This document provides an overview of revenue management for hotels. It begins with a brief history of revenue management and how it has evolved over time in response to various economic and industry trends. It then discusses common revenue management tactics like forecasting demand, setting room rates and minimum stay requirements, overbooking, and managing bookings across different distribution channels. Finally, it covers developing a revenue management strategy through differentiation, customer loyalty programs, and strategic packaging. The document aims to explain both the science and art of revenue management to improve hotel profits.
This document discusses revenue management in the airline industry. It explains that revenue management aims to sell airline seats to the right customers at the right time for the right price in order to maximize revenue. This is achieved through optimizing flight scheduling, pricing, and inventory control, including developing optimal overbooking policies. The document also discusses how yield management uses booking limits and protection of seats to optimize revenue from different fare classes and passenger types.
This document defines marketing channels and discusses their needs, functions, levels and factors affecting selection. It describes types of middlemen like merchant, agents and facilitators. It discusses evaluating channel alternatives based on economic, control and adaptive criteria. It also covers logistics and supply chain management concepts like transportation decisions, warehousing, inventory management and order processing.
cross ducking,milk run ,pricing and revenue managment in LSCMsravan reddy
Milk run is a distribution strategy where a single truck delivers products from one supplier to multiple retailers, reducing transportation costs. Cross docking consolidates inbound materials and sorts them for quick outbound shipment to fulfill replenishment orders, as seen in Walmart's system. Revenue management uses differential pricing based on customer segment, time, and availability to increase profits from limited supply chain assets, such as charging more for airline seats during peak seasons.
VideoEgg is an online video advertising company founded in 2004 by three Yale graduates. It delivers ads to social media, video, and gaming sites. VideoEgg created AdFrames that allow users to roll over ads to watch sponsored content. Unlike traditional CPM or CPC models, VideoEgg charges advertisers $0.75 per user roll over, splitting the fee with content sites. This innovative pricing scheme differs from standard online advertising models.
The winning team was Karan Sarao from India. They used a two-stage modeling process with five initial models that were then scored on a validation set and those scores incorporated into a final blended model. Key aspects of their solution included feature engineering by creating new derived variables and extensive model tuning using cross-validation. Their top-performing model was an extreme gradient boosted model.
Managerial Economics | Overview and SummaryMBA ASAP
Managerial economics deals with the application of the economic concepts, theories, tools and methodologies to solve practical problems in a business. It helps the manager in decision making and acts as a link between practice and theory.
This document provides an overview of techniques for airline revenue management that have been developed between 1982 and 2001. It summarizes the key concepts and approaches for the seat inventory control problem, which involves allocating a flight's finite seat capacity to maximize revenue over time. The document outlines static and dynamic, single-leg and network solution methods. Static single-leg approaches determine optimal booking limits for fare classes based on demand forecasts, while dynamic methods use booking data over time. Network solutions optimize across connected flight segments simultaneously.
IRJET- Credit Profile of E-Commerce CustomerIRJET Journal
This document proposes using RFM (Recency, Frequency, Monetary) variables and advanced k-means clustering to create positive and negative credit profiles for e-commerce customers. This will help minimize losses by identifying genuine versus fraudulent customers. The methodology calculates credit scores based on RFM and other factors. Advanced k-means clustering is then used to segment customers into clusters like excellent, good, average, and worst. Customers in different clusters will receive different benefits or restrictions based on their predicted reliability. The goal is to reduce losses from unwanted cancellations while retaining high value customers.
A Multi-Attribute Auction Mechanism based on Conditional Constraints and Cond...Shubhashis Shil
This document summarizes a research paper that proposes a multi-attribute reverse auction mechanism. The mechanism allows buyers to specify conditional constraints and conditional qualitative preferences over multiple attributes of auctioned items. It maps qualitative preferences to a multi-criteria decision model to determine winners in a computationally efficient way. The authors designed a multi-round auction protocol where buyer preferences are fully revealed. They conducted experiments on a 10-attribute auction to demonstrate the feasibility and performance of the proposed auction mechanism. User acceptance of the requirements specification and winner selection tools was also assessed.
The supply chain is defined as the network of organizations involved in processes and activities that produce value for the ultimate customer. Supply chain management (SCM) involves flows of materials, money, and information through procurement, manufacturing, distribution, and customer processes. Effective SCM can provide benefits like reduced costs, improved service levels, business growth opportunities, and preferred supplier status. Key elements of SCM include inventory management, warehousing, and transportation.
Short university lecture about how mobile Telco operators can improve their profitability leveraging a strategic and value based approach to Channel Management
The document discusses several factors to consider when determining the optimal logistics network for a company operating in Detroit. It examines the number, location, size, and transportation modes for various facilities including supplier plants, warehouses, distribution centers, and retail outlets. Key considerations include costs, customer service levels, demand patterns, product attributes, infrastructure, and regulatory environment. An effective logistics network requires analyzing multiple interrelated factors to maximize efficiency and minimize total costs.
This document discusses revenue management theory and practice. It summarizes that revenue management has established principles including demand forecasting, optimization, and overbooking models. However, airlines face many constraints applying these principles including strategic differences across airline types, legal/regulatory issues, technical system limitations, organizational challenges, and competitive pressures. Therefore, revenue management analysts must understand both the underlying science and its real-world implementation given the complex constraints faced by individual airlines.
Constraint and Qualitative Preference Specification in Multi-Attribute Revers...Shubhashis Shil
This document describes a multi-attribute reverse auction (MARA) system that allows buyers to specify both constraints and qualitative preferences for product attributes. The key aspects are:
1) The MARA system allows buyers to specify constraints and qualitative preferences for product attributes in an interactive way using simple syntax.
2) The system transforms the qualitative preferences into quantitative weights and utility values to evaluate bids using multi-attribute utility theory (MAUT).
3) The MARA system automatically calculates the weights and utility functions, eliminating the difficult task for buyers of determining these values themselves.
The document outlines a presentation on revenue management. It begins with defining revenue management as selling inventory, like hotel rooms, at the right price to the right customer. It then discusses how revenue management aims to maximize profit by forecasting demand and adjusting pricing and inventory levels. The presentation also covers how revenue management works in different industries and what factors like capacity, discounts, and length of stay control are important to the revenue management process.
The document outlines a presentation on revenue management. It begins with defining revenue management as selling inventory, like hotel rooms, at the right price to the right customer. It then discusses how revenue management aims to maximize profit by forecasting demand and adjusting pricing and inventory levels. The presentation also covers how revenue management works in different industries and what factors like capacity, discounts, and length of stay control are important to the revenue management process.
150 Business models and graphics for your business presentations.
Content:
Powerpoint, presentations, business, slides, diagrams, charts, Break-even, Financing Life Cycle, Economies of Scale, Elasticity, Sales Cycles Market Potential, Portfolio Matrix, Product Model, Four P's, Push/Pull Strategy, Marketing Mix, PDCA Cycle, SWOT, Value Chain, Ansoff Matrix, BCG Matrix, 7-S Modell, Core Competencies, GE Business Screen, Nine Cell Industry Risk/Reward Diagram, Porter's Five Forces, Industry Competition, Generic Strategies, Geobusiness Modell, Porter's Diamond, Matrix Design, PIMS, Leavitt's Diamond, Belbin's Team Roles, Theory X/Y, Maslow's Hierarchy, Herberg's Theory, Cultural Web, Pareto Curve, CIM Concept, Value Drivers
Download these diagrams on
http://www.drawpack.com
(try our free membership offer)
The document discusses different market structures including perfect competition, monopoly, monopolistic competition, and oligopoly. It provides details on the key features and assumptions of perfect competition, including that firms are price takers, there is freedom of entry and exit, products are homogeneous, and profits are only normal in the long run. It also discusses monopoly, including how monopolies obtain and maintain market power through barriers to entry, and how a monopolist determines its profit-maximizing price and output where marginal cost equals marginal revenue.
RepoRemarketing provides a managed liquidation solution for Credit Unions. Get more $ for you repossessed inventory. Leverage technology for transparency, tracking, and benchmark your results.
OverviewThe US is currently undergoing an energy boom largel.docxjacksnathalie
Overview
The US is currently undergoing an energy boom largely because of the development of the greatly expanded use of a well technique developed over 40 years ago - hydraulic fracking. It can be used for both oil and natural gas wells.. The technique allows previously unrecoverable oil and gas in old, played out wells to be accessed and increases the efficiency of recovery in new wells significantly. The current level of both recovery and new well drilling is dramatically higher than it has been for decades. The dramatic increase in well activity, some of which has been near towns and places no one thought drilling would ever occur. It has brought a great deal of attention to the technique and associated effects on everything from ground water and air pollution, to biodiversity disruption and earthquakes.
One important fact to weave into your opinion about fracking pro or con is that all of the sub-surface mineral rights in the US are owned by someone (a private individual, a business, or the state or federal government) but surface and mineral rights can be separated, i.e. sold. Originally, mineral rights were sold along with the land and then companies or individuals could decide if they wanted to keep or sell the mineral rights. Before mineral rights were so valuable, many people opted to sell their mineral rights to oil & gas companies. It never occurred to many people that someone would actually be drilling on their property or their neighbors. Oil and gas companies have a legal right to exercise their ownership options and if you are going to say "no" to them, then you owe them for what you are not letting them have, i.e. the money that would be produced if they were allowed to drill. This is not a trivial issue.
Instructions
This week’s discussion focuses on the pros and cons of hydraulic fracking and asks for your SCIENCE informed opinion on whether the economics and political fossil fuel issues justify the negative tradeoffs.
Address each of the following in your discussion:
How is fracking done and why are companies doing this action versus traditional drilling?
Are the environmental issues with fracking worse than conventional drilling? Why or why not?
Why are people along the Front Range and in other states where fracking is widespread, so upset about it now even though fracking has been occurring for a long time?
*In your initial post, please provide 3-4 references in APA format with in-text citations.
.
OverviewThe United Nations (UN) has hired you as a consultan.docxjacksnathalie
Overview
The United Nations (UN) has hired you as a consultant, and your task is to assess the impact that global warming is expected to have on population growth and the ability of societies in the developing world to ensure the adequate security of their food supplies.
Case Assessment
As the world’s population nears 10 billion by 2050, the effects of global warming are stripping some natural resources from the environment. As they diminish in number, developing countries will face mounting obstacles to improving the livelihoods of their citizens and stabilizing their access to enough food. The reason these governments are struggling even now is that our climate influences their economic health and the consequent diminishing living standards of their peoples. Climate changes are responsible for the current loss of biodiversity as well as the physical access to some critical farming regions. As such, these changes in global weather patterns diminish agricultural output and the distribution of food to local and international markets. These difficulties will become even more significant for these countries as the Earth’s climate changes for the worse. Temperatures are already increasing incrementally, and polar ice caps are melting, so the salient question is: what does this suggest for developing societies?
The issue before the developing world is not its lack of food, but rather how to gain access to food. Simply put, changes in our climate are affecting the global food chain, and hence, the living standards of entire populations. Added to this is the fact that food is not getting to where it is needed in time to prevent hunger or starvation. In many developing countries, shortages are due to governments’ control over distribution networks rather than an insufficient supply of food itself. In effect, these governments are weaponizing food by favoring certain ethnic or religious groups over others. When added to dramatic climate changes that we are experiencing even now, the future for billions of poor people looks increasingly dim.
Instructions
You are to write a minimum of a 5 page persuasive paper for the UN that addresses the following questions about the relationship between atmospheric weather patterns and food security in the developing world:
Climate change and global warming are often used interchangeably, but they are not the same phenomenon. What are the differences between the two concepts and what leads to the confusion between them?
In 1900, the average global temperature was about 13.7° Celsius (56.7° Fahrenheit) (Osborn, 2021), but as of 2020, the temperature has risen another 1.2°C to 14.9°C (58.9°F). According to the Earth and climate science community, if the Earth’s surface temperature rises another 2°C (3.6°F), we will suffer catastrophic weather patterns that, among other things, will raise sea levels, cause widespread droughts and wildfires, result in plant, insect, and animal extinctions, and reduce agricultura.
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This document discusses revenue management in the airline industry. It explains that revenue management aims to sell airline seats to the right customers at the right time for the right price in order to maximize revenue. This is achieved through optimizing flight scheduling, pricing, and inventory control, including developing optimal overbooking policies. The document also discusses how yield management uses booking limits and protection of seats to optimize revenue from different fare classes and passenger types.
This document defines marketing channels and discusses their needs, functions, levels and factors affecting selection. It describes types of middlemen like merchant, agents and facilitators. It discusses evaluating channel alternatives based on economic, control and adaptive criteria. It also covers logistics and supply chain management concepts like transportation decisions, warehousing, inventory management and order processing.
cross ducking,milk run ,pricing and revenue managment in LSCMsravan reddy
Milk run is a distribution strategy where a single truck delivers products from one supplier to multiple retailers, reducing transportation costs. Cross docking consolidates inbound materials and sorts them for quick outbound shipment to fulfill replenishment orders, as seen in Walmart's system. Revenue management uses differential pricing based on customer segment, time, and availability to increase profits from limited supply chain assets, such as charging more for airline seats during peak seasons.
VideoEgg is an online video advertising company founded in 2004 by three Yale graduates. It delivers ads to social media, video, and gaming sites. VideoEgg created AdFrames that allow users to roll over ads to watch sponsored content. Unlike traditional CPM or CPC models, VideoEgg charges advertisers $0.75 per user roll over, splitting the fee with content sites. This innovative pricing scheme differs from standard online advertising models.
The winning team was Karan Sarao from India. They used a two-stage modeling process with five initial models that were then scored on a validation set and those scores incorporated into a final blended model. Key aspects of their solution included feature engineering by creating new derived variables and extensive model tuning using cross-validation. Their top-performing model was an extreme gradient boosted model.
Managerial Economics | Overview and SummaryMBA ASAP
Managerial economics deals with the application of the economic concepts, theories, tools and methodologies to solve practical problems in a business. It helps the manager in decision making and acts as a link between practice and theory.
This document provides an overview of techniques for airline revenue management that have been developed between 1982 and 2001. It summarizes the key concepts and approaches for the seat inventory control problem, which involves allocating a flight's finite seat capacity to maximize revenue over time. The document outlines static and dynamic, single-leg and network solution methods. Static single-leg approaches determine optimal booking limits for fare classes based on demand forecasts, while dynamic methods use booking data over time. Network solutions optimize across connected flight segments simultaneously.
IRJET- Credit Profile of E-Commerce CustomerIRJET Journal
This document proposes using RFM (Recency, Frequency, Monetary) variables and advanced k-means clustering to create positive and negative credit profiles for e-commerce customers. This will help minimize losses by identifying genuine versus fraudulent customers. The methodology calculates credit scores based on RFM and other factors. Advanced k-means clustering is then used to segment customers into clusters like excellent, good, average, and worst. Customers in different clusters will receive different benefits or restrictions based on their predicted reliability. The goal is to reduce losses from unwanted cancellations while retaining high value customers.
A Multi-Attribute Auction Mechanism based on Conditional Constraints and Cond...Shubhashis Shil
This document summarizes a research paper that proposes a multi-attribute reverse auction mechanism. The mechanism allows buyers to specify conditional constraints and conditional qualitative preferences over multiple attributes of auctioned items. It maps qualitative preferences to a multi-criteria decision model to determine winners in a computationally efficient way. The authors designed a multi-round auction protocol where buyer preferences are fully revealed. They conducted experiments on a 10-attribute auction to demonstrate the feasibility and performance of the proposed auction mechanism. User acceptance of the requirements specification and winner selection tools was also assessed.
The supply chain is defined as the network of organizations involved in processes and activities that produce value for the ultimate customer. Supply chain management (SCM) involves flows of materials, money, and information through procurement, manufacturing, distribution, and customer processes. Effective SCM can provide benefits like reduced costs, improved service levels, business growth opportunities, and preferred supplier status. Key elements of SCM include inventory management, warehousing, and transportation.
Short university lecture about how mobile Telco operators can improve their profitability leveraging a strategic and value based approach to Channel Management
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This document discusses revenue management theory and practice. It summarizes that revenue management has established principles including demand forecasting, optimization, and overbooking models. However, airlines face many constraints applying these principles including strategic differences across airline types, legal/regulatory issues, technical system limitations, organizational challenges, and competitive pressures. Therefore, revenue management analysts must understand both the underlying science and its real-world implementation given the complex constraints faced by individual airlines.
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This document describes a multi-attribute reverse auction (MARA) system that allows buyers to specify both constraints and qualitative preferences for product attributes. The key aspects are:
1) The MARA system allows buyers to specify constraints and qualitative preferences for product attributes in an interactive way using simple syntax.
2) The system transforms the qualitative preferences into quantitative weights and utility values to evaluate bids using multi-attribute utility theory (MAUT).
3) The MARA system automatically calculates the weights and utility functions, eliminating the difficult task for buyers of determining these values themselves.
The document outlines a presentation on revenue management. It begins with defining revenue management as selling inventory, like hotel rooms, at the right price to the right customer. It then discusses how revenue management aims to maximize profit by forecasting demand and adjusting pricing and inventory levels. The presentation also covers how revenue management works in different industries and what factors like capacity, discounts, and length of stay control are important to the revenue management process.
The document outlines a presentation on revenue management. It begins with defining revenue management as selling inventory, like hotel rooms, at the right price to the right customer. It then discusses how revenue management aims to maximize profit by forecasting demand and adjusting pricing and inventory levels. The presentation also covers how revenue management works in different industries and what factors like capacity, discounts, and length of stay control are important to the revenue management process.
