key: cord-0197625-z3rvrx72 authors: Im, Ka Ian; Choi, In Kio; Lei, Pak Kio; Chan, Hou Fai; Ian, U In; Lee, Wei Shan title: Optimal Layout Plan of Stands at the Macao Food Festival via Minimizing the Electrostatic Potential Energy with the Effective Charge as Popularity of Stands date: 2022-01-17 journal: nan DOI: nan sha: 48ba65de193e6580546f3f8cda755886905eb13c doc_id: 197625 cord_uid: z3rvrx72 We proposed a mathematical model for designing the layout diagram of stand locations at the Macao Food Festival. The optimal layout diagram may be defined in such a way that, while requiring the distance between every pair of stands should not be too far away from each other, the crowd control is well managed so that people may patronize stands more effectively. More popular stands may have larger patronage, resulting in higher pedestrian flow nearby. Therefore, to avoid customers from packing shoulder to shoulder around more popular stands, we may treat every stand as a charged particle carrying an effective charge: the more popular a stand is, the higher the effective charge it carries. Under this assumption, the problem is then converted to the minimization problem of Coulomb electrostatic potential energy on a specific configuration of charge locations, with which the global minimum may be found by the Simulated Annealing and Metropolis Algorithm. Electrostatic energy density is interpreted as density of customers, while electric field the reversed crowd flow. Therefore, at a certain location we are able to predict the customer density by calculating the energy density and the net crowd flow with electric field lines. We also concluded that even though the required computation time to obtain a configuration of stand locations with the energy value close to the global minimum with a tolerable difference may be irrelevant to the randomly generated initial configuration of stand locations, setting up an appropriate initial configuration could be one of the key issues to find out the actual global minimum. Macau is a special administrative region of China and it was once a colony of the Portuguese Empire. Under the mixing of Portuguese and Chinese culture, Macau has become a diversified city that preserves both cultural heritage like traditional food, historic buildings, etc. This unique cultural background attracts a large number of tourists and it led tourism to become one of the important economic income of Macau. Therefore, different types of traditional festival celebrations and national events are held. Known as the capital of gastronomy, Macao has a wide variety of cuisines and retains the local cultural flavor, Coexisting and blending with China and the West, so to meet the taste buds of tourists around the enjoyment. Macau was named as a Creative City of Gastronomy by UNESCO in 2017 because of its 400-year culinary heritage [1] . It is noted that local festivals are widely used as tourism promoters and boost regional economies [2] . Also, the local festival scene does fit the pattern of local sustainable economic development activities better than other scenes [3] . Because of the uniqueness of cuisine has become one of the important factors to attract tourists to Macau, some annual national events such as Macau Grand Prix and Macau Food Festival are held each November [4] . Macau Food Festival which locates Sai Van Lake Square is one of the annual events that contains various culinary delights from Asia and Europe [5] . This landmark event gathers local and international chefs and key stakeholders from the sector from across Asia and Europe is a fine example of Macao's expertise in hosting large-scale gastronomy-related events [6] . Owing to the fact that the local cuisine is an important part of the travel experience [7] , Macau Food Festival takes food as an attraction to attract tourists to come specially to taste food and experience food culture [8] . Su [9] showed that the foreign tourists who flock to Macau's food festival prove that the government has come to see it as a tourist attraction. Since the utilization efficiency of regional tourism industry elements can be improved by a rational spatial organization of the tourism industry [10] , various places are using optimal layout trying to optimize the facilities, such as proposing a macro layout planning method to better serve self-driving tour travelers [11] , or layout optimization of tourist toilets [12] . It can even be used to solve problems like using mathematical optimization to determine the optimal layout of a dining room during the era of COVID-19 [13] , within the intended span of a metro station, address the layout planning of a public bicycle system [14] or even improve the damping effect of the MR damper [15] . However, no article to date has been done on optimal stands locations for the Macau Food Festival. Therefore, the purpose of this study was to optimize the booth layout of the Macau food Festival, with which 140 booths were allocated among 200 possible locations, by approaching through minimizing Coublumb electrostatic potential energy while treating the stand popularity as the effective charge (EffQ), which was acquired by questionnaire survey and data analysis, where we classified booths into more popular ones with higher values of EffQ and less popular ones with lower EffQs. Suppose there are  possible locations for the stands at the Macao food festival heritage labeled as [0, 1, 2, … ,  − 1], and there are possible stands, where <  . Also, let denote the effective charge (EffQ) of stand . We then imagined a set of charged particles dispersing around in the 2-dimensional space. At the food festival, customers desire to patronize those more popular stands, equivalently referring to the highest charged particles. If the popular stands are located very near at the festival, it would be much easier for customers to get huddled together and also less convenient for customers to shop. To avoid crowding, our first idea was that it would be preferable to separate popular stands from one another as far as possible; nonetheless, it would cause inconvenience for customers to patronize popular stands. Keeping these in mind, we have to find out a layout plan whose stands are not too far away from one another while preventing most of the popular stands from locating close to one another. To the best of our imagination, this situation resembles that of the minimum energy of a system of two-dimensional electric charges. In other words, we would like to find out the locations for every charged particle (stand) such that the overall Coulomb electrostatic potential energy would be minimized. This is to say, we would like to find out the configuration of locations { , ∀ }, such that the overall electric potential energy is minimized. We then tried to calculate the global minimum via the simulated annealing and Metropolis algorithm (SAMA) [16] . The idea of the algorithm may be summarized as follows. While the probability for the system with energy is described by the Boltzmann distribution with = 1∕ = 1∕ , being the Boltzmann constant: we arbitrarily generate an initial configuration of locations of our stands, then calculate the initial value of the quantity in Eq.(1). Then, we generate another trial configuration of locations of stands by randomly picking up two locations of stands to swap, calculating again the quantity after the change in Eq.(1), say . If the new quantity is smaller than the old one, we accept the swap. However, if the new quantity is larger than the old one, we may still try to accept the new configuration by introducing the Metropolis Algorithm with the following criteria [17] , [18] : In practice, we generated a random number ∈ [0, 1], then if satisfies Optimal Layout Plan via minimizing Electrostatic Energy T max T min = k b T 1.0 1 × 10 −2 1 × 10 4 Table 1 Parameters used in SAMA. we accept the swap; otherwise we reject the swap and go back to the previous configuration of stand locations. Throughout our study we have chosen values of the maximum temperature T max , minimum temperature T min , and = k b T as listed in Table 1 . It is interesting to indicate that Karabin and Stuart [19] demonstrated that one may improve the efficiency of classical SAMA by allowing the temperature-varying k b (T). Figure 1 indicates the map of the event. It shows the locations of all the 140 booths that were designated in the year 2020, together with 60 more empty locations we deliberately chose to allow possibilities for the stands to select. Each booth has a different ID number on it. Remade and redrawn were made on the original map acquired from the official Facebook account of Macau Food festival,[20] and stands were divided into eight types [21] . The red dot in Figure 1 represents the origin of the coordinate so we could get the coordinates of the booths in Table 2 . In addition, the arrows stand for the entrance of the event. In order to determine the value of of a stand in Eq. (1), we conducted a questionnaire survey. The first idea was that it would be better off if we may ask respondents to evaluate all 140 stands. However, it is not practical because it is time consuming for a person to answer. Therefore, we divided the questionnaire into eight parts. Besides a questionnaire of all types of stands, which