This section contains 1 compulsory question. You are required to answer this question. This section accounts for 25 marks.
All New Paper Recycling (hereafter, ANPR) is a large recycling factory located in North Melbourne, Victoria. The main business of ANPR is to recycle waste papers into three types of paper pulp that can be used to produce newsprint, packaging, and high-quality photographic print, respectively. The sources of waste papers for ANPR mainly come from newspaper, white office paper, and cardboard. The table below gives the yield for each pulp recycled from each ton of waste paper.
Recycling Yield Newspaper White Office Paper Cardboard
Newsprint 85% 90% 80%
Packaging 80% 85% 70%
Photographic print 0 80% 0%
To illustrate the data in the table, ANPR can recover 0.85 tons of newsprint pulp per ton of newspaper waste. It can also recover 0.8 tons of packaging paper per ton of newspaper waste. However, no photographic print pulp can be produced by using newspaper waste.
There are two types of costs involved in the paper recycling business: processing cost, and purchase cost of waste papers. The cost of processing each ton of waste paper into various types of pulp is given in the table below.
Processing Cost Newspaper White Office Paper Cardboard
Newsprint $6.5 $4.75 $7.5
Packaging $11 $7.75 $8.5
Photographic print — $8.5 —
The following table gives the capacity of each type of waste paper and its purchase cost per ton.
Newspaper White Office Paper Cardboard
Cost per ton $15 $19 $17
Capacity available (in tons) 600 800 600
ANPR must produce exactly 500 tons of newsprint pulp, at least 600 tons of packaging paper pulp, and exactly 300 tons of photographic print pulp. The objective of ANPR is to minimize the total costs to meet the production requirements by determining how many tones of each waste paper to be used for producing each paper pulp.
Below is a snapshot of the (partial) sensitivity analysis report generated by Solver.
Note that you do not need to solve the model in Excel to answer the following questions. Read the questions carefully as some questions require you to provide justifications for your answers.
i. Formulate an LP model for the above problem. Clearly define the decision variables, objective function, and constraints. (7 marks)
ii. What is the value for the cell D10 (i.e., row 10 column D) in the sensitivity analysis table? Provide a brief justification for your answer. (2 marks)
iii. What is the value for the cell E12 (i.e., row 12 column E) in the sensitivity analysis table? Provide a brief justification for your answer. (2 marks)
iv. What is the value for the cell D23 (i.e., row 23 column D) in the sensitivity analysis table? Provide a brief justification for your answer. (2 marks)
v. What is the value for the cell E24 (i.e., row 24 column E) in the sensitivity analysis table? Provide a brief justification for your answer. (2 marks)
vi. Suppose the objective function coefficient associated with “Packaging – Newspaper” is changed to 30. Is the current solution still optimal? And why? (3 marks)
vii. Provide the intuition of why the optimal value for the decision variable associated with “Newsprint – Cardboard” is 0. Also, do we expect to see a positive value for this variable if the corresponding objective function coefficient becomes 26? And why? (4 marks)
viii. If ANPR must produce at least 700 tons of packaging paper pulp, what would be the resulting change to the total cost? Provide justifications for your answer. (3 marks)
END OF SECTION
This section contains 4 selective questions. You are required to answer 3 questions. This section accounts for 75 marks.
An investment company, ‘Breed’ considers several proposed projects to undertake in the information technology (IT) sector in the next fiscal year. The demand for online activities has increased tremendously over the past year. The number of engineers, the number of technicians, hardware costs (in millions of dollars) required, and the expected profits (in millions of dollars) for each project are summarised in the following table.
