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Economics

CUNY Brooklyn College

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Crown virus outbreak, spread and casualties in city Mintaqa-5 of Dawla Mihwar, brought all the empires of the world to cooperatively work towards eradicating the virus. An organization named International Pandemic & Epidemic Corporation (IPEC) was formed to control and eradicate the virus. IPEC hired scientists and practitioners from all the disciplines, including Operations Research (OR). From the biochemistry research group at IPEC, it has been confirmed that the virus cannot live more than 12 days inside any human being, and it cannot stay alive for more than 4 days on any animal or non-living object. Thus, the current aim of IPEC is to Detect the virus, Dispense the supplies, and Delay the spread; in short, the 3D aims. The 3D aims ensure that the virus will not spread to new individuals, and will eventually die out. In parallel to the 3D aims, IPEC is also working towards identifying possible cure for the crown virus. Your team is part of OR group, hired by the IPEC to design a successful implementation of the 3D aims Relevant data and details are provided in the following pages of this document. Since the organization is unaware of the methodologies that you will be using to design the plan, you are allowed to make reasonable assumptions for the missing data/information. State clearly all the assumptions, use standard or references as much as possible. Your goal is to provide the organization with the best possible solution.

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Project No: ISE01920321 Start Date: Due Date: Apr 11, 2020 May 10, 2020 Crown virus outbreak, spread and casualties in city Mintaqa-5 of Dawla Mihwar, brought all the empires of the world to cooperatively work towards eradicating the virus. An organization named International Pandemic & Epidemic Corporation (IPEC) was formed to control and eradicate the virus. IPEC hired scientists and practitioners from all the disciplines, including Operations Research (OR). From the biochemistry research group at IPEC, it has been confirmed that the virus cannot live more than 12 days inside any human being, and it cannot stay alive for more than 4 days on any animal or non-living object. Thus, the current aim of IPEC is to Detect the virus, Dispense the supplies, and Delay the spread; in short, the 3D aims. The 3D aims ensure that the virus will not spread to new individuals, and will eventually die out. In parallel to the 3D aims, IPEC is also working towards identifying possible cure for the crown virus. Your team is part of OR group, hired by the IPEC to design a successful implementation of the 3D aims Relevant data and details are provided in the following pages of this document. Since the organization is unaware of the methodologies that you will be using to design the plan, you are allowed to make reasonable assumptions for the missing data/information. State clearly all the assumptions, use standard or references as much as possible. Your goal is to provide the organization with the best possible solution. Course No: Instructor: Contact: ISE-321 Syed Mujahid mnusyed@gmail.com Objective-1: Detecting The analysis done by biochemists shows that the DNA for Crown Virus (CrVi) consists of following three basic elements (bases): A, B & C. For example, a typical CrVi DNA structure looks like: 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 To detect the existence of CrVi in an individual, one needs to detect the above sequence in the individual’s DNA. However, extracting the whole DNA sequence is almost impossible from an individual. Thus, the practical approach is to extract large amount of short DNA sequences (shotguns), and reconstruct the full DNA sequence. For example, for the above DNA sequence, following strings could be extracted from individual (highlighted in red color): 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨 Now the question is, given an alphabet Ξ£ = {𝑨, 𝑩, π‘ͺ} and the following string set (𝑺): 𝑺 = {𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨, 𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨, 𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨, 𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨, 𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨, π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺ, 𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ, π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨, 𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩, 𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨, π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩, 𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩, 𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨}; can it be possible to identify the shortest superstring containing all the strings of 𝑺. It is believed that the shortest superstring may represent the actual DNA sequence. That is, the following superstring (𝑺𝑺) contains all the strings of 𝑺. 𝑺𝑺 = {𝑨𝑩𝑨𝑩𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩𝑨𝑨𝑩π‘ͺπ‘ͺ𝑨𝑨𝑨} There are many ways to formulate and solve the above problem. In the following paragraphs, some ideas that can be used in developing the solution method is presented. Idea-1: the higher the overlap between the end of one string and the beginning of another, the more the probability of them being connected in the prefix suffix order. Example: Let 𝑆 = {π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩, 𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩, 𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨}. Let 𝑠𝑖 represent the 𝑖 π‘‘β„Ž string of 𝑆. That is 𝑠1 = π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩, 𝑠2 = 𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩, and 𝑠3 = 𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨. All possible pairwise combinations Overlap from 𝑠1 to 𝑠2 = 5 characters. οƒ  Overlap from 𝑠1 to 𝑠3 = 1 character. Overlap from 𝑠2 to 𝑠1 = 0 characters. Overlap from 𝑠2 to 𝑠3 = 1 character. Overlap from 𝑠3 to 𝑠1 = 7 characters. Overlap from 𝑠3 to 𝑠2 = 3 characters. From 𝑠1 to 𝑠2 : 𝑠1 = π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩 𝑠2 = 𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩 The above implies, they should be connected as follows: π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩 The maximum total overlap that we can get by merging the above three strings will be 8 characters, and the merging will be in the following specific order: 𝑠3 βˆ’ 𝑠1 βˆ’ 𝑠2 . Notation: The overlap from 𝑠1 to 𝑠2 can be denoted as π‘œ(𝑠1 , 𝑠2 ). Idea-2: finding a shortest superstring is same as finding the sequence of strings such that the total overlap is maximized. Idea-3: if π‘ π‘Ž βˆ’ 𝑠𝑏 βˆ’ 𝑠𝑐 is the ordered sequence with the maximum total overlap, then the corresponding shortest superstring will be obtained as follows: Shortest superstring: pref(π‘ π‘Ž βˆ’ 𝑠𝑏 ) + pref(𝑠𝑏 βˆ’ 𝑠𝑐 ) + 𝑠𝑐 where pref(π‘ π‘Ž βˆ’ 𝑠𝑏 ) is the prefix or part of π‘ π‘Ž not in 𝑠𝑏 . Example: Let 𝑆 = {π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩, 𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩, 𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨}. Let 𝑠𝑖 represent the 𝑖 π‘‘β„Ž string of 𝑆. That is 𝑠1 = π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩, 𝑠2 = 𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩, and 𝑠3 = 𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨. Now, pref(𝑠3 βˆ’ 𝑠1) = 𝑩𝑨, pref(𝑠1 βˆ’ 𝑠2 ) = π‘ͺ𝑨𝑨π‘ͺ. This implies that the shortest superstring containing 𝑠1 , 𝑠2 & 𝑠3 will be: 𝑩𝑨π‘ͺ𝑨𝑨π‘ͺ𝑩π‘ͺ𝑨𝑨𝑩𝑨π‘ͺπ‘ͺ𝑩. The above ideas are enough to formulate the problem. Your task is to use Ideas-1, 2 & 3 and: (1a) Draw the TSP network for the following 5-strings data. The data is available in the excel workbook named Data-xx. xlsx (where β€œxx” stands for your group number), on the sheet titled Q1a. (1b) Formulate the problem as the travelling salesperson problem. Define all the indices, parameters and variables clearly before building the model. (1c) Solve the 5-strings data problem in GAMS (or any software of your choice). Use the data from Question-1a. Objective-2: Dispensaries IPEC plans to supply food, medication and other essential commodities to the people of Mintaqa-5. The idea is to send the items from drones, called Raha Tayara or RahaT, to all the houses of Mintaqa-5 every day. The drones will be launched from the base station located in a nearby safe & sanitized city. Currently, IPEC has the three types of drones and the details are provided in the following table: Type Available Numbers RahaT Alif RahaT Baa RahaT Taa 𝑁1 𝑁2 𝑁3 Max air-time Serving capacity Operation Costs per tour per tour per tour (minutes) (houses) (1000 dollars) 120 5 25 250 10 40 520 20 75 In the above table, air-time means the total time it can stay in air without refueling. The goal is to serve as many houses per day as possible. It is assumed that each drone will make only one tour per day. Your task is to: (2a) Build a mathematical model that will identify the maximum number of houses that can be served in a day for the data given in the above table. Use the following main indices and parameters to build your model: 𝐻: total number of houses in Mintaqa-5. β„Ž: index for houses, β„Ž = 1, … , 𝐻. 