Case Study: Modelling Requests to a Courier Service

Submitted: 9 Aug 2009

Operations Research Topics: SimulationModelling

Application Areas: Logistics

Contents

• Problem Description
• Problem Formulation
• Computational Model
• Results
• Conclusions

Problem Description

A courier company offers two services: an Inner City delivery service that delivers packages within the Inner City area in under an hour; a Metropolitan delivery service that delivers packages within the Metrolpolitan area in less than 4 hours.

An Inner City courier leaves the distribution centre if EITHER there are 10 deliveries to make OR it has been 15 minutes since the "oldest" delivery request arrived, i.e., the first request that arrived after the last delivery run departed. A Metropolitan courier leaves the distribution centre if EITHER there are 30 deliveries to make OR it has been 30 mins since the oldest delivery arrived. The courier company works for 8 hours a day and starts each day with no deliveries to make.

The courier company has collected data on the time between requests for both Inner City and Metropolitan deliveries as well as the time to make Inner City and Metropolitan deliveries. They want to know how many deliveries their Inner City and Metropolitan couriers make each day and also how long they are on the road during a day.

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Problem Formulation

To simplfy the modelling we will abstract 4 sections of the problem:

1. Requests for Inner City deliveries;
2. Requests for Metropolitan deliveries;
3. Delivery runs for Inner City deliveries;
4. Delivery runs for Metropolitan deliveries.

We start by assuming we can model these accurately using random variables, so we can concentrate on the remaining logic in the system. Figure 1 shows a flow diagram for the Inner City deliveries (note that a flowchart for the Metropolitan deliveries would be similar).

Figure 1 Flow Diagram for Inner City Deliveries

The way to implement this flow digram will differ depending on the simulation modelling tools used.

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Computational Model

To model the flow digram from Figure 1 we first abstract the delivery requests and request runs as Arena submodels. The submodel for requests only generate requests, so a single submodel with only an exit point is needed. The submodel for a delivery run has the load of deliveries as input and outputs the load of deliveries after the delivery run finishes, so it has a single entry and exit point. The following flash tutorial shows how to add the appropriate submodels in Arena:

Adding submodels for Courier Model

Next, we want queues that store delivery entities until a delivery run occurs. The next flash tutorial shows how to add a Hold module and a Variable to implement this queue for Inner City deliveries:

Adding waiting queue for Courier Model

Next, we want to create logical entities to take care of triggering delivery runs if the oldest delivery request has waited long enough. Each time a delivery request comes in and there are no other deliveries waiting we note down its arrival time and create a timing entity using a Delay module. When the timing entity leaves the system a calendar event will be generated. We will see how this calendar event is used later. The following flash tutorial shows how to create the timing entity for the Inner City deliveries:

Adding timing entity to Courier Model

Next, we create logical entities to trigger a delivery if:

1. The amount of time since the oldest delivery arrived has reached the maximum time;
2. The number of deliveries waiting has reached the maximum load.

The arrival of a new delivery creates a calendar event, as does the disposal of the timing entity created just before. Hold modules will scan the simulation state each calendar event, so we can use one of these to trigger a delivery. We will use a second Hold module to delay the logical entity until the deliveries have been loaded, it will then wait until the next trigger event:

Adding trigger entity to Courier Model

Now that we can trigger the loading of a delivery run, we want to combine all the deliveries into a single load, finish loading and make a delivery run. Once the run is finished we want to separate the deliveries and let them leave the system:

Adding deliveries to Courier Model

Now the logic of the model is complete except for the arrivals and delivery runs. You complete model should look like that in Figure 2.

Figure 2 Courier Model with complete logic

To start off with we will try some simple distributions. Use exponential interarrival times with mean 10 minutes and 20 minutes for Inner City and Metropolitan delivery requests (make sure to use EXPO(10) and EXPO(20) for the first arrival times too). We will use triangular distributions with min, mode and max of 10, 15, 20 and 25, 45, 90 for Inner CIty and Metropolitan delivery runs respectively. Add the appropriate Create and Process modules to your submodels. Connect them to the appropriate entry and exit points. Finally, make sure you have an infinite number of couriers by editing the capacity of your Resources.

Now run your model for 50 replications. Each replication should run for 8 hours (i.e., 1 day of the courier operation) and the Base Time Units are minutes.

Debugging

You should notice that your model appears to "stall". If you look carefully you will see the logic entity that triggers Inner City delivery runs in an infinite loop. This happens because our hold condition is not quite correct. We rely on a request arriving and resetting the oldest delivery time. However, this does not happen before the truck is loaded, so the initial trigger hold does not work. We need to wait for a delivery after we load the truck. By adding an extra condition to the timing trigger, we can fix the problem. Figure 3 shows this extra condition, that the delivery queue must have at least one delivery AND that the oldest delivery has been there for too long for a delivery run to take place.

Figure 3 Extra condition for timing trigger

After debugging and re-running you model you should get some output such as 0.5782 Inner City couriers being used on average and 18.84 Inner City delivery runs on average.

You can also use animation to check your model, such as animating queues and times to make sure the maximum load and maximum time restrictions are enforced. You can visualise the utilisation of the couriers and the number of each type of courier on the road at any one time:

Adding animation to Courier Model

Finding Input Distributions

As discussed previously, the courier company has recorded data on their delivery requests and delivery runs. This data is given courier.xls.

We can use Arena's built-in Input Analyzer to determine distributions that fit the data well. First, we need to break the data up into separate text files as shown in Figure 4.

Figure 4 Splitting data into text files

Then, we can use the Input Analyzer:

Analysing Inner City delivery requests

The Input Analyzer uses the Chi Square and Kolmogorov-Smirnov goodness-of-fit tests to detremine which distributions fit best.

Once you have determined the best distributions, you can cut-and-paste their expressions into the appropriate submodels and run your simulation model again. Some results are shown in Figure 5.

Figure 5 Results after input analysis

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Results

The results...

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Conclusions

In conclusion...

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• courier_prep_data.png:
  

• courier_input_usage.png: