A Monotonicity Result for Inspecting Independent Items

1989 ◽  
Vol 3 (1) ◽  
pp. 135-140
Author(s):  
F.K. Hwang ◽  
S.G. Papastavridis
Keyword(s):  
Cost Function ◽  
Failure Probability ◽  
General Model ◽  
Group Testing ◽  
Probabilistic Algorithms ◽  
Expected Number ◽  
Monotonicity Result ◽  
Crucial Part

Recently, the conjecture that the expected number of tests is nondecreasing in the failure probability for binomial group testing has been proved. The proof has also been extended to three models of multiaccess systems. However, probabilistic algorithms are used as a crucial part of these proofs. In this paper, we give conceptually simpler new proofs without using probabilistic algorithms. We also extend the result to a more general model where the number of tests is replaced by a cost function.

scholarly journals Step-Wise Group Screening Designs with Unequal A-Priori Probabilities and Errors in Observations

2003 ◽  
Vol 8 (2) ◽  
pp. 153
Author(s):  
M.M. Manene
Keyword(s):  
Cost Function ◽  
Group Size ◽  
A Priori ◽  
Expected Number ◽  
Total Cost ◽  
Screening Designs ◽  
Expected Total Cost

The performance of step-wise group screening with unequal a-priori probabilities in terms of the expected number of runs and the expected maximum number of incorrect decisions is considered. A method of obtaining optimal step-wise designs with unequal a-priori probabilities is presented for the case in which the direction of each defective factor is assumed to be known a -priori and observations are subject to error. An appropriate cost function is introduced and the value of the group size which minimizes the expected total cost is obtained.  

Asymptotics of pooling design performance

1999 ◽  
Vol 36 (04) ◽  
pp. 951-964
Author(s):  
J. K. Percus ◽  
O. E. Percus ◽  
W. J. Bruno ◽  
D. C. Torney
Keyword(s):  
Group Testing ◽  
Performance Criterion ◽  
Expected Number ◽  
Pooling Design ◽  
Pooling Designs ◽  
Parametric Control ◽  
Expected Performance ◽  
Qualitative Effect ◽  
Pooling Strategy ◽  
Asymptotic Forms

We analyse the expected performance of various group testing, or pooling, designs. The context is that of identifying characterized clones in a large collection of clones. Here we choose as performance criterion the expected number of unresolved ‘negative’ clones, and we aim to minimize this quantity. Technically, long inclusion–exclusion summations are encountered which, aside from being computationally demanding, give little inkling of the qualitative effect of parametric control on the pooling strategy. We show that readily-interpreted re-summation can be performed, leading to asymptotic forms and systematic corrections. We apply our results to randomized designs, illustrating how they might be implemented for approximating combinatorial formulae.

Asymptotics of pooling design performance

1999 ◽  
Vol 36 (4) ◽  
pp. 951-964 ◽  
Author(s):  
J. K. Percus ◽  
O. E. Percus ◽  
W. J. Bruno ◽  
D. C. Torney
Keyword(s):  
Group Testing ◽  
Performance Criterion ◽  
Expected Number ◽  
Pooling Design ◽  
Pooling Designs ◽  
Parametric Control ◽  
Expected Performance ◽  
Qualitative Effect ◽  
Pooling Strategy ◽  
Asymptotic Forms

We analyse the expected performance of various group testing, or pooling, designs. The context is that of identifying characterized clones in a large collection of clones. Here we choose as performance criterion the expected number of unresolved ‘negative’ clones, and we aim to minimize this quantity. Technically, long inclusion–exclusion summations are encountered which, aside from being computationally demanding, give little inkling of the qualitative effect of parametric control on the pooling strategy. We show that readily-interpreted re-summation can be performed, leading to asymptotic forms and systematic corrections. We apply our results to randomized designs, illustrating how they might be implemented for approximating combinatorial formulae.

scholarly journals An Improvement Over the Information Lower Bound in Binomial Group Testing

1988 ◽  
Vol 2 (3) ◽  
pp. 313-320 ◽  
Author(s):  
Y.C. Yao
Keyword(s):  
Lower Bound ◽  
Group Testing ◽  
Expected Number ◽  
Testing Problem

The group-testing problem for a binomial set of items is considered. It is desired to classify all items as good or defective with a minimum expected number of group tests. An improvement over the information lower bound, via a weak concavity property, is made for the minimum expected number of group tests.

Scheduling jobs on non-identical IFR processors to minimize general cost functions

1991 ◽  
Vol 23 (04) ◽  
pp. 909-924 ◽  
Author(s):  
Rhonda Righter ◽  
Susan H. Xu
Keyword(s):  
Cost Function ◽  
Optimal Policy ◽  
Expected Number ◽  
Expected Cost ◽  
Processing Times ◽  
Special Cases ◽  
The Cost ◽  
General Cost Functions ◽  
Tardy Jobs ◽  
Completion Times

We consider the problem of scheduling n jobs non-preemptively on m parallel, non-identical processors to minimize a weighted expected cost function of job completion times, where the weights are associated with the jobs. The cost function is assumed to be increasing and concave but otherwise arbitrary. Processing times are IFR with different distributions for different processors. Jobs may be processed on any processor and there are no precedences. We show that the optimal policy orders the jobs in decreasing order of their weights and then uses the individually optimal policy for each job. In other words, processors are offered to jobs in order, and each job considers its own expected cost function for its completion time to decide whether to accept or reject a processor. Therefore, the optimal policy does not depend on the weights of the jobs except through their order. Special cases of our objective function are weighted expected flowtime, weighted discounted expected flowtime, and weighted expected number of tardy jobs.