𝑑𝑖𝑗 : travel time from house 𝑖 to house 𝑗 in minutes. π‘žπ‘– : travel time from base station to house 𝑖 in minutes. You are allowed to use additional items like: indices, variables and/or parameters. Clearly define the additional items, and link them appropriately, before using them in the model. The model type should be MIP. (2b) Propose a heuristic that solves the above problem. That is propose: I. A solution structure that handles as many constraints a possible, II. A mechanism to randomly generate a solution, III. A mechanism to generate a neighbor solutions from a given solution, and IV. V. A corresponding objective function to minimize. Show an overview of two consecutive iterations of the proposed heuristic. Objective-3: Delaying In order to reduce the travel time and provide enough time to detect the virus, IPEC would like you to identify the roads that should be enlarged such that the shortest path to reach Airport from Mintaqa-5 is prolonged or maximized. Roads cannot be physically enlarged, so the idea is: 1) to place barricades on the road to reduce the travel time 2) to completely shut down some of the roads for public usage Let us say the travel time in minutes from city 𝑖 to city 𝑗 is 𝑑𝑖𝑗 . From the past experiences, IPEC estimates that it will take $𝑐𝑖𝑗 per day per barricade to increase the travel time by 𝛿𝑖𝑗 mins on a given road. Furthermore, multiple barricades can be placed on a road. For example, spending $2𝑐𝑖𝑗 on a road will result in two barricades, which will increase the total travel time on that road by 2𝛿𝑖𝑗 mins. That is, the travel time from City-i to City-j by building two barricades will be 𝑑𝑖𝑗 + 2𝛿𝑖𝑗 minutes. There is no actual cost in shutting down a road. However, it may create a panic and unruliness among the public. Thus, all the roads cannot be shut down from Mintaqa-5 to the airport in a given day. It is estimated that closing at most Ξ” number of roads in a day will be enough to mitigate the panic among the public. Your tasks are as follows: (3a) A toy example is given in the following figure, and the corresponding data is given in the excel workbook named Data-xx. xlsx on the sheet titled Q3a. From the toy example, study the mechanics of the problem. Specifically, identify the infeasible and the best scenarios. Figure: Toy example where movement from Node-5 to Node-1 is delayed. 3b) From the experience obtained from the toy example, build a general mathematical model that will maximize the shortest path from Mintaqa-5 (𝑖 = 5) to the airport (𝑖 = 1). Use the following main indices and parameters in the model: 𝑛: the total number of nodes (including cities and airport). 𝑖, 𝑗: the indices used for the cities, 𝑖, 𝑗 = 1, … , 𝑛. π‘Žπ‘–π‘— : is equal to 1 if there is an arc from city 𝑖 to city 𝑗, and otherwise 0. 𝑑𝑖𝑗 : travel time in minutes from city 𝑖 to city 𝑗 without any barricades. 𝛿𝑖𝑗 : additional travel time in minutes per barricade on the road from city 𝑖 to city 𝑗. Ξ”: maximum number of roads that can be shut down in a day. You are allowed to use additional items like: indices, variables and/or parameters. Clearly define the additional items, and link them appropriately, before using them in the model. The model type can be MIP, NLP or MINLP. (3c) Implement the mathematical model in GAMS (or any software of your choice). No need to solve the model. However, you must be able to compile the model to check for any syntax errors. Use the data given in the excel workbook named Data-xx. xlsx, on the sheet titled Q3c. Disclaimer: The information, data and problem statements given in the above case study is purely fictional. There resemblance to reality is coincidental. The purpose of the case study is to show the usage and applicability of ISE 321 (Optimization Methods) theory and tools in decision making. ...
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