Distributional and inferential properties of some new multivariate process capability indices for symmetric specification region

Abstract

Statistical quality control is used to improve performance of processes. Since most of the processes are multivariate in nature, multivariate process capability indices (MPCIs) have been developed by many researchers depending on the context. However, it is generally difficult to understand and calculate MPCIs, compared to their univariate counterparts like Cp, Cpk, and so on. This paper discusses a relatively new development in MPCIs, namely, CGfalse(u,vfalse), which is a multivariate analogue of Cpfalse(u,vfalse)—the celebrated superstructure of univariate process capability indices . Some statistical properties of CGfalse(u,vfalse) are studied, particularly of CGfalse(0,0false), a member MPCI of the superstructure, which measures potential capability of a multivariate process. A threshold value of CGfalse(0,0false) is computed, and this can be considered as a logical cut‐off for other member indices of CGfalse(u,vfalse) as well. The expression for the upper limit of the proportion of nonconformance is derived as a function of CGfalse(0,0false). Density plots of asymptotic distributions of four major member indices of CGfalse(u,vfalse), namely, CGfalse(0,0false), CGfalse(1,0false), CGfalse(0,1false), and CGfalse(1,1false), are made. Finally, a numerical example is discussed to supplement the theory developed in this paper.

Two‐phase multivariate supplier selection for symmetric specification region and correlated quality characteristics

“…Similar to the member indices of $$C_p(u, v)$$, Chakraborty and Chatterjee 9 have defined four major member indices of $$C_G(u, v)$$ as, $$C_G(0, 0)$$, $$C_G(1, 0)$$, $$C_G(0, 1)$$, and $$C_G(1, 1)$$, which are defined analogous to $$C_p$$, $$C_{pk}$$, $$C_{pm}$$, and $$C_{pmk}$$, respectively.…”

“…Unless this assumption is satisfied, 𝐶 𝑇 𝑝𝑘 will fail to assess the actual capability of a process. Similar to V ä nnman's 4 𝐶 𝑝 (𝑢, 𝑣), Chakraborty and Chatterjee 9 have studied the properties of a superstructure of MPCIs, viz. 𝐶 𝐺 (𝑢, 𝑣).…”

“…Here also 𝑢 = 0, 1 and 𝑣 = 0, 1 are considered to construct major member indices. Similar to the member indices of 𝐶 𝑝 (𝑢, 𝑣), Chakraborty and Chatterjee 9 have defined four major member indices of 𝐶 𝐺 (𝑢, 𝑣) as, 𝐶 𝐺 (0, 0), 𝐶 𝐺 (1, 0), 𝐶 𝐺 (0, 1), and 𝐶 𝐺 (1,1), which are defined analogous to 𝐶 𝑝 , 𝐶 𝑝𝑘 , 𝐶 𝑝𝑚 , and 𝐶 𝑝𝑚𝑘 , respectively.…”

“…Similar to V $$\ddot{a}$$ nnman's 4 $$C_p(u, v)$$, Chakraborty and Chatterjee 9 have studied the properties of a superstructure of MPCIs, viz. $$C_G(u, v)$$.…”

Testing Absolute Error Loss-Based Capability Index

“…New studies about process capability indices strongly continue for one-dimensional [5] and also multivariate quality characteristic [3]. Some inferential properties of new multivariate process capability indices are discussed in [1] and [8] from statistical point of view. We succinctly introduce only two capability indices in follows and the interested readers can refer to books [4] and [6] for more details on process capability indices.…”