hospitality-productivity assessment using data-envelopment analysis

Hospitality-productivity

using Ass essment

Data-envelopment Analysis

Using DEA, one can determine how effectively a restaurant or hotel is using resources- and also identify factors that are beyond managers’ control.

BY DENNIS REYNOLDS

s A hospitality businesses have focused on improving resource allocation, market penetration, and profit maximization, managers have sought ways to achieve

maximum operational efficiency. Moreover, the goal of maximizing unit-level productivity has developed from a matter of seeking competitive advantage into an ongoing management requirement. That is particularly true in the food- service segment, where the need for efficiency has been fit- eled by escalating customer expectations and expanding competition. Unfortunately, the often-used mechanisms for measuring and analyzing productivity in the service sector have remained too narrow to capture the broad spectrum of factors that actually contribute to a restaurant’s success.

One remedy for this shortcoming is an analytic technique developed in a study of educational programs’ efficiency.’ The

technique, known as data-envelopment analysis (DFA), inte- grates multiple input and output variables simultaneously- including discretionary input variables (i.e., those that are under managers’ control) and nondiscretionary variables (i.e., those that are beyond managers’ control). The technique produces a single productivity index that compares all units to the most-efficient units in the sample.

The restaurant industry is an excellent candidate for illus- trating the value of DEA in practice. Consider first how productivity has typically been assessed. Restaurateurs have usu- ally relied on labor-cost percentage as a measure of organizational efficiency. They did so on the theory that if you can minimize labor expenses, for instance, the effects on the bottom line should be readily evident. The shortcoming in that thinking is that decreasing labor costs-whether in

’ A.C. Charnes, W.W. Cooper, and E. Rhodes, “Measuring Effkiency of Decision-making Units,” European /ournal of Operations Research, Vol. 2 (19781, pp. 429449. 0 2003, CORNELL UNIVERSITY

130 Cornell Hotel and Restaurant Administration Quarterly APRIL 2003

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the total number of labor hours used or in the cost for each hour worked-can have a direct and dramatic negative effect on service qual- ity.2 Thus, reducing labor expense as a percentage of sales can maximize short-term profits, but the long-term effect could be reduced operational viability (if labor support is reduced to the point that service standards cannot be assured).

In the 1980s some restaurant operators be- gan to look at labor productivity and, in turn, unit-level efficiency by drawing from practices developed in manufacturing. Using the basic definition of productivity, which is a ratio of measurable output and input units, quick- service-restaurant managers, for example, analyzed ratios that focus on output as a function of a single input, including such partial-factor statistics as transactions per hour.3 A similar approach, particularly in midscale operations, was the analysis of sales per labor hour.*

Such partial-factor measures facilitated meaningful comparisons in other retail industries when applied to the same operation over multiple operating periods or when used to compare similar operations. Unfortunately, measures of that kind are of limited use for restaurants because they reflect only certain as- pects of the operation. Even worse, such measures may not correlate to actual operational efficiency5

Other partial-factor statistics have been used but suffer from the same limitations. For example, the ratio of sales per cover offers the advantage that the corresponding average- check measure is frequently assessed across restaurant segments and provides interesting comparative data. A related metric, revenue per available seat-hour, is arguably more useful, particularly for revenue management, but is simi-

2 J.B. Tracey and A. E. Nathan, “The Strategic and Opera- tional Roles of Human Resources: An Emerging Model,” Cornell Hotel and Restaurant Administration Quarter& Vol. 43, No. 4 (2002), pp. 17-26.

3 R.D. Filley, “Putting the ‘Fast’ in Fast Foods: Burger King,” IndustrialEngineering, Vol. 15, No. 1 (1983), pp. 4447.

* M. Jablomb, “productivity in Industry and Government,” Monthly Labor Review, Vol. 117, No. 8 (1994), pp. 49-57.

5 D. Reynolds, “ProductivityAnalysis in the On-site Food- service Segment,” Cornell Hoteland Restaurant Administra- tion Quarterb Vol. 39, No. 3 (June 1998), pp. 22-31.

larly inadequate as a comprehensive measure of operational efficiency.”