we called Grand questionnaire as shown in Table 3 , for every type of stands, we also made a questionnaire of the subcategory consisting of all stands for that particular type. The respondents may choose to answer either all of the eight questionnaires or only one of them. The popularity of every stand in the particular subcategory was denoted as average (together with the standard deviation) in the following tables: Chinese restaurant zone (Table 4) , European delicacies zone (Table 5) , Dessert zone (Table 6) , Asia delicicas zone (Table 7) , Local delicacies zone ( Table 8) , Sponsor and other (Table 9) , and Game booth( Table 10 ). Notice that the Type Sponsor and Type Other were grouped together in the survey and results were listed together in Table 9 . Afterward, the effective charge of each stand was obtained by multiplying the value of the type in the grand questionnaire with the value of that particular stand in the subcategory. The error propagation was also considered. Suppose that there are two mean values and from the grand questionnaire and one of the subcategories, with the standard deviation of mean and , respectively. The of the stand may then be understood as = = ⋅ , and the standard deviation of mean for After acquiring all the s, we normalized them with where is the mean of the effective charge and is the standard deviation for . By doing so, the ′s have properties that mean ′ = 0 and variance is equal to 1 [23] . As a result, the normalized effective charges ′ of all stands were listed in Table 11 . Optimal Layout Plan via minimizing Electrostatic Energy Table 2 3. Results and Discussions Figure 2 illustrated the initial locations of stands which was generated randomly at the very first. The color bar at the right-hand side indicated the normalized effective charge ′ mentioned in Eq.(6) with brighter color indicating higher ′. As the figure showed, most of the atoms (stands) were focused on the area of the y-axis 14 to 18, and the x-axis 15.0 to 22.5. As we could see around the two-column booths at = 20.0, the stands were quite close to each other. In spite of this, there was also a diffuse community: dispersing in the area of the x-axis from 7.5 to 15.0, and the y-axis 8 to 18. In addition, some booths formed an arch shape at the right-side of the diagram. The initial energy calculated by Eq.(1) was 477.55282. The complete list of stand locations were given in Table 12 . After randomly generating the above initial configuration of locations, we implemented the SAMA algorithm to try to find out the optimal value of energy by randomly picking up two locations to swap. In order to find out lower energy with some configuration of space, we allocated 60 more locations besides the original 140 ones which were already occupied by the stands. Under this circumstance, there were three possible cases when we randomly chose two locations to swap. The first case was that two of the locations were occupied by one booth each, therefore there was no problem for us to swap between them. Secondly, it could also be possible that only one of the two locations we picked up was occupied with a booth while the other was empty. In this case, we could still swap the two locations by carefully identifying the empty location. One more possibility that should be avoided was that we may also picked up two originally empty locations, which did not make any sense to swap between them. Figure 3 illustrated energy vs. iteration, which may be divided into eight periods along Iteration axis, with the interval of roughly 46 thousand iterations each. The first period started from the initial energy around 440. In the process of running the SAMA, it dropped very quickly from the initial value down to −200 within less than 2000 iterations, whereas it took 46 thousand more iterations to reduce energy down to around −299 with some shapes of serration, with the layout shown in Fig 4 . As we discussed earlier, this configuration may not be the global minimum. Therefore after 46 thousand iterations we randomly set up another initial configuration and started over the algorithm again, as shown in Fig 4 . In the second period within Iteration 46 thousand to 92 thousand, energy dropped from 520 to −294 (see also Fig 4 and Fig 4 ) before we set up another initial condition as in Fig 4 . Similar patterns occurred in the following consecutive five periods, as indicated through Fig 4 to Fig 4 . However, since SAMA was purportedly designed to continue running until it found out an energy lower than −300, it did not succeed until in the eighth period. We may see that even if figures at the left-hand sides of Fig.4 