Empirical Evidence of Revenue Management in the Cruise Line Industry

Abstract

Revenue management (RM) has received considerable attention from both academic and business professionals. It encompasses several techniques regarding capacity allocation, pricing, and resource management of fixed, time-sensitive capacity. RM can be roughly divided into two categories defined by the control mechanism that increases revenue: capacity allocation or price optimization. Our work falls in the latter category. In our model, we allow for partial substitutability among products (e.g., a customer making a purchase decision may consider multiple alternatives—different departure dates, different destinations, different cabin types). We also include marketing expense in addition to prices as a lever for increasing revenue. These features are relevant to dynamic pricing in practice. The method is illustrated with booking data from a cruise company, yielding optimal advertising and prices for 300 products. The application of the model results in an increase in revenue in the range of 8%–20%.

Keywords

revenue management cruise industry multinomial choice model empirical application

Introduction

Revenue management is defined as the “practice of controlling the availability and/or pricing of travel seats in different booking classes with the goal of maximizing expected revenues” (McGill and van Ryzin 1999). As is clear from the definition, revenue management has mainly been developed for the travel and hospitality industries. It was first developed in the airline industry after the Deregulation Act of 1978, which dramatically changed the industry’s rules of competition. The first revenue management methodologies were developed by large airline companies requiring a new strategy to compete with the smaller low-price, low-cost carriers. These methodologies proved to be so efficient that the many small firms who did not employ revenue management techniques could no longer compete with the major players of the industry and consequently went out of business in a short time (Cross 1997).

After its first applications in the 1980s, revenue management has received considerable attention from both academic and business professionals. It encompasses several techniques regarding capacity allocation, pricing, and resource management of fixed, time-sensitive capacity. Revenue management can be roughly divided into two categories defined by the control mechanism that increases revenue: capacity allocation or price optimization. The airline industry mainly uses capacity allocation to increase revenue. Prices for a flight’s different fare classes are fixed over the intermediate term, and optimization algorithms focus on the number of tickets to reserve for high-paying and late-purchase fare classes. Capacity allocation limits for a flight’s different fare classes are dynamically updated over time, up to flight departure.

With some exceptions (e.g., see Biehn 2006; Ji and Mazzarella 2007), revenue management for hotel and cruise line applications has mainly focused on optimizing prices of different rooms (hotels) or cabins (cruises). Prices are dynamically updated over time by date of stay (hotel) or by date of departure (cruise). Our work falls into this second category of revenue management focused on price optimization. We define and analyze a model that includes two features new to the literature on revenue management via dynamic price optimization: (1) we allow for partial substitutability among products (e.g., a customer making a purchase decision may consider multiple alternatives—different departure dates, different destinations, different cabin types), (2) we include marketing expense in addition to prices as a lever for increasing revenues. These features are relevant to dynamic pricing in practice. We illustrate the practicality of our approach by using data from a cruise line firm. In particular, we use the firm’s purchase transaction history to estimate parameters of a discrete choice model for an array of offered products, and we solve the dynamic price and marketing expense problem over a rolling horizon. After providing additional background on the cruise line industry, we expand on these points below.

The cruise line industry has become one of the fastest-growing sectors in the hospitality and tourism market (Petrick, Tonner, and Quinn 2006; Hyun and Han 2015). The cruise industry is a unique line of business as it combines travel, transportation, accommodation, dining, onboard entertainment, recreation, and shore excursions at a set of ports (Ahmed et al. 2002; Kwortnik, Piccoli, and Applegate 2005). Although the modern cruise industry started in the 1960s, it is the fastest-growing segment in the travel industry with more than 2,100% growth since 1970. It still has a strong growth potential. In 2013, 21.3 million passengers were carried globally by cruise line providers, and it is estimated that the cruise market will welcome more than 62 million guests in 2017 (CLIA Report 2016). These trends are motivating cruise companies to expand their capacities by increasing the number of ships, offering new itineraries, or increasing the frequency of already existing ports of call. Carnival and Royal Caribbean, the two largest North American cruise companies, have both added many ships with capacities of 5,000 or more passengers in their fleets over the past few years (USA Today 2015 [Sloan 2015]).

While it is one of the most dynamic sectors of the hospitality and tourism industry, the cruise industry has so far received comparatively little attention from researchers (Petrick 2004; Yiw. Li and Kwortnik 2016; Sun, Gauri, and Webster 2011). Perhaps as Talluri and van Ryzin (2004b) argue, cruise ships were essentially regarded as floating hotels, and it was assumed that the existing models in the hotel revenue management literature were also readily applicable to the cruise line companies. However, the cruise line pricing problem involves distinct elements that do not exist (or are more limited in scope) in other tourism and travel sectors (Biehn 2006; Ji and Mazzarella 2007; Maddah et al. 2010; Joo et al. 2014). For instance, in cruise ships, the types of cabins offered (such as balcony, interior, and ocean view) differ based on several characteristics including size, location in ship, internal amenities etc., and this makes the problem more complicated than the quantity-based revenue management of a single resource (Ji and Mazzarella 2007; Maddah et al. 2010). A “product” in the cruise industry is defined by various features such as the cabin type, the departure date, the itinerary of the cruise, etc., and the price of products is an important factor that could impact each customer’s final selection from a set of offered products. Therefore, the pricing problem of cruise lines, which involves setting the prices of a variety of products simultaneously and updating these prices according to the new sales figures and updated forecasts every period, should involve customer choice among alternatives. Hence, the cruise line problem is rather dynamic and involves interdependent relationships. This calls for incorporating a customer choice element in the dynamic pricing problem of cruise lines. However, to our knowledge, the dynamic pricing problem of a revenue-maximizing firm where customers can select from a number of products with different appeals has been understudied in the revenue management literature (Vulcano, van Ryzin, and Chaar 2010). All common properties of revenue management (RM) are reported in the cruise line industry—segmented market, fixed capacity, perishable inventory, finite selling period, advance sales and stochastic demand—which makes it a natural candidate for RM applications (Gallego and van Ryzin 1997; Hamzaee and Vasigh 1997; McGill and van Ryzin 1999; Talluri and van Ryzin 2004a). Moreover, the advertisement activities have a significant effect on the total sales and average prices of cruise line products, and advertising works to build category demand instead of promoting specific products, prices, or destinations (Joo et al. 2014). Hence, the advertising component, which can determine the size of the customer pool, should be simultaneously considered.

