MSCI512: Simulation and Stochastic Modelling

2024/25 Coursework Task

Task instructions (please read carefully)

•   The coursework consists of three problems that require the application of stochastic modelling or simulation.

•    Provide an answer for all three problems.

•   The maximum score is 100:

•    Question 1 is worth 25 marks

•    Question 2 is worth 20 marks

•    Question 3 is worth 45 marks

•   A further 10 marks will be awarded for the clarity of the report writing.

•   You are expected to use appropriate software (e.g. Microsoft Excel,R, Python, Simul8) to help you answer the questions.

•   This coursework is related to all parts of the module.  You may use any method from the course to approach the problems.

•    For all questions describe and justify the assumptions you are making with your model, as well as any potential limitations.

•   When providing a numerical answer that does not involve a confidence  interval, report it correct to at least 3 significant figures.  Also, when using non-exact numbers obtained from aprevious calculation, make sure these numbers are correct to at least

3 significant figures.

•   This coursework is to be done as part of a group of two or three. Collusion between groups and plagiarism are regarded as serious offences and will be penalised severely. Information about the university’s  plagiarism framework can be found   here: https://modules.lancaster.ac.uk/mod/resource/view.php?id=1384800 .

Submission instructions

•   The deadline for submission is 9:00AM on March 24th 2025.

•   Only one member of the group is required to make the submission.

•   Your answers to the questions can be either typed (e.g. using Microsoft Word) or handwritten and scanned.

•   You should include full details of your methods in your answers.   If you  calculate answers by hand, you should include all of your calculation steps.  If you use computer software (e.g. spreadsheets, computer programs) you should include the relevant files (e.g. .xlsx, .py, .txt, .s8) in your submission.  Please also include enough explanation in your answers to make it clear what you are doing and why.

Question 1 [25 marks]

A call centre operates 24 hours a day, 7 days a week. When a customer calls, they wait for one member of staff to take their call. All members of staff are cross-trained and so are able to handle all parts of the enquiry. After the enquiry has been fulfilled, the call ends and the staff member is free to take the next call.

The staff all work in 8-hour shifts in the following pattern:

•   6am - 2pm

•   2pm - 10pm

•    10pm - 6am.

The call centre managers are concerned about the current quality of service, measured by the number of calls waiting to be handled. They are therefore considering how many staff are required, and how best to allocate them between the shifts.

The call centre has hourly counts of the number of arrivals for 2 weeks. It also collected the time spent speaking to a member of staff for all customers in this period. Using this data, answer the following questions.

(a)  Using a stationary approach, estimate the average number of servers (per shift) that are required for this system such that at least 40% of calls are answered immediately, without having to wait.

(b)  Using a total of 3 times the number of servers identified in part (a), allocate these across the three shifts in away that controls the expected number of calls currently ongoing and the probability of having to wait in the best way.

In reality, the telephone network at the centre can only handle a finite number of calls at anyone time. If the network is full,a new call will be declined. The network could be setup to hold 15 calls, 25 calls, 50 or 100 calls. The cost is of each configuration is approximately linear in the number of calls. The managers will need to decide which togo for.

(c)  Explore how the different call capacities impact your answers from above. For each case, when does the probability of a call being declined reach its maximum, and what value does this take? What would your recommendation be in caller capacity? Provide a description of any models used and detail any assumptions that you make.

Question 2 [20 marks]

A local removal firm offers two short term services, a self-drive van-hire in which the customer hires the van but drives the van themselves, and a full removal service in which the firm also provides a team to move the customer’s belongings. All jobs generally take one day. The company will only accept up to 5 customers for each service at a time. If the number of customers on the books for the following working day goes above 5 for either service, no more work is accepted on that service.

The income from the self-drive service is £40, whilst for the full removal service, the income is £60. However, if the firm are notable to complete a job the next working day, the customer is offered a discount of £15 or £20 for the two services, respectively. If the customer still hasn’t been served, they are offered a further discount of equal amount.

