Indifference Zone Approach Research Articles

Unbiased decision making in location-routing problems with uncertain customer demands

In this paper, we focus on a location-routing problem (LRP) in the dairy industry. This problem consists of locating a cold storage warehouse, from which vehicles of limited capacity are dispatched to serve a given number of supermarkets with uncertain service requirements, and determining the order of supermarkets served by each vehicle. First, the LRP is solved by using a classical approach based on a deterministic model where the service requirements, i.e. customer demands, are defined through sample means. Second, we propose an indifference zone approach to the LRP. The indifference zone procedures are specific ranking and selection methods aimed at selecting the best option from a set of alternative configurations. In particular, they attempt to guarantee the probability of correct choice, while minimising the computational effort. The numerical results presented in the paper highlight the risk of biased decision making when mere sample means are used in a deterministic model. In addition, they show the effectiveness of indifference zone approaches to the dairy products distribution activity.

Lower bounds for the expected sample size of sequential procedures for selecting and ranking of binomial and Poisson populations

In this paper we consider selection and ranking problems for the cases of binomial and Poisson populations. Our main goal is to construct lower bounds for the expected sample size of sequential selection and ranking procedures for the aforementioned problems in the setting of the indifference zone approach. Established lower bounds are investigated and their explicit estimates are obtained. After we investigate the efficiency of some sequential selection and ranking procedures with respect to our lower bounds.

Selecting the best of two gamma populations having unequal shape parameters

Consider two independent gamma populations π1 and π2, where the population πi has an unknown scale parameter θi>0 and known shape parameter αi>0,i=1,2. Assume that the correct ordering between θ1 and θ2 is not known a priori and let θ[1]≤θ[2] denote the ordered values of θ1 and θ2. Consider the goal of identifying (or selecting) the population associated with θ[2], under the indifference-zone approach of Bechhofer (1954), when the quality of a selection rule is assessed in terms of the infimum of the probability of correct selection over the preference-zone. Under the decision-theoretic framework this goal is equivalent to that of finding the minimax selection rule when (θ1,θ2) lies in the preference-zone and 0–1 loss function is used (which takes the value 0 if correct selection is made and takes the value 1 if correct selection is not made). Based on independent observations from the two populations, the minimax selection rule is derived. This minimax selection rule is shown to be generalized Bayes and admissible. Finally, using a numerical study, it is shown that the minimax selection rule outperforms various natural selection rules.

A Simulated Annealing with Ranking and Selection for Stochastic Optimization

We consider the problem of stochastic optimization, where the objective function values are not available and need to be simulated to get their estimates. When the function values are available one can use the simulated annealing algorithm. In this paper, we modify an algorithm that uses the hill climbing feature of simulated annealing with fixed temperature to search the feasible solution set. The proposed algorithm uses indifference zone approach of ranking and selection method to compare the current optimal solution and the potential solution that guarantee the optimal solution with a pre specified level of confidence. The algorithm is tested on a (s, S) inventory problem and compared to other competing algorithm. The numerical results show that the proposed method outperforms the competing method and indeed locate the optimal solution quickly.

A Three-Stage Selection Procedure With an Exact Consistency

This article considers the problem of selecting the best population by using Bechhofer's indifference zone approach. Stein's two-stage procedure meets the probability requirement (exact consistency). However, it is shown that the procedure does not become asymptotically second-order efficient whatever the first sample sizes are chosen. Hall's three-stage procedure is asymptotically second-order efficient, but it loses the exact consistency. We propose a three-stage procedure that has an exact consistency and show that the procedure enjoys the asymptotic second-order efficiency.

Selecting the Best Component of a Multivariate Normal Population

This article considers the problem of selecting the best component of a mean vector of a multivariate normal distribution. Using Bechhofer's indifference-zone approach, Clark and Yang (1986) proposed a two-stage selection procedure to solve the problem when the covariance matrix is totally unknown. We are interested in the asymptotic performances of their sample size and show that their procedure becomes asymptotically first-order efficient, but is not asymptotically second-order efficient. We propose a three-stage procedure which enjoys an asymptotically second-order efficiency.

Finding probably best systems quickly via simulations

We propose an indifference-zone approach for a ranking and selection problem with the goal of reducing both the number of simulated samples of the performance and the frequency of configuration changes. We prove that with a prespecified high probability, our algorithm finds the best system configuration. Our proof hinges on several ideas, including the use of Anderson's probability bound, that have not been fully investigated for the ranking and selection problem. Numerical experiments show that our algorithm can select the best system configuration using up to 50% fewer simulated samples than existing algorithms without increasing the frequency of configuration changes.