There are several other distinctive features of cruise lines that add more complexity to the problem. For instance, the booking horizon for a typical cruise is long, comparable to those of airlines and hotels (up to a year in advance). For planning purposes, however, this horizon could be effectively reduced to a time window of 3–4 months, known as the “wave booking period,” the time interval in which the majority of cruise reservations arrive (CLIA Report 2006; Maddah et al. 2010). Moreover, cruise lines tend to sail at high occupancy rates (Maddah et al. 2010). In 2003, most major cruise lines were averaging an occupancy rate above 95%, whereas the occupancy rate for the hotel industry was only 59% (Toh, Rivers, and Ling 2005). Finally, for customer satisfaction, it is important to keep the prices within some limits and avoid extreme price movements over time. In practice, cruise companies try to avoid extreme price changes since this could result in the dilution of bookings and/or loss of goodwill (Talluri and van Ryzin 2004b). This is not the case for airline companies or hotels, and this feature of cruise lines has not been considered in previous revenue management literature.

This paper tries to narrow this gap in the literature by developing a pricing approach for cruise lines by mainly focusing on the customer choice element and three of the unique properties of this industry: (1) the long booking period, (2) the restriction on the price variations from week to week, and (3) the effect of promotion expense decisions (e.g., advertising) on the total revenues of the company. A substantial amount of proprietary booking data is obtained from a major cruise line company based in North America. As recognized by the company, prices and promotion interact to influence alignment between demand and capacity. In addition, products are substitutable across time (date of departure) and type (cabin characteristics), which leads to the natural choice of using the multinomial logit choice model to express the customer choices across various products. The problem is formulated as a nonlinear constrained optimization model with product prices and promotion expense as decision variables. An efficient solution method is presented and applied to the set of cruises offered by the company. The model is solved each period based on observed bookings. The method is illustrated with data from the company, yielding optimal advertising and prices for 300 products. The application of the model results in a considerable increase in revenues. The model is generally applicable to cruise companies and other companies operating in similar environments, namely, companies with long booking periods, low variances between prices in one period to the next period, and the combination of pricing and promotion decisions.

The contribution of this paper to the literature in revenue management is twofold. First, an optimization problem that involves the customer choice element is formulated and efficiently solved by building on recent theory of price setting in the widely used multinomial logit consumer choice model (Dong, Kouvelis, and Tian 2009; Vulcano, van Ryzin, and Chaar 2010; H. Li and Huh 2011). The results are illustrated on an example problem that is modeled using real-life data, contributing to the limited existent empirical research on the practicality of choice-based models. Considering the fact that there is still little empirical understanding of how choice behavior impacts RM settings, our model makes a significant contribution by studying the potential increase in revenues with the incorporation of the underlying choice behavior into the pricing model. Second, the advertisement element is embedded into the pricing problem, and to our knowledge, this type of joint pricing and advertising problem has not previously been considered in the literature, except for the studies concerned with the complementary effects of price reductions and increased advertising. Thus, the solution procedure and the results presented here could be insightful for the marketing and pricing teams of companies who need to work collaboratively in the pricing process.

The rest of the paper is organized as follows: the second section provides a literature review of relevant work on revenue management, dynamic pricing, and marketing analyses mainly focusing on the cruise line industry; the third section provides a description of the booking data obtained from a major North American cruise line operator and identifies patterns regarding the company’s pricing and advertising, as well as its number of bookings; then, in the fourth section, a model of the cruise line operator’s pricing and advertising problem is developed, and the solution procedure to obtain the best pricing and advertisement strategy is presented. The fifth section concludes with a summary and a discussion of future avenues for research.

Literature Review

Although the area of revenue management has received a considerable amount of attention from researchers of various backgrounds in the past two decades, the literature on the pricing and revenue management practices of cruise lines is still limited (Chiang, Chen, and Xu 2007; Maddah et al. 2010). The interested reader might find a broad review of the literature about the cruise industry from both marketing research and revenue optimization perspectives in the work of Sun, Gauri, and Webster (2011).

Most of the past work concerned with RM techniques in the cruise industry has focused on analytical models, with a few exceptions (Sun, Gauri, and Webster 2011; Joo et al. 2014). Hersh and Ladany (1988) employ a discrete dynamic programming formulation to determine the types of cruises offered and the optimal prices of each cruise. Leong and Ladany (2001) develop an integer-programming formulation to analyze the itinerary design problem of cruise operators while Ladany and Arbel (1991) characterize the optimal price differentiation strategy for passenger cabins on a cruise liner. Biehn (2006) develops a simple deterministic linear model for capacity control. Ji and Mazzarella (2007) examine how the nested class allocation and dynamic class allocation inventory control models can be applied to cruise inventory. Sun, Gauri, and Webster (2011) investigate the effectiveness of alternative forecasting methods for cruise lines. Uğurlu, Coşgun, and Ekinci (2012) consider the dynamic pricing optimization problem faced by a maritime transportation service provider company and aim to find optimal prices for each level of unsold seats in each journey by using probabilistic dynamic programming. Their results show the necessity of applying a dynamic pricing policy instead of fixed pricing and the diversification of optimal policies under different conditions. Chua et al. (2015) study cruise vacationers’ evaluations of onboard experiences with cruise lines in North America and their loyalty-formation process, and the researchers find that the perceived price was a significant predictor of the perceived value.

Among all papers focusing on the RM practices in the cruise industry, the ones most relevant to our work are those of Maddah et al. (2010) and Yih. Li, Miao, and Wang (2014). Maddah et al. (2010) develop a discrete-time dynamic capacity control model for a cruise ship characterized by multiple constraints on cabin and lifeboat capacities and propose heuristics for dynamic capacity management. The authors show that the opportunity cost of accepting a customer is not always monotone in the reservation levels or time, proving that “conventional” booking limits or critical time periods capacity control policies are not optimal. Despite the similarity of their problem to ours, their model differs from the model of this study in that they consider the revenue maximization of a single cruise ship and assume the prices are exogenous, whereas we consider the price-setting problem across multiple ships that compete for customers. A similar difference exists between our work and the work of Yih. Li, Miao, and Wang (2014), who develop an integer programming model with a linear objective function and constraints that can facilitate demand forecasting, which not only considers pricing and room assignment but also considers potential onboard expenses of customers.