were all very close to −300, for the SAMA to reach the optimal (or even more importantly, the global) minimum, it could be crucial for the system to start at the appropriate initial configuration, as indicated in Fig.4 . Snapshots in Fig.4 were taken from the animation videos[24] of stands locations in company with the aforementioned featuring iterations in Figure 3 . In addition, we also plotted the energy gradient (zoomed in between range ±15) vs. iterations in Figure 5 . Inset was the energy gradient for the whole range in the vertical axis. Envelopes formed by the curves in every period were almost very similar. This may demonstrate that the initial configuration of the system may be irrelevant for the algorithm to search for a configuration with tolerable energy difference to the desired optimal energy value. With the computer specs of Intel Core i7-4500U CPU @ 1.80 GHz 2.39 GHz and RAM 8GB, 64-bit operating system and x64-based processor, it took as about 5 hours to minimize energy from the initial 477.55282 to the optimal −300.00073. Codes may be accessed in Ref. [25] . Figure 6 showed the optional locations of stands. Compared with the initial locations of stands in Fig.2 , it had a significant difference. Generally speaking, the stands were well organized in the optimal layout as expected. Then, stands in Fig.6 were divided mainly into two parts apparently, mostly in the area of the x-axis 15.0 to 22.5, and the y-axis 14 to 18. One was that most of the stands were located on the upper right half the diagram, while lower left half part was composed of very few stands aligned with two columns, together with some sporadic booths surrounding around the Square holding the event. One interesting feature was that there were 2 parallel columns of stands, located along = 20.0 and the other skew one along near = 12, alternately consisting of booths with the higher ′ and the lower ones. This interesting arrangement demonstrated that it may be a good idea to allocate together popular booths with less popular ones. On the other hand, very few of the stands were dispersed in the area from x-axis 7.5 to 15.0, and the y-axis 10 to 18. This layout may accomplish our goal of allocating stands as close as possible while minimizing the overall electrostatic potential energy. The complete list of stand locations were given in Table 13 . Electrostatic theorem was further brought upon with real-world applications once we interpret that the electrostatic energy density is the density of customers, while the electric field line points to the reversed net crowd flow at the particular location. Fig. 7 showed the electric field vectors[26] of the optimal layout plan within 1000 × 1000 grid. Optimal Layout Plan via minimizing Electrostatic Energy Optimal Layout Plan via minimizing Electrostatic Energy Vectors indicated the reversed net crowd flows, with directions toward high ′ while outward away from the lower ′. Moreover, Fig. 8 showed the logarithmic energy density, which was the prediction of customer density at the Festival. By calculating the energy density and electric field line, we may be able to predict the customer density and net movement, making it possible to enhance customer service at the festival. Optimal Layout Plan via minimizing Electrostatic Energy We successfully built up a mathematical model to calculate the optimal layout plan for stands at the Macau Food Festival while treating the popularity of stands as the Effective Charges. The popularity of every stand was acquired by implementing questionnaire surveys. Stands were purportedly treated as charged particles carrying some effective charge with the concept of more popular stands carrying higher EffQs. Also, to avoid customers from gathering together around popular stands, as well as the requirements that stands should not be too far away from one another, it is convincing that the optimal layout plan could be justified by searching for the global minimum of Coulomb energy for a certain configuration of stand locations. The above reasoning may serve as a guideline for future similar applications of this method. Since the landmark annual event Macau Food Festival enhances the local economy effectively and attracts a large number of tourists, our model may help prevent customers from congregating around more popular stands effectively. Our SAMA accomplished the purpose and reduced the initial energy from 477.55282 to −300.00073 about 5 hours (computer specs: Intel Core i7-4500U CPU @ 1.80 GHz 2.39 GHz and RAM 8GB, 64-bit operating system and x64based processor). Moreover, we also proposed the optimal locations for booths in Figure 6 , with