Optimal strategy for time-limited sequential search
Introduction
Sequential search, where one candidate is evaluated at a time, is very common in many real-life search processes. The decision at each step is either to reject or to accept the candidate currently under consideration. The search is terminated if a candidate is accepted, or proceeds to the next candidate in the sequence if the current candidate is rejected [1]. Examples of sequential search are found in common applications in diverse fields such as consumer choice in economics [2], [3], habitat selection by territorial animals in biology [4], radar target detection in military science [5], visual search of moving scenes in vision science [6], [7], etc. In a realistic sequential search, the length of the sequence is always finite, but the selection criteria may vary from problem to problem. Sequential search processes have been broadly classified in the past into two categories. The first of these is where candidates are evaluated one at a time and the search is stopped as soon as a positive choice is made but, in the process of making the choice, the selector is allowed to revisit previously evaluated candidates. An example if this type of process would be visual recognition of a possible suspect in a crime by examining a set of photographs of potential suspects. The photographs themselves are evaluated sequentially, but at any step in the sequence the selector can decide to make a positive choice either of the current photograph or of any of the previously examined photographs. In either case, the search is concluded as soon as the selector makes a positive choice.
The second category of sequential search process is also one where the choice is made with respect to each candidate as the candidates are evaluated sequentially. As before, if the candidate currently being evaluated is accepted, the search is over and, if not, the search proceeds to the next candidate in sequence. However, there is no possibility of revisiting and accepting a previously rejected candidate. An example of this type of search is when one looks for a suitable exit to pull out in order to get something to eat when driving on the highway. Typically, there is a time limit imposed on the search, and only one exit is considered at a time. The choice as to whether or not to take the exit is made on the basis of visual screening of the billboards and the scenery from the road. Once a positive choice is made, the search process is terminated. Also, since it is extremely rare that one would go back to an exit previously considered and rejected, for all practical purposes it can be justifiably assumed that there is no revisiting of candidates considered in the previous steps of the search sequence. In both these categories of searches, information obtained from the evaluation of the candidate, even when a candidate is rejected, may be used in the evaluation of candidates further down in the search sequence. Thus, attribute information obtained on the candidate is used in updating the knowledge of the distribution of the particular attribute in the candidate pool [8]. This is true whether or not revisiting is allowed.
However, there are some real-life search processes that involve time-limited sequential searches with very low candidate encounter rates. In this type of time-limited sequential search, the search sequence is made up of time steps rather than candidates, and the overall duration of the search is limited by the specified time limit. Because the encounter rates are very low, in a given time-step there is only a very small chance of encountering a suitable candidate, and, indeed, only few candidates may be encountered over the entire duration of the search. Therefore, it is not possible in this type of searches to reliably update information on the distribution of candidate attributes during the search process, based on candidates encountered previously in the search. Thus, not only is there no revisiting, but there is effectively no memory of the candidates encountered and rejected at any time step previous to the current step. In this paper, we focus exclusively on such time-limited sequential search processes with very low encounter rates. As with other sequential search models, the search process is terminated either at the end of the specified time limit or as soon as a positive choice is made in one of the time steps.
Our goal in this study is to determine a search strategy which will maximize the expected benefit. There are k different types of potential candidates to choose from, and each of these types has a known benefit associated with it. The rates at which candidates of various types may be encountered during the search are also known ahead of time. The search strategy is to consider only candidates from the best type, i.e., the type with the highest associated benefit, as acceptable during the first steps of time, and then to lower the threshold so as to consider the best two types as acceptable until time step , and so on. Thus, until the next switching time, , the acceptable pool of candidates comes only from types 1 and 2. The thresholds are progressively lowered at so that candidates of type k will become part of the acceptable candidate pool only from time onwards. The objective is to find the optimal values of the switching times, such that the expected benefit from the search is maximized.
Section snippets
Model
The basic assumptions underlying the model of the time-limited sequential search process described in this paper are as follows:
- 1.
The search is carried out sequentially in time, with each unit of time representing a step in the sequence. The total search is limited to a predetermined number of time steps, T. No more than one candidate is encountered in a time step; indeed, during many of the time steps, no suitable candidate may be encountered at all.
- 2.
There are only k types of candidates
Discussion
One of the implicit assumptions related to the time-limited sequential search process is that initially one would restrict the search exclusively to candidates from the highest quality type, and subsequently lower the threshold progressively as time increases. Thus, if represents the ith switching time, i.e., the time at which we lower the selection threshold level to candidates of type , it follows that we should require . It is also obvious that we need to arrange the
Summary
We have considered the problem of time-limited sequential search processes with a candidate pool made up of k different types of candidates and very low candidate encounter rates. The strategy is requires that we start initially with a small candidate pool made up of the candidate type with the highest associated benefit and enlarge this pool at k specified switching times during the search by including candidate types with progressively lower associated benefits. The optimal strategy is the
Dr. V.V. Krishnan is a Professor of Mechanical Engineering at San Francisco State University. He received his undergraduate degree from the Indian Institute of Technology, Bombay, and his graduate degrees from the University of California at Berkeley. His research interests are in the areas of Biological Systems Modeling, Decision Analysis, and Fuzzy Systems.
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Dr. V.V. Krishnan is a Professor of Mechanical Engineering at San Francisco State University. He received his undergraduate degree from the Indian Institute of Technology, Bombay, and his graduate degrees from the University of California at Berkeley. His research interests are in the areas of Biological Systems Modeling, Decision Analysis, and Fuzzy Systems.