So far, we have looked at statistics that you perform on actual, real data that you have gathered.

On the other hand, simulation is a class of analysis based on hypothetical data that you make up, in order to study real life events and objects. This chapter explains this concept briefly, starting with an initial example.

To explain this concept of simulation with hypothetical made-up data, imagine you are an automobile company. You want to assess the production of a certain car, which is assembled on a production line on which employees and machines work in a given sequence of activities. There are various things you could change to improve the efficiency of the production line, such as the sequence of activities and the speed that the cars come down the line. However, actually changing the currently active production line to test out different options is not feasible.

You could simulate the production line changes to assess the most efficient workflow design. In other words you could create a hypothetical model of the production line using the known speeds, possible employee and machine combinations, and the like. These days, such models would be created by computer. Then, you could try out various options on the hypothetical model, such as varying speeds, number of employees performing certain actions, and the like. Modern computing power could enable many thousands – even far more – different combinations and possibly show management more efficient configurations for the factory.

The following sections discuss types of simulations, describe a broad approach to simulation, and give a specific example.

To better understand mathematical simulation, let us turn immediately to a simple example.

Say you want to model a queuing system in a retail environment because you wish to optimize how many tills to use and what type of queuing system you can use. You could build mathematical equations for each major part of the experience. For instance, the time to process each person at the till could be represented as a function of the number of items they are buying and the payment method (cash or card). A simple equation in this regard could be:

(Time to process person “i”) = Constant + β₁(No. items) + β₂(Card payment) + ϵ.

Here, the time to process a given person is defined as a linear function of:

Therefore, this is essentially the same as a regression line. Say that the constant is 23.56 seconds, β₁ = 4.3 (which means that a cashier processes an item every 4.3 seconds on average), and β₂= 10.85 (which means that it takes the cashier 10.85 seconds longer to process a card payment than a cash payment).

Then, if a simulated person has ten items and pays with a card, we can expect the time at the till to be:

(Time to process person “i”) = 23.56 + (4.3 seconds x 10 items) + (10.85 seconds to represent the extra time it takes to process a card) = 77.41 seconds.

So far, this is an example of a deterministic model, in that the parameters and data is set and we get a specific outcome.

However, simulation models are usually most useful when we build stochastic models, which include the element of probability. Say, for instance, that we program SAS to use the above equation but to generate a random, hypothetical sample of shoppers drawn from a specified distribution of number of items and with a certain randomized chance of being either a cash or card payer. The distribution of items could be specified as a particular shape, say the normal or lognormal distributions as seen in Chapter 7, so that a random person could have any number of shopping items in the range but the total distribution of shoppers is determined by the shape. (For example, you could specify that most simulated shoppers be allocated close to the average number of items and only a few shoppers be given an extremely low or high number of items.) Both the distribution of items and of payment methods could be drawn from observed, known, historical distributions seen in past shopper patterns.

To illustrate, say that we randomly simulate the five shoppers seen in Figure 18.3 Example of time at till for five simulated shoppers below, where the inputs (number of items and payment method) are randomly chosen. The equation above will dictate the time at till (we keep the parameters the same).

We see in Figure 18.3 Example of time at till for five simulated shoppers above that this results in a simulated distribution of time at till.

Now, what happens if we decide to look at multiple queues and types of tills, set up on different systems? For instance, do we have one queue feeding multiple queues, or one queue per till? What happens if we have cash-only tills, and if so, how many should we have? We could add some more elements and some more equations using the same principles, update the equations to express the interrelationships of elements across the whole experience of the queuing system, and perhaps compare different systems. Telling SAS to simulate distributions of shoppers would allow us to look for a system that – according to the simulated distribution – minimizes the average person’s time in the queue. We could also test the effect of changing the time parameters on these systems.

As you can see, the data and analyses elements exist in multiple places in this very simple example:

Such mathematical simulation models can be used to optimize many processes and systems across organizations.

Mathematical simulation models are incredibly useful for multiple business situations ranging from assessing logistics supply chain systems, to staffing call centers, to production problems, to financial analysis. To read more on simulation, see texts such as Greasley (2003) and Wicklin (2013).