150 Business models and graphics for your business presentations.
Content:
Powerpoint, presentations, business, slides, diagrams, charts, Break-even, Financing Life Cycle, Economies of Scale, Elasticity, Sales Cycles Market Potential, Portfolio Matrix, Product Model, Four P's, Push/Pull Strategy, Marketing Mix, PDCA Cycle, SWOT, Value Chain, Ansoff Matrix, BCG Matrix, 7-S Modell, Core Competencies, GE Business Screen, Nine Cell Industry Risk/Reward Diagram, Porter's Five Forces, Industry Competition, Generic Strategies, Geobusiness Modell, Porter's Diamond, Matrix Design, PIMS, Leavitt's Diamond, Belbin's Team Roles, Theory X/Y, Maslow's Hierarchy, Herberg's Theory, Cultural Web, Pareto Curve, CIM Concept, Value Drivers
Download these diagrams on
http://www.drawpack.com
(try our free membership offer)
The document discusses different market structures including perfect competition, monopoly, monopolistic competition, and oligopoly. It provides details on the key features and assumptions of perfect competition, including that firms are price takers, there is freedom of entry and exit, products are homogeneous, and profits are only normal in the long run. It also discusses monopoly, including how monopolies obtain and maintain market power through barriers to entry, and how a monopolist determines its profit-maximizing price and output where marginal cost equals marginal revenue.
RepoRemarketing provides a managed liquidation solution for Credit Unions. Get more $ for you repossessed inventory. Leverage technology for transparency, tracking, and benchmark your results.
Similar to Dynamic Pricing andRevenue ManagementIEOR 4601 Spring .docx (20)
OverviewThe US is currently undergoing an energy boom largel.docxjacksnathalie
Overview
The US is currently undergoing an energy boom largely because of the development of the greatly expanded use of a well technique developed over 40 years ago - hydraulic fracking. It can be used for both oil and natural gas wells.. The technique allows previously unrecoverable oil and gas in old, played out wells to be accessed and increases the efficiency of recovery in new wells significantly. The current level of both recovery and new well drilling is dramatically higher than it has been for decades. The dramatic increase in well activity, some of which has been near towns and places no one thought drilling would ever occur. It has brought a great deal of attention to the technique and associated effects on everything from ground water and air pollution, to biodiversity disruption and earthquakes.
One important fact to weave into your opinion about fracking pro or con is that all of the sub-surface mineral rights in the US are owned by someone (a private individual, a business, or the state or federal government) but surface and mineral rights can be separated, i.e. sold. Originally, mineral rights were sold along with the land and then companies or individuals could decide if they wanted to keep or sell the mineral rights. Before mineral rights were so valuable, many people opted to sell their mineral rights to oil & gas companies. It never occurred to many people that someone would actually be drilling on their property or their neighbors. Oil and gas companies have a legal right to exercise their ownership options and if you are going to say "no" to them, then you owe them for what you are not letting them have, i.e. the money that would be produced if they were allowed to drill. This is not a trivial issue.
Instructions
This week’s discussion focuses on the pros and cons of hydraulic fracking and asks for your SCIENCE informed opinion on whether the economics and political fossil fuel issues justify the negative tradeoffs.
Address each of the following in your discussion:
How is fracking done and why are companies doing this action versus traditional drilling?
Are the environmental issues with fracking worse than conventional drilling? Why or why not?
Why are people along the Front Range and in other states where fracking is widespread, so upset about it now even though fracking has been occurring for a long time?
*In your initial post, please provide 3-4 references in APA format with in-text citations.
.
OverviewThe United Nations (UN) has hired you as a consultan.docxjacksnathalie
Overview
The United Nations (UN) has hired you as a consultant, and your task is to assess the impact that global warming is expected to have on population growth and the ability of societies in the developing world to ensure the adequate security of their food supplies.
Case Assessment
As the world’s population nears 10 billion by 2050, the effects of global warming are stripping some natural resources from the environment. As they diminish in number, developing countries will face mounting obstacles to improving the livelihoods of their citizens and stabilizing their access to enough food. The reason these governments are struggling even now is that our climate influences their economic health and the consequent diminishing living standards of their peoples. Climate changes are responsible for the current loss of biodiversity as well as the physical access to some critical farming regions. As such, these changes in global weather patterns diminish agricultural output and the distribution of food to local and international markets. These difficulties will become even more significant for these countries as the Earth’s climate changes for the worse. Temperatures are already increasing incrementally, and polar ice caps are melting, so the salient question is: what does this suggest for developing societies?
The issue before the developing world is not its lack of food, but rather how to gain access to food. Simply put, changes in our climate are affecting the global food chain, and hence, the living standards of entire populations. Added to this is the fact that food is not getting to where it is needed in time to prevent hunger or starvation. In many developing countries, shortages are due to governments’ control over distribution networks rather than an insufficient supply of food itself. In effect, these governments are weaponizing food by favoring certain ethnic or religious groups over others. When added to dramatic climate changes that we are experiencing even now, the future for billions of poor people looks increasingly dim.
Instructions
You are to write a minimum of a 5 page persuasive paper for the UN that addresses the following questions about the relationship between atmospheric weather patterns and food security in the developing world:
Climate change and global warming are often used interchangeably, but they are not the same phenomenon. What are the differences between the two concepts and what leads to the confusion between them?
In 1900, the average global temperature was about 13.7° Celsius (56.7° Fahrenheit) (Osborn, 2021), but as of 2020, the temperature has risen another 1.2°C to 14.9°C (58.9°F). According to the Earth and climate science community, if the Earth’s surface temperature rises another 2°C (3.6°F), we will suffer catastrophic weather patterns that, among other things, will raise sea levels, cause widespread droughts and wildfires, result in plant, insect, and animal extinctions, and reduce agricultura.
OverviewThis project will allow you to write a program to get mo.docxjacksnathalie
Overview
This project will allow you to write a program to get more practice with object-oriented ideas that we explored in the previous project, as well as some practice with more advanced ideas such as inheritance and the use of interfaces.
Ipods and other MP3 players organize a user's music selection into groups known as playlists. These are data structures that provide a collection of songs and an ordering for how those songs will be played. For this assignment you will be writing a set of PlayList classes that could be used for a program that organizes music for a user. These classes will be written to implement a particular PlayList interface so that they can be easily exchange in and out as the program requires. In addition, you will also be using the SimpleTrack class you wrote for the closed lab on Interfaces - if you did not finish this class before the end of lab, you will need to finish it before starting on this project.
Objectives
Practice with programming fundamentals
Review of various Java fundamentals (branching, loops, variables, methods, etc.)
Review of Java File I/O concepts
Practice with Java ArrayList concepts
Practice with object-oriented programming and design
Practice with Java interfaces
Project Description
The SimplePlaylist Class
Once you have coded and tested your SimpleTrack class, you will need to write a SimplePlaylist class that implements the Playist interface given in the project folder.
The SimplePlayList class stores music tracks in order - the first track added to the play list should be the first one removed from the play list. You should recognize this data structure as a
queue
(or a
first-in, first-out queue
). You do not need to implement the equals, hashCode and toString methods for this class but if you choose to do so make sure you document your implementations properly!
The PlayList Management Program
Once you have written and tested a SimpleTrack class and a SimplePlaylist class, it is time to use them to write a program to manage playlists. This program will simulate the playing of songs from a play list. For the SimplePlaylist, the songs are removed from the playlist as they are played, so you know that you're at the end of the list when your list is empty. This program should be implemented in the file MusicPlayerSimulator.java. Note that we are not defining ANY of the methods you are using for this program - the design is all up to you. You must, however, practice good programming style - make sure you are breaking the program up into smaller methods and aren't just trying to solve everything with one monolithic main method. If you have fewer than 5 methods for this program you are probably trying to fit too much into a single method.
Here is a sample transcript of the output of this program:
Enter database filename:
input.txt
Currently playing: 'Elvis Presley / Blue Suede Shoes / Elvis Presley: Legacy Edition' Next track to play: 'The Beatles / Wit.
OverviewThis week, we begin our examination of contemporary resp.docxjacksnathalie
Overview
This week, we begin our examination of contemporary responses to youths’ illegal behaviors. The goal for this week is to assess pre-adjudication responses to youths’ illegal behavior. Primarily, our focus will be on nonformal responses or diversion. As a prelude to this discussion, we will consider the “school to prison pipeline” as it provides a good way to understand the need for diversion in juvenile justice.
Objectives
Upon completion of this week’s lesson, you should be able to:
Define what is meant by the “school to prison pipeline.”
Explain how the political economy contributes to the school to prison pipeline.
Explain how trends in education, policing, and juvenile justice contribute to the school to prison pipeline
Describe juvenile arrest trends and trends in the willingness of police to refer youths to juvenile court.
Define radical nonintervention or true diversion and assess the role in can play in juvenile justice.
Explain the rationale for diversion and its value in juvenile justice.
Describe diversion programs that appear to be effective and programs that are not effective
Assess arguments that are made in support of diversion.
Assess the potential problems that should be addressed when developing or operating diversion programs
Tasks
View Video Lecture (Part 1 and Part 2 below) on the School to Prison Pipeline. While viewing the videos, use the pause feature to stop the slides when needed so that you can examine the content.
Part 1
Part 2
Watch the video:
Rethinking Challenging Kids-Where There's a Skill There's a Way | J. Stuart Ablon | TEDxBeaconStreet
Read the material below, Juvenile Diversion.
View Video Lecture 3
.
OverviewProgress monitoring is a type of formative assessment in.docxjacksnathalie
Overview
Progress monitoring is a type of formative assessment in which student learning is evaluated
on a regular basis to provide useful feedback about performance to both students and
teachers. Though there are a number of methods for monitoring a student’s progress, the most
widely used is general outcome measurement, sometimes referred to as curriculum-based
measurement (CBM). Progress monitoring consists of the frequent administration (e.g., once
per month, every two weeks) of brief probes or tests, which include sample items from every
skill taught across the academic year. After each probe is scored, the teacher or student plots
the score on an individual CBM graph. The teacher can then use this data to determine a
student’s:
• Rate of growth — Average growth of a student’s mathematics skills over a period of time
• Performance level — An indication of a student’s current mathematics skills, often
denoted by a score on a test or probe.
You will determine the rate of growth for the two students listed on page 3 using the data provided.
.
OverviewThe work you do throughout the modules culminates into a.docxjacksnathalie
The document outlines the components of a customer service plan, including examining the customer perspective, quality recognition, and proactive practices. The plan incorporates analyzing the company, customer service, quality, and modern customer service practices. It provides instructions to observe aspects of the business from the customer's point of view like appearance, greeting speed, transaction pace, parking, hours and staff courtesy and knowledge. It also asks to identify important communication criteria, how staff are evaluated and trained, and expectations for technology interactions. Lastly, it prompts an evaluation of practices to respect customers' time, maintain positive attitudes, recognize regulars, communicate professionally, and show initiative.
OverviewThis discussion is about organizational design and.docxjacksnathalie
Overview
This discussion is about
organizational design and leadership
, as well as
global leadership issues and practices
. Conduct research on current events relating to one of the unit concepts of interest to you. Then, share your findings in an initial post. Try to choose a concept that has not been, or is rarely, addressed by your classmates. Review peers' findings and then engage in an active discussion to learn more about the topic at hand.
Resources
Park LibraryLinks to an external site.
Click on the Library Sources tab.
Enter your topic in the search box.
Click on full text, and you will find one, or several, articles to analyze.
.
OverviewScholarly dissemination is essential for any doctora.docxjacksnathalie
Overview
Scholarly dissemination is essential for any doctoral level student. Posters are often a way to ease into scholarly communication. Building a poster is one of the ways scholars participate in the dissemination of knowledge.
Instructions
1. Your poster submission must have a central focus, as developed from the topic selected in Module 2, and that focus must be evident throughout the poster. Specifically, your introduction, analysis, and results must be focused on a set of research questions and/or hypotheses that are obvious in your theoretical diagram.
2. The focus must comprehensively place the problem/question in appropriate scholarly context (scholarly literature, theory, model, or genre).
.
OverviewRegardless of whether you own a business or are a s.docxjacksnathalie
Overview:
Regardless of whether you own a business or are a stakeholder in a business, understanding basic contract terms is important. Businesses enter into contracts with many areas, from shipping to suppliers to customers. As a business owner or manager knowledge of these basic terms will assist you in the day to day operations of the business, regardless of the field.
Instructions:
• Fill in the attached template.
• For each term, define the term with citation to authority, define the term in your own words and provide an example of each term.
Requirements:
• Use APA format for non-legal sources such as the textbook. Use Bluebook citation format for any legal citations.
• Submit a Word document using the template.
• Maximum two pages in length, excluding the Reference page.
.
OverviewImagine you have been hired as a consultant for th.docxjacksnathalie
Overview
Imagine you have been hired as a consultant for the United Nations. You have been asked to write an analysis on how global population growth has caused the following problem and how it affects
TURKEY
A growing global population that consumes natural resources is partially to blame for the release of greenhouse gases since human consumption patterns lead to deforestation, soil erosion, and farming (overturned dirt releases CO2). However, the critical issue is the burning of fossil fuels (hydrocarbons) such as coal oil and natural gas to produce energy that is used for things like electricity production, and vehicle, heating, and cooking fuels.
Instructions
Content
The U.N. has asked that your paper contain three sections. It has asked that each section be one page (or approximately 300 words) in length and answer specific questions, identified in the outline below. It also asks that you use examples from Turkey when answering the questions.
Introduction
Provide an introduction of half a page minimum that addresses points
points
1–5 below:
Explain the problem the U.N. has asked you to address in your own words.
Identify the three sections your paper will cover.
Identify the developing country (TURKEY) you will consider.
Telly
the U.N. which causes of greenhouse gases you will explore.
Provide a one-sentence statement of your solutions at the end of your introduction paragraph.
Section I. Background
What are greenhouse gases?
How do greenhouse gases contribute to global warming?
Section II. How Emissions Causes Problems for the Developing World
Which countries produce the most greenhouse gases?
What are the economic challenges of these emissions in Turkey?
What are the security challenges of these emissions in Turkey?
What are the political challenges of these emissions in Turkey?
Section III. Causes and
Solution
s of Greenhouse Gases
Name two causes of greenhouse gases.
What are potential solutions to address each of the causes you identified?
What is the relationship between population control and greenhouse gases?
Conclusion
Provide a conclusion of half a page minimum that includes a summary of your findings that the United Nations can use to inform future policy decisions.
Success Tips
In answering each question, use examples from Turkey to illustrate your points.
The U.N. needs facts and objective analysis on which to base future policy decisions. Avoid
personal opinion
and make sure your answers are based on information you find through research.
Formatting Requirements
Make sure your paper consists of 4–6 pages (1,200 words minimum, not including the cover page, reference page, and quoted material if any).
Create headings for each section of your paper as follows:
Section I. Background.
Section II. How Emissions Causes Problems for the Developing World.
Section III. Causes and
.
OverviewDevelop a 4–6-page position about a specific health care.docxjacksnathalie
Overview
Develop a 4–6-page position about a specific health care issue as it relates to a target vulnerable population. Include an analysis of existing evidence and position papers to help support your position. Your analysis should also present and respond to one or more opposing viewpoints.
Note
: Each assessment in this course builds on the work you completed in the previous assessment. Therefore, you must complete the assessments in this course in the order in which they are presented.
Position papers are a method to evaluate the most current evidence and policies related to health care issues. They offer a way for researchers to explore the views of any number of organizations around a topic. This can help you to develop your own position and approach to care around a topic or issue.
This assessment will focus on analyzing position papers about an issue related to addiction, chronicity, emotional and mental health, genetics and genomics, or immunity. Many of these topics are quickly evolving as technology advances, or as we attempt to push past stigmas. For example, technology advances and DNA sequencing provide comprehensive information to allow treatment to become more targeted and effective for the individual. However as a result, nurses must be able to understand and teach patients about the impact of this information. With this great power comes concerns that patient conditions are protected in an ethical and compassionate manner.
By successfully completing this assessment, you will demonstrate your proficiency in the following course competencies and assessment criteria:
Competency 1: Design evidence-based advanced nursing care for achieving high-quality population outcomes.
Evaluate the evidence and positions of others that could support a team's approach to improving the quality and outcomes of care for a specific issue in a target population.
Evaluate the evidence and positions of others that are contrary to a team's approach to improving the quality and outcomes of care for a specific issue in a target population.
Competency 2: Evaluate the efficiency and effectiveness of interprofessional interventions in achieving desired population health outcomes.
Explain the role of the interprofessional team in facilitating improvements for a specific issue in a target population.
Competency 3: Analyze population health outcomes in terms of their implications for health policy advocacy.
Explain a position with regard to health outcomes for a specific issue in a target population.
Competency 4: Communicate effectively with diverse audiences, in an appropriate form and style, consistent with organizational, professional, and scholarly standards.
Communicate an initial viewpoint regarding a specific issue in a target population and a synthesis of existing positions in a logically structured and concise manner, writing content clearly with correct use of grammar, punctuation, and spelling.
Integrate .
Overview This purpose of the week 6 discussion board is to exam.docxjacksnathalie
Overview:
This purpose of the week 6 discussion board is to examine social class and global stratification. Answer prompt 1. Then select and answer one prompt from prompts 2-4. Refer to Chapters 7 and 8 to answer the prompts.
Instructions:
Respond to prompts in paragraph form (200-400 words
Prompt 1:
Describe 3 topics from Chapters 7 and 8 that you found interesting. Three topics I found interesting from Chapter 7 and 8 were the Dependency Theory, World Systems Theory, and Modernization Theory.
