Optimizing Linear Functions of Random Variables having a Joint Multinomial or Multivariate Normal Distribution

Abstract

A computer method to find vectors s that minimize (c_i>0 constants) subject to a probability constraint P{μ_i≤s_i, i=1,…,r}≥1-α (≤α≤1) where _v ,…,v_r have a joint multinomial distribution, is obtained by solving the corresponding optimization problem through the usual normal approximation. Thus vectors are sought that minimize (b_i>0 constants) subject to a multivariate normal probability constraint where v₁,…,v_r have a joint singular multivariate normal distribution.

The singular normal probability integral is expressed in various computer-ready formulas as: (a) one integral over a simplex, (b) a sum of integral over multidimensional rectangular regions, and (c) a sum of integrals over multidimensional right triangles or plane orthoschemes.

The optimization of F, and thereby of G, is accomplished using a known nonlinear program in conjunction with also known numerical multivariate normal distribution computer codes which work well for r=3. Binomial tables and a bisection method may be used for r=2. However for r≥4, the optimization routine requires many function evaluations of, making the solution somewhat difficult and expensive while theoretically simple and feasible.

In this regard an approximation with Bonferroni bounds to evaluate is derived and is shown to be accurate to within ±.005 for values of x_i such that , in the equicorrelated-equicoordinate case, namely, x_i=x and , i=1,…,r.

Keywords:

• multinomial
• limiting singular normal
• equicorrelated-equicoordinate probability point
• optimal upper α probability point
• probability-constrained programming
• single-period inventory model
• simplex
• inclusion-exclusion method
• orthoschemes
• numerical evaluation of the multivariate normal probability distribution
• approximation with Bonferroni bounds