A generic selection-with-constraints procedure

Abstract

This paper develops a generic procedure for selection with constraints. The bounds of current selection with constraints are based on user-specified values of the underlying performance measures. In some cases, users have no priori of what the values of the secondary performance measures may be, hence, the specified values may not be accurate. On the basis of the difference between comparison with a standard and comparison with a control, we propose using relative performance measures as constraints. That is, systems having each performance measure within a user-specified amount of the unknown best are considered as feasible systems. An experimental performance evaluation demonstrates the validity and efficiency of the selection-with-constraints procedure.

Appendices

Appendix A

The two-stage procedure of Dudewicz and Dalal (1975) to select the best of k systems has been widely studied and applied. They perform the selection on the weighted sample means X̃ _i =W_i1X̄ _i ⁽¹⁾+W_i2X̄ _i ⁽²⁾ for i=1,2,…, k. Here, the weights

and W_i2=1−W_i1, for i=1,2,…, k. The expression for W_i1 was chosen to guarantee (X̃ _i −μ _i )/(d^*/h₁) have a t distribution with n₀−1 degrees of freedom. Note that h₁ (which depends on k, n₀, and P^*) is a constant that can be found from the tables in Law (2007). The procedure proceeds as follows.

• 1) Simulate the initial n₀ samples for all systems. Compute the first-stage sample means

  

  and sample variances

  

  for i=1,2,…, k.

• 2) Compute the required sample sizes

  

  where ⌈z⌉ is the smallest integer that is greater than or equal to the real number z.

• 3) Simulate additional N _i −n₀ samples.

• 4) Compute the second-stage sample means

  

• 5) Compute the weighted sample means

  

  and select the system with the smallest X̃ _i .

Appendix B

Let T _i for i=1,2,…, k, be independent t-distributed random variables with n₀−1 d.f. and let F denote the cdf of the t distribution with n₀−1 d.f. Let , under the null hypothesis that , and we can write

Note that in this setting δ=d^*/2 instead of d^* needs to be used when computing the weight. We equate the right-hand side to P^* and solve for h₁.

Under the null hypothesis that , we can write

The last equality holds because is the sample range and the right-hand-side is the cdf of the sample range, see Tippett (1925). We equate the right-hand side to P^* and solve for h₃.