Some researchers have suggested that deli- ciencies associated with the aforementioned measures can be remedied by using multi- factor or total-factor productivity ratios.7 Such metrics integrate multiple operational variables linked, most frequently, to sales as the output variable. For example, a multi-factor indica- tor might include such input variables as food cost, labor cost, and related variable expenses for a specific period. While more robust than partial-factor measures, these metrics, when applied across multiple operations, are still limited in that they produce averages by which other units within a chain can be compared, but they fail to provide comparative information regarding high-performing operations. Similarly, even when paired with more elabo- rate models, such as those including regression analysis to create predictions of likely relationships, such approaches fail to create benchmark information.

In contrast to the methods of productivity assessment that I just described, DEA allows for assessment of contingent productivity, which takes into account each restaurant’s performance while controlling for differing envi- ronmental or situational factors. Thus, operators can use the best-performing units as a basis for evaluating other units. Furthermore, DEA allows for the inclusion of a number of diverse variables. The limit on the number of appli- cable input and output variables is based on the number of restaurants in the comparative set. Analyzing a chain with hundreds of restaurants would allow for the inclusion of doz- ens of variables, provided each input variable being tested has a relationship to at least one output variable.

6 For more on sales per available seat hour, see: S.E. Rimes, R.B. Chase, S. Choi, P.Y. Lee, and E.N. Ngonzi, “Restau- rant Revenue Management: Applying Yield Management to the Restaurant Industry,” Cornell Hotel and Restaurant Administration Quarter& Vol. 39, No. 3 (June 1998), pp. 32-39.

’ For example, see: M.D.M. Brown and L.W. Hoover, “Pro- ductivity Measurement in Food Service: Past Accomplish- ments-A Future Alternative,” Journal of the American Dietetic Association, Vol. 90 (1990), pp. 973-98 1.

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FOCUS ON RESEARCH I DATA-ENVELOPMENT ANALYSIS

Average sales and labor hours for a small restaurant chain

Location Monthly Monthly

labor-hours sales

1 2,483 $174,340 2 2,190 $209,876

7 1,523 $106,127 8 1 765 $171 654

10 2,108 $175,981 11 1,867 $164 235

13 2,159 $132.852 14 21434 $189:543

DEA Decoded Say that there’s a 15-unit midscale restaurant chain (see Exhibit 1). For a given month, the average sales per labor-hour is $78.53 (for front- of-house employees). Thus, any store with sales per labor-hour above this level might be considered efficient. Moreover, a regression analysis intended to predict what sales should be, based on the single independent variable (labor) and using the small data set, would return the line shown in black in Exhibit 2.

The regression equation describing that straight line would serve as a gauge to project what sales unit n might be expected to produce as a linear function of monthly labor hours (or the monthly labor hours that might be expected for a particular sales level). Typical of ordinary- least-squares regression, this parametric approach is intended to formulate a single regression plane that fits as closely as possible to the data points. Put another way, the operator would predict the typical value of the output variable, say sales, by using the values of the various input variables, say, labor.

Implicitly, however, productivity assessment doesn’t aim for an average. Rather, it is predi- cated on the related objective of maximizing output while using a limited number of inputs as effectively as possible. This is where the nonpara-

metric (mathematical programming) techniques such as DEA fit a need. Rather than compare a restaurant to the theoretical average, DEA looks to optimize the output measure for each restaurant given the inputs used. The same technique also facilitates the examination of how inputs might be minimized. In either case, the focus is where it should be-on the individual variable(s) for each individual restaurant that may be affect- ing that unit’s productivity

The other critical component of DEA, which is starkly underscored when DEA is compared with parametric approaches, is that DEA requires no assumptions about what form the function will take. Thus, where a linear-regression analysis produces a straight line (and requires assumptions about the data, including how erroneous the estimate is compared to the actual data), DEA requires no assumptions about whether the function that describes the data is a straight line, a bell curve, or any shape at all. As shown in Ex- hibit 2, the optimal frontier resulting from the DEA analysis is a curve (shown in color). This frontier line demonstrates which units (i.e., 2,4, and 8) are loo-percent efficient on the basis of sales and labor hours, given this population. The frontier is so called because it is the edge of efftciency and it is said to “envelop” the less- efficient units.