Currently, the firm has 5 vans, three of which are set up to be insured to be driven by hiring customers (and are used exclusively for this purpose), and two which are just used by the company for the full removal service.

The company has provided data showing the number of requests made in each day and the number of customers on the books (at the end of a day over a period of the last 6 months    (130 working days).

(a)  In the long run, what is the average operating profit per day in this way? On what proportion of the days are not all the vans busy? On what proportion of days might work be turned away?

Suppose that the company could hire up to an additional 2 vans on any given day. The cost of hiring a van is £50. These vans could be allocated to either of the services.

(b) Considering a time horizon of 28 days, determine a policy that states when it is best to hire one van or two vans, and to which services should they be allocated, based on the number of jobs currently in the books.

(c)  If this policy were to start 5 days after the end of the data period you have, what would the expected operating profit be?

(d) Suppose there was a chance of negotiating the van hire cost. Considering a range of values, how would this change the decisions and the expected operating profit?

Provide a description of any models used and detail any assumptions that you make.

Question 3 [45 marks]

A fast-food restaurant in Lancaster is currently going through a refurbishment and is looking to introduce two new features when it reopens. The first is that it will offer a delivery service, just open between 4pm and 9pm. It will also only be available to those as far south as Galgate, as far north as Slyne and Hest Bank and to the west into Morecambe. They believe this will help them to attract more customers and so increase revenue, complementing the people who arrive at the restaurant in person (some of whomeat in, whilst others take away). Their projection, based on a local survey, is that this could be as high as an increase of 10%.

The second is that it will replace its current ordering process with some large touchscreen self-service machines that send the order directly to the kitchen. They hope this will help them reduce costs and potentially improve the time it takes to serve a customer.

This will also mean that the layout of the restaurant will have to change. Currently there are 30 tables in the restaurant. However, for each automated system they install, they will need to lose 2 tables. Their experience is that people who sit in the restaurant to eat sometimes come back for an additional order, such as getting a dessert or a coffee. Whilst they want the time between the arrival and the food being prepared to be small, they don’twant to lose too many tables and therefore lose sales.

The current procedure is as follows. The customers arrive and order their food (currently by speaking to a member of staff, but this will move to the self-service machines on re-opening). After this, the food is prepared by one of 6 members of staff. Once the food is prepared, the customers can theneither take this food away and eat it elsewhere or find a   table and sit in the restaurant while they eat. If all the tables are taken, customers generally take their food out. As previously mentioned, some of the customers who eat in may come  up to reorder food and can theneither sit in (usually this second period of time is shorter than the first) or take away.

For food that is being delivered, the order will go straight to the kitchen. Once the food is ready, it will be set aside ready for a courier (employed by the restaurant) to deliver. They are thinking that a sensible target would be to deliver the food within an hour.

The managers need to decide:

1.   How many couriers should they employ?

2.   How many self-service order machines should be installed? The key performance measures to bear in mind are:

•   The percentage of food orders which are delivered within 1 hour of the order being submitted.

•   The time it takes between a customer arriving and receiving their food.

•   The expected number of orders completed per day. The data available are

•   The number of arrivals per hour to the restaurant over a two-week period.

•    Manufacturer time trial data on how long it takes to use the self-service system.

•    For some customers in one day, the following information was tracked:

o How long it took to prepare food for an order

o How long they stayed in the restaurant (if they did), following the order,

o If they went for a second order, how long it was before this food was ready and how long they stayed after that (if they did)

•    For the time to deliver, no data is yet available. However, the restaurant has asked local couriers to provide some data. They expect that collecting the order and going back to a vehicle will take at least five minutes. Travel times for the area identified are likely to be less than 15 minutes, often around the 8-12 minutes mark.

Produce a report to advise the restaurant owners on these issues. Consider how sensitive the results are to some of the assumptions made. Include details of any models developed. [25 marks for conceptual and computational model, 20 marks for analysis].