Instructions:
Please provide typed answers in a word document or in pdf. Please do
not provide handwritten answers.
If you have used program to compute the answer, please provide the code
in the final package with detailed comments.
This is an individual assignment, so no collaboration is allowed.
Question 1. (5 pts)
This is a game of identifying three kinds of tea, green, oolong and white.
60% of green teas I taste will taste green to me, and 30% of green teas I’llmisidentify as oolong and 10% as white.
70% of white teas I taste will taste white to me, and 20% will be misidentifiedas oolong and 10% as green.
80% of oolong teas I’ll correctly identify as oolong, and 10% will be misidentified as white and 10% will be misidentified as green.
One week, I tasted a tea, and it tasted oolong. Are you able to compute the probability that it is indeed oolong? If yes, what is the probability? If not, make the required assumptions to compute the probability? Explain your answer in detail.
Question 2. (5 points)
There is a trip being planned for 400 people by using a transportation company.
The transportation company has 10 large buses, each of which can hold 40 people and 8 small buses, each of which can hold 30 people. The rental cost for a large bus is 750 and for a small bus is 500. If there are only 13 drivers available , calculate how many buses of each type should be used for the trip with the least possible cost?
Provide the details of the method employed and explain if there are any drawbacks to the approach?
Question 3. (10 pts)
In the following question we consider a belief network related to COVID-19. Age and Risk factors impact whether a person has COVID. We can diagnose whether a person has COVID using multiple ways:
(i) Loss of taste or smell
(ii) Cough
(iii) Through testing
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In the attached file (simulated_data.csv), we give a simulated dataset of observations. We employ the following convention for the values assigned to the different variables.
1 Age (dimensionality 3)
a. < 15 yrs: 0
b. 15 yrs – 49 yrs: 1
c. > 49 yrs: 2
2. Risk factors (dimensionality 3)
a. Low: 0
b. Medium: 1
c. High: 2
3. Covid Status (dimensionality
4) a. None: 0
b. Asymptomatic: 1
c. Mild: 2
d. Severe: 3
4. Cough (dimensionality 2)
a. No: 0
b. Yes:
1 5. Loss of smell or taste (dimensionality
2) a. No: 0
b. Yes: 1
6. Tested result (dimensionality 3)
a. Negative: 0
b. Positive: 1
c. NA: 2
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Answer the following questions:
(i) What are the local probability tables for the nodes in belief network shown below? Please explain how you computed these tables.
(ii) What is P(Risk Factors | Loss of Taste or Smell = 1, Cough = 0)? Use likelihood weighting to compute the answer by generating 100000 samples. SMU Classification: Restricted
(iii) What is the most probable explanation for someone who has cough and is 35 years of age? Provide the code with detailed comments explaining what each part of the code does.
Question 4.
(10 pts) In this problem, we consider a set of blocks of various sizes, shapes, and colors. The blocks can be stacked into one or more piles, possibly with some constraints (depending, e.g., on their size); the goal is to form a target set of piles starting from a given initial configuration, by moving one block at a time, either to the table or atop another pile of blocks, and only if it is not under another block.
Consider a simple version of this problem, in which there are five blocks with identical shape and size, in different colours with labels A, B, C. Please note the following:
Relative position of the piles does not matter (Configuration 1 and Configuration 2 are the same).
Every block is either on the table (there are no constraints on number of piles) or atop any other block.
Only blocks at the top of the current piles can be moved, and can be placed only on the table or atop another pile.
Placing a block on another block costs 1, irrespective of the block label or colour. However, placing blocks on the table costs differently (Orange block costs 2, Gray block costs 3, Yellow block costs 1, Green block costs 4) .
The objective is to move to a given goal configuration (e.g., Configuration 3) from starting configuration (e.g., Configuration 1) with minimum cost.
1. Formulate the above version of the block’s world as a search problem, by precisely defining the state space, the initial state, the goal state, the set of possible operators and the path cost.
2. Please provide an admissible heuristic function (that is not a zero for every search node) for this search problem. Show how the A* search strategy expands the first three nodes of the search tree, avoiding repeated states. When drawing the search tree you should clearly indicate: the order of expansion of each node; the action corresponding to each edge of the tree; the state, the path cost and the value of the heuristic of each node.
