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Statistical Case Study: The Possibility

Statistical analysis may be a useful tool in examining the possibilities which are open to a business, and may be used to assist in the decision-making process. This paper will examine the use of the statistical technique known as linear programming in relation to two case studies based in the restaurant business.

The first case-study is based upon the restaurant "The Possibility," a new start-up venture in which two friends, Angela and Zooey, were aiming to provide a French dining experience where there were no other businesses doing so in the area. The problem to be solved in this case study involved determining how many meals to prepare each night in order to maximize their profitability from the venture. Additional parameters which were to be considered included staffing requirements, dietary concerns of customers, and reduction of waste to reduce overheads.

The second case-study also focused on "The Possibility," but centered around a set of slightly more complex decisions, related to changes in the business which may or may not increase profitability. The first of these related to an analysis of the cost-effectiveness of advertising in relation to boosting profits. The second question related to the impact which reducing labor input may have on profitability, and finally the impact of increasing the sale price of their dinners.

It is likely that operational research would be useful to the owners of the Possibility in finding the optimal solution to these problems. There is significant evidence available as to the utility of operations research in the restaurant industry, and a brief su

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mmary of that research is provided in the next section.

Operations Research in the Restaurant Industry

The use of 'scientific management' in the restaurant sector is a concept which stemmed originally from Taylor's Principles of Scientific Management, published in 1911. This used scientific and mathematical techniques to demonstrate how the different service elements could measured, predicted and therefore tailored to requirements. It is the application of these techniques from the 1950s onwards which has seen the successful development of many of the fast-food chains, such as McDonalds. This concept was further developed by Sasser in 1976, who described the importance of these scientific techniques for balancing supply and demand (Chase & Apte, 2007).

The use of scientifically-based research into business strategies for improvement of service provision is now more commonly known as 'operations research'. This operations research may be used in many areas of the restaurant business. For example it is may be possible to use these techniques to determine pricing strategies, ways of improving customer satisfaction, how to model staff schedules, and so on. There are many tools which may be used in operations research, although this paper is concerned with one specific tool - linear programming.

Linear programming works as a means of predicting the impact which different decisions will have on the profitability of a particular firm, by assuming that certain inputs are used to create certain outputs. The inputs may be either exogenous (uncontrollable) or controllable factors (Sanjeev, 2007), although it is impact of changes to the controllable factors which are likely to be of most interest. For example in the restaurant industry the input factors which would be considered may include the cost of food which is used for meal preparation and staffing factors, while the outcome is most likely profits generated, but may be customer satisfaction or other such factors of interest.

In particular, it has been shown that linear programming is an effective way to measure the staffing needs of a particular service, particularly where there are different skill levels and different roles to take into account. Evidence from nursing shows that linear programming may be used to take account of these factors and create a far more efficient staff schedule than would be possible through manual allocation alone (Jaumard et al., 1998). A similar situation exists in the restaurant industry, where there are many different roles to schedule, for example chefs, waitresses, kitchen assistants, and so on. Therefore linear programming is likely to be beneficial here also, and in fact Brusco and Johns (1998) show that linear programming not only eases the task of producing a schedule with an adequate skill mix, but also allows for the optimal productivity to be gained.

Data envelopment analysis (DEA) is a technique which has been used since the 1970s, when it was developed by Charnes and colleagues (1978, cited in Chase & Apte, 2007). This technique is aimed at measuring the overall operating efficiency of a service enterprise, although it is usually applied to organizations operating multiple units, for example restaurant chains. A DEA model utilizes linear programming within, but the complexity which is added by considering multiple units means that more complex operations are also included. The end product is a measurement of which units in the organization are operating efficiently and which are not. The analysis also allows for identification of how efficiency may be improved in these units.

There have been numerous studies which have shown that this is an effective method for improvement of overall organizational profitability. Some of these studies have focused on one specific organization, and have been conducted in an attempt to highlight strategies specific to that organization. For example Gimenez-Garc'a et al. (2007) used the linear programming-based DEA approach to determine which units in a Spanish restaurant chain were not performing to the same efficiency standards as the others. Other studies have focused on a much larger sample, for example the study by Sanjeev (2007) examined the efficiency of the entire restaurant and hotel industry in India.

