2023-228

A company launched a new gadget C to be produced globally for North American and South American markets.

The company uses just-in-time as a logistic strategy with no warehouse. Still, it has a quality control standard with the assistance of laboratories in the United Kingdom, Spain, and South Africa.

The company has some contractors in the Asia-Pacific region that make two types of components, A and B, to create the C product.  In order to produce one container of product C, two containers of component A and three containers of component B are necessary.

In the graph above, we show the logistic circuit of factories, ensemble centers (located in Singapure, Dubai , and Egypt), quality control labs, and consumer centers for this operation.  The production flow of component A (blue arrows), component B (red arrows), and the distribution flow (purple arrows) is shown. The numbers near the arrows show the capacity to transport from one location to another.

Which of the statements below is compatible with a production and distribution scheme in the manufacturing supply chain that delivers the maximum quantity of gadgets while producing the minimum amount of components?

1. New York gets 20 gadgets, Japan produces 42 components B, and Australia produces 22 components A.
2. Sao Paulo gets 9 gadgets, India produces 75 components B, and Hong Kong produces 50 components A.
3. Sao Paulo gets 10 gadgets, India produces 74 components B, and Hong Kong produces 45 components A
4. New York gets 18 gadgets, Japan produces 40 components B, and Australia produces 20 components A

Chose the right answer.

1. I
2. I and II
3. II and IV
4. III and IV
5. None of the above.

Original idea by: Alexander Valle Rey

2023-227

Consider a flow network for the company "A", which needs to deliver products from the capital to 10 cities. Three of these cities are small and don't have direct delivery routes from the capital. However, each of these small cities can receive products from the other 7 cities.

Due to a special event, City "T", one of the three small cities, requires a delivery of 650 products. The flow capacities of the network are defined as follows:

• Company "A" can send a maximum of 150 products to each city that has a direct connection from the capital.
• Two of the 7 cities that can receive products directly from the capital can forward a maximum of 200 products each to City "T".
• The remaining 5 of the 7 cities can forward a maximum of 50 products each to City "T".
• The other two small cities don't require any products.

Given these conditions, how many of the 650 requested products can be delivered to City "T" for the event?

1. 500
2. 550
3. 600
4. 650
5. None of the above

Original idea by: Thaysa Bello

2023-226

 

Imagine a Cayley Tree, constructed starting from a central node of degree \( k \). Each node at distance \( d \) from the central node has degree \( k \), until the nodes reachable at distance \( P \) that have degree one, as shown in figure above. Consider \( k+1 \) communities, with communities corresponding to the subtrees rooted at the children of the central node, and the central node by itself being a community. What is the modularity of this partition if \( k=3 \) and \( P=4 \)? Round it up to three decimal places.

1. 0.523
2. 0.621
3. 0.785
4. 0.891
5. None of the above.

Original idea by: Germán Darío Buitrago Salazar

2023-225

Agglomerative hierarchical clustering can be applied to community detection in graphs. By using dendrograms, one can visualize the order in which nodes merged. This visualization enables the extraction of the underlying community organization.

Which of the presented dendrograms and respective cuts would faithfully represent the given graph and its three colored communities?

1. IV;
2. III;
3. II;
4. I;
5. None of the above.

Original idea by: Fillipi Valadares

2023-224

Consider the following network. The fuchsia nodes are all part of a weak community.

Which option presents a modification that makes this community a strong one?

1. Remove node 8 from the community.
2. Add node 9 to the community.
3. Add node 5 to the community.
4. Remove node 6 from the community.
5. None of the above

Original idea by: Christian Konishi

2023-223

Considering the following undirected graph:

I. {1, 2, 3, 4} is a clique.

II. {1, 2, 3, 4} is a strong community.

III. {5, 6, 7} is a weak community.

IV. {5, 6, 7} is a clique.

V. {4, 5, 6, 7} is a weak community.

Which statements from the options above are correct?

1. I, III and V
2. II, III and V
3. II, IV and IV
4. II, III, IV and V
5. none of above

Original idea by: Anderson Nogueira Cotrim

2023-222

Consider a scale-free network with average degree 2.9, second moment of the degree distribution 32.3, and degree exponent 2.89. It is correct to state that:

1. The preferential attachment assumption is valid, with a probability Π(k) that a link of a new node links to another node of degree k proportional to the square of k.
2. This network is perfectly assortative, with a degree correlation coefficient equal to 1.
3. This network displays enhanced robustness, since its breakdown threshold is greater than the breakdown threshold of a random network with the same average degree.
4. This network is in a random network regime, since the degree exponent is close to 3.
5. None of the above.

Original idea by: Vitoria D. M. Pinho