The line of broader RM literature relevant for our problem is choice-based RM, which only started to receive attention recently. One of the pioneering works in the area is that of Talluri and van Ryzin (2004a), who provide an exact analysis of the optimal control policy for an airline company under a general discrete choice model of demand. Zhang and Cooper (2006) analyze choice among parallel flights having different departure times between the same O-D pair. They developed bounds and approximations to the resulting dynamic program. E. A. Boyd and Kallesen (2004) illustrate the effect of considering demand models for airlines where customers are price-sensitive and not perfectly segmented. Gallego et al. (2004) and Liu and van Ryzin (2008) study a deterministic network RM problem using a customer choice–based linear programming model. The dynamic programming (DP) approximation proposal of Zhang and Adelman (2009) and the DP decomposition scheme of Kunnumkal and Topaloglu (2010) also belong to this line of research.

Choice-based models are appealing for revenue management since they offer strong theoretical promise as a means to incorporate a realistic customer behavior directly into RM systems. Moreover, they provide the freedom to work with unrestricted fare class structures. However, most firms and software providers have been reluctant to make the multimillion-dollar investments required to transition to these new systems, because of difficulties associated with estimating choice model parameters. As quoted from Newman et al. (2014), “In a talk at the 2011 INFORMS annual meeting, Scott Nason, the former vice president of revenue management at American Airlines, said that the airlines will have to move to choice-based RM systems, but the current methodologies are not yet sufficient to allow airlines to implement choice-based RM systems at this time.”

Moreover, there is not much empirical evidence regarding the practicality and effectiveness of customer choice models in the RM literature. Even in the densely studied airline RM literature, the answers to the questions like “How significant is choice behavior in real markets? Can it be estimated well using available data? What are the potential revenue improvements from using choice-based RM?” are not explored in great detail, except for a few examples. Within this line of research lies the work of Vulcano, van Ryzin, and Chaar 2010), who use a simulation study to show the potential revenue improvements resulting from the use of choice-based RM models over traditional models by using the data from a major airline, as well as Farias, Jagabathula, and Shah (2013), who develop a data-driven, nonparametric approach to predict expected sales/revenues from a particular assortment of products. We refer the reader to Garrow (2016) for a broad review of theoretical and practical applications of the discrete-choice models in the airline industry. Bodea, Ferguson, and Garrow (2009) describe data collected from five U.S. properties of a major hotel chain that can be used to benchmark the performance of choice-based revenue management (RM) algorithms, and later Newman et al. (2014) use the same publicly available data to illustrate their model, which reexamines Talluri and van Ryzin’s (2004a) estimation procedure for the multinomial logit (MNL) discrete-choice model. Both papers draw attention to the difficulties in estimating choice parameters using real-life data.

Researchers in marketing literature have also explored the effect of advertising on revenues (Aaker 1991; Keller 1993; Dekimpe and Hanssens 1999). For a competitive market like the cruise market, the models involving rivalry are particularly relevant. As an example to this line of research, Clarke (1973) focuses on the analysis of advertising competition in an industry by providing estimates of the sales-advertising cross-elasticities from a system of market-share equations, and thus provides a means of explicitly measuring the effect of the advertising of one brand on the sales of another. Karnani (1985) is the first to investigate the practical implications of market share attraction models. Using a dynamic model of advertising rivalry between competitors in a duopoly, Erickson (1985) obtains analytical results for the case of pure market share rivalry in a mature market and provides a numerical analysis for a more general model, allowing for market expansion as well as market share rivalry. Bass et al. (2005) examine whether, when, and how much brand advertising versus generic advertising should be used. In a recent meta-analytic study about advertising brand elasticities, Sethuraman, Tellis, and Briesch (2011) find that mean short-term and long-term advertising elasticities are 0.12 and 0.24, respectively. Another relevant article that combined marketing and RM practices is the work of Mathies, Gudergan, and Wang (2013), who examine how the simultaneous use of customer-centric marketing (CCM) and revenue management (RM) affects travelers’ perceptions of fairness and, ultimately, travelers’ purchasing choices. The findings from these two empirical studies suggest that firms need to account not only for the effects of RM and CCM attributes but also for the corresponding reference-dependent fairness adjustments relating to those attributes.

In summary, this article offers two main contributions to narrow this gap in the literature. First, an optimization problem that involves customer choice element is formulated and efficiently solved by building on recent theory on the tractability of price setting in the widely used multinomial logit consumer choice model. Furthermore, the results are illustrated on an example problem that is modeled using real-life data. Second, the advertisement element is embedded into the pricing problem, and to our knowledge, this type of joint pricing and advertising problem has not previously been considered in the literature.

Data and Current Practice Description

In this section, the historical price and booking data are described to gain insights on the patterns relating to consumer purchase behavior and to identify the appropriate mathematical model for supporting cruise line pricing and advertising decisions.

Data

This paper uses proprietary booking data from a leading North American cruise line. The data consists of booking reservations for seven-day duration cruises that depart between December 2003 and January 2005. The data include sales of every two-person cabin on the focal company’s cruises departing from Miami, Florida, its most active market with approximately half a million bookings. Each booking record includes price paid, purchase date, departure date, itinerary, cabin type, consumer age, and zip code. A cruise itinerary is the specific sequence of sites/ports that are visited during the cruise.

The seven-day cruises sail on five distinct itineraries, and the operation of these cruises follows a pattern over periods of two weeks. All cruises depart from and return to Miami but follow different itineraries, visiting different islands in the Caribbean region. The different itineraries tend to be popular among the same set of consumers; for example, the products are substitutes. Tickets go on sale one year in advance, and the company offers up to six types of double-berth cabins. However, 84% of bookings take place within 20 weeks of departure, and 85% of all double-berth bookings are for one of three cabin types (interior, ocean view, and balcony). To be able to obtain enough observations to compute reasonably accurate parameter values, this analysis is limited to a time window of 20 weeks before departure (i.e., the wave booking period) and to the three most popular cabin types.

