BANA 7030: Simulation Modeling and Method-Individual Project Report

“On my honor, I have neither given nor received unauthorized aid in completing this academic work.”

- Aditya Nakate (M10947951)

A. Topic : Conducted a simulation study on ‘Canes’ restaurant near Calhoun street. This restaurant gets flooded with customers at night, especially on weekends and there are long queues outside the restaurant. Customers’ waiting time is quite high on these timings, also some customers leave the queues without taking service due to the long waiting times. In this study, I was planning to see if adding a server at the peak timings would help reduce customer waiting time significantly and thereby help restaurant serve more customers and make more profit. But after my conversation with the manager of the restaurant, I came to know that adding one more server just to reduce customers’ waiting time is not a beneficial option for the restaurant as they will need to pay the additional server. But after the close observation of the system, I realized that some customers decide their order after arriving at the counter, mainly due to unavailability of menu (Though menu is available on the screen it is very inconvenient to read it from a distance). Also, those customers who come in groups discuss their order after reaching to an order counter, which consumes a lot of time and other customers have to spend more time waiting in queue. In this study, I have analyzed two scenarios:

1) ‘Original’ scenario: Where some percentage of customers decide their orders after arriving at the counter

2) ‘Whatif’ scenario: Where customers have already decided their orders

I checked if there is a significant difference in the metrics like average waiting time, average number of customers waiting in a queue etc. between these two scenarios.

B. Model design and Data collection: In this restaurant, there is one server for taking orders from the customers, one server for delivering the food and four servers work in the kitchen. Out of these four servers, three servers work on the individual food items for the combo, for example, one server will work on preparing fingers, one will work on preparing fries etc. And one server will just produce a combo from all the individual food items. Following data was collected on this system:

1. Interarrival time of the customers2. Service Time at the order counter3. Time required for food preparation and delivery4. Number of customers who decide their order after arriving at the order counter

B. Part (1). Interarrival time distribution:Data for interarrival times (in minutes) was collected for three days (one hour on each day) between 8:30 to 10:30 PM as this is the peak time for the restaurant.

Following is the data for the same:

Day 1 Day 2 Day 30.47 2.11 4.16 2.61 0.37 2.35 0.03 4.70 0.385.17 2.89 0.11 0.01 1.12 0.87 1.41 1.08 2.773.53 0.23 0.33 0.23 4.29 0.92 1.26 0.24 1.780.48 0.22 3.26 2.50 0.01 1.35 0.16 0.04 1.811.90 0.64 6.52 0.61 1.64 0.66 5.35 1.08 5.970.94 0.82 1.39 4.98 1.37 0.07 1.30 1.69 0.094.78 0.65 4.21 0.47 0.70 1.56 1.92 2.33 0.111.55 0.03 1.37 1.88 4.99 1.81 2.02 0.12 0.650.54 2.11 0.24 1.47 0.56 1.47 1.29 0.19 2.401.00 0.39 0.98 0.08 0.14 0.73 3.18 0.40 0.652.01 1.07 1.22 0.31 0.74 0.23 8.93 1.044.71 0.43

Using Input Analyzer interarrival distribution was decided for the data:

Fig. 1

Figure 1 shows the fit for the data. Let us check the statistics for different distributions.Squared errors after fitting different distributions:

Above table shows that Exponential and Erlang distributions have the smallest squared error. Now let us look at the p values for both:

Erlang distribution: Exponential distribution:

As we can see in above tables, p-values for both the distributions and both the tests is greater than 0.05, which means that we fail to reject the null hypothesis and we can use one of the two distributions for our study. But as p-value for exponential distribution is greater than p-value for erlang distribution, we will choose exponential distribution: EXPO (1.62)

B. Part (2). Service time distribution for Order placement counter:Data was collected separately for the customers who had already decided their orders (before arriving at the order counter) and customers who decided their order after arriving at the counter. For convenience, this data was collected on separate days (i.e. not on the same days when interarrival time data was collected):

B. Part (2a). Service Time data for customers who had already decided their order:

Service Time (min)1.38 1.20 1.251.23 1.47 1.291.30 1.45 1.481.19 1.00 1.331.18 1.35 1.411.11 1.42 1.421.18 1.21 1.431.64 1.32 1.510.91 1.16 1.041.33 1.06 1.26

1.09 1.03 1.271.38 1.26

Using Input Analyzer, service time distribution was decided for the data:

Fig. 2

Figure 2 shows the fit for the data. Let us check the statistics for different distributions.Squared errors after fitting different distributions:

Above table shows that Normal distribution has the smallest squared error. But, Normal distribution can’t be used for service time distribution as it does create negative values also. Now let us look at the p values for Weibull and Beta distribution:

Weibull distribution: Beta distribution:

As we can see in above tables, p-values for both the distributions are lesser than 0.05 for chi-square test, which means that we reject the null hypothesis and we cannot use both the distributions for our study. Now, let us look at the p-values for triangular distribution:

As we can see p-values for both the tests are greater than 0.05 in case of triangular distribution. So, we will use triangular distribution to approximate the service time for the customers who have already decided their order (before arriving at the counter): TRIA(0.8, 1.3, 1.7)

B. Part (2b). Service Time data for customers who had not decided their order before reaching to the order counter:

Service Time (min)1.61 2.34 2.352.06 2.39 2.561.85 1.77 2.092.15 2.14 2.482.23 2.20 2.191.93 2.26 2.421.68 1.97 2.02

Using Input Analyzer, service time distribution was decided for the data:

