DETECTING A RANDOMLY MOVING COVID-19 USING WEIGHT TCHEBYCHEFF OPTIMIZATION TECHNIQUE

This paper completely characterizes the search for randomly moving COVID-19 among a finite set of different states (cells). A search effort at each fixed number of time intervals is a random variable with a given distribution. Both of probability of detecting the COVID-19 in a certain state j at a certain time i and the detection functions are supposed to be known to the searcher or robot. We seek for the optimal distribution of the effort to deduce an explicit formula for the optimality distribution of the random variable effort using weighted Tchebycheff optimization technique. We find the necessary conditions for the optimal search technique. An algorithm is constructed for generated weighted Tchebycheff optimization technique. The effectiveness of these techniques is illustrated by numerical results which indicate that the proposed techniques are promising.


INTRODUCTION
The Searching Process for lost COVID-19 that moves randomly becomes very interesting contemporary problem. Before starting the search for the missing goal, the search team should put some questions under considerations to facilitate detecting the COVID-19 correctly and quickly. Is this target valuable or not? Is this target static or moving? What is the nature of the location where the target is expected to be lost? What is the way to minimize the cost of searching for the target to be detected quickly? Can the missing target be detected quickly by using only one or more researchers? Using more than one researcher will affect your search cost or not? Most of previous questions have already been answered by presenting different models that have been studied previously using different strategies and are also interesting at the same time. To mention some important models, better to start with a linear search strategy, which has many life and mission applications, for example searching for a damaged unit in a large linear organ (electrical power lines, telephone lines and petrol or gas support lines), whether this linear organ is independent or intersecting has been studied by El-Rayes, Mohamed, and Fergani [2], El-Rayes, Mohamed [1], Mohamed [3] and Mohamed, and Abou Gabal [4] on the other side Balkhi [20] obtained the generalized optimal search paths for continuous univariate Random variable Also, the coordinated search technique has been studied in case of linear search has been analyzed by Mohamed and Afifi [5], and with different models by Mohamed, Abou Gabal, and Afifi [6] and [7], the authors discussed the coordinated search technique for a located target on two intersected lines, when the located target has symmetric and unsymmetrical distribution Mohamed, Abou Gabal, and Afifi [8], presented more advanced work when they applied the coordinated search technique for a moving target on one of two independent lines. The coordinated search technique in the plane has been discussed by Afifi, El-Bagoury and AL-Aziz [19]. Recently, El-Hadidy, and Fakharany [9], Afifi and EL-Bagoury et al. [10] proposed and studied a modern search model in the three-dimensional space to find a 3-D randomly located target by one searcher, two searchers and four searchers.
Also, recently Afifi, El-Bagoury and AL-Aziz [17] illustrated a random walker target on one of n disjoint lines. Also, In the discrete search problems W. Afifi et al. [18]. obtained an optimal discrete search for a randomly moving COVID-19 between several cells in the part in human nasopharynx using monitoring system. More recently Afifi, El-Bagoury calculated the optimal search plan of a discrete Randomly Located Target [16].
In this article, the authors mainly discuss the following answers using weighted Chebycheff optimization technique, the location of the COVID-19 which assumed to be in consists of several cells in additions the searcher must distributes his effort (for time, energy,..., etc.) among the cells, the probability of being a COVID-19 in a certain cell at a certain time, and the detection function are supposed to be known to the searcher, the optimal distribution of the effort over the set of possible cells such that the overall probability of detecting the target is maximal.
The case of the target moving between two states according to a discrete parameter Markov chain has been studied by Kress,Kyle and Szechhtman [11]. Also, Stone [13] and BROWN [14], the authors discussed the case of the target moving among a finite number of states according to a Markov chain with a discrete parameter. El-Rayes and Mohamed [1], calculated the optimal distribution of the effort in problems, where the target moves randomly from state to state following any process, and the effort has a fixed. The distribution of effort which made the probability of undetected for unrestricted effort when the states are not identical and the cost of finding the target are minimum has been studied by Bahgat [12]. Moshe Kress, Kyle and Roberto [15] and Afifi [18] studied problem of located and a randomly moving target among a finite set of different states and used a surveillance organ in this case to search the lost target which hidden in one of n cells in each fixed number of time intervals m. the target located in a certain state j at a certain time i. and move to a new state in each time interval. They introduced the expected time of the search in each time interval. Also, they found the optimal search plan which minimizes the expected time of the search in each time interval for more studies.