Prompt 2:
Describe 3 different social classes and criteria for membership in each.
Prompt 3:
Describe the effect of social inequality upon dominant and minority groups.
Prompt 4
: Describe social mobility regarding how to rise up the social class ladder, if it is possible.
Prompt 5:
Apply a functionalist or conflict theory perspective to social inequality.
.
Overall Scenario Always Fresh Foods Inc. is a food distributor w.docxjacksnathalie
Overall Scenario
Always Fresh Foods Inc. is a food distributor with a central headquarters and main warehouse in Colorado, as well as two regional warehouses in Nevada and Virginia. The company runs Microsoft Windows 2019 on its servers and Microsoft Windows 10 on its workstations. There are 2 database servers, 4 application servers, 2 web servers, and 25 workstation computers in the headquarters offices and main warehouse. The network uses workgroups, and users are created locally on each computer. Employees from the regional warehouses connect to the Colorado network via a virtual private network (VPN) connection. Due to a recent security breach, Always Fresh wants to increase the overall security of its network and systems. They have chosen to use a solid multilayered defense to reduce the likelihood that an attacker will successfully compromise the company’s information security. Multiple layers of defense throughout the IT infrastructure makes the process of compromising any protected resource or data more difficult than any single security control. In this way, Always Fresh protects its business by protecting its information.
Scenario 1
Assume you are an entry-level security administrator working for Always Fresh. You have been asked to evaluate the option of adding Active Directory to the company’s network.
Tasks
Create a summary report to management that answers the following questions to satisfy the key points of interest regarding the addition of Active Directory to the network:
1. System administrators currently create users on each computer where users need access. In Active Directory, where will system administrators create users?
2. How will the procedures for making changes to the user accounts, such as password changes, be different in Active Directory?
3. What action should administrators take for the existing workgroup user accounts after converting to Active Directory?
4. How will the administrators resolve differences between user accounts defined on different computers? In other words, if user accounts have different settings on different computers, how will Active Directory address that issue? (Hint: Consider security identifiers [SIDs].)
.
OverviewCreate a 15-minute oral presentation (3–4 pages) that .docxjacksnathalie
Overview
Create a 15-minute oral presentation (3–4 pages) that examines the moral and ethical issues related to triaging patients in an emergency room.
By successfully completing this assessment, you will demonstrate your proficiency in the following course competencies and assessment criteria:
· Competency 1: Explain the effect of health care policies, legislation, and legal issues on health care delivery and patient outcomes.
. Explain the health care policies that can affect emergency care.
. Recommend evidence-based decision-making strategies nurses can use during triage.
· Competency 3: Apply professional nursing ethical standards and principles to the decision-making process.
. Describe the moral and ethical challenges nurses can face when following hospital policies and protocols.
. Explain how health care disparities impact treatment decisions.
· Competency 4: Communicate in a manner that is consistent with expectations of nursing professionals.
. Write content clearly and logically, with correct use of grammar, punctuation, and mechanics.
. Correctly format citations and references using APA style.
Context
Working in an emergency room gives rise to ethical dilemmas. Due to time restraints and the patient's cognitive impairment and lack of medical history, complications can and do occur. The nurse has very little time to get detailed patient information. He or she must make a quick assessment and take action based on hospital protocol. The organized chaos of the emergency room presents unique ethical challenge, which is why nurses are required to have knowledge of ethical concepts and principles.
Questions to consider
To deepen your understanding, you are encouraged to consider the questions below and discuss them with a fellow learner, a work associate, an interested friend, or a member of your professional community.
· How does a triage nurse decide which patient gets seen first?
· How does health disparity affect the triage nurse's decision making?
· What ethical and moral issues does the triage nurse take into account when making a decision?
· What are triage-level designations?
Resources
Suggested Resources
The following optional resources are provided to support you in completing the assessment or to provide a helpful context. For additional resources, refer to the Research Resources and Supplemental Resources in the left navigation menu of your courseroom.
Capella Resources
· APA Paper Template.
· APA Paper Tutorial.
Library Resources
The following e-books or articles from the Capella University Library are linked directly in this course:
· Tingle, J., & Cribb, A. (Eds.). (2014). Nursing law and ethics (4th ed.). Somerset, NJ: John Wiley & Sons.
· Cranmer, P., & Nhemachena, J. (2013). Ethics for nurses: Theory and practice. Maidenhead, UK: Open University Press.
· Aacharya, R. P., Gastmans, C., & Denier, Y. (2011). Emergency department triage: An ethical analysis. B MC Emergency Medicine, 11(1), 16–29.
· Guidet, B., H.
Overall CommentsHi Khanh,Overall you made a nice start with y.docxjacksnathalie
Overall Comments:
Hi Khanh,
Overall you made a nice start with your U06a1 assignment; however, many of the required objectives have not been addressed in the first version of your assignment. Please carefully review the scoring guide, and review my feedback below, and be sure to contact me if you have any questions about my comments. You can reach me at: [email protected] or 813-417-0860.
Sincerely,
Dr. Marni Swain
COMPETENCY: Assess approaches for recruiting, selecting, and retaining talent.
CRITERION: Explain why and when candidate background checks will be authorized.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Basic
Explains why but not when candidate background checks will be authorized.
Faculty Comments:“
You made a nice start with this discussion; however, it is important to develop your content further to address the legalities involving when a background check can be conducted during the interview process, and the other steps employers have to follow to be in compliance with the law.
”
CRITERION: Identify the top three candidates to interview for the position.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Non-Performance
Does not identify the top three candidates to interview for the position.
Faculty Comments:“
Please develop your content further to address this topic in your assignment.
”
CRITERION: Explain rationale for why the selected candidates should be interviewed.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Non-Performance
Does not explain rationale for why the selected candidates should be interviewed.
Faculty Comments:“
Please develop your content further to address this topic in your assignment.
”
CRITERION: Identify pre-employment screening tests for the position being recruited.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Basic
Identifies a pre-employment screening test for the position being recruited.
Faculty Comments:“
I would like to see your content developed further to clearly identify your rationale for the pre-employment screening tests you selected, as this is not clear based on the limited information provided.
”
CRITERION: Select assessment methods to use based on the job being recruited and the budget available.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Non-Performance
Does not select assessment methods to use based on the job being recruited and the budget available.
Faculty Comments:“
I would like to see your content developed further to clearly identify the assessment methods you will use for CapraTek's Regional Sales positions based on the available budget, as this is not identified in your work.
”
CRITERION: Develop the sequence in which methods will be used to screen applicants.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Non-Performance
Does not develop the sequence in which methods will be used to screen applicants.
Faculty Comments:“
Please develop your content further to address this topic in your assignment.
”
CRITERION: Design a final candidate selection process for the CapraTek.
Overall CommentsHi Khanh,Overall you made a nice start with.docxjacksnathalie
Overall Comments:
Hi Khanh,
Overall you made a nice start with your U03a1 assignment; however, your content still does not address the required objectives. For this assignment you will need to focus the content on Capra Tek's regional sales position, and for objective #1 analyze the KSAs for this position, and for objective #2 you will need to analyze wage trends related to this position as well. Objectives 3 & 4 focus on job description and the job analysis so please carefully review what is required for these two objectives.
Please see my feedback below and be sure to let me know if you have any questions about my comments.
Sincerely,
Dr. Marni Swain
COMPETENCY: Describe how hiring practices support an organization's strategy.
CRITERION: Articulate the components of a job description for a position.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Non-Performance
Does not articulate the components of a job description for this position.
Faculty Comments:“
Please see feedback above.
”
COMPETENCY: Assess approaches for recruiting, selecting, and retaining talent.
CRITERION: Identify the knowledge, skills, and abilities required for this position.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Non-Performance
Does not identify the knowledge, skills, and abilities required for this position.
Faculty Comments:“
Please see feedback above.
”
COMPETENCY: Explore technology tools that support recruiting and staffing management.
CRITERION: Identify wage information and employment trends for this position in a selected state.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Non-Performance
Does not identify wage information and employment trends for this position in a selected state.
Faculty Comments:“
Please see feedback above.
”
COMPETENCY: Analyze the impact of legal and regulatory issues on staffing management.
CRITERION: Explain why a job analysis is a requirement for any recruiting and selecting process.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Non-Performance
Does not explain why a job analysis is a requirement for any recruiting and selecting process.
Faculty Comments:“
Please see feedback above.
”
COMPETENCY: Communicate in a manner that is scholarly and professional.
CRITERION: Communicate in a professional manner that is appropriate for the intended audience.
DISTINGUISHED
PROFICIENT
BASIC
NON-PERFORMANCE
Non-Performance
Does not communicate in a professional manner that is appropriate for the intended audience.
Faculty Comments:“
Please see feedback above.
”
Dysphagia .
Dysphagia is a serious problem and contributes to weight loss, malnutrition, dehydration, aspiration pneumonia, and death. Careful assessment of risk factors, observation for signs and symptoms, and collaboration with speech-language pathologists on interventions are essential.
Dysphagia, or difficulty swallowing, is a common problem in older adults. The prevalence of swallowing disorders is 16% to 22% in adults older than 50 years of age, and up to 60% of nursing ho.
Overall feedbackYou addressed most all of the assignment req.docxjacksnathalie
The document provides feedback on an assignment submitted by a student. It notes that while the student addressed most requirements, the introduction could have better identified the key areas to be covered. Additionally, only one scholarly peer-reviewed journal article was included when two were required. The feedback recommends reviewing instructions carefully and including an introduction describing coverage areas and the required number of scholarly sources in the future.
Overall Comments Overall you made a nice start with your U02a1 .docxjacksnathalie
This document provides feedback from a faculty member on an assignment analyzing legal and regulatory issues related to staffing management. For most criteria evaluated, the faculty member provided basic or non-performance feedback, noting the student did not sufficiently analyze the key aspects of the case such as important issues, outcome, evidence of discriminatory effects, and how guidelines help avoid issues. The faculty member recommended developing more in-depth content on the case analyzed and ensuring it is a disparate impact case. Minor errors in formatting references were also noted.
Overview This purpose of the week 12 discussion board is to e.docxjacksnathalie
Overview:
This purpose of the week 12 discussion board is to examine health, healthcare, and disability status. Answer prompt 1. Then select and answer one prompt from prompts 2-4. Refer to Chapter 13 to answer the prompts.
Instructions:
Respond to prompts in paragraph form (200-400 words)
Prompt 1:
Describe 3 topics from Chapter 13 that you found interesting.Three topics I found interesting in Chapter 14 was "A Functionalist Perspective: The Sick Role", "A Symbolic Interactionist Perspective:
Prompt 2:
Describe how stereotypes regarding disability status may lead to prejudice and discrimination.
Prompt 3:
Describe how access to healthcare is associated with social class location (e.g., socioeconomic status).
Prompt 4:
How is culture associated with attitudes towards health and healthcare.
Prompt 5:
Compare how the United States pays for health care with how other nations provide health services for their citizens.
.
Over the years, the style and practice of leadership within law .docxjacksnathalie
Over the years, the style and practice of leadership within law enforcement agencies has gradually changed. In the past, leadership was primarily relegated to one individual within the department. However, there has been a transformation in leadership theory resulting in a more dynamic, multifaceted nature of teamwork, inclusion, and dispersed leadership. More and more, police chiefs are being encouraged to move toward a more participatory leadership style of management, one that encourages collaboration and cooperation in the decision-making process.
Based on your readings in the text and credible Internet research, respond to the following:
What does the term
shared leadership
mean? What advantages or disadvantages do you see in this leadership approach?
What direction should law enforcement leaders take for the future, related to leadership styles?
What does the term
visionary leadership
mean?
2-3 pages
.
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
How to Setup Default Value for a Field in Odoo 17Celine George
In Odoo, we can set a default value for a field during the creation of a record for a model. We have many methods in odoo for setting a default value to the field.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
How to Download & Install Module From the Odoo App Store in Odoo 17Celine George
Custom modules offer the flexibility to extend Odoo's capabilities, address unique requirements, and optimize workflows to align seamlessly with your organization's processes. By leveraging custom modules, businesses can unlock greater efficiency, productivity, and innovation, empowering them to stay competitive in today's dynamic market landscape. In this tutorial, we'll guide you step by step on how to easily download and install modules from the Odoo App Store.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
CapTechTalks Webinar Slides June 2024 Donovan Wright.pptxCapitolTechU
Slides from a Capitol Technology University webinar held June 20, 2024. The webinar featured Dr. Donovan Wright, presenting on the Department of Defense Digital Transformation.
Dynamic Pricing andRevenue ManagementIEOR 4601 Spring .docx
1. Dynamic Pricing and
Revenue Management
IEOR 4601 Spring 2013
Professor Guillermo Gallego
Class: Monday and Wednesday 11:40-12:55pm
Office Hours: Wednesdays 3:30-4:30pm
Office Location: 820 CEPSR
E-mail: [email protected]
Why Study Dynamic Pricing and Revenue
Management?
¡� Revenue Management had its origins in the
airline industry and is one of the most
successful applications of Operations
Research to decision making
¡� Pricing and capacity allocation decisions
directly impact the bottom line
¡� Pricing transparency and competition make
2. pricing and capacity allocation decisions
more difficult and more important
Applications of Dynamic Pricing and
Revenue Management
¡� Capacity allocation of limited, perishable,
resources to different fare classes
l� Airlines, hotels, car rentals, cruises, travel packages, tickets
for events
¡� Design and pricing of products
l� Fare restrictions and pricing
l� Consumption and fulfillment options
l� Upgrades, downgrades and upsells
¡� Pricing under competition
l� Electronic-commerce
Objectives of this course
¡� Understand the critical tradeoffs and
decisions in Revenue Management
3. ¡� Learn how to
l� Monitor and control product availability for single and
multiple resources
l� Overbook limited resources when customers
shows are random
l� Use upgrades, upsells and real options to improve revenues
l� Price under competition
l� Improve on the current practice of Revenue
Management
“Physics should be explained as simply as possible, but no
simpler.”
Albert Einstein
Professor Gal ego’s experience and
background on subject
¡� Author of seminal papers on dynamic
pricing and revenue management
¡� Winner of several prices from academia
and industry related to work on Revenue
Management
4. ¡� Consultant for airlines and RM solution
providers
¡� Consultant for other users of Revenue
Management
Readings
¡� Class Notes
l� I will provide with notes of the different topic we cover in
class
¡� Textbook
l� R.L. Phillips, Pricing and Revenue
Optimization, Stanford University Press,
2005, ISBN 0-8047-4698-2.
¡� References
l� K.T. Talluri and G.J. Van Ryzin, The Theory and Practice of
Revenue Management,
Springer, 2005, ISBN 0-387-24376-3.
l� Assigned papers
Prerequisites and Grading
5. ¡� Prerequisites
l� Probability and Statistics at the level of IEOR
4150
l� Deterministic Models at the level of IEOR 4003
¡� Corequisites: Stochastic Models IEOR 4106
¡� Grading
l� Assignments
20%
l� Midterm
35%
l� Final
45%
Introduction to Revenue Management
¡� Revenue Management refers to the
strategy and tactics used by perishable
capacity providers to allocate capacity to
different fare classes or market segments
6. to maximize expected revenues. (See
Chapter 6 in Phillips.)
¡� RM is often practice when
l� Sellers have fixed stock of perishable capacity l�
Customers book capacity prior to usage
l� Seller offers a sets of fare classes
l� Seller can change the availability of fare
classes
History of Revenue Management
¡� Prior to 1978 the Airline Industry was
heavily regulated
¡� In the early 80’s the industry was
deregulated to encourage new entrants
¡� Low-cost carriers such as People Express
started encroaching into key markets of
large carriers
¡� American Airline dilemma:
l� Match fares and lose money
7. l� Keep fares and lose customers
AA’s Response to People Express
¡� Ultimate Super Saver Discount Fare
l� Same fare as People Express
l� Passenger must buy at least two weeks
prior to departure
l� Stay at his destination over a Saturday
night
¡� AA restricted the number of
discount seats sold on each flight to
save seats for full-fare passengers
Rational and Impact of Strategy
¡� Product Design
l� Imposing restrictions that appealed to the
leisure segment without cannibalizing the
business segment
¡� Capacity Allocation
l� Carefully control capacity to maximize
8. revenues
¡� Strategy started in January 85
l� PE was struggling by March
l� PE was at the verge of bankruptcy by August
Post-mortem
¡� People Express was bought by
Texas Air for 10% of the market
value it had enjoyed a year before
¡� “We had great people, tremendous
value, terrific growth. We did a lot
of things right. But we didn’t get
our hands around the yield
management and automation
issues.” Donald Burr CEO of PE
RM: The System Context
¡� AA was based on a computerized
reservation system (CRS) called Sabre
9. developed in 1963. This system:
l� Replaced index cards to manage reservations
l� Sabre is also a GDS (global distribution
system) that allowed AA to distribute its
products and fares globally
¡� Other GDSs: Amadeus, Galileo, Worldspan.
RM Constraints Imposed by Systems
¡� AA’s used Sabre’s Computerized
Reservation System as the backbone to
Revenue Management
l� The reservation system served as a repository of all the
bookings that have been accepted for
future flights
l� The CRS also contains the controls that specify how many
bookings from different fare classes
the airline will accept on future flights
¡� Remember: RM systems were developed
in the context of existing CRSs.