The real advantage of DEA lies in the technique’s ability to integrate several relatively disparate inputs and outputs-while still allow- ing analysts to calculate which unit is most efficient given its own set ofvariables, which are then compared with others in the set. For example, consider a hypothetical chain of restaurants where size (as measured in number of seats) and parking availability vary considerably among units. A simple partial-factor productivity analysis using sales per seat for each restaurant might yield results that would not reflect the effects of parking availability What this means is that a unit could appear to be running efficiently compared to others, but that outcome might be only because it has no parking limitations and not because its managers are using their resources with full efficiency. If one uses DEA to assess each unit by examining both unit size and parking availability as a function of sales, however, the analysis will provide information as to which units are


DATA-ENVELOPMENT ANALYSIS I FOCUS ON RESEARCH

more productive given the constraint of number of seats in concert with parking availability. Thus, the multiunit operator could assess unit-level productivity for each unit based on the specific unit’s parameters.

Ideal for Food-service Operations As mentioned earlier, the application of DEL4 to the food-service industry is particularly advanta- geous because the method accommodates both controllable and uncontrollable factors. Control- lable factors are those within a manager’s purview, such as labor hours, number of servers during a given shift, or time spent training each employee. Uncontrollable factors might include a restaurant’s maximum seating capacity, its location, and number of nearby competitors. Criti- cal though these factors might be to a restaurant’s efficiency, they typically are ignored in other methods of productivity assessment, owing to the difftculty in making comparisons across units (particularly when units possess numerous dissimilar uncontrollable characteristics). Nonethe- less, such variables are often the first cited by unit- level managers in explaining why sales are not as high as they might be.8

The handful of studies applying DEA to multiunit food-service operations offer provocative results and underscore the utility of DEA-based productivity analysis for restaurateurs. For example, a study of 38 midscale, corporately owned and operated restaurants contrasted the DEA results with the company’s ranking of the units in terms of perceived efficiency.g The analysis demonstrated that while seven of the 38 units were loo-percent efficient, the restaurant that the company considered to be the most productive was operating at only 88-percent efficiency. Sub- sequent analyses brought to bear a number of issues with the operations that had previously been considered company stars.

* R.D. Banker and R.C. Morey, “Efficiency Analysis for Exogenously Fixed Inputs and Outputs,” Operations Research, Vol. 34, No. 4 (1986), pp. 513-521.

lo D. Reynolds and G. Thompson, “Multiunit Restaurant Productivity Assessment: A Test of Data-envelopment Analysis,” The Center for Hospitality Research at Cornell University (CHR report), 2002.

9 D. Reynolds, “An Exploratory Investigation of Multiunit- t1 The value of focusing exclusively on uncontrollable vari- restaurant-productivity Assessment Using Data Envelop- ables is explained more fully in: M. Norman and B. Stoker, ment Analysis,” /ownal of Travel and Tourism Marketbag, Data-envelopment Analysis: The Assessment of Performance 2003 (in press). (New York: John Wiley and Sons, 1991).

Simplified comparison of DEA and regression analysis

$250,000 ata-envelopment fro tier

$200,000

a z $150,000 v)

$100,000

$50,000 I 1,500

Note: Data are from Exhibit 1

2,000

Labor Hours

2,500

In the study, outpur variables included sales and customer satisfaction. Controllable input variables that were analyzed were front-of-the- house hours worked during lunch, front-of-the- house hours worked during dinner, and average wage. Of these, aggregated front-of-the-house hours worked per day proved to be a viable measure of efficiency. (Average wage was not used because all of the company’s restaurants paid their servers the same hourly rate regardless of location or employee tenure.) Uncontrollable variables used in the analysis were number of competitors within a two-mile radius and seating capacity. Both of these factors proved important in the final analysis.

Another restaurant-efficiency study that applied DEA focused on uncontrollablevariables.1° The primary purpose of this study was to clarify the importance of variables outside of unit managers’ purview before considering the effects of variables that managers can control.” Using 60


FOCUS ON RESEARCH I DATA-ENVELOPMENT ANALYSIS

Mathematical Underpinnings of DEA DEL4 Model Specification

As an example of how DEA-related values are calculated, consider a restaurant that is evaluated using two output variables, Y, and Y,, and three input variables, X,, X,, and Xs. Its efficiency (P,) is calculated as:

P, = “I yi + “2Y2

v,x, + v,x2 + v3x3

In applying DEA, the weights (U, and V,) are estimated sepa- rately for each restaurant such that the efficiency is the maximum attainable. Moreover, the weights estimated for the first restaurant are such that when they are applied to corresponding outputs and inputs from other units in the analysis, the ratio of weighted outputs to weighted inputs is less than or equal to 1. On a more general basis, assuming that the number of outputs and inputs is infinite, the maximum efficiency of restaurant c as compared with n other restaurants is calculated as follows:

Up y>O;r= l,..., s; i= l,..., m

where

Yti is the rrh output for the jrb restaurant,

Xy is the irh input for the fh restaurant,

.!I, and Vi are the variable weights estimated and used to determine the relative efficiency of c,

s is the number of outputs, and

m is the number of inputs.