PRELIMINARIES
The COVID-19 moves randomly in the respiratory system which divided into organs (states), we will suppose every organ represents a searching state. Here, we aim to find the optimal distribution 5655 DETECTING A RANDOMLY MOVING COVID19 of the random variable effort using weighted Tchebycheff optimization technique, which provides a clear interpretation of minimizes the largest difference between the probability of undetected the lost COVID-19 and its utopia point.  The searcher must distribute his effort among the states, the probability that the COVID-19 is in is denoted by ij P , during the i th time interval, the effort is given by 0 ( ) ii L Z v  which will be divided among the state.
In this paper, we will assume that the lost COVID-19 is the target and the states are the cells which in organs of the human body. The COVID-19 moves from one cell to another every time interval, and due to the importance of our target the value of effort is bounded by a random variable.

THE SEARCHING TECHNIQUE AND MAIN RESULTS
In this technique, we have only one searcher (robot), the main task is to distribute his effort in an ideal way to discover the lost goal as soon as possible with skill and speed.
The allocation of search effort is ij Z where 1, 2, ..., in = and 1, 2, ..., jm = , which gives the effort to be put into state j at time i, we call ij ZZ = a search plan, the conditional probability of finding the target at time given that it is located in state j, is given by the detection function . We assume that the searches at distinct time intervals are independent of searcher's action.
The most importance factor in discovering the COVID-19 in the cells of the human body is calculation the following weight Tchebycheff optimization technique HZ is an utopia point, and we can defined it as the following: , 2…, n, and n is the number of the time intervals.
The probabilities of undetection of the target over the whole time (see El-Rayes, Mohamed (1986)).

DETECTING A RANDOMLY MOVING COVID19
And the effort is Where i v is a random variable whose probability density function () fx and distribution function is (  Our aim is to minimize the value of U(Z), so we need to find the optimal value of () If the detection function is exponential function that is 1 ( , , ) − and the target is located in one of several states no transitions we will consider the following stochastic nonlinear programming problem: Definition 2: is said to be an optimal solution for problem (6) if there does not exist The corresponding chance-constraints problem is: The constrain ( ( ) ) 1 ii P L Z v    − has to be satisfied with an un probability of at least (1 )  − and can be restated as is a standard normal random variable.
If p K represent the value of the standard normal random variable at () p K  = , then this constraint can be expressed as Thus, the probabilistic programming problem stated (7) Which written in the form The optimal solution of problem (4) can be characterized in terms of optimal solution of the problem (10). () HZ is a convex function and the using Kuhn-Tucker theorem, we obtain the following results, where i G is the geometric mean, and by substituting from equation (12) Since 0 ij Z  and 0 ( ) 1 i HZ  we can obtain the following conditions 11 1 ln ln ln ( ) ln  Step 6. Calculate ( ) HZ the probability of undetected the lost COVID-19 from (2).
Step 7. Compute min U from (1), and then go to step 8.
Step 8. End (stop).  (13) and (16) . Also, from (11) we obtain a different value of i r , which must satisfy conditions (18) and (22), in this example we use weighted Tchebycheff optimization technique from relation (1) and satisfy the optimal condition when 0.1  = , which minimizes the largest difference between the probability of undetected the lost COVID-19 and its utopia point to see (Table 1).

DETECTING A RANDOMLY MOVING COVID19
( Table 1): The value of minimizing the largest difference between the probability of undetected the lost COVID-19 and its utopia point.

SPECIAL CASE
The optimal solution of problem (6) when the lost COVID-19 is located in one of m cell and the effort a random variable given by:

CONCLUSION
In this paper, we investigated the search model for a randomly moving the lost COVID-19 among a finite set of different states, a search effort is available at each fixed number of time intervals and the bounded of it is a random variable with a given distribution. The probability of finding the lost COVID-19 in a certain state j at a certain time i and the detection functions are supposed to be known to the searcher. We got the optimal distribution of the effort, which minimizes the probability of undeception the lost COVID-19 and searching effort, and therefore we minimized the largest difference between the probability of undetected the lost COVID-19 and its utopia point using weighted Tchebycheff optimization technique.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.