Advertising and Booking Pattern

According to the research reports on the cruise industry, the number of global passengers increased from 9.83 million in 2003 to 10.85 million in 2004 (CLIA Report 2006), which is the time frame of our data. Data on cruise company advertising expenditures were obtained from TNS Media Intelligence. The overall advertisement expenditures of the focal company reached a total of $87 million within a period of 52 weeks from 2003 to 2004. With this amount of advertisement expenditure, the company sold 230,000 tickets across the five itineraries.

Advertising at the company is largely generic and not specific to a particular itinerary. The purpose is to raise consumer awareness—to spur consumers to consider a cruise with the company for their next vacation. The firm invests in a variety of media and methods to increase awareness including TV, targeted print mailings (magazine and newspaper), and advertising on various sites on the Internet. The company does offer targeted discounts on specific cruises to certain segments of customers through travel agencies. The information about these discounts is not available, and hence this promotional aspect is not considered in the analysis.

Figure 1 shows the company’s number of bookings, the amount of advertisement expenditure, and the “average price” for each biweekly period between the weeks of January 2004 and August 2004. The average price in a period is computed as the weighted average of the prices obtained on each itinerary–cabin type pair, and the weight associated with each product is the ratio of the number of bookings on that particular itinerary–cabin type pair to the total number of bookings in that period.

Figure 1.

Total number of bookings, average prices, and total advertising expense of the firm for a two-week period.

The data of the total number of bookings show a comparably high number of bookings between the first week of January until the last week of March. These bookings are advance bookings, mostly for the summer-month cruises (departing from the last week of May to the last week of August). After March, a decreasing pattern of bookings for the cruises departing in September through December is seen until the next high season, around Christmas time. Moreover, an increasing average price suggests a decrease in the total number of bookings. Since we have a limited data set spanning about a year, it is not possible to remove the seasonality component from the bookings data. However, taking these factors into consideration, it is still possible to observe a positive correlation between the advertisement expenditure and the total amount of bookings in the above graph. For instance, in the time period between early January and the end of March, where the average price is relatively stable, the correlation coefficient between advertisement and the number of bookings is 0.76 with a coefficient of determination 0.57, indicating a positive correlation at a 95% significance level.

Overview of the Price-Setting Process at the Company

The company follows a two-step process for setting prices in the upcoming period for each itinerary, cabin type, and date of departure. In the first step, prices that maximize revenue are obtained. Parameter estimates in price–demand functions are updated at the end of each period using historical data. The price–demand model predicts demand in each period for each product (itinerary, cabin type, departure date) as a function of prices. The capacity of the product in the upcoming period is calculated as the total remaining capacity divided by the remaining number of periods before departure. For example, if prices are set such that predicted demand for a product is equal to capacity, then predicted sales from the current period to the time of departure exactly match the remaining capacity of the product. Prices are set to maximize revenue in the upcoming period subject to the capacity constraints and limits on the amount of change in the price during the previous period. The optimal prices from this step are inputs to the second step of the process wherein pricing analysts at the firm set the final prices using intuition and external factors not captured by the model. The firm does not include advertising decisions in the pricing model.

For computational reasons, the company makes two major simplifications in their price-setting optimization model: (1) demand is approximated with a linear price–demand function, and (2) prices for each product are set independently (e.g., changes in the price of one product do not affect the demand of any other product). The consequence of (1) and (2) is that the model can be efficiently solved—the objective function is quadratic and the constraints are linear. The company recognizes (1) and (2) as limitations of their approach, and to some extent, the intervention by pricing analysts in the second step of their process helps to mitigate the consequences of these simplifications. The model described in the fourth section eliminates these simplifications and incorporates the effect of advertising expenses, thus providing a stronger starting point for the pricing analysts at the firm.

Pricing and Booking Pattern

As noted in the previous section, the company sets prices by itinerary, cabin type, and date of departure. Based on our 20-week horizon, the company sets prices on around 300 (= 10 cruises on 5 itineraries × 3 cabin types × 10 departure dates) products at the end of each period and determines the advertising expense for the upcoming period.

Figures 2 –5 show the total bookings and the average prices observed for interior cabin bookings in four selected cruises sailing on a Western Caribbean itinerary over the wave booking period. The x-axis shows the amount of time left until departure (the initial value zero indicates a last-minute booking for a cruise that is to leave within the current period).

Figure 2.

Total number of bookings and average prices for interior cabin type for the cruises sailing on a Western Caribbean itinerary during March–April 2004.

Figure 3.

Total number of bookings and average prices for interior cabin type for the cruises sailing on a Western Caribbean itinerary during April–May 2004.

Figure 4.

Total number of bookings and average prices for interior cabin type for the cruises sailing on a Western Caribbean itinerary during May–June 2004.

Figure 5.

Total number of bookings and average prices for interior cabin type for the cruises sailing on a Western Caribbean itinerary during June–July 2004.

When analyzed altogether, the figures show an interesting pattern in the data set. For instance, although the average price for a summer season cruise is typically much higher than the average price for a non–summer season cruise, the price for a particular cruise does not vary much from week to week. This observation is consistent with the claim by managers at the company (noted earlier) that cruise companies tend to keep the prices within some limits for the same product and refrain from extreme price reductions in order to avoid dilutions in bookings and loss of customer goodwill. However, the total bookings across various cruises on the same itinerary may change considerably. Another observation is that the majority of bookings take place within 3–10 weeks before the departure of the cruise.

Insights from the Booking Data with Implications for Model Development

The patterns discussed above offer insights regarding the pricing and advertising models appropriate for this industry. These insights are stated below. First, the fact that the majority of bookings occur within particular time periods suggests that parameter values exhibit time dependency. The proposed model should take this into consideration. Second, although it is difficult to measure the exact relationship between sales and advertisement expenditures because of various unaccounted factors such as seasonality and availability of discounts, the data suggest a positive correlation between them. Moreover, the fact that the advertisement activities in cruise companies are not specific to a particular itinerary should be accounted for in the suggested model. Third, the average prices on a particular cruise do not change much across time periods. However, the prices of a summer-season and a non–summer season cruise on a particular itinerary may be considerably different. This observation suggests departure date–dependent values for problem parameters along with time (i.e., time until departure) dependency.

