OR_Project-Gavazzi-Tardito.pdf

Course: STRATEGOS - Operation Research 2019/2020 Exam Date: 09/07/2020 Students: Alberto Michele Gavazzi 4799050 Giorgio Tardito 4026585 INTRODUCTION During all outbreaks of infectious diseases, especially in the case of Covid - 19, the access to sanitizing products is essential in order to protect ourselves and the ones around us. So, it is well known that ensuring safety and sanitation practices, managing waste products from prevention products in universities, homes, scho ols, markets and healthcare facilities will help prevent the human - to - human transmission of the virus causing Covid - 19. The main means of transmission are respiratory droplets and direct contact. Droplets can also land on surfaces where the virus may rem ain viable; therefore, the immediate environment of an infected individual can serve as a source of transmission. Recent evidence indicates that the survival of the COVID - 19 virus (SARS - CoV - 2) on surfaces is similar to that of SARS - CoV - 1, the virus that causes severe acute respiratory syndrome (SARS), with survival on surfaces that goes from 2 hours to 9 days. The same study also found that effective inactivation could be achieved within 1 minute using common disinfectants, such as 70%. In this proj ect we focused on the university environment. The best way to remind students to use hand sanitizer is to make it easily accessible and always put it in view. It is important to place the hand sanitizer near and around high - touch surfaces and common areas. We believe it is fundamental to find the location of the dispensers in order to optimize the distance from each room and make a sanitizer easily accessible from any point of the area considered by us. So, as first point, we tackled the problem of the location of the dispensers in a university floor and then we wanted to find the best route in order to sanitize the floor considered. Related to that problem we also wanted to see which types of dispenser could be the most effective to use. Link to Prezi Presentation : https://prezi.com/p/0kfswcjvvjjb/?present=1 DISPENSER LOCATION The problem that we wanted to tackle first is the location of hand sanitizer dispensers. Since we do not have the possibility to access the real blueprint of a floor of our un iversity, we created a layout of an alleged university floor with classrooms, bathrooms and study halls. In order to do that, we used the website Planner 5d. In this floor there are 7 classrooms, 2 study halls and 2 bathrooms (we considered man and woman bathrooms closed to one another as one single bathroom). In order to calculate the distances from one place to another we choose to measure the distance between one door and another following the path coasting the walls. The measures of the walls wer e taken from the data available on the website. In order to solve this problem, we have formulated this model. The objective function is to minimize the distance from the dispenser to the rooms. 𝑛 𝑛 𝑛 𝑚𝑖𝑛 ∑ 𝑦 𝑗 + ∑ ∑ 𝑑 𝑖𝑗 𝑥 𝑖𝑗 𝑗 =1 𝑖 =1 𝑗 =1 Where y represents the dispensers, d are the distances between each room and x stands for the rooms. The constraints are: • Each room must be connected to only one dispenser. 𝑛 ∑ 𝑥 𝑖𝑗 = 1 𝑗 =1 • Max number of rooms that can be supplied by each dispenser. 𝑛 ∑ 𝑥 𝑖𝑗 ≤ 11 𝑦 𝑖 𝑖 =1 • Number of dispensers (We suppose that given the number of rooms present on this floor we should test the model with the following number of dispensers: 2, 3, 4, 5, 6). 𝑛 ∑ 𝑦 𝑗 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑖𝑠𝑝𝑒𝑛𝑠𝑒𝑟𝑠 𝑗 =1 In this case we have n equal to 11 that are the rooms we take into consideration. In order to solve this model, we created an Excel file and we solved it by using the Resolver. First, we created a table in which are stored all the distances from one place to another. Then we created other two tables that are used by the Resolver as variables, thanks to these tables we are able to see where the dispensers should be located and whi ch dispenser the people from each room must use. Then in other cells we created some formulas in order to implement every constraint in the Resolver. In this Excel file all the data are in the cells with the colour blue, the variables in the cells with the colour yellow and the green cells are the ones where is present a formula. The results that we have found with Excel are: NUMBER OF DISPENSER DISTANCE POSITION 2 104 C, D 3 72 A, D, G 4 56.5 A, D, F, G 5 41.7 A, C, E, F, G 6 33.5 B, C, E, F, G, STUDY HALL A In order to see if the resolver of Excel gave us good results we created also the model on Lindo. With the model done on Lindo we found these results. NUMBER OF DISPENSER DISTANCE POSITION 2 94.70 C (X33), Study Hall B (X99) 3 68.15 A (X11), Study Hall B (X99), Bathroom A (X1010) 4 53.48 A (X11), C (X33), Study Hall B (X99), Bathroom A (X1010) 5 40.25 A (X11), D (X44), E (X55), F (X66), Bathroom B (X1111) 6 31.68 A (X11), C (X33), E (X55), F (X66), G (X77), Study Hall B (X99) Despite there is little discrepancy form these results and the ones found with Excel, we can assume that the model is good. This kind of difference can be due to the fact that Lindo performs better that Excel in this type of problem because is able to manage a greater number of variables in a more efficient way. The advance of using Excel is that we are able to identify immediately and easily which dispenser must be used by the people of a specific r oom. NUMBER OF DISPENSER EXCEL LINDO 2 104 94.70 3 72 68.15 4 56.5 53.48 5 41.7 40.25 6 33.5 31.68 First of all we can see that, as we expected, with an increase in the number of dispensers, we have a decrease of the distances. Then we also see that we have a decrease in the differences between the results found in Excel and the one found with Lindo. DISPENSER OPTIONS Now we want to see which one of that configurations