Selecting the Normal Population with the Smallest Coefficient of Variation

SYNOPTIC ABSTRACTWe consider the problem of selecting the normal population with the smallest coefficient of variation, which is a natural goal when the means as well as the variances of the populations are unknown and unequal. The indifference-zone approach (Bechhofer 1954) to this problem has been previously considered by Choi, Jeon and Kim (1982). We review their selection procedure and provide tables of sample sizes for it. Next we consider the subset selection approach of Gupta (1956, 1965). We propose a natural selection procedure, derive its least favorable configuration and provide tables of critical constants. An example is given to illustrate the two procedures.

A Two-Stage Procedure with Elimination for Partitioning a Set of Normal Populations with Respect to a Control

We consider the problem of partitioning a set of given normal populations, with respect to a control population, into two subsets according to their unknown means. In this article, using the subset selection approach in the first stage and indifference zone approach in the second stage, we propose a two-stage procedure. The procedure partitions “too superior or inferior” treatments after the first stage, thereby reducing the average sample size and making the procedure more attractive for practitioners. The proposed procedure is studied and compared via the Monte Carlo simulation studies with other competitive procedures known in the literature.

Selection Procedures with Frequentist Expected Opportunity Cost Bounds

Selection procedures help identify the best of a finite set of simulated alternatives. Most work has measured the quality of a selection with the probability of correct selection, P(CS), but the expected opportunity cost of a potentially incorrect decision makes more sense in business contexts. This paper analyzes the first selection procedures that guarantee an upper bound for the expected opportunity cost, in a frequentist sense, of a potentially incorrect selection. The paper therefore bridges a gap between the indifference-zone approach (with frequentist guarantees, but for the P(CS)) and the Bayesian approach to selection procedures (which has considered the opportunity cost). We also provide “unexpected” expected opportunity cost guarantees for several existing indifference-zone selection procedures that were originally designed to provide P(CS) guarantees.

Solution quality of random search methods for discrete stochastic optimization

In this paper, we propose a framework for selecting a high quality global optimal solution for discrete stochastic optimization problems with a predetermined confidence level using general random search methods. This procedure is based on performing the random search algorithm several replications to get estimate of the error gap between the estimated optimal value and the actual optimal value. A confidence set that contains the optimal solution is then constructed and methods of the indifference zone approach are used to select the optimal solution with high probability. The proposed procedure is applied on a simulated annealing algorithm for solving a particular discrete stochastic optimization problem involving queuing models. The numerical results indicate that the proposed technique indeed locate a high quality optimal solution.

Confidence Intervals for the Ratios of Ranked Scale Parameters for Censored Data

SYNOPTIC ABSTRACTConsider k(k ≥ 2) independent populations or treatments such that the data from the ith population or treatment follow a distribution with cumulative distribution function (cdf) Fi(x) = F((x- μi)/θi characterized by the location parameter μi,(-∞ < μi < ∞) and scale parameter θi (θi >0), i = 1,…,k Here F(.) is any absolutely continuous cdf. Let θ[1] ≤…θ≤[k] denote the unknown ordered values of θs and the population associated with θ[1] is labeled as the best. Suppose the interest of the experimenter is to know “how the remaining populations depart from the best population?” or “how many populations are tied with the best population?” or “how many populations are close to the best?” The parameters of interest, in such situations, are θ[1]θ[i], i = 2,…,k. In this paper, we discuss simultaneous confidence intervals, based on the sample quasi ranges, for the ranked parameters θ[1]/θ[i], i = 2,…,k, useful in the situations where the experimenter has censored data or there is suspicion that the data contain outliers. The proposed confidence intervals are conservative and their usefulness to obtain the confidence intervals for the true probability of correct selection, under the indifference-zone approach of Bechhofer (1954) and the subset approach of Gupta (1965), is discussed. The constants needed for obtaining the confidence intervals are also presented.

One-Stage and Two-Stage Designs for Clinical Trials Using an Indifference-Zone Approach

In new drug development we often need to select experimental treatments which are better than a control treatment for further development. Many statistical selection procedures are proposed for choosing optimal treatments in clinical trials since such procedures are very simple and the required number of patients is smaller than that of test-based methods. However most of these methods focus on selecting the best treatment or on selecting a subset containing all treatments better than the control. It may sometimes be more reasonable to select all treatments better than the control and no treatment worse than the control in order to avoid assigning patients to inferior treatments in later trials. In this article, we consider the one-stage and two-stage designs for choosing all the treatments whose response probabilities are greater than a control. Some numerical results and examples are also shown and the advantages of the two-stage design in decreasing the number of patients are discussed.

A Multiple Attribute Utility Theory Approach to Ranking and Selection

Managers of large industrial projects often measure performance by multiple attributes. For example, our paper is motivated by the simulation of a large industrial project called a land seismic survey, in which project performance is based on duration, cost, and resource utilization. To address these types of problems, we develop a ranking and selection procedure for making comparisons of systems (e.g., project configurations) that have multiple performance measures. The procedure combines multiple attribute utility theory with statistical ranking and selection to select the best configuration from a set of possible configurations using the indifference-zone approach. We apply our procedure to results generated by the simulator for a land seismic survey that has six performance measures, and describe a particular type of sensitivity analysis that can be used as a robustness check.