Model Development

A model of the cruise operator’s pricing and advertising problem, based on the company’s current pricing and advertising practice, is developed in this section. At the end of each period, the company updates its estimates of demand function parameters and sets the prices and total advertising expense for the upcoming period. Prices and advertising expense are set to maximize the revenue (net of advertising) in the upcoming period. This strategy maximizes revenue over the wave booking period under two main assumptions. First, there are no predictable changes in the price–demand function of a product over time (e.g., due to either seasonality effects or due to time-to-departure effects). Second, each product that departs in the previous period is replaced in the upcoming period with a product that has the same supply and demand characteristics.

There are N cruises that return at the end of each period and that are scheduled to depart at the beginning of the next period. Tickets are put on sale T periods in advance of departure for L different types of cabins. Let K denote the number of products, which is the number of combinations of itineraries and departure periods on the books, that is, K = L × N × T. For the data set, these parameters take the values L = 3 (namely, interior, ocean view, and balcony), N = 10 (on five different itineraries), T = 10 periods (of two weeks), and thus, at the end of each period, 270 product prices are updated and prices for 30 new products are set.¹

Let p_i denote the price of product i and x denote the planned total advertising expense. Let B denote the upper limit on advertising expense in the upcoming period. The demand per period of product i is denoted d_i(p, x) where p = (p₁, …, p_K). Note that the dependency on p allows for the possibility that all K products are substitutes; for example, a customer may consider all products when making a purchase decision. The company sets a lower bound on a new price as a fraction of the price in the previous period. This policy is observed in practice at the company motivating this study. As noted above, the company is concerned about the potential loss of goodwill or even cancellation if a customer discovers that his or her ticket could have been purchased in a subsequent period at a much lower price. Let p′ denote the vector of product prices in the previous period and let α denote the minimum allowable value of the new price as a fraction of the old price, that is, p_i/p_i′ ≥ α. (If product i does not exist in the previous period, which is the case for cruises that depart in T periods, then p_i′ can be set to zero or according to a desired lower bound on price L_i, i.e., p_i′ = L_i/α.) The remaining capacity per period of product i is C_i. For example, if there are 200 unsold cabins of product 1 with five periods remaining before departure, then C₁ = 200/5 = 40. The company’s pricing and advertising problem for the upcoming period is

Π^{*} = \max_{p, x} {\begin{cases} \sum_{i = 1}^{K} p_{i} d_{i} (p, x) - x | x \leq B, d_{i} \\ (p, x) \leq C_{i}, p_{i} \geq α p_{i}^{'} \forall i \end{cases}} .

Demand Function

Many empirical studies have shown that demand function parameters are relatively insensitive to the market size assumption (e.g., Berry, Levinsohn, and Pakes 1995; Nevo 2001; Sudhir 2001). Consequently, it is also assumed by us that the weekly market size, M, is fixed, and that it takes the following form:

M = k^{'} (\begin{array}{l} total number of cruise tickets \\ sold in the U . S . in a year \end{array}) / 52,

where k is a parameter that is estimated using historical booking data. Let q_i(p, x) denote the fraction of M consumers who purchase a ticket in the upcoming period. Thus, the demand function can be expressed as

d_{i} (p, x) = M q_{i} (p, x) .

The company assumes linear price–demand functions (and that products are not substitutes). This approach is generalized and enriched by using a popular nonlinear model that accommodates product substitution. In particular, the multinomial logit (MNL) model for the function q_i(p, x) is used. The MNL model is widely used in practice and is empirically well supported (Dong, Kouvelis, and Tian 2009; Talluri and van Ryzin 2004a; Vulcano, van Ryzin, and Chaar 2010, etc.).² The expected utility of product i is

A (x) + a_{i} - b_{i} p_{i},

where the function A(x) captures how consumer utility is enhanced by advertising (e.g., via greater awareness), a_i is expected valuation of attributes of product i exclusive of price and advertising, and b_i captures how changes in price affect the expected utility of product i. Note that the advertisement expense in our model, x, represents the spending on generic promotion (e.g., print, TV, Internet, etc.). In addition, note the possibility that different products have different weights attached to price in the determination of consumer utility (e.g., as in Erdem, Swait, and Louviere 2002), which is allowed in the model (see (3)). The expected fraction of consumers who purchase product i in a period is

q_{i} (p, x) = \frac{e^{A (x) + a_{i} - b_{i} p_{i}}}{1 + \sum_{k = 1}^{K} e^{A (x) + a_{k} - b_{k} p_{k}}}

or in inverted form (see the appendix for details),

p_{i} (q, x) = (A (x) + a_{i} - \ln q_{i} + \ln (1 - \sum_{k = 1}^{K} q_{k})) / b_{i} .

It is assumed that A(x) is an increasing concave function to reflect a diminishing marginal return to advertising expenditure.

By substituting (1) and (4) into. (1), the problem can be rewritten as

Π^{*} = \max_{x} {Π (x) = M ρ^{*} (x) - x},

where

ρ^{*} (x) = \max_{q} {\begin{cases} ρ (q, x) = \sum_{i = 1}^{K} p_{i} (q, x) q_{i} | q_{i} \leq C_{i} / M, - p_{i} \\ (q, x) \leq - α p_{i}^{'} \forall i \end{cases}} .

The problem is re-solved at the end of each period based on revised estimates of demand parameters and remaining capacity per period. Demand parameters are updated using a rolling history that includes realized sales data from the most recent period.

Solving the Pricing and Advertising Problem

For a given level of advertising x, the solution to the unconstrained pricing problem

\max_{q} {ρ (q, x) = \sum_{i = 1}^{K} p_{i} (q, x) q_{i}}

is as follows:

p_{i}^{*} (x) = r^{*} (x) + 1 / b_{i},

q_{i}^{*} (x) = \frac{e^{A (x) + a_{i} - b_{i} ρ^{*} (x) - 1}}{1 + \sum_{k = 1}^{K} e^{A (x) + a_{k} - b_{k} ρ^{*} (x) - 1}},

where ρ^*(x) is the unique solution to

ρ (x) = e^{A (x) - 1} \sum_{k = 1}^{K} \frac{e^{a_{k} - b_{k} ρ (x)}}{b_{k}}

(see the appendix for details). The above expressions in conjunction with the following lemma allow us to obtain an optimal solution to the constrained problem (7) efficiently.

Lemma 1. The function g_i(q, x) = – p_i(q, x) is quasiconvex in q.