10. Levels of Revenue Management
¡� Strategic:
l� Market segmentation (leisure vs business)
l� Product design (restrictions, fares, options) l� Pricing
(Static vs. Dynamic)
¡� Tactical:
l� Calculate and updated booking limits
¡� Booking Control:
l� Determine which booking to accept and which
to reject based on booking limits
Strategic Revenue Management
¡� Design low fares to increase sales without
cannibalizing full fare demand
l� Time of Purchase Restrictions
¡� Advance purchase requirements
l� Traveling Restrictions
¡� Saturday night stays
l� High cancellation and change penalties
11. ¡� Other opportunities
l� Contingent options on capacity
l� Flexible and callable products
Tactical Revenue Management
¡� Resources
l� Units of capacity
¡� Seats on a flight
¡� Hotel capacity for a specific night
¡� Products
l� What consumers seek to purchase
¡� May involve one or more resources
¡� Fares
l� A combination of a price and a set of
restrictions
Tactical Definition of RM
¡� A supplier controls a set of resources with
fixed and perishable capacity, a portfolio
12. of products consisting of combinations of
one or more of the resources, and a set of
fare classes associated with each of the
products. The tactical revenue
management problem is to chose which
fare classes should be open and which
closed for sale at each moment to
maximize total revenue.
Components of Tactical RM
¡� Capacity Allocation
l� How many customers from different
fare classes should be allowed to book?
¡� Network Management
l� How should bookings be managed
across a network of resources?
¡� Overbooking
l� How many total bookings should be
13. accepted for a product when there are
cancellations and show uncertainty?
Booking Controls
¡� Limits on bookings for different fare
classes:
l� Example: An airline receives a B-class
request for 3 seats departing in two
weeks. The current B-class booking
limit is two seats. As a result, the
request is rejected
Booking Limits
¡� Nesting Controls:
l� Label the fare classes so that 1 is the highest fare class and
n is the lowest fare class. For any i let bi
denote the nested booking limit for class i.
b 1 ≥ b 2 ≥�≥ bn
l� Protection Levels:
y
�
14. i = b − b
, i
i
= ,
1 2, n −1
1
1
+
l� Updates: If x units are sold in a transaction b
�
i ← max( bi − x
),
0
,
i = ,
1 2, n −1
Nested Booking Limits (Example)
21. 2
3 Accept
8
89
62
1
0
0
27
88
89
89
89
Is RM successful?
¡� By most measures (revenues relative to
resources) it has been a success at most
major airline, hotel, rental car companies.
22. l� Why have major airlines have been losing
money?
¡� Costs are 25-30% higher per mile, so even though larger
carriers bring in about 25% more revenue
per mile the cost disadvantage is overwhelming
l� What can be done?
¡� Cost costs
¡� Improve RM systems
l� Big move from independent to dependent demand models
RM and Price Discrimination
¡� Price discrimination exists when sales of identical goods or
services are transacted at different
prices from the same provider.
¡� A feature of monopolistic or oligopolistic markets where
market power can be exercised.
¡� Requires market segmentation and means to discourage
discount customers becoming resellers.
l�
This is achieved by fences to keep segments separate.
¡� Price discrimination is more common in services where
resale is not possible.
23. ¡� Price discrimination can also be seen when the requirement
of identical goods is relaxed.
l�
Premium products have price differential not explained by
production costs.
Taxonomies of Price Discrimination
¡� First degree: requires selling at maximum willingness to pay
¡� Second degree: quantity discounts (sellers not able to
differentiate consumer types)
¡� Third degree: Prices vary by attributes (e.g., senior
discounts)
¡� Fourth degree: Prices are the same but costs are different
(reverse discrimination)
¡� Alternative taxonomy:
l�
Complete discrimination (like first degree)
l�
Direct discrimination: seller conditions price on some attribute
(like third degree)
l�
24. Indirect discrimination: the seller relies on some proxy such as
quantity discounts (like second degree)
RM and Price Discrimination
¡� Differentiating by time-of-purchase and imposing traveling
restrictions like Saturday night stays is a
form of second degree or indirect discrimination.
¡� Selling premium seats is another form of second degree or
indirect discrimination.
l� Eg., uncomfortable second class seats on trains to entice
wealthier people to purchase first class
seats.
l� Advance seat selection, mileage accrual, use of lounge, and
priority boarding may be forms of second
and/or fourth degree discrimination.
Other Examples of Price Discrimination
¡� Retail price discrimination is in violation of the Robinson-
Patman Act (1936)
¡� Coupons
¡� Segmentation by age group and student status
¡� Discounts for members of certain occupations
¡� Employee discounts
¡� Retail incentives (rebates, seasonal discounts, quantity
discounts)
25. ¡� Gender based examples
¡� College financial aid
¡� User-controlled price discrimination
¡� See http://en.wikipedia.org/wiki/Price_discrimination Static
and Dynamic Pricing
¡� Pricing is studied by people in Economics
and Marketing
¡� Economist look at equilibrium prices
¡� Marketing focuses on demand estimation
¡� We focus on more tactical aspects of
pricing
l� Customer arrival rates
l� Capacity constraints
l� And increasingly on choice models and
competition
Static Pricing
¡� d(p) demand at p
¡� z unit cost or dual of capacity constraint
26. ¡� r(p,z) = (p-z)d(p) profit function
¡� Find p to maximize r(p,z)
¡� Is there a finite maximizer p(z)?
¡� Is p(z) monotone?
¡� Is r(z) = r(p(z),z) monotone? Convex?
¡� Multiple market segments with limited
price menus
Dynamic Pricing
¡� Finite sales horizon
¡� Customers arrive stochastically over time
¡� State (t,x)
l� t time-to-go
l� x remaining inventory
¡� What price p(t,x) should be charged at
state (t,x)?
¡� Are there simple and effective pricing
heuristics?
27. ¡� What about strategic customers?
¡� What about competition?
Topics to be Covered
¡� Single Resource RM
l� Independent Demands
¡� Dynamic Programming, Bounds and Heuristics
l� Dependent Demands based on choice models
¡� Static and Dynamic Pricing
¡� Network RM
l� Independent Demands, Choice Models
¡� Overbooking
¡� Service Engineering
l� Design and pricing of service features
Useful Techniques you wil Learn
¡� Dynamic Programming (DP)
l� Tool for sequential decision making
l� Optimal Control (continuous time DP)
28. ¡� Approximate Dynamic Programming
l� Tool to approximate DPs
¡� Bounds and Heuristics Techniques
¡� Choice Modeling
¡� Game Theory
Single Resource Revenue Management with Independent
Demands
c
Guillermo Gallego
Updated Spring 2013
Abstract
Providers of fixed perishable capacity, such as airline seats and
hotel rooms use price discrimination
to improve revenues; in practice, this discrimination is typically
achieved by imposing booking and usage
restrictions or including ancillary services such as mileage
accrual and luggage handling, to sell the same
capacity to different customers at different prices. We will
assume that the set of fare classes (a menu
29. of prices, restrictions and ancillary services) is given, and that
the capacity provider’s goal is to allocate
capacity among the different fare classes to maximize expected
revenues. The problem of designing and
pricing fare classes is treated in a separate chapter. We analyze
the two fare class problem under the
assumption that the lower fare class books first. We use
marginal analysis to informally derive
Littlewood’s
rule and then show that Littlewood’s rule is in fact optimal.
Spill rates, spill penalties and callable
products are discussed next. A dynamic programming
formulation for the multiple fare class problem is
then introduced under the assumption that lower fare classes
book first. Commonly used heuristics as well
as bounds on the value function are presented. Dynamic models
that explicitly take time into account,
allow for more general fare arrival patterns and for randomness
in the size of the requests. We compare
the performance of static and dynamic policies and find that
dynamic policies have a real advantage when
the fare arrivals patterns are not low-to-high. We finalize the
chapter with a model where fare classes are
30. not allowed to reopen after they are closed for the first time.
1
Introduction
This chapter considers the simplest and best known revenue
management problem, the single resource,
inde-
pendent demand problem. We assume that the capacity provider
is trying to maximize the expected
revenues
from a sunk investment in c units of capacity. We assume that
capacity is sold through a reservation
system
and that capacity cannot be modified or replenished during the
booking horizon. We also assume that
unsold
capacity has no salvage value. Later we will see that the zero
salvage value assumption is made without
loss
of generality as any problem with positive salvage value can be
transformed into a problem with zero
salvage
value. We assume that the set of fare classes (a menu of prices
and restrictions) is given, and that the
demands
31. for the different fare classes are statistically independent. In
particular, we assume that if a customer finds
his
preferred fare class closed, he will leave the system without
purchasing. This assumption holds
approximately
if the difference in fares is large so that demands are decoupled
or if customers can find alternative
sources
of capacity for their preferred fare class. In some cases,
however, part of the demand may be recaptured
by
other available fare classes. In such cases, the independent
demand assumption is too strong and needs to
be
relaxed. We address this issue in a separate chapter where we
discussed demand models based on
discrete
choice theory.
In this chapter, we present a variety of models that have been
developed in industry and in academia.
There has been a preference in industry for models that suppress
the time dimension and assume that the
arrival pattern of the fare classes is low-to-high. We call these
class of models static to distinguish them
32. from the dynamic models, favored by academics, that model
time explicitly. Both models have advantages
and disadvantages as we will soon see. Static models are
relatively easy to understand. Also, good
heuristics
1
were developed before optimal solutions based on dynamic
programming were discovered. Bringing in
the
time dimension helps deal with more general fare arrival
patterns, but specifying the model requires a
more
detailed estimation of demand. This Chapter starts with a
review of the two fare class problem in §2
where we
present a heuristic derivation of Littlewood’s rule via marginal
analysis. Littlewood’s rule is formally
derived
in §2.3 where a formal DP for the two fare class problem is
presented. The dynamic program for multiple
fare
classes is presented in §3. Commonly used heuristics are
presented in §4 and bounds on the optimal
expected
revenue are presented in §5. The dynamic model is presented in
§6, for the Poisson case and for the
33. compound
Poisson case in §7, where each request is for a random demand
size. In §8, we restrict fares so that they
cannot
be opened once they are closed.
2
Two Fare Classes: Marginal Analysis
The product can be sold either at the full-fare p1 or at a
discounted-fare p2 < p1. The discounted-fare
typically
has advance purchasing and usage restrictions. Let D1 and D2
denote respectively the random demand for
the two fare classes for a specific instance of the problem, e.g.,
for a specific flight for an airline or a
specific
night for a hotel.
We assume that all booked customers will actually travel. This
avoids the need to overbook capacity and
allow us to focus on the problem of allocating capacity between
the two fares. We will discuss how to
deal
with pre-travel cancellations and day-of-travel no shows in a
separate chapter on overbooking models.
34. Fare class arrival order is an important part of the model. We
assume what is commonly known as the
low-to-high fare class arrival order, which implies that demand
for the discounted fare book earlier than
for
full-fare. This arrival pattern holds approximately in practice,
and it is encouraged by advance purchase
restrictions imposed on lower fare classes. Notice that this is a
worst case arrival pattern. Indeed, if full-
fare
class customers arrived first then we would accept them up to
capacity and use residual capacity, if any,
to
satisfy demand from the discounted-fare class. We will relax the
low-to-high fare order arrival
assumption
after we solve the multi-fare problem via dynamic
programming.
Under the low-to-high arrival pattern, discount-fare customers
may exhaust capacity, say c, unless part
of it is protected for later-booking by full-fare customers.
Consequently, booking limits (known as
discount
authorizations) are placed on the discount sales. Suppose we
protect y ∈ {0, 1, . . . , c} units of capacity
for
35. the full-fare demand, D1, before observing the actual demand
for the discount-fare, D2. This results in a
booking limit c − y on the discounted-fare, so sales at the
discounted-fare class are given by min(c − y,
D2).
The remaining capacity is equal to c − min(c − y, D2) = max(y,
c − D2) and it is all made available to the
full-fare class. Consequently, sales at the full fare equal
min(max(y, c − D2), D1). The total expected
revenue
is
W (y, c) = p2E min(c − y, D2) + p1E min(max(y, c − D2), D1)
and the goal is to find a protection level y that maximizes W (y,
c). The extreme strategies y = 0 and y = c
correspond, respectively, to the case where no capacity is
protected and all of the capacity is protected.
We
will later come back and discuss when these extreme strategies
are optimal. In most cases, however, an
intermediate strategy is optimal.
The fare ratio r = p2/p1 plays an important role in determining
optimal protection levels. If the ratio is
36. very small then we would be inclined to protect more capacity
for the full-fare demand. If the ratio is
close
to one, we would be inclined to accept nearly all discount-fare
requests since we can get almost the same
revenue, per unit of capacity, without risk. The distribution of
full-fare demand is also important in
deciding
how many units to protect for that fare. If, P (D1 ≥ c) is very
large, then it makes sense to protect the
entire
capacity for full-fare sales as it is likely that the provider can
sell all of the capacity at the full-fare.
However,
if P (D1 ≥ c) is very low then it is unlikely that all the capacity
can be sold at the full-fare, so fewer units
should be protected. It turns out that the demand distribution of
the discount-fare D2 has no influence on
the
optimal protection level under our assumption that D2 and D1
are independent. A formula for the optimal
protection level, involving only P (D1 ≥ y) and r, was first
proposed by Littlewood [14] in 1972. His
arguments
were not formal; however, they were later justified by Bhatia
and Prakesh [1] in 1973, and Richter [17]
in 1982.
37. 2
One can obtain Littlewood’s formula intuitively by using
marginal analysis. The advantage of marginal
analysis is that it allows us to quickly derive the solution for the
two fare class problem. The marginal
analysis
argument goes as follows: Suppose we have y > 0 units of
capacity, and that we receive a request for the
discounted-fare. Consider the marginal revenue associated with
accepting and rejecting this request. If we
accept, we obtain p2. If we close down the discount-fare then
we will be able to sell the yth unit at p1
only
if the full-fare demand D1 is at least as large as y, so it is
intuitively optimal to reject the discount fare if
p1P (D1 ≥ y) > p2. This suggests that an optimal protection
level y1 should be given by:
y1 = max{y ∈ N : P (D1 ≥ y) > r},
(1)
where N = {0, 1, . . . , } is the set of non-negative integers.
Equation (1) is known as Littlewood’s rule.
Example 1. Suppose D1 is Poisson with parameter 80, the full
38. fare is p1 = $100 and the discounted fare
is
p2 = $60, so r = 60/100 = 0.6. We are interested in the
cumulative tail distribution P (D1 ≥ y) = 1 − P
(D1 ≤
y − 1). Since most statistical software packages return the value
of P (D1 ≤ y), we see that y1 satisfies
P (D1 ≤ y1 − 1) < 1 − r ≤ P (D1 ≤ y1). Since P (D1 ≤ 77) =< 0.4
≤ P (D1 ≤ 78) we conclude that y1 = 78.
Consequently, if c = 200 then the booking limit for the discount
fare is 122. However, if c < y1, then all
units
should be protected for the full-fare resulting in a booking limit
of zero.
Remarks:
• y(c) = min(y1, c) is also an optimal protection level. If y(c) =
c, or equivalently if y1 ≥ c, then all the
capacity should be reserved for sale at the full-fare.
• The quantity b2 = max(c − y1, 0) is known as the optimal
booking limit for the discount fare. It is the
maximum number of discount-fare customers that we will book.
• y1 is independent of the distribution of D2.
• If P (D1 ≥ y1 + 1) = r, then y1 + 1 is also optimal protection
level, so both y1 and y1 + 1 result in the
39. same expected revenue. Protecting the y1 + 1 unit of capacity
increases the variance of the revenue, but
it reduces the probability of rejecting requests from full-fare
customers.
From Littlewood’s rule (1), we see that the extreme strategy y =
0 is optimal when P (D1 ≥ 1) ≤ r and
the extreme strategy y = c is optimal when P (D1 ≥ c) > r.
2.1
Continuous Demand Model
Although revenue management demands are actually discrete,
continuous distributions can be easier to
work
with and are often employed in practice. If we model D1 as a
continuous random variable with
cumulative
distribution function F1(y) = P (D1 ≤ y), then
y1 = F −1(1 − r)
1
where F −1 denotes the inverse of F . In particular, if D
1
40. 1 is Normal with mean µ1 and standard deviation σ1
then
y1 = µ1 + σ1Φ−1(1 − r)
(2)
where Φ denotes the cumulative distribution function of the
standard Normal random variable.
This formula allows for comparative statics as given in Table 1:
Example 2. Suppose that D1 is Normal with mean 80 and
standard deviation 9, the full-fare is p1 = $100
and the discount-fare is p2 = $60. Then y1 = F −1(1 − 0.6) =
77.72 < 80 since r > 1/2. Notice that the
1
solution is quite close to that of Example 1. This is because a
Poisson random variable with mean 80 can
be
√
well approximated by a normal with mean 80 and standard
deviation
80
9.
3
41. Fare Ratio
Dependence of protection level
r > 1
y
2
1 < µ1 and y1 decreases with σ1
r = 1
y
2
1 = µ1 independent of σ1
r < 1
y
2
1 > µ1 and y1 increases with σ1
Table 1: Comparative Statics for Normal Full Fare Demand
2.2
Connection with the Newsvendor Problem
42. There is a close connection between the classical Newsvendor
Problem and the two-fare Revenue
Management
Problem that we will briefly explore here. In the classical
Newsvendor Problem a manager must decide
how
many units, say y, to stock for random sales D1 at p1 assuming
a unit cost p2 < p1. The solution is to stock
y1
units where y1 is the largest integer such that P (D1 ≥ y) > r =
p2/p1. We can think of the two-fare
Revenue
Management Problem as a situation where capacity c is pre-
decided, at a possible sub-optimal level,
there is
random demand D1 at ”salvage value” p1 > p2 that arrives after
demand D2 at p2. The revenue
management
problem is to determine how many units to allow to be sold at
p2. We know that the solution is to allow
(c − y1)+ units to book at p2, reserving max(y1, c − D2) units
for sale at p1.
