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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:
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
To detect the existence of CrVi in an individual, one needs to detect the above sequence in the
individualβs 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):
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨
Now the question is, given an alphabet Ξ£ = {π¨, π©, πͺ} and the following string set (πΊ):
πΊ = {π¨πͺπ©πͺπ¨π¨, π©πͺπͺπ¨π¨π¨, π©π¨π©π©π¨πͺπ¨, π©π¨πͺπͺπ©π¨, π¨π©π¨π©π©π¨πͺπ¨, πͺπͺπ©π¨π¨π©πͺ, π¨πͺπ¨π¨πͺπ©πͺ,
πͺπ¨π¨π©π¨πͺπͺπ©π¨, π¨π©π©π¨πͺπ¨π¨πͺπ©, π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨,
πͺπ¨π¨πͺπ©πͺπ¨π¨π©, π©πͺπ¨π¨π©π¨πͺπͺπ©, π©π¨πͺπ¨π¨πͺπ©πͺπ¨};
can it be possible to identify the shortest superstring containing all the strings of πΊ. It is believed that
the shortest superstring may represent the actual DNA sequence. That is, the following superstring
(πΊπΊ) contains all the strings of πΊ.
πΊπΊ = {π¨π©π¨π©π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©π¨π¨π©πͺπͺπ¨π¨π¨}
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 π = {πͺπ¨π¨πͺπ©πͺπ¨π¨π©, π©πͺπ¨π¨π©π¨πͺπͺπ©, π©π¨πͺπ¨π¨πͺπ©πͺπ¨}. Let π π represent
the π π‘β string of π. That is π 1 = πͺπ¨π¨πͺπ©πͺπ¨π¨π©, π 2 = π©πͺπ¨π¨π©π¨πͺπͺπ©, and π 3 =
π©π¨πͺπ¨π¨πͺπ©πͺπ¨.
All possible pairwise combinations
Overlap from π 1 to π 2 = 5 characters. ο
Overlap from π 1 to π 3 = 1 character.
Overlap from π 2 to π 1 = 0 characters.
Overlap from π 2 to π 3 = 1 character.
Overlap from π 3 to π 1 = 7 characters.
Overlap from π 3 to π 2 = 3 characters.
From π 1 to π 2 :
π 1 = πͺπ¨π¨πͺπ©πͺπ¨π¨π©
π 2 =
π©πͺπ¨π¨π©π¨πͺπͺπ©
The above implies, they
should be connected as
follows:
πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©
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: π 3 β π 1 β π 2 .
Notation: The overlap from π 1 to π 2 can be denoted as π(π 1 , π 2 ).
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 β π π β π π is the ordered sequence with the maximum total overlap, then the
corresponding shortest superstring will be obtained as follows:
Shortest superstring: pref(π π β π π ) + pref(π π β π π ) + π π
where pref(π π β π π ) is the prefix or part of π π not in π π .
Example: Let π = {πͺπ¨π¨πͺπ©πͺπ¨π¨π©, π©πͺπ¨π¨π©π¨πͺπͺπ©, π©π¨πͺπ¨π¨πͺπ©πͺπ¨}. Let π π represent
the π π‘β string of π. That is π 1 = πͺπ¨π¨πͺπ©πͺπ¨π¨π©, π 2 = π©πͺπ¨π¨π©π¨πͺπͺπ©, and π 3 =
π©π¨πͺπ¨π¨πͺπ©πͺπ¨. Now, pref(π 3 β π 1) = π©π¨, pref(π 1 β π 2 ) = πͺπ¨π¨πͺ. This implies that the
shortest superstring containing π 1 , π 2 & π 3 will be: π©π¨πͺπ¨π¨πͺπ©πͺπ¨π¨π©π¨πͺπͺπ©.
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 βxxβ 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
π1
π2
π3
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:
π»: total number of houses in Mintaqa-5.
β: index for houses, β = 1, β¦ , π».
πππ : travel time from house π to house π in minutes.
ππ : travel time from base station to house π 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 π to city π is π‘ππ . From the past experiences, IPEC
estimates that it will take $πππ per day per barricade to increase the travel time by πΏππ mins on a given
road. Furthermore, multiple barricades can be placed on a road. For example, spending $2πππ on a
road will result in two barricades, which will increase the total travel time on that road by 2πΏππ mins.
That is, the travel time from City-i to City-j by building two barricades will be π‘ππ + 2πΏππ 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 (π = 5) to the airport (π = 1). Use the
following main indices and parameters in the model:
π: the total number of nodes (including cities and airport).
π, π: the indices used for the cities, π, π = 1, β¦ , π.
πππ : is equal to 1 if there is an arc from city π to city π, and otherwise 0.
π‘ππ : travel time in minutes from city π to city π without any barricades.
πΏππ : additional travel time in minutes per barricade on the road from city π to city π.
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.
...