Since DEA seeks optimization contingent on each individual restaurant’s performance in relation to the performance of all other units, those with the greatest productivity have a productivity score (p) of 1, suggesting loo-percent efficiency. These optimal units lie on a multidimensional frontier, which is shown as the curve in Exhibit 2 (on the previous page). As mentioned in the accompanying article, the efficiency frontier “envelops” the less-efficient units within. Furthermore, the DEA quantifies that inefficiency by giving each unit a relative score of less than 100 percent and providing a relational measure on each output and input. Referring to Exhibit 2, then, the two-dimensional frontier would transcend only a single plane. However, the number of dimensions would increase for every additional variable added during the analysis. Although it is difficult to conceptualize a model with a frontier that occupies a dozen or more dimensions, such an approach provides useful efficiency information.

The final technical piece of information that is important in understanding the subtleties of DEA pertains to the model specification. There are four general model types. While the mathematical nuances underlying each is beyond the scope of this paper, the major differences are worth noting.

The first and most commonly used model type is the CCR model, which was named for those who proposed it- Charnes, Cooper, and Rhodes.’ This model yields an objective evaluation of overall efficiency on the basis of the population being considered and identifies values associated with the variables corresponding to the inefficient units. The key assumption of the CCR model is that the associated frontiers have constant return-to-scale characteristics. Thus, such a model might be most appropriate when this assumption is reasonable given the variable specifications.

Also frequently applied, the BCC model, named for Banker, Charnes, and Cooper, distinguishes between technical and scale inefficiencies by estimating pure technical efficiency given the various scale inefficiencies. It also identifies whether increasing, decreasing, or constant returns to such scales are possible.2 In contrast to the CCR model, the BCC model is appropriate when output maximization is achieved through a proportional (but not necessarily constant) reduc- tion in inputs. This method, therefore, requires mathematical constructs not present in the CCR model.

The third type, referred to as multiplicative model, is distinct from the other two because a model of this type provides a piecewise log-linear or a piecewise Cobb-Douglas envelopment. Multiplicative models can include a convexity constraint, which is sometimes useful given the transformed data space. Additive models, the fourth and final type, relate the efficiency results to concepts found in econometrics, including the concept of Pareto optimality. At the risk of oversimplifying things, this model type uses unique geometric interpretations of the data and, as a result, depends on the units of measurement. As with the other models, the decision to choose such a model specification should pair with the variable definitions as well as the problem parameters.-D.R.

1 M.D.M. Brown and L.W. Hoover, “Productivity Measurement in Food Service: Past Accomplishments-A Future Alternative,” Journal of the American Dietetic Association, Vol. 90 (1990), pp. 973-981.

* R.D. Banker, A. Charnes, and W.W. Cooper, “Some Models for Estimating Technical and Scale Inefficiencies in Data-envelopment Analysis,” Management Science, Vol. 30, No. 9 (1984), pp. 1078- 1092.



FOCUS ON RESEARCH

restaurants operated by a single chain located across the continental United States, the researchers explored such inputs as server wage (the chains policy was to pay minimum wage for all food servers, but that minimum wage differed by state), number of seats, square footage, penetration within a given geography (as measured by number of stores per state), time each unit had been open, whether the restaurant had dedicated parking, whether each restaurant was constructed as a stand alone or was adjacent to other businesses (such as within a mall), and the number of competitors within two miles. As with the other study, sales and customer satisfaction were used as output variables.

The study demonstrated that, for this chain, only server wage, number of seats, and stand alone versus adjacent were influential. Moreover, the analysis demonstrated that the average effr- ciency score across all 60 restaurants was 0.820, suggesting that revenue and customer satisfaction could be increased by as much as 22 percent. The lowest-scoring restaurant had an effr- ciency score of 0.527.