Proof. Note that

g_{i} (q, x) = g_{i} (q) = (\ln q_{i} - \ln (1 - \sum_{k = 1}^{K} q_{k}) - a_{i}) / b_{i}

is quasiconvex if and only if for any quantity vectors q¹ and q²,

g_{i} (q^{1}) \leq g_{i} (q^{2}) \Rightarrow \nabla g_{i} {(q^{2})}^{T} (q^{1} - q^{2}) \leq 0

(see S. Boyd and Vandenberghe 2004, first-order conditions for quasiconvex functions, section 3.4.3). Letting q₀ = $1 - \sum_{k = 1}^{K} q_{k}$ , the inequality in the left-hand side of (11) can be interpreted as

[\ln q_{i}^{1} - \ln q_{0}^{1} - (\ln q_{i}^{2} - \ln q_{0}^{2})] / b_{i} \leq 0 \Leftrightarrow \frac{q_{i}^{1}}{q_{i}^{2}} \leq \frac{q_{0}^{2}}{q_{0}^{1}} .

Noting ∇g_i(q)^T = $\frac{1}{q_{0}} + \frac{e_{i}}{q_{i}}$ where 1 is a vector of 1’s and e_i is a vector of zeros except for a 1 in the i^th position, it follows

\begin{array}{l} \nabla g_{i} {(q^{2})}^{T} (q^{1} - q^{2}) = \frac{1}{q_{0}^{2}} \sum_{k = 1}^{K} (q_{k}^{1} - q_{k}^{2}) \\ + \frac{1}{q_{i}^{2}} (q_{i}^{1} - q_{i}^{2}) = \frac{1}{q_{0}^{2}} (q_{0}^{2} - q_{0}^{1}) + \frac{1}{q_{i}^{2}} (q_{i}^{1} - q_{i}^{2}) = \frac{q_{i}^{1}}{q_{i}^{2}} - \frac{q_{0}^{1}}{q_{0}^{2}}, \end{array}

and next it can be seen that equivalently, g_i(q¹) ≤ g_i(q²)) implies ∇g_i(q²)^T(q¹ – q²) ≤ 0.

The capacity constraints are linear functions of q and thus are quasiconvex. From Lemma 1, the price constraints in (7) are quasiconvex, and thus the optimal solution can be obtained by Karush-Kuhn-Tucker (KKT) conditions (see S. Boyd and Vandenberghe 2004, section 4.2.5.) In particular, the optimal sales quantities (therefore, prices) are obtained by solving the following equations (which involve 3K unknowns, q’s, and the Lagrangian multipliers λ’s and θ’s) simultaneously:

\begin{array}{l} λ_{i} (C_{i} / M - q_{i}) = 0, \begin{matrix} λ_{i} \geq 0, \begin{matrix} \forall i = 1, 2, \dots K \end{matrix} \end{matrix} \\ θ_{i} (p_{i} (q, x) - α p_{i}^{'}) = 0, \begin{matrix} θ_{i} \geq 0, \begin{matrix} \forall i = 1, 2, \dots K \end{matrix} \end{matrix} \\ p_{i} (q, x) + \sum_{j = 1}^{K} \frac{\partial p_{j} (q, x)}{\partial q_{i}} [q_{j} - θ_{j}] + λ_{i} = 0, \begin{matrix} \forall i = 1, 2, \dots K \end{matrix} \end{array}

where p_i(q, x) = $(A (x) + a_{i} - \ln q_{i} + \ln (1 - \sum_{k = 1}^{K} q_{k})) / b_{i}$ .

This set of equations is solved via Matlab version 2011B on a computer with 8 GB of RAM in very short times for realistic state space sizes (for the details of this solution procedure on a real-life problem, please see above). Note that Π(x) is not concave. Consequently, the solution procedure requires a line search over x:

Solution Procedure

Obtain ρ^*(x) via KKT conditions and compute Π(x) = Mρ^*(x) – x over the range of viable x values to obtain Π^* = max_x≤BΠ(x).

Empirical Application of the Model

In this section, the empirical application of our model is illustrated in order to assess its viability for addressing real-life problems. We develop a simulation study using the real-life data of the cruise line described above. The data encompasses seven-day-duration cruises on the selected set of five itineraries that tend to be considered by the same set of consumers (i.e., the products are substitutes). While tickets go on sale 52 weeks prior to departure, as noted earlier, sales tend to be concentrated in the wave booking period, which goes from 20 weeks before departure up to departure in our data set. This is also the case in the cruise industry, that is, majority of reservations arrive in 3 to 4 months before departure (CLIA Report 2006). We test our model using the three double-berth cabin types on these itineraries.³ The pricing procedure for three cabin types on five itineraries over 20 weeks sets prices for 300 products at the beginning of each two-week booking period.

Recall that the weekly market size M takes the following form:

M = k^{'} (\begin{array}{l} total number of cruise tickets \\ sold in the U . S . in a year \end{array}) / 52 .

According to the research reports in the cruise industry, the number of global passengers increased from 9.83 million in 2003 to 10.85 million in 2004 (CLIA Report 2006). This calculates to 208,650 potential cruise customers per week. Taking into account the market share of the cruise firm under consideration, k is set to 0.1 and the assumption is M = 20,865 (≈20,000) per week.

The next issue is to find an appropriate form of the function A(x). To this end, the overall advertisement expenditure of the cruise company is known to reach a total of $87 million within a period of 52 weeks from 2003 to 2004. It is also known that with this amount of advertisement expenditure, the company sold 230,000 tickets across the five itineraries. Taking into account the size of the cruise industry and the relative market share of the company under consideration, the concave and increasing advertisement function is estimated as

A (x) = c_{1} \ln (x) + c_{2},

where c₁ and c₂ are appropriate constants. These constants are estimated from company data (i.e., parameters selected to minimize the sum of the squared errors); the values are c₁ = 1, c₂ = 1.2. The parameters are consistent with the market share of the company and no-purchase rates.

Having described the underlying model, we next focus our attention on the traditional revenue maximization problem of the company for the cruises under consideration. Guided by company policy, the restriction is imposed that the average prices cannot be less than 80% of the previous period’s prices, and a total marketing budget of $2 million per week is assumed for the entire set of cruises of the company.