Fig. 3

Figure 3 shows the fit for the data. Let us check the statistics for different distributions.Squared errors after fitting different distributions:

Above table shows that Triangular distribution has the smallest squared error. Now, let us look at the p-values for triangular distribution:

As we can see p-values for both the tests are greater than 0.05 in case of triangular distribution. So, we will use triangular distribution to approximate the service time for the customers who had not already decided their order (before arriving at the counter): TRIA (1.5, 2.2, 2.7)

B. Part (3). Percentage of customers who decide their order after arriving at the order counter:This percentage was estimated on two facts:

1. After spending around one and half hours collecting the data for service time at the order counter, I could collect this data for 35 customers who had already decided their order and for 21 customers who decided their order after arriving at the counter.So, % of people who did not decide their order before coming to the counter: (21/35+21): 37.5%

2. After talking to the server at order counter and manager I got the estimate of around 25-30%.So, I decided to conduct the analysis with 30%.

B. Part (4). Service time distribution for food preparation and serving counter:In this case, all the individual items in the combo are already prepared by the three servers who are just working on individual food items. The fourth server just needs to prepare combo out of these individual items, so service time in this case is quite low. Based on few readings taken for the same, we will assume that it follows uniform distribution: UNIF (0.5,1)

Based on the collected data and the distributions estimated, following model was built in ARENA:

Fig. 41. Create Module: Customers arrive at the rate of: EXPO (1.62)2. Decide Module: Customers are divided in the group of 70-30% (randomly, as we have estimated

that 70% of customers already decide their order and 30% do not3. Two Assign Modules: These assign modules are for assigning different service time distributions

to the customers A. Customers who have already decided their order will have lesser service times:

TRIA (0.8, 1.3, 1.7)B. Customers who have not already decided their order will have higher service times:

TRIA (1.5, 2.2, 2.7)4. Two Process Modules: Two process modules are for order placement and food preparation and

delivery process. Service time for food preparation and delivery: UNIF (0.5,1)

C. Model verification and Primary Analysis:

C. Part (1). Model Verification:

1. Single entity was released in the system and checked if it is following the expected path2. Output values like waiting time and total time were quite high when service time was

increased and interarrival time was decreased significantly3. 100 replications were run on the model and performance metrics like wait time, total

time, average number in queue and average number in system made sense

C. Part (2). Primary Analysis: Models for both the scenarios were run for 3 hours (as 8-11PM are the peak hours for the restaurant), 100 replications and following stats were obtained:

Average Max Average

Original Scenario

What-if scenario

Original Scenario

What-if scenario

Number of customers served 105 109 122 131Total Time in system 7.91(min) 3.95 (min) 18.8(min) 7.38(min)Average number of customers in system 4.98 2.47 12.99 5.26Average wait time in order counter queue 5.67(min) 1.94(min) 16.51(min) 5.26(min)Average number waiting in order counter queue 3.64 1.25 11.51 3.81

Comparison of metrics in graphs:Original Scenario:

Fig. 5

What-if scenario:

Fig. 6

Total customers in system and total customers waiting in queue vs time:

Original scenario:

Fig. 7What-if scenario:

Fig. 8

Checking the precision values for 100 replications:

Metric Original Scenario Whatif scenario

Output performance metric:Number seen

Average length of stay

Number seen

Average length of stay

Initial number of replicatins (N0) 100 100 100 100Initial mean 106.7 7.9114 108.65 3.95Initial 95% half width (h0) 4.66 0.74 2.1 0.19Initial relative precision 0.0437 0.0935 0.0193 0.0481Intial % relative precision 4.3674 9.3536 1.9328 4.8101Desired relative precision 0.09 0.09 0.09 0.09Desired half width (h) 9.603 0.7120 9.7785 0.3555Approximate total n required 24 109 5 29Approximate additional replications required -76 9 -95 -71

As we can see from above table, our output values are quite precise with 100 replications.From the above analysis, we see that performance metrics of what-if scenario are comparatively better than the original scenario. Now, let’s check these differences statistically and see if the difference is significant.

D. Statistical Analysis:

Statistical analysis was conducted in ARENA output analyzer and ARENA process analyzer to check if the difference is significant. Twenty replications of each scenario were run to collect the data.

1. Analysis using ARENA output analyzer: Using the output from twenty replications of each scenario two sample t-test with 95% confidence interval was run on the samples using output analyzer to check if the difference is significant:

As we can see from above output, the difference in all the metrics are significant with alpha value=0.05 level. This proves that alternate scenario is better than the original one.

2. Analysis using ARENA process analyzer:

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Above figures also give the same conclusion, that the difference between all the metrics is significant.

E. Conclusion: From the ARENA process and output analyzer, we can conclude that in alternate scenario average wait time, average number of customers in queue, total time in system and average number of customers in system(WIP) has decreased compared to the original scenario and the number of customers served has increased and these differences are statistically significant. So, our alternate system is better than the original system and is more beneficial for both customers and restaurant.

F. Recommendation: As we saw, in alternate scenario customer waiting time, total time in system has decreased significantly. Also, the average number of customers in system and in the queue at the time has also come down. If, restaurant can implement the alternate scenario it will result in better customer experience and more profit for the restaurant. It is not difficult to implement this scenario, as it can be done in following ways:

1. Restaurant can put menu boards just besides the queue, so that customers don’t have any difficulty in reading it and can decide their menu while standing in a queue.

2. Also, it would be great if restaurant can keep that board updated in the current time per the availability of the food item, so that customers don’t have to rethink about their order while at the order counter

These options are not very costly and will be very beneficial to customers as well as restaurant.