These studies have overall shown that linear programming-based approaches are simpler than many other approaches. They have also been shown to be at least as effective at analyzing business efficiency in the restaurant sector as more traditional financial ratio analysis approaches (Sanjeev, 2007). Aside from the lower complexity in the use of linear programming and DEA, there are also other advantages over financial ratio analysis. This includes the fact that it focuses predominantly on the technical aspects of production, and is not based on assumptions on the cost of inputs and outputs (Banker & Morey, 1986). This basically means that rather than estimating the likely profit which will result from making a change, the estimate is instead of which action is likely to produce the most profit.

Although linear programming-based approaches have been an integral part of the operational research toolbox for a number of years now, there are still many researchers attempting to develop the capabilities of the technique further. This is evident in the vast amount of literature which appears regularly in the major peer-reviewed journals related to operations management (Gattoufi et al., 2004). Therefore it is likely that in time there will become ever more sophisticated tools available for managers in the restaurant industry, to assist them in making complex business decisions. However for many small restaurants, it is possible that the simplest linear programming techniques which are already available would provide a sufficient tool to allow them to better judge the outcomes of their business options.

Given the application which linear programming may have for the restaurant industry, it may be seen that each of these problems detailed in the introduction lends itself well to analysis with a linear programming model. This technique is therefore demonstrated in this paper using Microsoft Excel, and is used to model and solve the problems in the case studies.

Determining the Number of Meals to Prepare

The problem in the first case study relates to the number of meals which should be prepared each night at the restaurant if the profitability is to be maximized. The first step is to establish the constraints on the linear programming problem. It was already determined that there would be only two types of dishes served at the restaurant - beef dishes and fish dishes. Therefore the beef dishes are labeled as the first variable, X1, and the fish dishes are labeled as the second variable, X2. Now the profit from each fish dinner sold is around $12, and the profit from each beef dinner sold is around $16. This means that the overall profits which are made, which will be termed Z, would be given by:

16X1 + 12X2

The first constraint which must be taken into account in this decision is the labor requirements for the dishes. It is estimated that each beef dish requires 1/2 an hour to prepare, while each fish dish requires 1/4 an hour to prepare. Therefore the total labor required is given by:

Labour = 1/2 X1 + 1/4 X2

As there are only 20 hours of labor available for all the food preparation each day, this therefore gives the constraint:

X1 + 1/4 X2 ? 20

It has also been determined that the total number of dinners prepared each night will be less than 60, which gives:

X1 + X2 ? 60

However it has also been identified that at least 10% of… READ MORE

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Quoted Instructions for "Statistical" Assignment:

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This graduate level term paper/case study should be written in APA style, 10 pages and is centered around *****The Possibility***** Restaurant case studies (included below). While the case study will focus on the mathematical solution, you are also expected to bring value-added material in the way of outside research. The paper should adhere to the following outline.

1. Introduction

a. Start with the problem statements from the two case studies. Don*****t simply copy from the text, but briefly restate the problem in your own terms. This should take no more than one page.

b. Provide your value-added material from your research here. You are to research the application of operations research as applied to restaurant operations and provide a brief write-up. The write-up should take at least 3 pages. The material you provide should refer to an area of restaurant operations to which linear programming is applied, such as staff scheduling, food selection and pricing, etc.

2. Solve the problem from the first case study.

a. Formulation of the linear programming problem with each constraint labeled.

b. A graph of the feasible region with each extreme point labeled (X1, X2, Z). You can label the graph directly or provide a separate table of points.

3. Solve the problem from the second case study.

a. A copy and paste of the computer solution.

b. A copy and paste of the sensitivity report.

c. Answer the first question from the last paragraph of the first case study.

d. Answer the second question from the last paragraph of the first case study.

e. Answer question A of the second case study.

f. Answer question B of the second case study.

g. Answer question C of the second case study.

4. Conclusion and Recommendations

a. Recommendations on improvements or changes to the solution model based on your interpretation of the data from the sensitivity report. Examples are, what prices you might charge in keeping with the optimal product mix, the addition of resources to increase profits, etc. I am looking for your value-added input or perspective on this problem.

THIS PAPER MUST BE SUBMITTED AS A SINGLE MS WORD DOCUMENT, EITHER IN MS WORD 2003 OR EARLIER, OR AS AN .RTF FILE.