The following is a summary of how we estimate initial demand model parameters and simulate the performance of the proposed and traditional models.

The “true” demand model in our simulation is of the form given in (2). We use the entire data set to set the values of product utility parameters a_i and b_i (minimizing the sum of squared errors), which is similar to previous studies estimating choice parameters (Vulcano, van Ryzin, and Chaar 2010; Bodea, Ferguson, and Garrow 2009).

In each simulation period, demand is randomly generated using the true demand model according to a Poisson arrival process as follows: (1) The number of arrivals in a period is a Poisson random variable with mean M = 20,000, and (2) each arrival randomly chooses an alternative among 301 choices (300 products and a no-purchase alternative) according to the price-and-advertising-dependent purchase probabilities of the proposed and traditional models, that is, the purchase probabilities are q_i(p,x) = $\frac{e^{A (x) + a_{i} - b_{i} p_{i}}}{1 + \sum_{k = 1}^{K} e^{A (x) + a_{k} - b_{k} p_{k}}}$ , q₀(p, x) = 1 – Σ_iq_i(p, x), where a_i’s and b_i’s are the true parameter values while p and x are set according to the traditional and proposed models.

Recall that demand of product i in the traditional model has the following form: d_i = A_i – B _ip_i. Prior to the first simulated period, the parameters of the demand function used in the proposed model (a_i, b_i) and traditional model (A_i, B_i) are estimated from the most recent eight periods in the data set.

For the first simulated period, the total remaining capacity of each product is set to be proportional to the remaining number of periods before departure, that is, $total capacity \times \frac{remaining periods}{total periods}$ . In each subsequent period, the remaining capacities are updated according to simulated sales.

The demand functions are used to set prices for the 300 products for the upcoming simulated period. In the traditional model, advertising expense is set to $2 million in every period, which is the company average for the entire data horizon. In the proposed model, advertising expense is optimized using the estimated parameters of the demand function.

Using the prices and advertising expense for traditional and proposed models, random demands for each product in a period are generated using the true demand model, and revenues are recorded.

At the end of the period, the demand parameter values a_i, b_i, A_i, and B_i are updated. The parameters a_i, b_i for the proposed model are updated using the average of the prior period’s parameter values. A new set of estimated values are found by maximum likelihood estimation to fit the sales data of the most recent period to the prices set by the proposed model. Similarly, the parameters A_i and B_i of the traditional model are updated using the average of the prior period estimated parameter values and a new set of estimated values are found in a similar way for the traditional model.

Steps 5 through 7 are repeated over 20 sales periods (equivalent of 40 weeks), which corresponds to one trial of the simulation. The summary statistics that are reported below are based on 1,000 simulated trials of 20 periods each.

Optimizing the prices according to the proposed model takes less than 60 seconds to solve via Matlab version 2011B on a computer with 8 GB of RAM. A sample of the results involving the average optimal prices, forecasted sales and realized booking requests under the traditional model and under the proposed model for the itinerary in Western Caribbean with an interior cabin type for a randomly selected period is tabulated in Table 1. The comparison of the estimated values of the modeling parameters a of the three products on the itinerary Eastern Caribbean / balcony cabin type with the respective real parameter values (where Product 1 corresponds to the cruises with 1 period left until departure, Product 2 corresponds to the cruises with 5 periods left until departure, and Product 3 corresponds to the cruises with 9 periods left until departure) is displayed in Figure 6.

Table 1.

The Average Optimal Prices and the Predicted and Actual Number of Bookings (under the Traditional and the Proposed Models) of the Firm for an Interior Cabin Type.

	Proposed Model			Traditional Model
Periods until Departure	Optimal Prices	Predicted Sales	Simulation Sales	Optimal Prices	Predicted Sales	Simulation Sales
0	425	29	30	385	203	30
1	477	125	109	470	171	78
2	577	128	121	559	157	84
3	625	123	139	611	158	93
4	663	125	109	622	167	106
5	667	113	104	647	162	91
6	681	110	106	689	158	71
7	699	123	112	704	145	68
8	647	113	126	729	136	55
9	653	120	107	746	126	43
10	601	118	132	741	118	41

Figure 6.

Estimated and real parameters a of the three products on the itinerary Eastern Caribbean/balcony cabin type.

From Table 1, it can be observed that the proposed model is much more successful than the traditional model at estimating the sales levels, and sets the prices accordingly. There is a significant discrepancy between the predicted and the actual sales levels under the traditional model. Another observation is that the proposed model usually sets the prices at higher levels than the traditional model for the products that have fewer periods left until departure, while the pattern is reversed for the products that have more than 6 periods left until departure. A reasonable explanation for this observation is that in an attempt to create high last-minute sales, the traditional model lowers the prices of the soon-to-be-perishable products that have ample capacity left because of a mismatch between predictions and real sales. However, this could only result in high fluctuations in product prices, which is certainly not desirable for the company. In general, the prices suggested by the proposed model exhibit fewer up-and-down fluctuations in period-to-period prices, creating a smooth pricing policy as desired.

Figure 6 suggests that, except for the initial period which could be considered as a “warm-up period” in the simulation, the proposed model estimates the updated parameter values quite closely to the real values of the parameter. Another observation from the picture is the fact that the attraction parameter for the cruises that are to leave in a moderate amount of time (i.e., in 5 periods) is higher than both the cruises that are to leave soon (i.e., in 1 period) and those with a considerably long time until departure (i.e., to leave in 9 periods), which is consistent with real-world occurrences.

Finally, in more than 1,000 simulation trials, we compare the revenues obtained under the proposed model and the traditional model. We find that the proposed model brings an average of an 11.3% increase in net revenues at a market size level of M = 20,000, which is similar to the improvement reported in Yih. Li, Miao, and Wang (2014). We also find that the proposed model advises higher advertisement expenses in each period (the advertising expense in the proposed model starts at $3.5 million per period in the first period, then remains constant at the level of $3 million for the rest of the simulation periods, while the advertising expense for the traditional model was fixed at a real-life average of $2 million per period). When the market size is M = 25,000, the proposed model contributed to an average 9.8% more revenues than the traditional pricing model of the firm, and when M = 15,000, the revenue increases by 20.8%. Moreover, the optimal advertising expenses become $3.5 and $2.5 per period in these two occasions, respectively. The results for various values of M within the range of 15,000 and 35,000 are summarized in Table 2.