2.3
Two Fare Classes: Dynamic Programming
43. In this section we formulate and analyze the two fare class
problem using dynamic programming and
present
a formal proof of the optimality of Littlewood’s rule. We will
from now on refer to the full-fare class as
fare
class 1 and to the discounted fare class as fare class 2. Dynamic
programming starts by solving the
problem
at the last stage, just before demand for fare class 1. Let V1(y)
be the optimal expected revenue that can
be
obtained from fare class 1 when capacity is y. Since it is
optimal to allow fare class 1 customers to book
all
of the available capacity, sales are equal to min(D1, y) and the
optimal expected revenue is
V1(y) = p1E min(D1, y).
Our next task is to find V2(c), the optimal expected revenue
that can be obtained from c units of ca-
pacity. Suppose that y ∈ {0, 1, . . . , c} units are protected for
fare class 1 demand. This results in
revenues
p2 min(D2, c − y) from sales to fare class 2 and remaining
inventory max(c − D2, y) available for fare
class
44. 1. Notice that we can obtain expected revenue EV1(max(c − D2,
y)) from this inventory from fare class 1
customers. Then
W (y, c)
=
p2E min(c − y, D2) + p1E min(max(y, c − D2), D1)
=
E{p2 min(D2, c − y) + V1(max(c − D2, y))}
is the expected revenue associated with protecting y ∈ {0, 1, . .
. , c} units for the full-fare. V2(c) can be
obtained by maximizing W (y, c) over y. More precisely,
V2(c) =
max
E{p2 min(D2, c − y) + V1(max(c − D2, y))}.
(3)
y∈ {0,1,...,c}
The key to Dynamic Programming is that it involves a recursive
equation (3) linking the expected
revenues
45. V2(c), at stage 2, to the expected revenue function V1 at stage
1. To solve for V2(c) we first need to solve
for V1(y) for y ∈ {0, 1, . . . , c}. Before moving on to the
multi-fare formulation we will provide a formal
proof of Littlewood’s rule (1), and discuss the quality of service
implications of using Littlewood’s rule
under
competition.
4
2.4
Formal Proof of Littlewood’s Rule
For any function f (y) over the integers, let ∆f (y) = f (y) − f (y
− 1). The following result will help us to
determine ∆V (y) and ∆W (y, c) = W (y, c) − W (y − 1, c).
Lemma 1 Let g(y) = EG(min(X, y)) where X is an integer
valued random variable with E[X] < ∞ and G
is an arbitrary function defined over the integers. Then
∆g(y) = ∆G(y)P (X ≥ y).
Let r(y) = ER(max(X, y)) where X is an integer valued random
variable with E[X] < ∞ and R is an
arbitrary
function defined over the integers. Then
46. ∆r(y) = ∆R(y)P (X < y).
An application of the Lemma 1 yields the following proposition
that provides the desired formulas for
∆V1(y) and ∆W (y, c).
Proposition 1
∆V1(y) = p1P (D1 ≥ y)
y ∈ {1, . . . , }
∆W (y, c) = [∆V1(y) − p2]P (D2 > c − y)
y ∈ {1, . . . , c}.
The proof of the Lemma 1 and Proposition 1 are relegated to the
Appendix. With the help of Proposition 1
we can now formally establish the main result for the Two-Fare
Problem.
Theorem 1 The function W (y, c) is unimodal in y and is
maximized at y(c) = min(y1, c) where
y1 = max{y ∈ N : ∆V1(y) > p2}.
Moreover, V2(c) = W (y(c), c).
Proof: Consider the expression in brackets for ∆W (y, c) and
notice that the sign of ∆W (y, c) is
determined
47. by ∆V1(y) − p2 as P (D2 > c − y) ≥ 0.Thus W (y, c) ≥ W (y − 1,
c) as long as ∆V1(y) − p2 > 0 and W (y,
c) ≤
W (y − 1, c) as long as ∆V1(y) − p2 ≤ 0. Since ∆V1(y) = p1P
(D1 ≥ y) is decreasing1 in y, ∆V1(y) − p2
changes
signs from + to − since ∆V1(0) − p2 = p1P (D1 ≥ 0) − p2 = p1 −
p2 > 0 and limy→∞[∆V1(y) − p2] =
−p2.
This means that W (y, c) is unimodal in y. Then
y1 = max{y ∈ N : ∆V1(y) > p2}.
coincides with Littlewood’s rule (1).
When restricted to {0, 1, . . . , c}, W (y, c) is maximized at y(c)
=
min(c, y1). Consequently, V2(c) = maxy∈ {0,1,...,c} W (y, c) =
W (y(c), c), completing the proof.
2.5
Quality of Service, Spill Penalties, Callable Products and
Salvage Values
Since max(y1, c − D2) units of capacity are available for fare
class 1, at least one fare class 1 customer
will be
denied capacity when D1 > max(y1, c − D2). The probability of
this happening is a measure of the quality
of
48. service to fare class 1, known as the full-fare spill rate.
Brumelle et al. [4] have observed that
P (D1 > max(y1, c − D2)) ≤ P (D1 > y1) ≤ r < P (D1 ≥ y1).
(4)
They call P (D1 > y1) the maximal spill rate. Notice that if the
inequality y1 ≥ c − D2 holds with high
probability, as it typically does in practice when D2 is large
relative to c, then the spill rate approaches
the
1We use the term increasing and decreasing in the weak sense.
5
maximal flight spill rate which is, by design, close to the ratio
r. High spill rates may lead to the loss of
full-fare customers to competition. To see this, imagine two
airlines each offering a discount fare and a
full-
fare in the same market where the fare ratio r is high and
demand from fare class 2 is high. Suppose
Airline
A practices tactically optimal Revenue Management by applying
Littlewood’s rule with spill rates close
to
49. r. Airline B can protect more seats than recommended by
Littlewood’s rule. By doing this Airline B will
sacrifice revenues in the short run but will attract some of the
full-fare customers spilled by Airline A.
Over
time, Airline A may see a decrease in full-fare demand as a
secular change and protect even fewer seats
for
full-fare passengers. In the meantime, Airline B will see an
increase in full-fare demand at which time it
can
set tactically optimal protection levels and derive higher
revenues in the long-run. In essence, Airline B
has
(correctly) traded discount-fare customers for full-fare
customers with Airline A.
One way to cope with high spill rates and its adverse strategic
consequences is to impose a penalty cost
ρ for each unit of full-fare demand in excess of the protection
level. This penalty is suppose to measure
the
ill-will incurred when capacity is denied to a full-fare customer.
This results in a modified value function
V1(y) = p1E min(D1, y) − ρE[(D1 − y)+] = (p1 + ρ)E min(D1,
y) − ρED1. From this it is easy to see that
50. ∆V1(y) = (p + ρ)P (D1 ≥ y), resulting in
∆W (y, c) = [(p1 + ρ)P (D1 ≥ y) − p2]P (D2 > c − y)
and
p2
y1 = max y ∈ N : P (D1 ≥ y) >
.
(5)
p1 + ρ
Notice that this is just Littlewood’s rule applied to fares p1 + ρ
and p2, resulting in fare ratio p2/(p1 + ρ)
and,
consequently, lower maximal spill rates. Obviously this
adjustment comes at the expense of having higher
protection levels and therefore lower sales at the discount-fare
and lower overall revenues.
Consequently, an
airline that wants to protect its full-fare market by imposing a
penalty on rejected full-fare demand does it
at
the expense of making less available capacity for the discount-
fare and less expected revenue. One way to
avoid
51. sacrificing sales at the discount-fare and improve the spill rate
at the same time is to modify the discount-
fare
by adding a restriction that allows the provider to recall or buy
back capacity when needed. This leads to
revenue management with callable products; see Gallego, Kou
and Phillips [12]. Callable products can
be
sold either by giving customers an upfront discount or by giving
them a compensation if and when
capacity
is recalled. If managed correctly, callable products can lead to
better capacity utilization, better service
to
full-fare customers and to demand induction from customers
who are attracted to either the upfront
discount
or to the compensation if their capacity is recalled.
The value function V1(y) may also be modified to account for
salvage values (also known as the
‘distressed
inventory problem’). Suppose there is a salvage value s < p2 on
excess capacity after the arrival of the
full-fare
demand (think of standby tickets or last-minute travel deals).
52. We can handle this case by modifying V1(y)
to
account for the salvaged units. Then V1(y) = p1E min(D1, y) +
sE(y − D1)+ = (p1 − s)E min(D1, y) + sy,
so
∆V1(y) = (p1 − s)P (D1 ≥ y) + s, resulting in
∆W (y, c) = [(p1 − s)P (D1 ≥ y) − (p2 − s)]P (D2 > c − y)
and
p2 − s
y1 = max y ∈ N : P (D1 ≥ y) >
.
(6)
p1 − s
Notice that this is just Littlewood’s rule applied to net fares p1
− s and p2 − s. This suggests that a
problem
with salvage values can be converted into a problem without
salvage values by using net fares pi ← pi −
s,
i = 1, 2 and then adding cs to the resulting optimal expected
revenue V2(c) in excess of salvage values.
3
53. Multiple Fare Classes: Exact
Solution
In this section we present an exact solution to the muli-fare
class problem using dynamic programming.
We
assume that the capacity provider has c units of perishable
capacity to be allocated among n fares indexed
6
so pn < . . . < p1. Lower fares typically have severe time of
purchase and traveling restrictions and may
have
restricted advanced selection that denies access to the more
desirable capacity. Given the time-of-
54. purchase
restriction, it is natural to assume that demands for fare classes
arrive in n stages, with fare class n
arriving
first, followed by n − 1, with fare class 1 arriving last. Let Dj
denote the random demand for fare class
j ∈ N = {1, . . . , n}. We assume that, conditional on the given
fares, the demands D1, . . . , Dn are
independent
random variables with finite means µj = E[Dj] j ∈ N . The
independent assumption is approximately
valid
in situations where fares are well spread and there are
alternative sources of capacity. Indeed, a customer
who finds his preferred fare closed is more likely to buy the
same fare for an alternative flight (perhaps
with
a competing carrier) rather than buying up to the next fare class
55. if the difference in fare is high. The case
of
dependent demands, where fare closures may result in demand
recapture, will be treated in a different
chapter.
The use of Dynamic Programming for the multi-fare problem
with discrete demands is due to Wollmer
[22].
Curry [6] derives optimality conditions when demands are
assumed to follow a continuos distribution.
Brumelle
and McGill [5] allow for either discrete or continuous demand
distributions and makes a connection with
the
theory of optimal stopping.
Let Vj(x) denote the maximum expected revenue that can be
obtained from x ∈ {0, 1, . . . , c} units of
capacity from fare classes {j, . . . , 1}. The sequence of events
56. for stage j are as follows:
1. Decide the protection level, say y ≤ x, for fares j − 1, j − 2, .
. . , 1 thus allowing at most x − y units of
fare j demand.
2. The realization of the demand Dj occurs, and we observe
min(Dj, x − y) sales at fare j.
3. The revenue pj min(Dj, x−y) is collected, and we proceed to
the beginning of stage j −1 with a
remaining
capacity of max(x − Dj, y).
The revenue from this process is
Wj(y, x) = pjE min(Dj, x − y) + EVj−1 (max(x − Dj, y)) .
(7)
57. We can think of Wj(y, x) as the expected revenue from x units
of capacity prior to seeing the demand for
fare
class j when up to x − y units are allowed to book at fare j and
an optimal policy is followed thereafter.
This
leads to the dynamic programming recursion
Vj(x)
=
max
Wj(y, x)
y∈ {0,1,...,x}
=
max
{pjE min(Dj, x − y) + EVj−1(max(x − Dj, y))} .
58. (8)
y∈ {0,1,...,x}
The dynamic program simply states that the optimal value
function is the sum of the expected revenues
from fare class j plus the expected revenues from fare classes j
− 1, . . . , 1 evaluated at the protection
level that
maximizes this sum. Notice that once we are at the beginning of
stage j − 1 we face a similar problem
over the
remaining j − 1 fare classes. Vj(x) is then the maximum
expected revenue that can be obtained from x units
of
capacity for the j-fare problem. Consequently Vn(c) is the
maximum expected revenue for the n-fare
59. problem
with capacity c. The recursion can be started with V0(x) = 0 if
there are no salvage values or penalties
for
spill. Alternatively, the recursion can start with V1(x) = p1E
min(D1, x) + sE[(x − D1)+] − ρE[(D1 −
x)+] for
x ≥ 0 if there is a salvage value s per unit of excess capacity
and a penalty ρ per unit of fare class 1
demand
that is denied.
3.1
Structure of the Optimal Policy
In order to analyze the structure of the optimal policy, we begin
by describing a few properties of the
value
function.
60. As a convention we set V0 ≡ 0. A function V (y) defined on y ∈
N is concave if ∆V (y) =
V (y) − V (y − 1) is decreasing in y ∈ N+.
7
Lemma 2 For any j ≥ 1,
a) ∆Vj(y) = Vj(y) − Vj(y − 1) is decreasing in y ∈ N+, so the
marginal value of capacity is diminishing.
b) ∆Vj(y) is increasing in j ∈ {1, . . . , n} so the marginal value
of capacity increases when we have more
stages to go.
The proof of Lemma 2 is in the Appendix. Using the Lemma we
can characterize an optimal policy as
stated in the following theorem. For the purpose of simplifying
notation we will extend the definition of
∆Vj(y)
61. to y = 0 by setting ∆Vj(0) = ∆V1(0) = p1 just as we did for j =
1.
Theorem 2 The function Wj(y, x) is unimodal in y and it is
maximized at min(yj−1, c), where the nested
protection levels 0 = y0 ≤ y1 ≤ y2 ≤ · · · ≤ yn−1 are given by
yj = max{y ∈ N : ∆Vj(y) > pj+1} j = 1, . . . , n − 1.
(9)
The optimal value functions are given by
Vj(x) = Wj(min(x, yj−1), x) j = 1, . . . , n, x ∈ N .
(10)
Moreover, Vj(x) is concave in x ∈ N for each j = 1, . . . , n.
Proof: An algebraic argument similar to that used to justify
Littlewood’s rule for n = 2, reveals that for
62. y ∈ {1, . . . , x}
∆Wj(y, x) = Wj(y, x) − Wj(y − 1, x) = [∆Vj−1(y) − pj] P (Dj > x
− y).
Let yj−1 = max{y ∈ N : ∆Vj−1(y) > pj}. By part a) of Lemma 2,
∆Vj−1(y) is decreasing in y so ∆Vj−1(y)
−
pj > 0 for all y ≤ yj−1 and ∆Vj−1(y) − pj ≤ 0 for all y > yj−1.
Consequently, if x ≤ yj−1 then ∆Wj(y, x) ≥
0
for all y ∈ {1, . . . , x} implying that Vj(x) = Wj(x, x).
Alternatively, if x > yj−1 then ∆Wj(y, x) ≥ 0
for y ∈ {1, . . . , yj} and ∆Wj(y, x) ≤ 0 for y ∈ {yj−1 + 1, . . . ,
x} implying Vj(x) = Wj(yj−1, x). Since
∆Vj(x) = ∆Vj−1(x) on x ≤ yj−1, it follows that ∆Vj(yj−1) =
∆Vj−1(yj−1) > pj > pj+1, so yj ≥ yj−1. The
concavity of Vj(x) is is equivalent to ∆Vj(x) decreasing in x,
and this follows directly from part a) of
Lemma 2.
63. Remarks:
1. Notice that the unconstrained protection level yj−1 is
independent of the demands Dk, k ≥ j as observed
before in the two fare setting (Littlewood’s Rule).
2. We can think of yj, j = 1, . . . , n − 1 as the unconstrained
protection levels. If we start stage j with xj
units of capacity, the constrained protection level for fares {j −
1, . . . , 1} is min(xj, yj−1). Thus capacity
is made available to fare j only if xj > yj−1.
3. The policy is implemented as follows. At stage n we start
with xn = c units of inventory, and we protect
yn−1(xn) = min(xn, yn−1) units of capacity for fares {n − 1, . . .
, 1} by allowing up to (xn − yn−1)+ units
to be sold at fare pn. Since min(Dn, (xn −yn−1)+) units are sold
during stage n, we start stage n−1 with
64. xn−1 = xn − min(Dn, (xn − yn−1)+). We protect yn−2(xn−1) =
min(xn−1, yn−2) units of capacity for
fares {n − 2, . . . , 1} and thus allow up to (xn−1 −yn−2)+ units
of capacity to be sold at pn−1. The
process
continues until we reach stage one with x1 units of capacity and
allow (x1 − y0)+ = (x1 − 0)+ = x1
to be sold at p1. Assuming discrete distributions, the
computational requirement to solve the dynamic
program for the n stages has been estimated by Talluri and van
Ryzin [20] to be of order O(nc2).
4. The concavity of Vn(c) is helpful if capacity can be procured
at a linear or convex cost because in this
case the problem of finding an optimal capacity level is a
concave problem in c.