Project A B C D E F
Engineers 30 60 50 70 100 80
Technicians 50 90 80 100 170 120
Hardware costs 0.2 0.1 0.3 0.4 0.2 0.5
Profit 1.5 1 3 2 4 5
Breed is subject to some restrictions presented by the annual budget approval committee. Breed cannot use more than 300 engineers and 400 technicians. However, the company must spend at least 1 million dollars on IT investments to meet the company targets. To maintain the diversity of the investments, management recommends the following tactics:
1. If either project A or project C is selected, both must be selected.
2. Project B can be selected only if project E is selected
3. If project D is selected, project F must not be selected and vice versa
4. No more than five projects are to be selected.
Formulate an ILP that maximises the company’s profits subject to the constraints mentioned
above. (10 marks)
Broadmeadow is a 100-room hotel located near the Niagara Falls, Ontario in Canada. When a special event is in town, Broadmeadow increases its normal room rates and takes reservations based on a revenue management system. Academy of Management (the largest academic community in the discipline of business management) scheduled its annual conference in Ontario for the first weekend in August. Broadmeadow agreed to make at least 60% of its rooms available for conference attendees at a special rate in order to be listed as a recommended hotel for the conference. Although a majority of attendees request a Friday and Saturday two-night package, some attendees may select a Friday night only or a Saturday night only reservation. Customers not attending the conference may also request a Friday and Saturday two-night package or make a Friday night only or a Saturday night only reservation. The costs for each type of reservation are shown below.
Two-night package Friday night only Saturday night only
Conference attendee 200 120 130
Regular customer 300 150 160
The anticipated demand for each type of reservation is as follows.
Two-night package Friday night only Saturday night only
Conference attendee 50 30 20
Regular customer 30 40 30
Broadmeadow wants to determine how many rooms to make available for each type of reservation to maximise total revenue. Formulate an ILP model for this revenue management application.
Demand for disposable masks has tremendously increased due to the spread of a global pandemic. An Asian country, X contracted a national company, Ximon to fulfill the increased demand for masks from the country, Y. Ximon proposed to set up five manufacturing plants to cater to the demand for masks from the country, Y as well as to keep a buffer stock at the proposed plants. As part of the strategic plan, these proposed plants are to rent three warehouses annually at the country, Y. Manufacturers decided to keep some buffer stock at these warehouses as well. Manufacturers organised these warehouses to directly supply to six supermarkets and pharmaceutical chains in country Y.
The following table provides the fixed costs in millions of dollars, annual capacities in thousands, and annual buffer stocks in thousands at the proposed plants by Ximon company. (Hint: the net capacity at each plant is the difference between annual capacity and buffer stock).
Proposed Plant Fixed cost (millions) Annual capacity
(thousands) Buffer stock
Juju $2.5 500 50
Haze $1.8 1000 100
Kale $1.5 800 80
Cumu $2.8 900 90
Opaz $3.0 750 80
The following table indicates the annual rents in thousand dollars and annual buffer stocks at the warehouses in thousands. (Hint: the buffer stock at each warehouse can be considered as its demand requirement.)
Warehouse Rental fee (thousands) Buffer stock (thousands)
Elite 300 80
Poll 250 70
Globe 280 100
Looking into the sales data of the country, Y and the pandemic spread data from WHO (World Health Organisation), predicted annual demand for masks by major supermarket and pharmaceutical chains for the next year is shown in the table below.
Supermarket/Pharmacy Demand (thousands)
The unit shipping cost from each proposed plant to each rental warehouse is below.
Elite Poll Globe
Juju 1.3 2 1.5
Haze 1.5 1.6 2
Kale 2 1.5 2
Cumu 2 1 1.4
Opaz 1.5 2 1
The unit shipping cost from each rental warehouse plant to each supermarket/pharmacy is below.
Boles Cashworth Ruldi Chem PriceChem Fair
Elite 0.5 0.6 0.4 0.3 0.2 0.8
Poll 0.4 0.4 0.5 – 0.5 0.7
Globe 0.5 0.3 0.2 0.4 0.8 0.6
Warehouses have some specific requirements due to budget constraints at the Ximon company. If Cumu was selected, then Haze could not be selected, and vice versa. If Juju is selected, then Kale must be selected.