Table 2.

The Expected Revenue and the Ratio of Increased Returns under Proposed and Traditional Models.

Market Size (M)	Proposed Model		Traditional Model	Ratio of Net Revenues, %(Proposed Model/Traditional Model)
Market Size (M)	Average Advertisement Expense, million USD	Expected Net Revenues, million USD(Per Period)	Expected Net Revenues, million USD(Per Period)	Ratio of Net Revenues, %(Proposed Model/Traditional Model)
15,000	2.5	1.431	1.119	120.8
20,000	3.0	2.679	2.405	111.3
25,000	3.5	3.625	3.320	109.8
30,000	3.5	4.307	3.978	108.9
35,000	3.5	4.832	4.484	108.4

Note: USD = US dollars.

As shown in Table 2, the proposed model leads to higher revenues than the traditional model. The last column shows that the percentage gain decreases as market size increases. One reason for this is the fact that at higher values of market size, even the traditional model reaches high occupation rates at relatively higher prices compared to the case of low market size. Thus, the effect of applying the optimal advertisement price is diluted. However, the percentage gain in revenues decreases by relatively insignificant amounts and at a diminishing pace (i.e., 0.9% for an increase of 5,000 from M = 25,000 to 30,000, and 0.5% for an increase of 5,000 from M = 30,000 to 35,000).

Conclusion

Revenue management has received considerable attention from both academic and business professionals. It encompasses several techniques regarding the capacity allocation, pricing, and resource management of fixed, time-sensitive capacity. Revenue management can be roughly divided into two categories defined by the control mechanism to increase revenue: capacity allocation or price optimization. With some exceptions (e.g., see Biehn 2006; Ji and Mazzarella 2007), revenue management for hotel and cruise line applications has mainly focused on optimizing prices of different rooms (hotels) or cabins (cruises). Prices by date of stay (hotel) or date of departure (cruise) are dynamically updated over time. Our work falls into this second category of revenue management. We define and analyze a model that includes two features that are new to the literature on revenue management via dynamic price optimization: (1) we allow for partial substitutability among products (e.g., a customer making a purchase decision may consider multiple alternatives—different departure dates, different destinations, different cabin types), and (2) we include marketing expense in addition to prices as a lever for increasing revenue. These features are relevant to dynamic pricing in practice. We illustrate the practicality of our approach by using data from a cruise line firm.

In recent years, the cruise line industry has become one of the fastest growing sectors in the hospitality and tourism market experiencing an annual growth rate of 7.2% in terms of the total number of passengers and continues to show strong growth potential. Despite the importance of the cruise industry, cruise line revenue management has received relatively little attention in the literature so far (Sun, Gauri, and Webster 2011). Moreover, the customer choice factor, a core element in the cruise industry as well as in other industries for which RM methods are applicable, remains largely understudied in the RM literature. This article offers three main contributions to narrow this gap. First, a cruise line revenue management problem is described and illustrated using knowledge about pricing/advertising procedures and proprietary booking data from a cruise line company. At the beginning of each period, the company determines the prices of products that are scheduled to depart within the wave booking period subject to constraints on remaining capacity and a limit on the price reduction from the previous period price. A separate group in the company makes decisions on advertising expense. The current practice of pricing and advertisement decisions is discussed in the paper, with the hope of drawing further attention to this understudied problem of cruise lines.

Second, a revenue management decision model is presented that includes two meaningful extensions over current practice: (1) product substitution effects (i.e., customer choice element) and (2) an advertising expense decision. Clearly, in practice both price and advertising interact to influence the alignment between demand and capacity, which impacts the revenues of the firm. However, to our knowledge, the joint pricing and advertising problem has not previously been considered in the literature

As a final contribution, a widely used and empirically well-supported model of consumer choice (i.e., the MNL model) is applied, adding to the limited literature of empirical applications of choice models in real-life problems, and a solution method for obtaining optimal prices and advertising expense is presented. The model and solution method are illustrated using actual proprietary booking data from a major company. The method is computationally efficient, yielding the optimal level of advertising and optimal prices for 300 products in a short period of time and producing higher net revenues compared to the traditional pricing model currently in use.

This work can be viewed as a starting point for addressing an important and scantily researched problem. As such, the greatest need at this point in time is additional empirical research both in the area of cruise line revenue management and in the choice-based RM. The model and method described in this article are based on the problem from the perspective of one cruise-line company. A broader base of perspectives from different companies across industries such as hotels, airlines, etc., will be useful for shaping future work on choice-based RM in industry. One promising area within this line is research studying the effect of advertisement expenditure in more detail. The model presented in this article could be further expanded to incorporate complexities like the diminishing effects of advertisement expenditure on the booking levels of future periods and the effect of uncertainty of demand parameters. Furthermore, the model presented in this article does not involve the competition in the industry. The problem can be enriched further by incorporating the effects of the marketing activities of the competitors.

One other limitation of our work is the fact that the data span a time frame of around one year. Working with a more comprehensive data could lead to identifying the seasonality element in the cruise line pricing problem. By taking into account the seasonality factors along with the potential capacity shifts induced by the cruise company to more effectively align the demand and the capacity, more realistic values of prices and advertisement expense could be proposed.

In summary, our paper attempts to illustrate the pricing problem of cruise lines and present the opportunities involved in approaching this problem from a revenue management perspective taking into account the customer choice element. Through a description of key challenges of cruise line revenue management in practice, we hope to spur additional research on this important real-world problem that has been scantily treated in the literature.

Footnotes

Appendix

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

Author Biographies

Nur Ayvaz-Cavdaroglu is an assistant professor of Operations Management at Business Administration Department of Kadir Has University, in Istanbul, Turkey. Her research interests involve applications of revenue management and pricing, and behavioral operations management.

Dinesh Gauri is a professor of Marketing and Wal-Mart Chair in Marketing in Sam M. Walton College of Business at University of Arkansas in Fayetteville. His research interests are in the areas of pricing, branding, retailing and revenue management.

Scott Webster is the Bob Herberger Arizona Heritage Chair in Supply Chain Management at the W.P. Carey School of Business, Arizona State University. His research focuses on improving competitiveness through logistics, especially managing risk and uncertainty in supply chains.

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