detailed coordinates listed in Table 13 . Besides, electrostatic energy density is interpreted as density of customers, and electric field is the reversed crowd flow. Therefore, our model could be used to predict the density of customers and net crowd flow at any specific locations at the event, making it possible to assist a better tourist experience. The effect of initial configuration of stands on the search of optimal energy value was further investigated by resetting the system several times until the desired energy value was found. We may observe that it could be significant for the system to be initialized with a proper configuration for SAMA to attain a global minimum, whereas computation time was irrelevant to the initial configuration if one only requested an approximate configuration energy close enough to the global minimum. All in all, it is our pleasure that the model may be executed in future events to providing an optimization of the booth layout, stimulating consumption and boost economic development for Macau. We thank Pui Ching Middle School in Macau PRC for the kindness to support this research project. 3 3 3 3 3 3 3 3 3 3 3 3 3 10 3 3 3 average 6.875 6.594 6.813 6.906 6.531 6.125 6.688 7.125 6.594 6.281 6.063 6.906 6.438 6.719 6.250 6.594 6.813 standard deviation of mean 0. Time stamp StandID 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 5 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2021/11/26 pm 2:27:37 4 10 4 7 2 4 3 7 10 4 8 4 7 6 6 3 4 5 4 3 2 3 5 6 4 7 6 4 2 8 6 8 5 4 2 8 2021/11/26 pm 2:28:16 6 4 5 7 5 5 5 4 4 7 2 5 4 7 5 8 4 5 4 5 6 4 2 5 6 4 4 6 4 6 5 5 5 4 8 4 2021/11/26 pm 2:29:11 10 10 8 8 6 7 7 8 10 10 10 10 9 8 8 7 9 10 8 7 8 9 8 10 10 9 9 8 9 7 10 10 10 10 10 10 2021/11/26 pm 2:29:53 8 8 9 8 9 9 8 9 10 9 9 8 8 8 8 8 8 8 8 8 8 8 8 9 9 8 10 8 8 8 8 8 9 10 8 8 2021/11/26 pm 2:32:29 3 2 7 7 8 8 8 1 10 1 1 1 10 7 5 7 6 7 6 1 6 1 6 6 1 7 10 6 6 5 4 1 2 9 5 2 2021/11/26 pm 10:34:16 8 8 8 9 8 8 8 8 8 9 9 8 9 8 8 9 8 8 8 8 8 8 8 8 9 8 8 8 8 8 9 8 8 9 8 8 2021/11/26 pm 11:30:17 6 1 8 6 6 6 6 6 6 9 10 6 6 6 6 6 6 6 6 8 7 7 6 6 8 6 10 6 6 6 8 6 6 9 6 6 average 6.811 6.703 6.919 6.405 6.108 6.568 6.405 6.811 7.027 6.946 7.216 6.541 7.189 7.081 6.541 6.568 6.622 6.649 6.730 6.324 6.297 6.297 5.973 6.784 6.486 6.486 6.973 6.865 6.405 6.405 6.568 6.730 6.676 6.514 6.568 6.378 standard deviation of mean 0. Appendix J Normalized effective charges ′ of stands Table 11 : The normalized effective charges ′ of stands were obtained as follows. First, we calculated and , referring to the mean of the popularity of the sub-categorical stand, and the standard deviation of mean for the particular stand, respectively. Second, we also calculated and , respectively referring to the mean of the popularity from the grand questionnaire for a specific type of stands, and the standard deviation of mean for that particular type of stands. Third, we calculated the final mean of each stand by calculating = , together with the error propagation described in Eq.(5). Finally we also calculated ′s by Eq.(6) at the last column, which were the effective charges of stands in our algorithm. Appendix K Coordinates of stands for the initial layout plan Local Festivals and Tourism Promotion: The Role of Public Assistance and Visitor Expenditure Festival Tourism: A Contributor to Sustainable Local Economic Development? Tourists' Approach to Local Food Macau Gourmet Tourism Development Review and Gourmet Festival Customer Satisfaction Survey Constructing a Model of Food Festival Visitor Behavior. Su WX(Doctoral thesis, Ph. D. Division of hospitality management and education, Department of human development and family studies Study on City-Level Optimization of Tourism Industry Spatial Organization Nodes and Organization Mode for Tourist Destinations Macro-Layout Planning of Tourism Highway Camp Service Facilities: Taking Jilin Province as an Example Optimal layout of tourist toilets using resilience theory: An empirical study on Dunhua City in ethnic region of China On the optimal layout of a dining room in the era of COVID-19 using mathematical optimization Determining the Optimal Layout Design for Public Bicycle System within the Attractive Scope of a Metro Station Study on the Semiactive Control and Optimal Layout of a Hydropower House Based on Magnetorheological Dampers Optimization by Simulated Annealing Statistical Mechanics: Algorithms and Computation Simulated Annealing with Adaptive Cooling Rates Introduction to Probability Time stamp StandID 118 119 120 121 122 123 124 125 137 138 139 140 2021/11/3 pm 7:49:23 3 4 4 3 4 7 5 6 6 5 8 8 2021/11/5 am 11:01:25 10 8 8 7 9 7 10 8 9 9 7 9