Formulate the integer programming model to determine which plant(s) and warehouse(s) should the Ximon company set up AND how many products should be shipped from plant i to supermarket/pharmacy chain j via warehouse k to reach the minimum total costs.
TaxiCity, an automobile company that designs and manufactures taxi cars, is considering what products to introduce to the market next year. The company can introduce Luxury cars or Functional cars. For each Luxury taxi, the profit is $20,000. For each Functional taxi, the profit is
$15,000. The demands for Luxury and Functional cars depend on economic conditions. In a highly growing economy, passengers are more interested in Luxury cars, while in a low growth economy, Functional cars tend to have a stronger demand. The demands for Luxury and Functional taxi can be summarized in the following table. Suppose TaxiCity has the production capacity to meet all the demands.
States of natur e
Decision alternatives Low Medium High
Luxury taxi 100 200 400
Functional taxi 250 300 350
(i) Apply the Opportunity Loss Approach (with the minimax regret criterion) to recommend a decision to TaxiCity to maximize its profit. (2 marks)
The management team made an initial assessment of the economic condition of next year. The probabilities are estimated as follows:
P(low) = 0.3, P(medium)=0.45, P(high)=0.25
(ii) Draw a decision tree and recommend a decision strategy to TaxiCity on what cars to introduce to the market to maximize its profit. Clearly show your calculations and justify your decision strategy. (5 marks)
For a consulting fee of $100,000, the Melbourne Institute (MI) will use their model to predict the economic condition of next year. Denote favourable prediction by F, and unfavourable prediction U. TaxiCity believes that the following conditional probabilities are realistic appraisals of MI’s prediction accuracy.
P(F|Low) = 0.2, P(F|Medium) = 0.5, P(F|High) =0.7
(iii) Apply Bayes’ Theorem to compute the posterior probabilities for both the Favourable and Unfavourable predictions. (6 marks)
(iv) Draw the decision tree, determine the recommended decision strategy(ies), and the expected value for the profit of next year, assuming that TaxiCity will conduct the prediction. Draw the decision tree with clear labels, show all your calculations, and provide justifications for your decision strategy to TaxiCity. (10 marks)
(v) Construct a risk profile for the optimal decision strategy from part (iv). (2 marks)
In semester 1, Max enrolled in two subjects, the MGMT1 and MGMT2. For the 100 students who enrolled in these two subjects, the average score for MGMT1 is 75, with a standard deviation of 5. The average score for MGMT2 is 72 with a standard deviation of 6. Assume that the final scores in these two subjects are normally distributed.
(i) What is the probability that a student enrolled in both subjects will have a final score of at least 80 for each subject? (4 marks)
(ii) What is the probability that a student’s final mark is between 70 and 82 for subject MGMT1? (4 marks)
(iii) How high does a final score have to be to put the student in the top 5% in MGMT2? (4 marks)
A medical team would like to know the factors affecting the infection rate of COVID-19. The team collected data on the number of new cases of 20 cities and other relevant data in the past month. The team would like to know how the number of new cases is related to the medical expenditure of the city (in millions), the average temperature (in degree centigrade), and whether a lockdown was conducted in the city. For qualitative variable – lockdown or not, it is coded as lockdown = 1 and no lockdown =0. The following table shows the results of the multiple regression analysis.
(iv) Develop an estimated regression equation that can be used to predict the number of new cases by using factors such as the medical expenditure, average temperature of the city, and whether there was a lockdown. (2 marks)
(v) Is the lockdown effective in reducing the spread of COVID-19? And why (Use α = 0.05) (2 marks)
(vi) Based on the regression ANOVA table, calculate the coefficient of determination and adjusted coefficient of determination for this model. (4 marks)
(vii) What is your opinion about the overall model significance? (Use α = 0.05) (2 marks)
(viii) The following table indicates the correlation statistics of the model variables. Briefly define
what multicollinearity is, whether it exists in this model, and why. (3 marks)
END OF SECTION
END OF EXAMINATION PAPER