73. Example 3. Suppose there are five different fare classes. We
assume the demand for each of the fares is
Poisson. The fares and the expected demands are given in the
first two columns of Table 2. The third
column
includes the optimal protection levels for fares 1, 2, 3 and 4.
j
pj
E[Dj]
yj
1
$100
15
75. 169
5
$15
120
Table 2: Five Fare Example with Poisson Demands: Data and
Optimal Protection Levels
Table 3 provides the expected revenues for different capacity
levels as well as the corresponding demand
factors (
5
E[D
j=1
j ])/c = 280/c.
76. These results should be intuitive. Greater revenue potential is
seen as
capacity increases (since potentially more demand can be
accepted). Further, the effect of restrictions on
discounted fares is apparent in the pattern of revenue across
classes; e.g. revenue V2(50) through V5(50)
is
$3,426.8 because fare classes 3,4, and 5 are rationed since y2 =
54 > c = 50 units are protected for fare 1
and
2. However, V1(350) through V5(350) vary from $1,500 to
$9,625 because there is sufficient capacity to
accept
sales in all fare classes.
c
DF
83. 80
100
120
140
160
Figure 1: ∆Vj(x), x = 1, . . . , 350, j = 1, 2, 3, 4, 5 for Example
3
9
3.2
Speeding up the Computation of the Value Function
While the value functions Vj(x), j ∈ {1, . . . , n}, x ∈ {1, . . .
c}, can be computed recursively there are
some
84. tricks to speed up the computations. Here we focus on how to
efficiently update ∆Vj+1(x) from ∆Vj(x).
The
key idea is to express ∆Vj+1(x) for x > yj in terms of previously
computed values of ∆Vj(x). The proof of
Proposition 2 is in the Appendix.
Proposition 2
∆V
∆V
j (x)
if x = 1, . . . , yj
j+1(x) =
E min(∆Vj(x − Dj+1), pj+1)
if x = yj + 1, . . ..
85. Since ∆Vj+1(x) = ∆Vj(x) for x ≤ yj, we only need to worry
about ∆Vj+1(x) for x > yj. The following
corollary to Proposition 2 makes the formula for ∆Vj+1(x), x >
yj more explicit.
Corollary 1
k−1
∆Vj+1(yj + k) = pj+1P (Dj+1 ≥ k) +
∆Vj(yj + k − i)P (Dj+1 = i)
k ∈ {1, . . . , c − yj}.
i=0
3.3
Linear and Convex Procurement Costs
Suppose that capacity c can be procured at a linear cost kc
86. before observing demands for the n fares: pn <
pn−1 < . . . < p1. How much capacity should be procured? The
objective is to find c to maximize Πn(c, k)
=
Vn(c) − kc. Let c(k) be the smallest optimizer of Πn(c, k) as a
function of the marginal cost k. The
following
Proposition characterizes c(k), shows that c(k) is decreasing in
k and relates c(pj+1) to protection level
yj for
j < n.
Proposition 3 The optimal procurement quantity at linear cost
kc is given by
c(k) = max{c ∈ N : ∆Vn(c) > k}.
Moreover, c(k) is decreasing in k, and yj = c(pj+1) for all j ∈
{1, . . . , n − 1}.
Clearly c(0) = ∞ since ∆Vn(c) ≥ ∆V1(c) = p1P (D1 ≥ c) > 0 for
87. all c ∈ N . At the other extreme,
c(p1) = y0 = 0 since ∆Vn(1) = p1P (D1 ≥ 1) < p1 = ∆Vn(0), so
no capacity would be purchased if k ≥ p1.
If the cost of capacity k(c) is increasing convex then Π(c, k(c))
is concave in c and
c(k) = max{c ∈ N : ∆Vn(c) − ∆k(c) > 0}.
Consider the convex cost function k(c) = K if c ≤ ¯
c and k(c) = ∞ for k > ¯
c. This may reflect the situation
where there is a fixed cost to leasing a resource with capacity ¯
c. The optimal choice is then to lease the
resource if Vn(¯
c) > K and not lease it otherwise.
88. 3.4
Relaxing the Monotonicity of Fares
We continue to assume that demands arrive in the order Dn,
Dn−1, . . . , D1 but will relax the assumption
that
the fares are monotone p1 > p2 > . . . > pn. All of the results
work as stated, except the monotonicity of the
protection levels, if we redefine ∆Vj(0) = max{p1, . . . , pj}. It
is also possible to skip some of the
optimization
10
optimal capacity
450
400
350
90. 60
70
80
90
100
cost
Figure 2: Optimal Capacity as a Function of Cost for the Data
of Example 3
steps as it is clear that yj−1 = 0 whenever pj > max(p1, . . . ,
pj−1) since it is optimal to allow all
bookings at
fare pj. The reader is referred to Robinson [18] for more details.
As an example, suppose that p3 < p2 >
p1.
91. Then V1(x) = p1E min(D1, x) and at stage 2 the decision is y1 =
0, so
V2(x) = p2E min(D2, x) + p1E min(D1, (x − D2)+).
Notice that
∆V2(x) = p2P (D2 ≥ x) + p1P (D2 < x ≤ D[1, 2])
so
y2 = max{y ∈ N : ∆V2(y) > p3}.
Notice that the capacity protected for fare 2 is higher than it
would be if there was no demand at fare 1.
4
Multiple Fare Classes: Commonly Used Heuristics
Several heuristics, essentially extensions of Littlewood’s rule,
were developed in the 1980’s. The most
important
92. heuristics are known as EMSR-a and EMSR-b, where EMSR
stands for expected marginal seat revenue.
Credit
for these heuristics is sometimes given to the American Airlines
team working on revenue management
problems
shortly after deregulation. The first published account of these
heuristics appear in Simpson [19] and
Belobaba
[2], [3]. For a while, some of these heuristics were even thought
to be optimal by their proponents until
optimal
policies based on dynamic programming were discovered in the
1990’s. By then heuristics were already
part of
implemented systems, and industry practitioners were reluctant
to replace them with the solutions
provided
93. by dynamic programming algorithms. There are several reasons
for this. First, people feel more
comfortable
with something they understand. Also, the performance gap
between the heuristics and the optimal
dynamic
program tends to be small. Finally, there is a feeling among
some users that the heuristics may be more
robust
to estimates of the mean and variance of demand.
Version a of the heuristic, EMSR-a, is based on the idea of
adding protection levels produced by applying
Littlewood’s rule to successive pairs of classes. At state j, we
need to decide how much capacity to
protect
for fares j − 1, . . . , 1. We can use Littlewood’s rule to decide
how much capacity to protect for fare k
demand
94. against fare j for k = j − 1, . . . , 1 and then add the protection
levels. More precisely, let rk,j = pj/pk and
set
yk,j = max{y ∈ N : P (Dk ≥ y) > rk,j}.
11
Then the EMSR-a heuristic will protect
j−1
ya
=
y
j−1
k,j
k=1
95. units of capacity for fares j − 1, . . . , 1 against fare j.
In particular, if Dk is Normal with mean µk and standard
deviation σk, then
j−1
ya
= µ[1, j − 1] +
σ
j−1
k Φ−1(1 − rk,j ),
k=1
where for any j, µ[1, j − 1] =
j−1 µ
96. k=1
k and sums over empty sets are zero.
Notice that the EMSR-a heuristic involves j − 1 calls to
Littlewood’s rule to find the protection level for
fares j − 1, . . . , 1. In contrast, the EMSR-b heuristic is based
on a single call to Littlewood’s rule for
each
protection level. However, using the EMSR-b heuristic requires
the distribution of D[1, j − 1] =
j−1 D
k=1
k .
This typically requires computing a convolution but in some
cases, such as the Normal or the Poisson, the
distribution of D[1, j − 1] can be easily obtained (because sums
of independent Normal or Poisson
97. random
variables are, respectively, Normal or Poisson). The distribution
of D[1, j − 1] is used together with the
weighted average fare
j−1
µk
¯
pj−1 =
pk µ[1, j − 1]
k=1
and calls on Littlewood’s rule to obtain protection level
yb
98. = max{y ∈ N : P (D[1, j − 1] ≥ y) > rb
}
j−1
j−1,j
where rb
= p
j−1,j
j / ¯
pj−1. Notice that the weighted average fare assumes that a
proportion µk/µ[1, j − 1] of the
protected capacity will be sold at fare pk, k = 1, . . . , j − 1. In
the special case when demands are Normal
we
obtain
99. yb
= µ[1, j − 1] + σ[1, j − 1]Φ−1(1 − rb
).
j−1
j−1,j
Recall that for the Normal, variances are additive, so the
standard deviation σ[1, j − 1] =
j−1 σ2.
k=1
k
4.1
Evaluating the Performance of Heuristics
100. While heuristic protection levels are easy to compute,
evaluating them is as hard as solving for the
optimal
policy. The expected return of a heuristic policy based on
protection levels 0 = yh ≤ yh ≤ . . . ≤ yh
can be
0
1
n−1
computed exactly or via simulation. To compute the expected
revenues exactly, let V h(x) be the expected
profit
j
from x units of capacity from fares {j, . . . , 1} under the
heuristic policy h. Then V h(x) = W h(min(x, yh
), x)
101. j
j
j−1
where for all y ≤ x we define W h(y, x) = p
(max(y, x − D
j
j E min(x − y, Dj ) + EV h
j−1
j )). The following result
is a direct consequence of Proposition 2 and its corollary.
Proposition 4
102. ∆V h(x)
if x = 1, . . . , yh
j
j
∆V h (x) =
j+1
x−yh−1
= p
j
j+1P (Dj+1 ≥ x − yh) +
∆V h(x − i)P (D
j
i=0
103. j
j+1 = i)
if x > yh
j
Thus, if V h(x) has already been computed then V h (x) = V
h(x) for x = 1, . . . , yh. For x > yh we can
j
j+1
j
j
j
use the recursion V h (x) = V h (x − 1) + ∆V h (x) starting with
x = yh + 1 in conjunction with the second
104. j+1
j+1
j+1
j
part of Proposition 4.
12
To estimate the expected revenue and other measures of
performance, such as the variance, we can also
use Monte Carlo simulation. Suppose we generate many random
copies of simulated demands (D1, . . . ,
Dn).
For each copy we compute sales (sh, . . . , sh) and revenues Rh
=
n
105. p
under heuristic h. Averaging
1
n
i=1
ish
i
over all the values of Rh gives an estimate of V h(c). Simulated
sales can be generated sequentially via sh
=
n
j
min(Dj, (xj −yh
106. )+) starting with j = n and x
.
n−1
n = c and using the capacity update formula xj = xj+1 − sh
j+1
Example 3 (continued) We have applied the EMSR-a and
EMSR-b heuristics to the data of Example 3.
Table
4 repeats the data and reports the heuristic protection levels ya,
yb as well as the optimal protection
levels.
Table 5 reports V a(c) and V b(c) as well as V
5
5
107. 5(c) for values of c ∈ {50, 100, 150, 200, 250, 300, 350}.
j
pj
E[Dj]
ya
yb
y
j
j
j
1
$100
114. under a low-to-high arrival pattern. The heuristics continue to
perform well if demands are compound
Poisson
and aggregate demands are approximated by the use of a
Gamma distribution.
However, EMSR based
heuristics can significantly underperform relative to models that
allow more general fare arrival rates.
We will
have an opportunity to revisit this issue in Section 7.
5
Bounds, Revenue Opportunity Model, and New Heuristics
In this section we develop bounds on Vn(c) which may be
useful in evaluating the potential of applying
revenue
115. management solutions. To obtain an upper bound, consider the
perfect foresight problem where the
demand
vector D = (D1, . . . , Dn) is known in advance. This demand
knowledge allows us to optimally allocate
capacity
by solving the following knapsack type problem
n
V U (c, D)
=
max
p
n
k xk
(11)
117. xk
≥
0
∀
k = 1, . . . , n.
13
Clearly, for each realization of D, advance knowledge results in
revenues that are at least as high as the
optimal
dynamic policy that does not have perfect foresight. As a result,
Vn(c) ≤ EV U (c, D). For convenience,
we
n
will denote this upper bound as V U (c) = EV U (c, D).
118. n
n
The solution to (11) can be written explicitly as xk = min(Dk, (c
− D[1, k − 1])+), k = 1, . . . , n where
for convenience we define D[1, 0] = 0. The intuition here is that
we give priority to higher fares so fare
k ∈ {1, . . . , n} gets the residual capacity (c − D[1, k − 1])+.
The expected revenue can be written more
succinctly after a few algebraic calculations:
n
V U (c)
=
p
n
119. k E min(Dk , (c − D[1, k − 1])+)
(12)
k=1
n
=
pk (E min(D[1, k], c) − E min(D[1, k − 1], c))
k=1
n
=
(pk − pk+1)E min(D[1, k], c)
(13)
120. k=1
where for convenience we define pn+1 = 0. Moreover, since V
U (c, D) is concave in D, it follows from
Jensen’s
n
inequality that V U (c) = EV U (c, D) ≤ V U (c, µ) where V U
(c, µ) is the solution to formulation (11)
with
n
n
n
n
µ = E[D] instead of D. More precisely,
n
V U (c, µ)
122. ∀
k = 1, . . . , n
n
xk
≤
c
k=1
xk
≥
0
∀
k = 1, . . . , n.
The linear program (14) is known as the fluid model or the
123. deterministic capacity allocation problem. It
is
essentially a knapsack problem whose solution can be given in
closed form xk = min(µk, (c − µ[1, k −
1])+) for
all k = 1, . . . , n. Consequently, V U (c) ≤ V U (c, µ) =
n
(p
n
n
k=1
k − pk+1) min(µ[1, k], c).
A lower bound can be obtained by assuming a low to high
arrival pattern with zero protection levels.
124. This gives rise to sales min(Dk, (c − D[k + 1, n])+) at fare k =
1, . . . , n and revenue lower bound V L(c,
D) =
n
p
k=1
k E min(Dk , (c − D[k + 1, n])+).
Taking expectations we obtain
n
V L(c)
=
p
n
125. k E min(Dk , (c − D[k + 1, n])+)
(15)
k=1
n
=
pk (E min(D[k, n], c) − E min(D[k + 1, n], c))
k=1
n
=
(pk − pk−1)E min(D[k, n], c)
(16)
k=1
126. where p0 = 0. Notice that all of the terms in the sum are
negative except for k = 1. Clearly V L(c) ≤ V
n
n(c)
since the expected revenue is computed under sub-optimal
protection levels. The above arguments justify
the
main result of this section.
14
Proposition 5
V L(c) ≤ V
(c) ≤ V U (c, µ)
(17)
n
127. n(c) ≤ V U
n
n
Of course, the bounds require the computation of E min(D[1, k],
c), k = 1, . . . , n. However, this is often
an easy computation. Indeed, if D[1, k] is any non-negative
integer random variable then E min(D[1, k], c)
=
c
P (D[1, k] ≥ j). If D[1, k] is Normal we can take advantage of
the fact that E min(Z, z) = z(1 − Φ(z)) −
j=1
φ(z) when Z is a standard Normal random variable and φ is the
standard Normal density function. If
follows
128. that if D[1, k] is Normal with mean µ and variance σ2, then
E min(D[1, k], c) = µ + σE min(Z, z) = µ + σ [z(1 − Φ(z)) −
φ(z)]
where z = (c − µ)/σ.
Tables 6 and 7 report V L(c), V
(c) and V
n
n(c), V U
n
n(c, µ) for the data of Examples 4 and 5, respectively.
Notice that V U (c) represents a significant improvement over
the better known bound V
n
129. n(c, µ), particularly
for intermediate values of capacity. The spread V U (c) − V
L(c) between the lower and upper bound is a
n
n
gauge of the potential improvements in revenues from using an
optimal or heuristic admission control
policy.
When capacity is scarce relative to the potential demand, then
the relative gap is large, and the potential
for applying revenue management solutions is also relatively
large. This is because significant
improvements
in revenues can be obtained from rationing capacity to lower
fares. As capacity increases, the relative
gap
130. decreases indicating that less can be gained by rationing
capacity. At very high levels of capacity it is
optimal
to accept all requests, and at this point there is nothing to be
gained from the use of an optimal admission
control policy.
c
V L(c)
V
(c)
V U (c, µ)
n
n(c)
139. $106,370
160
$104,385
$104,390
$105,368
$106,370
Table 7: Optimal Revenue and Bounds for Example 5.
5.1
Revenue Opportunity Model
The bounds presented here can help with the so called Revenue
Opportunity Model (ROM). The revenue
opportunity is the spread between the optimal revenue, obtained
by hindsight using the estimated
140. uncensored
15
demand, and the revenue that results from not applying booking
controls. Demand uncensoring refers to a
sta-
tistical technique that attempts to estimate actual demand from
the observed sales which may be
constrained
by booking limits. The ex-post optimal revenue is a hindsight
optimization and is equivalent to our perfect
foresight model, resulting in revenue V U (c, D), where D is the
uncensored demand. On the other hand,
the
n
revenue based on not applying booking controls is just V L(c,
D), so a measure of the revenue opportunity
is
141. n
V U (c, D) − V L(c, D). The achieved revenue opportunity is the
difference between the actual revenue
from
n
n
applying optimal or heuristic controls and the lower bound. The
ratio of the achieved revenue opportunity
to
the revenue opportunity is often called the percentage achieved
revenue opportunity. The revenue
opportunity
V U (c, D) − V L(c, D) is sometimes approximated by V U (c) −
V L(c) to get an idea of the revenue
opportunity.
n
142. n
n
n
Table 6 and 7 shows there is significant revenue opportunity,
particularly for c ≤ 140. Thus, one use for
the
ROM is to identify situations where RM has the most potential
so that more effort can be put where is
most
needed. The ROM has also been used to show the benefits of
using leg-based control versus network-
based
controls. The reader is refer to Chandler and Ja ([8]) and to
Temath et al. ([21]) for further information on
the uses of the ROM.