is the best by taking into account the amount of hand sanitizer a person uses each day and the cost of different type of dispenser. We suppose that our floor welcomes 170 people composed by students, professors and other personnel and that each person used the dispenser around 10 times. The dis pensers we took into account are the model M, L, XL and XXL produced by Astreea. In our case we need eat least 5.1 l of sanitizer gel, since normally each time a hand sanitizer is used, a person utilize 3 ml of gel to disinfect at least 90% of the hand (based on the article by Alex Berezow, PhD on American Council of Science and Health). The model that we created in order to solve this problem is the following. The objective function: 𝑛 𝑚𝑖𝑛 ∑ 𝑐 𝑖 𝑥 𝑖 𝑖 =1 Where c is the cost, x the type of dispenser. The constraint is that the total number of the capacity of the dispenser must be at least equal to 5.1 l 𝑛 ∑ 𝑉 𝑖 𝑥 𝑖 ≥ 5.1 𝑖 =1 Where V is the capacity of the dispenser. Also, we have to take into account that 𝑥 ∈ ℕ With the purpose of solving this problem we created a model on MPL. These are the results we found. COSTS ($) MODEL 2 818 XL(x2) 3 1097 M, L, XL 4 1366 M(x3), XL 5 1645 M(x4), L 6 1914 M(x6) Now we want to see which configuration we should choose based on if we gave more importance to the cost or if we gave more importance to the vicinity of the dispensers. In order to do that we created an Excel file. For the distances we use the values find in Lindo since we think that it gives us a more accurate results than the Resolver of Excel. The results from the Excel file tell us that if we give more importance to the distance with a weight of 0.9 and 0.8, the best option is the case B. If we give a weight of 0.7 or less the best case will be the first one. CLEANERS BEST PATH After this we decided to see what the shortest path should be to sanitize all the places present in our blueprint. In order to do that, first of all, we had to update the table of all distances adding the exit and the entrance to the f loor and also the corridors (the distance from every room and the corridors had been calculated from the door of the room taken into consideration to the centre of the corridor). Then we created a matrix with the time required to move from one room to anther taking into consideration that a person moves at 1.3 m/s We also choose the time, in seconds, required to sanitize each place. A B C D E F G STUDY HALL A STUDY HALL B SANIFICATION TIME 2400 1200 1800 2400 2400 2400 2400 2100 3600 BATHROOM A BATHROOM B CORRIDOR A CORRIDOR B SANIFICATION TIME 1200 1500 600 900 In order to compute the right path and the amount of time required, we created an algorithm in Python based on the Travelling Salesman Problem. We based our code on the example algorithm present on Google Developer OR - Tools website. We modified it in order to be useful to solve our problem. When we chose to use only one cleaner moving from the entrance to the exit, the result is: When we us e two cleaners one from the entrance to the exit and one in the opposite direction the paths are: If we use three cleaners, two starting from the exit and one from the entrance, the result is: With two cleaners stating from the entr ance and one from the exit, the resulting paths are: But if we use 4 cleaners, we have this result: As we can see, in this case, the path is not optimized since the cleaner number two goes directly form the entrance to the exit. So, we can conclude that we found the best path using three cleaners: two that start from the entrance and one from the exit. CONCLUSION The results that we have found in this project are not representative of a real case scenario. First of all, the dimensions and the areas that we created in the blueprint do not take into account any type of norms nor regulation s. This is due to the fact that we do not have any knowledge in this field and we did not had time to do any type of research about that. All the measures used in this project are taken coasting the walls since the software gave us that type of data. So, they do not reflect the actual movement of a person in this space. Also, we tested only our second problem with the dispenser of a specific company, different dispenser could generate different result. Moreover, the cost is computed based on dollars because it was the only information we were able to find. More likely tho se dispensers will have a different value in euros and also if they are bought in big quantities the cost will change. We also suppose that in our floor could be present 170 people and that each person will use a hand sanitizer roughly 10 times. We chose t hese values in order to make the problem more similar to what we think could be a real case, but they are not based on any type of data. In the last problem we supposed the time required to clean each room without any type of data since we were not able to find any information about that. In the end we think that our project can be useful to understand how to behave in this new condition w e are currently living and that with more reliable data, based on real scenarios, our project is able to give useful information in order to plan the reopening of the university to students. BIBLIOGRAPHY Water, sanitation, hygiene, and was te management for the COVID - 19 virus, World Health Organization and the United Nations Children’s Fund (UNICEF), 2020. https://planner5d.com/view/?key=5b2bbd3b73c5dd34acf72d8441474825 https://astreea.com/italiano/ https://1525.co/project/astreea/ https://developers.google.com/optimization/routing/routing_tasks#setting - start - and - end - locations - for - routes http s://www.acsh.org/news/2017/01/10/you - arent - using - enough - hand - sanitizer - 10717 Orthotics and Prosthetics in Rehabilitation, Chapter 5: Clinical Assessment of Gait, Olfat Mohamed, Heather Appling, Elsevier, 2020