BUSI2194 Introduction to Modeling and Simulation 2023-2024 Semester Two
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2023-2024 Semester Two
BUSI2194 Introduction to Modeling and Simulation
Coursework
Please answer the following Questions 1 to 3. Discuss each of the sub-questions clearly and show your reasoning and calculation procedures (including tables and figures if any). If you have any supporting documents (e.g., excel spreadsheets and codes), please upload together with your report in a single zip file. NOTE: the submission deadline is 3pm, April 18, 2024.
1. Consider the initial value problem of the continuous logistic growth model for the population dynamics: 
(a) Show that the problem has the following solution: 
[10 marks]
(b) Compare the above solution at time tf with the result of the Improved Euler,s Method you learned assuming that x0 = 1, r = 0.5, K = 10, t0 = 0, and tf = 10. [12 marks]
(c) How does the error scale in the step size parameter ℎ? You may plot the errors against various ℎ values in logarithm scale and compare it with the results using Matlab ode45 function. [8 marks]
2. Consider a mini-mall design problem where four stores are aligned as shown in the figure below. To analyse the flow of a group of a large number of customers, we consider an ‘average, customer,s shopping behaviour by a discrete-time model with its location as the state. The two virtual stores i = 1 and i = 5 are added to represent the entry and exit. We assume the travel time between stores can be neglected. The probability that a customer turns to visiting store j after visiting store i is given by the entry in the (j + 1)-throw and the (i + 1)-th column of the state transition matrix P, which is show below:
(a) Suppose that at time 0, the customer is located at the entry, what is the probability that this customer appears at each of the stores or exit at time 1, time 2 and time 3? [8 marks]
(c) What is the expected total number of time periods a customer will spend (untill exiting) in the mall? You may compute the result either by simulating the dynamics or applying the theory you learned in the readings. [10 marks]
(c) Suppose all the stores sell products with similar types and prices. If you were to invest in renovating one store for increasing its attractiveness, which of the four will you choose? Why? The estimated impact of renovating store, is that all the entries in the (j+1)-th row of P will increase by 20%. You may assume that the probabilities for other stores will reduce proportional to their original values to make P remain a valid state transition matrix. [12 marks]
(d) Discuss one aspect that you feel unrealistic regarding the model assumptions. Accordingly, what modification you will propose to the model? [5 marks]
3. Consider an inventory management problem where the inventory level is checked at the end of each day, if it drops to no more than level R, a fixed-size order Q is placed immediately which will be delivered one day later. Assume that the daily demand is normally distributed as N(μ, σ 2 ). At the beginning of each day, the demand is realized and inventory is consumed. Unfulfilled demand at each day is simply lost.
(a) Denote Dt and It the demand of day t and the inventory level at the end of day t , respectively. Express It+1 in terms of Dt and It as a set of difference equations. [6 marks]
(b) Setup a simulation forT = 60 days with Q = 20, μ = 10, σ = 4 and R = 16. You may start with inventory level I0 = 10 at the beginning of day 1. Define your own requirement of the half-width of the confidence interval (CI) as β . Then choose the proper number of simulation replications and report the 95% confidence interval of the estimate of the stock-out rate (i.e., the number of days inventory reduces to zero divided by T). [13 marks]
(c) Suppose you need to pay 250RMB for the delivery fee of each order and holding each unit of inventory costs you 50RMB per day. Will it be better to reduce Q to 10 for saving the sum of delivery and inventory costs? Create a reasonable criterion for this comparison and justify your answer using simulation. [10 marks]
(d) Present and validate a strategy that can help you reach the same conclusion as in (c) with fewer number of simulations. [6 marks]
2024-04-29
Business simulation and modeling