That information proves useful in a number of ways. First, it suggests that there is considerable room for improvement in a number of restaurants and that many of the managers’ expla- nations for why sales at various units were weak were unfounded. Second, the findings provide objective information pertaining to where best to locate future units-an issue important to this particular organization given its growth plans. Finally, the analysis demonstrated what was possible in terms of system-wide revenue if more units approached the optimal efficiency frontier. It appears that a top-line increase of more than I2 percent is possible.

DEL4 as an Avenue to Best-practice Identification One of the most valuable outcomes of DEA is the creation of a best-practice frontier. In the exarnple in Exhibit 2, this frontier is occupied by the three units operating at loo-percent efficiency. A restaurant’s presence on the frontier indicates that a unit is delivering maximum output given the available resources or, conversely, is maximizing its resources such that outcomes are proportional.

Other units can be compared to the exem- plars relative to individual variables, as indicated by the improvement-opportunity values calculated for each input and output variable associated with inefficient operations. Stated another way, the calculated relational-improvement scores (i.e., values assigned to variables that result from the DEA) show how unit A might improve regarding input Xi. A source showing what is lead- ing to greater efficiency on that given dimension can be found in unit B, where efficiency is 100

Data-envelopment analysis focuses managers’ attention on specific actions that will improve productivity.

percent. For example, in the study of the 38-unit chain described earlier, relational-improvement information suggests that the principal areas of potential productivity enhancement for underperforming units are improving customer satisfaction during dinner and reducing hours worked.

Using productivity analysis as a means to identify best practices is nothing new.” Many multiunit operators currently use performance-related indices that focus on the best performers and that integrate key operating measures. Such benchmarking is useful particularly when indi- cators span operations that are dissimilar but compete for similar customers.13 DEA-generated productivity indices aptly facilitate such comparison. Furthermore, because DEA makes the identification more explicit, the analysis can help focus specific management actions.

Consider a case where a firm launches a new promotional campaign, complete with a marketing blitz. Sales increase substantially as a conse- quence. Partial-factor productivity indices, such as sales per labor hour, would suggest that managers were scheduling staff more effectively, but

I2 D. Reynolds, “Productivity Analysis in the On-site Food- service Segment,” Cornell Hotel and RestaurantAdminkra- tion Qtutrter& Vol. 39, No. 3 (June 1998), pp. 22-31.

I3 K.W. Wcber, Benchmarking in Tourism and Hospitaliy Industries (London: CABI, 2002).

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the real driver would be the increased sales (because of the marketing blitz). One would make incorrect inferences if the results from that market-blitz period were compared with time- series data. Using a total-factor-productivity measure might also produce erroneous productivity statistics since the effect of the marketing would not be assessed relative to all other factors weighted relative to each specific unit. Another possibility is that certain combinations of variables create greater efficiency gains-a situation

DEA holds great promise for studies aimed at enhancing productivity in hospitality-related operations.

that could not be discerned using conventional ratio analysis. Productivity assessment using DFA, on the other hand, would provide empiri- cal evidence of how great an effect marketing expenditures had per store relative to other inputs and outputs for each of the chain’s restaurants. Thus, managers could readily learn the effects of incremental marketing expenditures while integrating operational and customer- demographic information that might be differ- ent among the various restaurants.

Regarding best-practice identification, DEA distinguishes the most productive unit or units within the competitive set, describes the relatively less-productive restaurants (as compared with those displaying best practices), and calculates the excess resources used by each of those less- productive operations. In addition, the envelopment analysis calculates precisely the amount of excess capacity-or the ability to increase outputs-in the restaurants showing lower efficiency (without adding additional resources to those operations) . Finally, it identifies which of the most- efficient restaurants are most similar to the less- productive units (based on similar resource use). The peer group so identified can be used in a variety of ways, not the least of which is as exem- plars for chain expansion.