5.2
143. Bounds Based Heuristic
It is common to use an approximation to the value function as a
heuristic. To do this, suppose that ˜
Vj(x) is
an approximation to Vj(x). Then a heuristic admission control
rule can be obtained as follows:
˜
yj = max{y ∈ N : ∆ ˜
Vj(y) > pj+1} j = 1, . . . , n − 1.
(18)
Suppose we approximate the value function Vj(x) by ˜
Vj(x) = θV L(x) + (1 − θ)V U (x) for some θ ∈ [0, 1]
144. j
j
and V L(x) and V U (x) are the bounds obtained in this section
applied to n = j and c = x. Notice that
j
j
∆V L(x) = p
(p
(x) =
j−1 (p
j
1P (D[1, j] ≥ x) +
j
145. k=2
k − pk−1)P (D[k, j] ≥ x), while ∆V U
j
k=1
k − pk+1)P (D[1, k] ≥
x) + pjP (D[1, j] ≥ x).
6
Multiple Fare Classes with Arbitrary Fare Arrival Patterns
So far we have suppressed the time dimension; the order of the
arrivals has provided us with stages that
are
a proxy for time, with the advance purchase restriction for fare j
serving as a mechanism to end stage j. In
this section we consider models where time is considered
explicitly. There are advantages of including
146. time as
part of the model as this allows for a more precise formulation
of the customer arrival process. For
example,
we can relax the low-to-high arrival assumption and allow for
overlapping or concurrent arrival rates. On
the
other hand, the flexibility advantage comes at the cost of
estimating arrival rates for each of the fare
classes
over the sales horizon. If arrival rates are not estimated
accurately, then adding the time dimension may
hurt
rather than help performance. In addition, the formulations
presented in this section assumes that demand
for each fare class follows a Poisson process, whereas our
earlier models based on sequential fare
147. arrivals do
not have this restriction. We will extend the formulation in this
section to the case of compound Poisson in
§7.
We assume that customers arrive to the system according to a
time heterogeneous Poisson process with
intensity λjt, 0 ≤ t ≤ T where T is the length of the horizon, t
represents the time-to-go and j ∈ {1, . . . , n}.
Then the number of customers that arrive during the last t units
of time and request product j, say Njt, is
t
Poisson with mean Λjt =
λ
0
jsds. For simplicity we will write Λj , instead of ΛjT , to denote
148. the expected
number of requests for fare j over the entire horizon [0, T ]. The
low-to-high arrival pattern can be
embedded
into the time varying model by dividing the selling horizon into
n sub-intervals [tj−1, tj], j = 1, . . . , n with
tj = jT /n, and setting λjt = nΛj/T over t ∈ [tj−1, tj] and λjt = 0
otherwise.
Let V (t, x) denote the maximum expected revenue that can be
attained over the last t units of the sale
horizon with x units of capacity. We will develop both discrete
and continuous time dynamic programs to
compute V (t, x). To construct a dynamic program we will need
the notion of functions that go to zero
faster
16
149. than their argument. More precisely, we say that a function g(x)
is o(x) if limx↓0 g(x)/x = 0. We will show
that the probability that over the interval [t − δt, t] there is
exactly one request and the request is for
product
j is of the form λjtδt + o(δt). To see this notice that the
probability that a customer arrives and requests
one
unit of product j over the interval [t − δt, t]is
λjtδ exp(−λjtδt) + o(δt) = λjtδt[1 − λjtδt] + o(δt) = λjtδt + o(δt),
while the probability that there are no requests for the other
products over the same interval is
exp(−
λktδt) + o(δt) = 1 −
λktδt + o(δt).
150. k=j
k=j
Multiplying the two terms and collecting terms we obtain λjtδt
+ o(δt) as claimed.
Recall that some fares have embedded time-of-purchase
restrictions. Let Nt ⊂ N = {1, . . . , n} to be the
set of allowable fares at time-to-go t. Usually Nt = N for large t,
but low fares are dropped from Nt as the
time-of-purchase restrictions become binding.
We can now write
V (t, x)
=
λjtδt max(pj + V (t − δt, x − 1), V (t − δt, x)) + (1 −
λjtδt)V (t − δt, x) + o(δt)
151. j∈ Nt
j∈ Nt
=
V (t − δt, x) + δt
λjt[pj − ∆V (t − δt, x)]+ + o(δt)
(19)
j∈ Nt
with boundary conditions V (t, 0) = 0 and V (0, x) = 0 for all x
≥ 0, where ∆V (t, x) = V (t, x) − V (t, x −
1) for
x ≥ 1 and t ≥ 0.
Subtracting V (t − δt, x) from both sides of equation (19),
dividing by δt and taking the limit as δt ↓ 0, we
152. obtain the following equation, known as the Hamilton Jacobi
Bellman (HJB) equation:
∂V (t, x) =
λjt[pj − ∆V (t, x)]+
(20)
∂t
j∈ Nt
with the same boundary conditions. The equation tells us that
the rate at which V (t, x) grows with t is the
weighted sum of the positive part of the fares net of the
marginal value of capacity ∆V (t, x) at state (t, x).
While the value function can be computed by solving and
pasting the differential equation (20), in practice
it is easier to understand and compute V (t, x) using a discrete
time dynamic programming formulation. A
153. discrete time dynamic programming formulation emerges from
(19) by rescaling time, setting δt = 1, and
dropping the o(δt) term. This can be done by selecting a > 1, so
that T ← aT is an integer, and setting
λjt ← 1 λ
λ
a
j,t/a, for t ∈ [0, aT ]. The scale factor a should be selected so
that, after scaling,
j∈ N
jt << 1,
t
e.g.,
154. λ
j∈ N
jt ≤ .01 for all t. The resulting dynamic program, after rescaling
time, is given by
t
V (t, x) = V (t − 1, x) +
λjt[pj − ∆V (t − 1, x)]+.
(21)
j∈ Nt
with the same boundary conditions. Computing V (t, x) via (21)
is quite easy and fairly accurate if time is
scaled appropriately. For each t, the complexity is order O(n)
for each x ∈ {1 . . . , c} so the complexity
per
period is O(nc), and the overall computational complexity is
155. O(ncT ).
A formulation equivalent to (21) was first proposed by Lee and
Hersh [13], who also show that ∆V (t, x)
is
increasing in t and decreasing in x. The intuition is that the
marginal value of capacity goes up if we have
more
time to sell and goes down when we have more units available
for sale. From the dynamic program (21),
it is
optimal to accept a request for product j when pj ≥ ∆V (t − 1, x)
or equivalently, when pj + V (t − 1, x −
1) ≥
V (t − 1, x), i.e., when the expecte revenue from accepting the
request exceeds the expected revenue of
denying
the request. Notice that if it is optimal to accept a request for
fare j, then it is also optimal to accept a
request
156. 17
for any higher fare. Indeed, if pk ≥ pj and pj ≥ ∆V (t − 1, x),
then pk ≥ ∆V (t − 1, x). Assuming that the
fares
are ordered: p1 ≥ p2 ≥ . . . ≥ pn, then it is optimal to accept all
fares in the active set A(t, x) = {1, . . . ,
a(t, x)}, where
a(t, x) = max{j ∈ Nt : pj ≥ ∆V (t − 1, x)}
t ≥ 1, x ≥ 1},
and to reject all fares in the complement R(t, x) = {j ∈ {1, . . . ,
n} : j > a(t, x)}. For convenience we
define
a(t, 0) = a(0, x) = 0 and A(t, 0) = A(0, x) = ∅ . For each time-to-
go t let the protection level for fares in
{1, . . . , j} be
157. yj(t) = max{x : a(t, x) = j},
so if x ≤ yj(t) then fares j + 1 and higher should be closed.
Proposition 6 The active set A(t, x) is decreasing in t and
increasing in x. Moreover, yj(t) is increasing in
j and increasing in t.
Proof: Both results follow directly from the fact that ∆V (t, x) is
increasing in t and decreasing in x.
That intuition is that A(t, x) is decreasing in t because it is
optimal to open fewer fares when we have
more
time to sell capacity at higher fares. The intuition that A(t, x) is
increasing in x is that it we may need open
more fares when we have more inventory. The intuition for yj(t)
to be monotone in j is that we should
protect
at least as many units for sales of fares in {1, . . . , j + 1} than
158. for sales of fares in {1, . . . , j}, so yj+1(t) ≥
yj(t).
The intuition for yj(t) to be monotone in t is that with more time
to sell, say t > t, we have the potential to
sell more from set {1, . . . , j} so at least as many units should
be protected: yj(t ) ≥ yj(t).
6.1
A Pricing Formulation with Broader Interpretation
At any time t, let λt =
n
λ
λ
j=1
159. jt be the overall arrival rate at time t.
Define πjt =
j
k=1
kt/λt and
rjt =
j
p
k=1
k λkt/λt.
We can think of πjt and rjt as the probability of sale and the
average revenue rate,
per arriving customer, when we offer all the fares in the
consecutive set Sj = {1, . . . , j}. For
160. convenience,
let π0t = r0t = 0 denote, respectively, the sales rate and the
revenue rate associated with S0 = ∅ . Now let
qjt = rjt/πjt be the average fare per unit sold when the offer set
is Sj. If πjt = 0, we define qjt = 0. This
implies that πjt[qjt − ∆V (t − 1, x)] is zero whenever πjt = 0,
e.g., when j = 0.
Let N + = N
t
t ∪ {0}. With this notation, we can write formulation (21) as
V (t, x)
=
V (t − 1, x) + λt max [rjt − πjt∆V (t − 1, x)]
j∈ N +
161. t
=
V (t − 1, x) + λt max πjt[qjt − ∆V (t − 1, x)].
(22)
j∈ N +
t
The reason to include 0 as a choice is that for j = 0, the term
vanishes and this allow us to drop the
positive
part that was present in formulation (21). The equivalent
formulation for the continuous time model (20)
is
∂V (t, x) = λt max πjt[qjt − ∆V (t, x)].
(23)
162. ∂t
j∈ N +
t
These formulation suggest that we are selecting among the
actions S0, S1, . . . , Sn to maximize the sales
rate
πjt times the average fare qjt net of the marginal value of
capacity (∆V (t − 1, x) for model (22) and ∆V (t,
x)
for model (23)). In essence, the problem has been reduced to a
pricing problem with a finite price menu.
Formulations (22) and (23) can be interpreted broadly as the
problem of optimizing the expected revenue
from state (t, x) where there are a finite number of actions.
These actions can be, as above, associated
with
163. offering products in the sets S0, S1, . . . , Sn, but other
interpretations are possible. For example, the
different
actions can be associated with offering a product at different
prices qjt, each price associated with a sales
rate
18
πjt for all j ∈ N+. This turns the capacity allocation problem
into a pricing problem with a finite price
menu.
We will come back to this pricing formulation when we discuss
the dynamic capacity allocation problem
with
dependent demands. We will see there that essentially the same
pricing formulation works for dependent
demands. For the case of dependent demands, the set N+ will be
164. the index corresponding to a collection
of
sets that is efficient in a sense that will be made precise later.
7
Compound Poisson Demands
The formulations of the dynamic programs (20) and (21),
implicitly assume that each request is for a
single
unit. Suppose instead, that each arrival is for a random number
of units. More specifically, suppose that
request for fare j are of random size Zj, and that the probability
mass function Pj(z) = P (Zj = z), z ≥ 1 is
known for each j. As before, we assume independent demands
for the different fare classes j ∈ N . We
seek
to generalize the dynamic programs (20) and (21) so that at each
state (t, x) we can decide whether or not
165. to
accept a fare pj request of size Zj = z. The expected revenue
from accepting the request is zpj + V (t − 1, x
− z)
and the expected revenue from rejecting the request is V (t − 1,
x). Let ∆zV (t, x) = V (t, x) − V (t, x − z)
for
all z ≤ x and ∆zV (t, x) = ∞ if z > x. We can think of ∆zV (t, x)
as a the sum of the the z marginal values
∆V (t, x) + ∆V (t, x − 1) + . . . + ∆V (t, x − z + 1).
The dynamic program (20) with compound Poisson demands is
given by
∞
∂V (t, x) =
λjt
Pj(z)[zpj − ∆zV (t, x)]+,
166. (24)
∂t
j∈ Nt
z=1
while the dynamic program (21) with compound Poisson
demands is given by
∞
V (t, x) = V (t − 1, x) +
λjt
Pj(z)[zpj − ∆zV (t − 1, x)]+,
(25)
j∈ Nt
167. z=1
with boundary conditions V (t, 0) = V (0, x) = 0. Notice that the
sums in (24) and (25) can be changed to
x
z=1
∞
instead of
as the terms z > x do not contribute to the sum given our
convention that ∆
z=1
z V (t, x) = ∞ for
x > z. The optimal policies for the two programs are,
respectively, to accept a size z request fare pj, j ∈
Nt,
168. if zpj ≥ ∆zV (t, x), and to accept a z request fare pj, j ∈ Nt, if
zpj ≥ ∆zV (t − 1, x). The two policies
should
largely coincide time is scaled correctly so that
λ
j∈ N
jt << 1 for all t ∈ [0, T ].
t
For compound Poisson demands, we can no longer claim that
the marginal value of capacity ∆V (t, x) is
decreasing in x, although it is still true that ∆V (t, x) is
increasing in t. To see why ∆V (t, x) is not
monotone in
x, consider a problem where the majority of the requests are for
two units and request are seldom for one
169. unit.
Then the marginal value of capacity for even values of x may be
larger than the marginal value of capacity
for
odd values of x. Consequently, some of the structure may be
lost. For example, it may be optimal to
accept a
request of a single unit of capacity when x is odd, but not if x is
even, violating the monotonicity of ∆V (t,
x).
However, even if some of the structure is lost, the computations
involved to solve (25) are
straightforward as
long as the distribution of Zj is known. Airlines, for example,
have a very good idea of the distribution of
Zj
for different fare classes that may depend on the market served.
Example 4. Consider again the data of Examples 3 with fares p1
170. = $100, p2 = $60, p3 = $40, p4 = $35
and
p5 = $15 with independent compound Poisson demands, with
uniform arrival rates λ1 = 15, λ2 = 40, λ3 =
50, λ4 = 55, λ5 = 120 over the horizon [0, 1]. We will assume
that Nt = N for all t ∈ [0, 1]. The aggregate
arrival rates are given by Λj = λjT = λj for all j. We will assume
that the distribution of the demand sizes
is
given by P (Z = 1) = 0.65, P (Z = 2) = 0.25, P (Z = 3) = 0.05
and P (Z = 4) = .05 for all fare classes.
Notice
that E[Z] = 1.5 and E[Z2] = 2.90, so the variance to mean ratio
is 1.933. We used the dynamic program
(25) with a rescaled time horizon T ← aT = 2, 800, and rescaled
arrival rates λj ← λj/a for all j. Table 8
provides the values V (T, c) for c ∈ {50, 100, 150, 200, 250,
300, 350}. Table 8 also provides the
171. values ∆V (t, x)
19
for t = 207 in the rescaled horizon for x ∈ {1, . . . , 6} to
illustrate the behavior of the policy. The reader
can
verify that at state (t, x) = (208, 3) it is optimal to accept a
request for one unit at fare p2, and to reject the
request if it is for two units. Conversely, if the state is (t, x) =
(208, 4) then it is optimal to reject a request
for one unit at fare p2, and to accept the request if it is for two
units. The reason for this is that the value
of
∆V (t, x) is not monotone decreasing at x = 4.
c
50
174. 54.62
50.41
Table 8: Value function V (T, c) and marginal revenues ∆V (t,
c) for Example 4: Compound Poisson
7.1
Static vs Dynamic Policies
Let Nj be the random number of request arrivals for fare j over
the horizon [0, T ], Then Nj is Poisson
with
T
parameter Λj =
λ
0
jtdt. Suppose each arrival is of random size Zj . Then the
175. aggregate demand, say Dj , for
fare j is equal to
Nj
Dj =
Zjk,
(26)
k=1
where Zjk is the size of the kth request. It is well known that
E[Dj] = E[Nj]E[Zj] = ΛjE[Zj] and that
Var[Dj] = E[Nj]E[Z2] = Λ
], where E[Z2] is the second moment of Z
j
176. j E[Z 2
j
j
j . Notice that Jensen’s inequality
implies that the the variance to mean ratio E[Z2]/E[Z
j
j ] ≥ E[Zj ] ≥ 1.
In practice, demands D1, . . . , Dn are fed, under the low-to-
high arrival assumption, into static policies to
compute Vj(c), j = 1, . . . , n and protection levels yj, j = 1, . . .
, n − 1 using the dynamic program (8), or
to
the EMSR-b heuristic to compute protection levels yb, . . . , yb
. Since the compound Poisson demands are
177. 1
n−1
difficult to deal with numerically, practitioners often
approximate the aggregate demands Dj by a Gamma
distribution with parameters αj and βj, such that αjβj =
E[Nj]E[Zj] and αjβ2 = E[N
], yielding
j
j ]E[Z 2
j
αj = ΛjE[Zj]2/E[Z2], and β
]/E[Z
j
178. j = E[Z 2
j
j ].
We are interested in comparing the expected revenues obtained
from static policies to those of dynamic
policies. More precisely, suppose that that demands are
compound poisson and Dj is given by (26) for
every
j = 1, . . . , n. Suppose that protection levels y1, y2, . . . , yn−1
are computed using the low-to-high static
dynamic program (8) and let yb, yb, . . . , yb
be the protection levels computed using the EMSR-b heuristic.