Question 5.
(10 pts) In image processing, Markov Random Fields (MRFs) are often used for denoising images by modelling the relationships between neighbouring pixels. Consider a simple binary image denoising problem, where each pixel in a noisy image is either black (B) or white (W). The task is to recover the original image using an MRF model based on noisy observations.
The image consists of four pixels arranged in a 2×2 grid, where each pixel can either be in state black (B) or white (W). Let the original states of the pixels be denoted by �!, �”, �#, �$ and let the observed (noisy) states be denoted by �!, �”, �#, �$. The Markov Random Field models the relationships between neighbouring pixels based on the following factors:
Pixels that are neighbours in the grid have pairwise potentials ?(?%, ?&), which encourage neighboring pixels to have the same color (smoothness). The potential function is defined as:
?(?%, ?&) = 2 ?? ?% = ?&,
= 0.5 ?? ?% ≠ ?&
The neighborhood structure in the 2×2 grid is as follows:
o ?! is connected to ?” and ?#,
o ?” is connected to ?! and ?$,
o ?# is connected to ?! and ?$,
o ?$ is connected to ?” and ?#.
2. Observation potential based on noisy pixels: Each noisy observation �% influences the corresponding pixel �% through an observation potential �(��, ��). This potential represents how likely the observed noisy pixel matches the true pixel value. The observation potential is defined as:
?(?%, ?%) = 4.0 ?? ?% = ?%
= 0.5 ?? ?% ≠ ?%
Calculate ‘((*!+,,*”+. | 1!+.,1″+,,1#+.,1$+,) ‘((*!+.,*”+, |1!+.,1″+,,1#+.,1$+,)
Question 6. (10 pts)
A university campus is served by a fleet of autonomous delivery robots that transport packages between various campus buildings. The campus is divided into four major zones: A1, A2, A3, and A4. Delivery robots can either wait in a zone to pick up a package or move to a different zone to look for a package to deliver.
The demand for deliveries varies by zone, and at any given time, the probability of a robot finding a package for delivery in a specific zone is shown in Table 1 below:
Zone Probability of finding a package
A1 0.25
A2 0.3
A3 0.35
A4 0.1
Once a delivery robot picks up a package, the destination is determined based on student or faculty requests. Historical data on deliveries between zones is presented in Table 2 below:
Source → Destination Probability
A1 → A2 0.3
A1 → A3 0.4
A1 → A4 0.3
A2 → A1 0.5
A2 → A3 0.2
A2 → A4 0.3
A3 → A1 0.4
A3 → A2 0.3
A3 → A4 0.3
A4 → A1 0.5
A4 → A2 0.25
A4 → A3 0.25
The delivery fee charged between different zones is shown in Table 3 below:
Source → Destination Delivery Fee
A1 → A2 $12
A1 → A3 $8
A1 → A4 $6
A2 → A1 $10
A2 → A3 $7
A2 → A4 $9
A3 → A1 $14
A3 → A2 $9
A3 → A4 $5
A4 → A1 $15
A4 → A2 $7
A4 → A3 $4
The cost incurred for a delivery robot moving between different zones (without a package) is listed in Table 4 below:
Source → Destination Travel Cost
A1 → A2 $1.5
A1 → A3 $2.0
A1 → A4 $2.5
A2 → A1 $1.0
A2 → A3 $1.8
A2 → A4 $2.2
A3 → A1 $1.8
A3 → A2 $1.5
A3 → A4 $1.2
A4 → A1 $2.0
A4 → A2 $1.8
A4 → A3 $1.0
Required:
(a) Formulate an MDP model that optimizes the robot’s strategy. The MDP is defined by the tuple <S, A, P, R>, i.e., the states, actions, transition probability matrices (for different actions), and reward functions.
(b) Solve the MDP model using policy iteration with a discount factor of 0.95. Provide: 1. The code you used to solve it. 2. The value of the first two iterations of the policy iteration process.
(c) Solve the MDP model using a linear programming formulation with a discount factor of 0.95.
Provide:
1. The LP formulation.
2. The code for solving it.
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