Applications in Other Hospitality Segments DEA has been applied in a variety of fields, including banking and public libraries.14 One study focused on the retail industry by applying data from a number of fast-food restaurants.15

Despite the apparent benefits, however, researchers have yet to apply DEA to more than one restaurant chain or to on-site food service (often referred to as contract food service), although the call for such research has been made.r6 Furthermore, DEA’s applicability to lodging op-

erations appears readily apparent and is underscored by one of the few papers on DEL4 that include hotel-related data. Among the notewor- thy findings of that study was “much higher lev- els of inefftciency in the hotel industry” than in the restaurant industry, and that individual ho- tels “are not using resources poorly or operating at the wrong output level, but rather failing to use the best input mix to produce revenue for the hotel.“”

DFXs Limitations While DEA offers considerable utility for hospitality operators and addresses many of the prob- lems associated with conventional productivity measures, it does have shortcomings. Foremost among them is that DEA is extremely sensitive to outliers, as these serve to influence the optimal frontier. Thus, it is possible that one restaurant or hotel could anomalously create a benchmark-potentially resulting from a variable not included in the productivity analysis-that no other operation can match. Arguably, this shortcoming can be mitigated by careful consideration

I4 For example, see: I. Jemric and B. Vujcic, “Efficiency of Banks in Croatia: A DEA Approach,” Comparative Economic Studies, Vol. 44, No. 2/3 (2002), pp. 169-193; and C. J. Hammond, “A Data-envelopment Analysis of U.K. Public- library Systems,” AppliedEconomics, Vol. 34, No. 5 (2002), pp. 649-657.

I5 N. Donthu and B. Yoo, “Retail-productivity Assessment: Using Data-envelopment Analysis,” Journal of Retailing, Vol. 74, No. 1 (1998), pp. 89-105.

l6 D. Reynolds, On-site Foodservice Management-A Best Practices Approach (New York: Wiley & Sons, 2003).

I7 R.I. Anderson, R. Fok, and J. Scott, “Hotel-industry Efficiency: An Advanced Linear-programming Examina- tion,” American Business Review, Vol. 18, No. 1, pp, 4048.

136 Cornell Hotel and Restaurant Administration Quarterly APRIL iOOj


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of what variables should be included, and test- ing the relationships between the input and output variables as noted earlier. In addition, a careful strategic analysis could help identify the crucial variables. Nonetheless, this is an integral concern for any DEA application.

Also related to the methodology, DEA is not stochastic in nature, which means that it does not allow for an error structure. Hence, there is no goodness-of-fit information as is found in tra- ditional statistical techniques such as regression analysis. Consequently, there is no objective way to evaluate the accuracy of the analysis. Again, a thoughtful assessment ofwhat variables are critical can help ameliorate that potential difficulty.

Another problem that must be recognized is that in a case where the number of operating units is too small relative to the number of inputs and outputs, the efficiency scores may be artificially inflated. In such instances, it is likely that many or all units will reside on the efficiency frontier. Heuristics exist that reduce the likelihood of this, and procedures have also been developed for bounding the multipliers, which in turn increases the sensitivity of the technique to the small number of inputs and outputs.” Still, it is an important aspect of DEA that must be recognized.

Finally, I must acknowledge the mathematical complexity associated with DFA. While a handful f k o pat ages exist for conducting envel-

I8 See, for example: R.G. Thompson, L.N. Langemeier, C.T. Lee, and RM. ThraU, “The Role of Multiplier Bounds in Effkiency Analysis with Application to Kansas Farming,“]ournal ofEconometrics, Vol. 46, No. l/2 (1990), pp. 93-108.

opment analyses (e.g., Frontier Analyst@), a reasonable level of understanding of the underlying concepts is important. This is readily apparent when specifying the model, as discussed in the sidebar (see page 134). Granted, the ease with which users can apply DEA will increase as the software becomes more user friendly than it is right now. At present, however, some people may be frightened away by the method’s complexity, despite the good software that is available.

Conclusion Despite the limitations that I just mentioned, DFA holds great promise for practitioners and researchers alike as we look to measure, assess, and enhance productivity in hospitality-related operations. It is also likely that as more DFA analyses are performed in one segment (e.g., restaurants), the value of DEA for operators in other segments will become more apparent. Moreover, as an increasing number people apply the technique, the identification of variables and model development will also improve, thereby providing a greater quantity of useful information.

Most operators are interested in enhancing productivity. The trick, however, is first to assess it accurately. Already, those who have embraced DEA as a useful tool in assessing productivity are seeing the benefits. Thus, for those operators who harness the utility of such a technique, there exists considerable business-intelligence oppor- tunities that are likely to provide an array of competitive advantages. At minimum, such information will enable hospitality operators to improve their use of resources to maximize profits.


hospitality-productivity assessment using data-envelopment analysis

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