An Improved Lower Bound on the Minimum Expected Number of Binomial Group Tests

1993 ◽  
Vol 7 (1) ◽  
pp. 121-124 ◽  
Author(s):  
Julia Abrahams
Keyword(s):  
Lower Bound ◽  
Group Testing ◽  
Huffman Coding ◽  
Expected Number ◽  
The Cost

The minimum expected number of binomial group tests is lower bounded by the cost of a particular Huffman coding problem whose solution is known. Thus, the information lower bound in binomial group testing is improved when the probability that each item is defective is small.

Scheduling jobs on non-identical IFR processors to minimize general cost functions

1991 ◽  
Vol 23 (4) ◽  
pp. 909-924 ◽  
Author(s):  
Rhonda Righter ◽  
Susan H. Xu
Keyword(s):  
Cost Function ◽  
Optimal Policy ◽  
Expected Number ◽  
Expected Cost ◽  
Processing Times ◽  
Special Cases ◽  
The Cost ◽  
General Cost Functions ◽  
Tardy Jobs ◽  
Completion Times

We consider the problem of scheduling n jobs non-preemptively on m parallel, non-identical processors to minimize a weighted expected cost function of job completion times, where the weights are associated with the jobs. The cost function is assumed to be increasing and concave but otherwise arbitrary. Processing times are IFR with different distributions for different processors. Jobs may be processed on any processor and there are no precedences. We show that the optimal policy orders the jobs in decreasing order of their weights and then uses the individually optimal policy for each job. In other words, processors are offered to jobs in order, and each job considers its own expected cost function for its completion time to decide whether to accept or reject a processor. Therefore, the optimal policy does not depend on the weights of the jobs except through their order. Special cases of our objective function are weighted expected flowtime, weighted discounted expected flowtime, and weighted expected number of tardy jobs.

ASYMPTOTIC ANALYSIS OF OPTIMAL NESTED GROUP-TESTING PROCEDURES

2016 ◽  
Vol 30 (4) ◽  
pp. 547-552 ◽  
Author(s):  
Nabil Zaman ◽  
Nicholas Pippenger
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Group Testing ◽  
Fractional Part ◽  
Expected Number ◽  
Small Oscillations ◽  
Testing Procedures ◽  
Average Value ◽  
Positive Individual ◽  
Image Position

We analyze a construction for optimal nested group-testing procedures, and show that, when individuals are independently positive with probability p, the expected number of tests per positive individual, F*(p), has, as p→0, the asymptotic behavior $$F^{\ast}(p) = \log_2 {1\over p} + \log_2 \log 2 + 2 + f\left(\log_2 {1\over p} + \log_2 \log 2\right) + O(p),$$ where $$f(z) = 4\times 2^{-2^{1-\{z\}}} - \{z\} - 1,$$ and {z}=z−⌊z⌋ is the fractional part of z. The function f(z) is a periodic function (with period 1) that exhibits small oscillations (with magnitude <0.005) about an even smaller average value (<0.0005).

scholarly journals Model Predictive Control for a Recover Water Process

2018 ◽  
Author(s):  
Paolo Mercorelli
Keyword(s):  
Model Predictive Control ◽  
Cost Function ◽  
Computer Simulations ◽  
Lower Bounds ◽  
Predictive Control ◽  
General Model ◽  
Constructive Condition ◽  
The Cost

The goal of this contribution is an application of the Linear General Model Predictive Control (LGMPC). In this paper, stability of the LGMPC is proven by means of a demonstration of a Theorem stating a sufficient and constructive condition. This condition can be applied for calculating the weight matrices of the cost function in the optimisation problem in LGMPC. Lower bounds conditions are found for one of these matrices and then a system with saturation is taken into consideration. The conditions could be interpreted and discussions through physical aspects. The obtained results were tested by means of computer simulations and an example with a recover water process is considered.

Individual Testing of Independent Items in Optimal Group Testing

1988 ◽  
Vol 2 (1) ◽  
pp. 23-29 ◽  
Author(s):  
Y. C. Yao ◽  
F. K. Hwang
Keyword(s):  
Group Testing ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Necessary And Sufficient Condition ◽  
Expected Number ◽  
Testing Problem ◽  
The Individual ◽  
Testing Algorithm ◽  
Necessary And Sufficient ◽  
Special Case

We consider the group testing problem for a set of independent items I = [I1,… In] where Ii, has probability pi, of being defective and probability qi = 1 – pi of being good. The problem is to classify all items as good or defective with a minimum expected number of group tests where a group test is a test on a subset S of I with two possible outcomes: either S is pure (contains no defective) or S is contaminated (contains at least one defective, with no information provided about which or how many). No polynomial-time algorithm is known for the group testing problem even for the special case pi = p for all i. Hence, any method that reduces the size of the problem is very helpful. In this paper, we give such a method by providing a simple condition to screen items that should be tested (only) individually. This condition leads to a necessary and sufficient condition for the individual testing algorithm to be optimal, generalizing a result of Unger [1] for the special case of identical pi.

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