Past Issue

Numbers Game - Modeling Helps It Add Up

By Robert Roman


Car washing is a “numbers game.” The more cars processed the more money to be made. The greater the production is on best days the healthier the bottom line.

Consider a wash facing below zero temperatures. Depending on circumstances, it may not make sense to deal with the multitude of issues involved with operating during extended periods of duress. Unfortunately, there is no way to know this except to try it and measure outcomes.

A way to capture behavior of a system is to create a model that explains previous behavior and can give confidence in its outcomes.

Consider a history of 100 washes a day when the temperature is between 5 and 10 degrees, 50 washes when zero, and 30 washes or fewer when below zero. If average cost of providing service is $80 an hour or $960 for 12 hours, average sales and trade-off analysis can be used to help decide whether to open or close.

Another example is the queuing model. Experience shows very long waiting lines cause motorists to balk and leave the line or refuse to join the line. This means the wash is losing business because it is not processing all of the cars possible.

Since customers arrive randomly, their behavior cannot be predicted exactly but it can be modeled. If it takes four minutes to wash and cars arrive with random probability of 25 percent, how many cars would get washed in 600 minutes, and how long would each car wait in line?

Answers can be obtained by varying wash time and probability of arrival to simulate the optimal state of the system. Here, the queuing model says 150 cars were washed and average wait was 13.5 minutes. With this data, the operator can perform benefit cost of reducing waiting time.

Another example is a system for creating “smart” models constructed by a computer expert that provides a set of objects from which to build car wash models that allow for constraints, queries, and rules for translating the model into one the system can solve.

There are also non-mathematical models like operations models, which describe the car wash system in terms of product standardization, process, and production of goods versus services.

For example, providing assisted services involves a higher degree of customer contact than exterior washing. However, both are sensitive to demand variability and alternate between lines of customers waiting for service, idle pay terminals, and employees waiting for customers.

Service providers require higher labor content and have greater variability of inputs whereas exterior washing is capital intensive due to automation and technology. Because machines tend to be smooth and efficient, assisted-services often appear awkward and slow.

Output-per-unit time is straightforward for exterior whereas assisted-services have variations that make productivity measurement difficult. In the tunnel, production errors are programmed out. Quality assurance is more difficult in services because production and consumption occurs at the same time.

Consequently, there are times when math and non-math models could be combined to solve problems.

For example, what is the availability of a wash if the machinery breaks down or employees call out? Here, the user of the system would query “availability,” and an object resolves the model. The “smart” system reinterprets the car wash model with a reliability model using solution techniques (e.g., Markov chains).

The expert who constructed this smart system imagines this model extended to handle profit-and-loss queries. This might include modeling the effect of throughput on reliability as in service technicians who become tired easily when above some threshold or adding people who are slower but are healthier.

This is one of the ideas behind models. The user should not have to be concerned about the details, such as how to choose the best solutions method, but rather just ask for a solution and have the appropriate solver.


Bob Roman is president of RJR Enterprises – Consulting Services ( You can reach Bob via e-mail at


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