You may use QM for Windows or Excel to solve your problem. I highly recommend QM for Windows since it will produce a graph of the feasible region for linear programming problems with two decision variables.

If using QM for Windows:

*****¢ The computer solution is in the solution window.

*****¢ The sensitivity report is in the range window.

*****¢ The graph of the feasible region is in the graph window.

*****¢ To copy and paste any of these windows, select the window you want to copy and then select FileÃ Save As HTML. You can then insert the HTML file into your MS Word document.

If using Excel:

*****¢ The computer solution is in the original worksheet.

*****¢ The sensitivity report is in its own worksheet.

*****¢ There is no graph of the feasible region available. There is freeware and shareware on the internet you can download to produce a graph of the feasible region given the constraint lines you feed it.

To copy and paste the solution and sensitivity report, you highlight the region you wish to copy and select File-->Copy. Then go to your Word document and select FileÃ Paste Special, and paste it as a picture. Do not directly copy the cells into the document because you will wind up embedding the MS Excel application into the Word document, causing your file size to increase dramatically..

Case Study #1

Angela Fox and Zooey Caulfield were food and nutrition majors at State University, as well as close friends and roommates. Upon graduation Angela and Zooey decided to open a French restaurant in Draperton, the small town where the university was located. There were no other French restaurants in Draperton and the possibility of doing something new and somewhat risky intrigued the two friends. They purchased an old Victorian home just off Main Street for their new restaurant, which they named *****The Possibility.*****

Angela and Zooey knew in advance that at least initially they could not offer a full, varied menu of dishes. They had no idea what their local customers***** tastes in French cuisine would be, so they decided to serve only two full-course meals each night, one with beef and the other with fish. Their chef, Pierre, was confident he could make each dish so exciting and unique that two meals would be sufficient, at least until they could assess which menu items were most popular. Pierre indicated that with each meal he could experiment with different appetizers, soups, salads, vegetable dishes, and desserts until they were able to identify a full selection of menu items.

The next problem for Angela and Zooey was to determine how many meals to prepare for each night so they could shop for ingredients and set up the work schedule. They could not afford too much waste. They estimated that they would sell a maximum of 60 meals each night. Each fish dinner, including all accompaniments, requires 15 minutes to prepare, and each beef dinner takes twice as long. There is a total of 20 hours of kitchen staff labor available each day. Angela and Zooey believe that because of the health consciousness of their potential clientele, they will sell at least three fish dinners for every two beef dinners. However, they also believe that at least 10% of their customers will order beef dinners. The profit from each fish dinner will be approximately $12, and the profit from a beef dinner will be about $16.

Formulate a linear programming model for Angela and Zooey that will help them estimate the number of meals they should prepare each night and solve this model graphically.

If Angela and Zooey increased the menu price on the fish dinner so that the profit for both dinners was the same, what effect would that have on their solution? Suppose Angela and Zooey reconsidered the demand for beef dinners and decided that at least 20% of their customers would purchase beef dinners. What effect would this have on their meal preparation meal?

Case Study 2 (*****The Possibility***** Restaurant cont)

In *****The Possibility***** Restaurant, Angela and Zooey opened a French restaurant called *****The Possibility*****. Initially, Angela and Zooey could not offer a full, varied menu, so their chef, Pierre, prepared two full-course dinners with beef and fish each evening. In the case problem, Angela and Zooey wanted to develop a liner programming model to help determine the number of beef and fish meals they should prepare each night. Solve Zooey and Angela*****s liner programming model by using the computer.

A. Angela and Zooey are considering investing in some advertising to increase the maximum number of meals they serve. They estimate that if they spend $30 per day on a newspaper ad, it will increase the maximum number of meals they serve per day from 60 to 70. Should they make the investment?

B. Zooey and Angela are concerned about the reliability of some of their kitchen staff. They estimate that on some evenings they could have a staff reduction of as much as 5 hours. How would this affect their profit level?

C. The final question they would like to explore is raising the price of the fish dinner. Angela believes the price for a fish dinner is a little low and that it could be closer to the price of a beef dinner without affecting customer demand. However, Zooey has noted that Pierre has already made plans based on the number of dinners recommended by the linear programming solution. Angela has suggested a price increase that will increase profit for the fish dinner to $14. Would this be acceptable to Pierre, and how much additional profit would be realized?

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