Protection
1
2
179. n−1
levels like these are often used in practice in situations where
the arrival rates λjt, t ∈ [0, T ], j = 1, . . . ,
n, are not necessarily low-to-high. Two possible
implementations are common. Under theft nesting a size
z request
for fare class j as state (t, x) is accepted if x − z ≥ yj−1. This
method is called theft nesting because the
remaining inventory x at time-to-go t is x = c − b[1, n] includes
all bookings up to time-to-go t, including
bookings b[1, j − 1]. Standard nesting counts only bookings for
lower fare classes and is implemented by
accepting a size z request for fare j at state (t, x) if x − z ≥
(yj−1 − b[1, j − 1])+, where b[1, j − 1] are the
observed bookings of fares [1, j − 1] up to state (t, x). When c >
yj−1 > b[1, j − 1], this is equivalent to
accepting a request for z units for fare j if c − b[j, n] − z ≥ yj−1,
or equivalently if b[j, n] + z ≤ c − yj−1,
180. so
only bookings of low fares count. In practice, standard nesting
works much better than theft nesting when
the arrival pattern is not low-to-high.
Notice that the expected revenue, say V s(T, c), resulting from
applying the static protection levels y1, . . .
, yn−1
with theft nesting is not, in general, equal to Vn(c), the optimal
expected revenue when the arrivals are
low-
to-high. Similarly, the expected revenue, say V b(T, c),
resulting from applying the EMSR-b protection
levels
yb, . . . , yb
with theft nesting is not, in general, equal to V b(c), the
181. expected revenue when the arrivals are
1
n−1
n
low-to-high.
The next proposition shows that V s(T, x) ≤ V (T, x), where V
(T, x) is the optimal expected revenue for
the
compound Poisson Dynamic Program. The same simple proof
can be used to show that V b(T, x) ≤ V (T,
x).
20
In fact, a proof is hardly needed as we are comparing heuristics
to optimal dynamic policies.
Proposition 7
182. V s(T, x) ≤ V (T, x) ∀ x ∈ {0, 1, . . . , c}.
Proof: Clearly for V s(0, x) = V (0, x) = 0 so the result holds for
t = 0, for all x ∈ {0, 1, . . . , c}. Suppose
the result holds for time-to-go t − 1, so V s(t − 1, x) ≤ V (t − 1,
x) for all x ∈ {0, 1, . . . , c}. We will show
that it also holds for time-to-go t. If a request of size z arrives
for fare class j, at state (t, x), the policy
based on
protection levels y1, . . . , yn−1 will accept the request if x − z
≥ (yj−1 − b[1, j − 1])+ and will rejected
otherwise.
In the following equations, we will use Qj(z) to denote P (Zj >
z). We have
x−(yj−1−b[1,j−1])+
V s(t, x)
=
λjt[
183. Pj(z)(zpj + V s(t − 1, x − z))
j∈ Nt
z=1
+
Qj(x − (yj−1 − b[1, j − 1])+)V s(t − 1, x)] + (1 −
λjt)V s(t − 1, x)
j∈ Nt
x−(yj−1−b[1,j−1])+
≤
λjt[
Pj(z)(zpj + V (t − 1, x − z))
185. ∞
≤
V (t − 1, x) +
λjt
Pj(z)(zpj − ∆zV (t − 1, x))+
j∈ Nt
z=1
=
V (t, x),
where the first equation follows from the application of the
protection level policy, the first inequality
follows
186. from the inductive hypothesis V s(t − 1, x) ≤ V (t − 1, x). The
second equality collects terms, the second
inequality follows because we are taking positive parts, and the
last equality from the definition of V (t,
x).
While we have shown that V s(T, c) ≤ V (T, c), one may wonder
whether there are conditions where
equality
holds. The following results answers this question.
Corollary 2 If the Dj’s are independent Poisson random
variables and the arrivals are low-to-high then
Vn(c) = V s(T, c) = V (T, c).
Proof: Notice that if the Djs are Poisson and the arrivals are
low-to-high, then we can stage the arrivals
so that λjt = nE[Dj]/T over t ∈ (tj−1, tj] where tj = jT /n for j =
1, . . . , n. We will show by induction in
j that Vj(x) = V (tj, x). Clearly y0 = 0 and V1(x) = p1E min(D1,
187. x) = V (t1, x) assuming a sufficiently
large
rescale factor. Suppose, by induction, that Vj−1(x) = V (tj−1,
x). Consider now an arrival at state (t, x)
with
t ∈ (tj−1, tj]. This means that an arrival, if any, will be for one
unit of fare j. The static policy will accept
this
request if x − 1 ≥ yj−1, or equivalently if x > yj−1. However, if
x > yj−1, then ∆(t − 1, x) ≥ ∆V (tj−1, x) ≥
pj, because ∆V (t, x) is increasing in t and because yj−1 =
max{y : ∆Vj−1(y) > pj} = max{y : ∆V (tj−1, x)
> pj},
by the inductive hypothesis. Conversely, if the dynamic
program accepts a request, then pj ≥ ∆V (t, x) and
therefore x > yj−1 on account of ∆V (t, x) ≥ ∆V (tj−1, x).
We have come a long way in this chapter and have surveyed
188. most of the models for the independent
demand case. Practitioners and proponents of static models,
have numerically compared the performance
of
static vs dynamic policies. Diwan [7], for example, compares
the performance of the EMSR-b heuristic
against
the performance of the dynamic formulation for Poisson
demands (21) even for cases where the aggregate
21
demands Dj, j = 1, . . . , n are not Poisson. Not surprisingly, this
heuristic use of (21) can underperform
relative
to the EMSR-b heuristic. However, as seen in Proposition 7, the
expected revenue under the optimal
dynamic
program (25) is always at least as large as the expected revenue
generated by any heuristic, including the
189. EMSR-b. In addition, the dynamic program does not require
assumptions about the arrival being low-to-
high
as the EMSR-b does. Even so, the EMSR-b heuristic performs
very well when the low-to-high
assumptions
hold. However, when the low-to-high assumptions are relaxed,
then the performance of the EMSR-b
heuristic
suffers relative to that of the dynamic program as illustrated by
the following example.
Example 5. Consider again the data of Example 4 with uniform
arrival rates. Table 9 compares the
perfor-
mance V (T, c) of the compound poisson formulation (25) to the
performance of the EMSR-b under
standard
nesting. Part of the gap between V b(T, c) and V (T, c) can be
190. reduced by frequently recomputing the
booking
limits applying the EMSR-b heuristic during the sales horizon.
c
50
100
150
200
250
300
V b(T, c)
$3,653
193. n(c), developed in Section 5 for the static multi-fare
model is still valid for V (T, c). The random revenue associated
with the perfect foresight model is Vn(c,
D)
and can be obtained by solving the linear program (11). Notice
that for all sample paths, this revenue is
at least as large as the revenue for the dynamic policy. Taking
expectations we obtain V (T, c) ≤ V U (c) =
n
EVn(c, D) =
n
(p
k=1
k − pk+1)E min(D[1, k], c), where for convenience pn+1 = 0.
Moreover, since dynamic
194. policies do at least as well as static policies, the lower bounds
obtained in Section 5 also apply to
dynamic
policies.
8
Monotonic Fare Offerings
The dynamic programs (20) and (21) and their counterparts (22)
and (23), all implicitly assume that fares
can be opened and closed at any time. To see how a closed fare
may reopen, suppose that a(t, x) = j so set
A(t, x) = {1, . . . , j} is offered at state (t, x), but an absence of
sales may trigger fare/action j + 1 to open
as
a(s, x) increases as the time-to-go s decreases. . This can lead to
the emergence of third parties that
specialize
195. on inter-temporal fare arbitrage. To avoid this capacity provider
may commit to a policy of never opening
fares
once they are closed. To handle monotonic fares requires
modifying the dynamic programming into
something
akin to the dynamic program (8) where time was handled
implicitly. Let Vj(t, x) be the maximum expected
revenue from state (t, x) when we can offer any consecutive
subset of open fares Sk = {1, . . . , k}, k ≤ j
and are
not allowed to reopen fares once they are closed. Let Wk(t, x)
be the expected revenue from accepting
fares
Sk at state (t, x) and then following an optimal policy. More
precisely,
k
197. k
λ
p
i=1
it and rkt =
j
i=1
iλit and pkt = rkt/πkt when
πkt > 0 and pkt = 0 otherwise.
22
Then Vj(t, x) satisfies the dynamic program
Vj(t, x) = max Wk(t, x) = max{Wj(t, x), Vj−1(t, x)}
198. (27)
k≤j
with the boundary conditions Vj(t, 0) = Vj(0, x) = 0 for all t ≥ 0
and all x ∈ N for all j = 1, . . . , n. Notice
that the optimization is over consecutive subsets Sk = {1, . . . ,
k}, k ≤ j. It follows immediately that Vj(t,
x)
is monotone increasing in j. An equivalent version of (27) for
the case n = 2 can be found in Weng and
Zheng [23]. The complexity to compute Vj(t, x), x = 1, . . . , c
for each j is O(c) so the complexity to
compute
Vj(t, x), j = 1, . . . , n, x = 1, . . . , c is O(nc). Since there are T
time periods the overall complexity is
O(ncT ).
While computing Vj(t, x) numerically is fairly simple, it is
satisfying to know more about the structure of
199. optimal policies as this gives both managerial insights and can
simplify computations. The proof of the
structural results are intricate and subtle, but they parallel the
results for the dynamic program (8) and
(21).
The following Lemma is the counterpart to Lemma 2 and uses
sample path arguments based on ideas in
[23]
to extend their results from n = 2 to general n. The proof can be
found in the Appendix.
Lemma 3 For any j ≥ 1,
a) ∆Vj(t, x) is decreasing in x ∈ N+, so the marginal value of
capacity is diminishing.
b) ∆Vj(t, x) is increasing in j ∈ {1, . . . , n} so the marginal
value of capacity increases when we have
200. more
stages to go.
c) ∆Vj(t, x) is increasing in t, so the marginal value of capacity
increases as the time-to-go increases.
Let
aj(t, x) = max{k ≤ j : Wk(t, x) = Vj(t, x)}.
In words, aj(t, x) is the index of the lowest open fare that is
optimal to post at state (t, x) if we are
allowed
to use any fares in Sj. Let
Aj(t, x) = {1, . . . , aj(t, x)}.
Then Aj(t, x) is the optimal set of fares to open at state (j, t, x).
Clearly Vi(t, x) = Vj(t, x) for all i ∈
{aj(t, x), . . . , j}. The following Lemma asserts that aj(t, x) is
monotone decreasing in t (it is optimal to
have
201. fewer open fares with more time-to-go and the same inventory),
monotone increasing in x (it is optimal to
have more open fares with more inventory and the same time-to-
go) and monotonically increasing in j.
Lemma 4 aj(t, x) is decreasing in t and increasing in x and j.
Moreover, aj(t, x) = k < j implies ai(t, x) = k
for all i ≥ k.
It is possible to think of the policy in terms of protection levels
and in terms of stopping sets. Indeed, let
Zj = {(t, x) : Vj(t, x) = Vj−1(t, x)}. We can think of Zj as the
stopping set for fare j as it is optimal to close
down fare j upon entering set Zj. For each t let yj(t) = max{x ∈
N : (t, x) ∈ Zj+1}. We can think of yj(t) as
the protection level for fares in Sj against higher fares. The
following result is the counterpart to Theorem
2.
202. Theorem 3
• Aj(t, x) is decreasing in t and increasing in x and j.
• Z1 ⊂ Z2 ⊂ . . . ⊂ Zn.
• yj(t) is increasing in t and in j.
• If x ≤ yj(t) then Vi(t, x) = Vj(t, x) for all i > j.
Proof: The properties of Aj(t, x) follow from the properties of
aj(t, x) established in Lemma 4. Zj = {(t, x)
:
aj(t, x) < j}. From Lemma 4, aj(t, x) < j implies that ai(t, x) < i
for all i > j, so Zj ⊂ Zi for all i > j. This
23
implies that yj(t) is increasing in j for any t ≥ 0. If t > t, then
aj+1(t , yj(t)) ≤ aj+1(t, yj(t)) < j + 1, so
yj(t ) ≥ yj(t). Since yj(t) ≤ yi(t) for all i > j, then x ≤ yj(t)
203. implies Vi+1(t, x) = Vi(t, x) for all i ≥ j and
therefore Vi(t, x) = Vj(t, x) for all i > j.
The policy is implemented as follows: The starting state is (n,
T, c) as we can use any of the fares {1, . . .
, n},
we have T units of time to go and c is the initial inventory. At
any state (j, t, x) we post fares Aj(t, x) =
{1, . . . , aj(t, x)}. If a unit is sold during period t the state is
updated to (aj(t, x), t − 1, x − 1) since all
fares in the set Aj(t, x) are allowed, the time-to-go is t − 1 and
the inventory is x − 1. If no sales occur
during period
t the state is updated to (aj(t, x), t − 1, x). The process
continues until either t = 0 or x = 0.
Example 6. Consider Example 1 again with 5 fares p1 = $100,
p2 = $60, p3 = $40, p4 = $35 and p5 =
$15
with independent Poisson demands with means Λ1 = 15, Λ2 =
204. 40, Λ3 = 50, Λ4 = 55 and Λ5 = 120 and
T = 1. The scaling factor was selected so that
5
Λ
i=1
i/a < .01 resulting in T ← aT = 2, 800.
We also
assume that the arrival rates are uniform over the horizon [0, T
], i.e., λj = Λj/T . In Table 10 we present
the expected revenues Vj(T, c), j = 1, . . . , 5 and V (T, c) for c
∈ {50, 100, 150, 200, 250}. The first row
is V5(c)
from Example 1. Notice that V5(c) ≤ V5(T, c). This is because
we here we are assuming uniform, rather
than
205. low-to-high arrivals. V (T, c) is even higher because we have
the flexibility of opening and closing fares
at
will. While the increase in expected revenues [V (T, c) − V5(T,
c)] due to the flexibility of opening and
closing
fares may be significant for some small values of c (it is 1.7%
for c = 50), attempting to go for this extra
revenue may invite strategic customers or third parties to
arbitrage the system. As such, it is not generally
recommended in practice.
c
50
100
150
211. Table 10: Expected Revenues V (T, c) with uniform arrival rates
To obtain a continuous time formulation, we can use the same
logic that lead to (20) to obtain
∂Vj(t, x)
rjt − πjt∆Vj(t, x)
if (t, x) /
∈ Zj−1
=
(28)
∂t
∂Vj−1(t,x)
if (t, x) ∈ Z
212. ∂t
j−1
with the same boundary conditions.
8.1
Mark-up and Mark-down Policies
We now go back to the broader pricing interpretation coupled
with the monotonic fare formulation (27).
In
many applications the price menu pjt, j = 1, . . . , n is time
invariant, but the associated sales rates πjt, j =
1, . . . , n are time varying. In addition, we will assume that
there is a price p0t such that π0t = 0 for all t.
This technicality helps with the formulation as a means of
turning off demand when the system runs out of
inventory. The case p1t ≥ p2t ≥ . . . ≥ pnt and π1t ≤ π2t ≤ . . . ≤
πnt is known as the mark-up problem,
213. while
the case p1t ≤ p2t ≤ . . . ≤ pnt and π1t ≥ π2t ≥ . . . ≥ πnt is
known as the mark-down problem. The former
model is relevant in Revenue Management while the second is
relevant in Retailing.
For the RM formulation, the problem can be viewed as
determining when to mark-up (switch from action
j to j − 1). The optimal mark-up times are random as they
depend on the evolution of sales under the
optimal policy. Suppose that the current state is (j, t, x), so the
last action was j, the time-to-go is t and the
inventory is x. We want to determine whether we should
continue using action j or switch to action j − 1.
We know that if x > yj−1(t), then we should keep action j and if
x ≤ yj−1(t) then we should close action
24
214. j. Let Zj = {(t, x) : x ≤ yj−1(t)}, then it is optimal to stop action
j upon first entering set Zj. Notice
that a mark-up occurs when the current inventory falls below a
curve, so low inventories trigger mark-
ups,
and mark-ups are triggered by sales. The retailing formulation
also has a threshold structure, but this time
a
mark-down is triggered by inventories that are high relative to a
curve, so the optimal timing of a mark-
down
is triggered by the absence of sales. Both the mark-up and the
mark-down problems can be studied from
the
point of view of stopping times. We refer the reader to Feng and
Gallego [9], [10], and Feng and Xiao
[11] and
215. reference therein for more on the markup and markdown
problems.
9
Acknowledgments
I acknowledge the feedback from my students and collaborators.
In particular, I would like to recognize
the
contributions and feedback from Anran Li, Lin Li, and Richard
Ratliff.
25
10
Appendix
Proof of Lemma 1. Notice that g(y) = G(y)P (X ≥ y) +
G(j)P (X = j), while g(y − 1) = G(y −
j≤y−1
216. 1)P (X ≥ y) +
G(j)P (X = j). Taking the difference yields ∆g(y) = G(y)P (X ≥
y). Notice that
j≤y−1
r(y) = R(y)P (X < y) +
R(j)P (X = j) while r(y − 1) = R(y − 1)P (X < y) +
R(j)P (X = j).
j≥y
j≥y
Taking the difference we see that ∆r(y) = ∆R(y)P (X < y).
Proof of Proposition 1. Let G(y) = p1y, then V1(y) = g(y) =
EG(min(D1, y)), so ∆V1(y) = ∆g(y) =