For this post I
am going to show with further detail how simulation techniques used in forecasting
can also be used to make predictions to understand and improve operations. In
particular, I will use the ten-step sequential process example (single-piece flow) provided in the core reading “Designing, Managing and Improving
Operations” [1] as a starting point.

As stated in this
example, we can imagine a ten-step sequential process in which each task takes
3 minutes per unit to complete. Therefore, a process with single-peace flow
takes 30 minutes to complete the first unit. However, this does not necessarily
happen in practice because instead of having specific times for each operation
we have a distribution of values. In other words, a task could take in average
3 minutes to complete but there is a probability it would take 3 minutes and 10
seconds or even less than the average. Statistics plays a vital role to
understand and improve timing in operations. I will show using basic Monte
Carlo simulations[1] how
this is possible.

To start with, we could restate the example
shown above having instead each task associated to a probability distribution,
for instance, a normal distribution with mean 3 and standard deviation 0.1 (see
Figure 1). We can assume these values were obtained from timing workers to
complete their tasks a statistically significant number of times. In particular,
it would be useful to know the distribution for the time to make the first unit
and the probability of taking less than 31 minutes to complete it -let us
assume this is a critical time for the process-.

Figure 1: Ten-step
sequential process using probability distributions. Adapted from [1]

The advantage of applying Monte Carlo in this case is that we can do repeated calculations
using random values that satisfy the distributions for each task for a very
large number of times (10,000 times in these simulations) to calculate the
values we are interested in. I programmed this simulation using Matlab (a script
with explanations is available at the end of the document, see Appendix 1) but
it could also be done in Excel.

From this
simulation, the estimated time to make the first unit gave a normal
distribution with mean 30 (an expected result) and standard deviation ~0.3. The
probability to make the first unit in less than 31 minutes was more than 0.999,
i.e., the probability of ending the first unit in more than 31 minutes is less
than 1 in 10,000.

Now, let us
assume that worker in charge of task 5 is retiring and will be replaced for a
new inexperienced worker. The time it takes to perform this task on average is
only 30 more seconds than the experienced worker (3.5 minutes), but is not
accustomed to work on this task so the time it takes varies more from average
(0.7 minutes). In other words, the distribution for this task now will be
assumed to be a normal distribution with mean 3.5 and standard deviation 0.7.
Then, again, what would be the distribution for the time to make the first unit
and the probability of taking less than 31 minutes to complete the first unit?

From this second
simulation, the estimated time to make the first unit gave a normal
distribution with mean 30.5 and standard deviation ~0.8. Although in average it
takes less than 31 minutes to make the first unit, the probability to make the
first unit in less than 31 minutes was ~0.75, i.e., the probability of ending
the first unit in more than 31 minutes is in the order of 1 in 4. This change could have a significant
impact in the overall process if this time were critical as was previously
assumed.

To sum up, I
tried to show in this example how a simulation technique that is used in
forecast (Monte Carlo simulation) can also be used to design, manage or improve
operations. This technique was used to calculate the probability distribution of the time
to make a single unit in a single-piece flow process but could be further
exploited to calculate the distribution time to make the batch, calculate dead
times, and also be used for other flow processes. Moreover, this is not the
only technique that can be used for this purpose, e.g. Markov Chains can be
used to forecast demand (a practical example is provided in [3]) but also for
inventory management [4].

[1] Roy D. Shapiro (September 10

^{th}, 2013), “Core Curriculum. Operations Management Reading: Designing, Managing, and Improving Operations”, Harvard Business Publishing, US.
[2]See for further information: http://www.vertex42.com/ExcelArticles/mc/MonteCarloSimulation.html (Accessed 8/31/2014)

[3] N.A. (n.d.), “Markov Chains”, Temple University,
Pennsylvania, USA. Retrieved from: http://astro.temple.edu/~hweiss/emba/markov.ppt
(Accessed 8/31/2014)

[4] Erhan Bayraktar, Michael Ludovski (n.d.) “Inventory
Management with Partially Observed Nonstationary Demand”, University of
California, California, US. Retrieved from: http://www.pstat.ucsb.edu/faculty/ludkovski/InvControl.pdf
(Accessed 8/31/2014)

APPENDIX 1

SIMULATION SCRIPT

%SIMULATION 1

%The following line
generates 10,000 examples for

%the 10 tasks of the
process with a normal distr.

%with mean 3 and standard
deviation 0.1

r1 = 3 + 0.1.*randn(10000,10);

%The following lines
calculate the total time

%for each of the 10,000
examples, the mean

%and the standard deviation of total times

totalr1=sum(r1,2);

meanr1=mean(totalr1);

stdr1=std(totalr1);

%The following line gives a
histogram using 21 bars

hist(totalr1,21);

%the following line
calculates the probability

%of taking less than 31
minutes for the total time

p1=sum(totalr1<=31)/10000;

%SIMULATION 2 (see
SIMULATION 1 for explanations

% of similar commands)

r2 = 3 + 0.1.*randn(10000,10);

% The following line
modifies task 5 distribution

% i.e., normal distribution
with mean 3.5 and std 0.7

r2(:,5) = 3.5 +
0.7.*randn(10000,1);

totalr2=sum(r2,2);

meanr2=mean(totalr2);

stdr2=std(totalr2);

hist(totalr2,21);

p2=sum(totalr2<=31)/10000;

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