Determination of sample size for selecting the smallest of k poisson population means

A procedure for selecting a Poisson population with smallest mean is considered using an indifference zone approach. The objective is to determine the smallest sample size n required from k ≥ 2 populations in order to attain the desired probability of correct selection. Since the means procedure is not consistent with respect to the difference or ratio alone, two distance measures are used simultaneously to overcome the difficulty in obtaining the smallest probability of correct selection that is greater than some specified limit. The constants required to determine n are computed and tabulated. The asymptotic results are derived using a normal approximation. A comparison with the exact results indicates that the proposed approximation works well. Only in the extreme cases small increases in n are observed. An example of industrial accident data is used to illustrate this procedure.

Adaptive selection of the best population

Assume k independent populations are given which are distributed according to Q ϑ 1 , …, Q ϑ k . Taking samples of size n the population with the largest ϑ-value is to be selected. For the sequence of models localized at ϑ 0 and any sequence of selection rules an asymptotic upper bound is derived for the minimum probability of correct selection in the indifference zone approach. Asymptotically optimal selection rules are constructed using the central sequence of the sequence of experiments and by substituting ϑ 0 by an √ n-consistent estimator. In the semiparametric location model f( x − ϑ) with unknown density f an asymptotically optimum selection rule is constructed where the score function −f′ f is estimated by kernel estimators.

Duality between multiple testing and selecting

Selection problems are considered for which it is possible to formulate a correct selection via a so-called correct selection indicator. The correct selection indicator can be used to define a related multiple hypotheses testing problem. It is shown that there exists a bijective mapping from the class of all selection rules which guarantee a specified confidence level P∗ for a correct selection into a subfamily of all multiple tests at multiple level 1 − P∗. First of all, this correspondence may be of interest from a pure theoretical viewpoint. Moreover, methods developed for multiple hypotheses testing, such as the closure principle, sometimes allow considerable improvements of existing rules. Last but not least, the duality between multiple testing and selecting may yield a better understanding of the structure of the underlying decision problem. Three selection problems will be considered in more detail: subset selection of Gupta's type, best-or-all selection, and unrestricted subset selection. Application of the closure principle to Gupta's selection problem indicates that Gupta's rule cannot be improved. In fact, Gupta's rule turns out to be admissible in a certain sense. It is shown how the best-or-all selection problem can be treated by applying the duality results, and its connection to Bechhofer's indifference zone approach is pointed out in detail. The idea of stepwise multiple tests is adopted for the unrestricted subset selection problem. The primary task in the derivation of step-down procedures is the determination of least favorable parameter configurations with respect to certain hypotheses. The validity of the suggested step-up selection procedures depends on the validity of a conjecture concerning the critical values for these procedures. The step-up approach can be considered as a more informative extension of the best-or-all selection rule.

A note on the conditional probability of correct selection under the two-element partition

Suppose there are k normal populations with unknown means, and the goal is to select the population having the largest mean based on independent observations from the k populations. Bechhofer (Ann. Math. Statist. 25 (1954), 16–39) proposed the indifference zone approach for this problem. The disadvantage of this approach is that often the same decision (which population to be selected) and the same probability of correct selection are assigned to two different sets of observations, one of which intuitively conveys a much stronger feeling of correct selection. Using Kiefer's (Proc. 4th Dayton Multivariate Conf., North-Holland, Amsterdam, 1975, 143–158; J. Amer. Statist. Assoc. 72 (1977), 789–827) conditional approach, the whole sample space can be partitioned into a set of lucky observations and a set of unlucky observations. In this note, we study the conditional probability of correct selection under this two-element partition. Results on least favourable configuration and ?-correct selection are established if we have lucky observations. Some unsolved problems are pointed out.

Two- and multi-stage selection procedures for Weibull populations

Abstract This paper considers two- and multi-stage procedures for selecting the most reliable of several Weibull populations, with a common known shape parameter but unknown scale parameters. Under an indifference zone approach, the least favourable configuration for a multi-stage procedure has been worked out. Also, explicit comparisons are made between two- and single-stage procedures.

On the estimation of the true probability of a correct selection

For ranking and selection problems, the true probabiIity of a correct selection P(CS) is unknown even if a selection is made under the indifference-zone approach. Thus to estimate the true P(CS) some Bayes estimators and a bootstrap estimator are proposed for two normcal populations with common known variance. Also a bootstrap estimator and a bootstrap confidence interval are proposed for normal populations with common unknown variance. Some comparisons between proposed estimators and some other known estimators are made via Monte Carlo simulations.