Visualization is an important approach to understanding the complex relationships between variables and objective functions to improve data analysis ( Wang et al., 2010 ; Goel et al., 2001 ; Wang & Saunders, 1999 ). For combinatorial problems, visualization can improve the understanding of the static and dynamic characteristics of their solution spaces ( Caballero-Morales, 2014 ; Caballero-Morales & Rahim, 2015 ). Consequently, visualization can be applied to understand the search mechanisms of solving algorithms and improve their performance ( Halim et al., 2006 ; Halim & Yap, 2007 ; Pérez et al., 2013 ).

In this work, the economic statistical design (ESD) problem for X-bar control charts is considered. A control chart is defined by three main parameters: the sample size (n), the length of the sampling interval (h) and the coefficient of the chart’s control limits (L). The ESD combinatorial problem consists in finding the optimal values for n, h and L, that globally minimize costs (modeled by a cost function) considering economic and statistical restrictions ( Rahim & Banerjee, 1993 ).

Previously, full three-dimensional (3D) visualizations were performed to understand the dynamics of cost function models with general failure distribution ( Caballero-Morales, 2014 ) and non-normality ( Caballero-Morales & Rahim, 2015 ) for the ESD problem. In this work, a general contour approach is proposed to identify bi-dimensional (2D) minimum regions for the ESD problem. The estimation of minimum cost scores (MCS) is defined to obtain the 2D visualization of the minimum regions. It is expected that this approach can lead to improvements in the search mechanisms of solving algorithms to obtain the optimal chart parameters. In contrast to a 3D visualization ( Caballero-Morales, 2014 ; Caballero-Morales & Rahim, 2015 ) the proposed MCS 2D visualization can be computed faster, improving its practical use.

The present paper is structured as follows: in Section 2 an overview of the cost function model for the ESD of X-bar control charts is presented. This overview includes a description of reference data sets for the assessment of the MCS 2D visualization approach which is described in Section 2. Then, in Section 3 the contour-based MCS 2D visualizations are presented and discussed for each reference data set. Finally, in Section 4 the conclusions and future work are presented.

In this work the cost function model of Rahim and Banerjee ( Rahim & Banerjee, 1993 ) with Gamma failure distribution and uniform sampling was considered. The objective function of this cost model consists of minimizing F(n, h, L) = E(C)/E(T), where E(T) is the Expected Cycle Length, and E(C) is the total Expected Cost per Cycle ( Rahim & Banerjee, 1993 ; Chih et al., 2011 ). These costs are defined as follows:

$E\left(T\right)=h+\left(\alpha Z_0+h\right)\left(\frac{e^{-\lambda h}}{1-e^{-}{}^{\lambda h}}\right)\left(1+\frac{\lambda h}{1-e^{-}{}^{\lambda h}}\right)+\frac{h\beta }{1-\beta }+Z_1,$(1)

$\begin{array}{c}E\left(C\right)=\left(\alpha +bn+\alpha Y+D_{1}h\right)\left(\frac{e^{-\lambda h}}{1-e^{-\lambda h}}\right)\left(1+\frac{\lambda h}{1-e^{-\lambda h}}\right)\\ +\frac{\alpha +bn}{1-\beta }+\frac{2D_0}{\lambda }+D_1\left(\frac{h}{1-\beta }-\frac{2}{\lambda }\right)+W,\end{array}$(2)

where $\alpha =2\varphi \left(-L\right)$ and $\beta =1-\left[\varphi \left(\delta \sqrt{n-L}\right)+\varphi \left(-\delta \sqrt{n-L}\right)\right]$. If α_U and p_L represent the maximum and minimum values for the type I and type II error probabilities respectively, the minimization of F(n, h, L) is subject to the following restrictions: α ≤ α_U, 1 − β ≥ p_L, n ≥ 1, h ≥ 0 and L ≥ 0. Constant elements of the cost function identify time and economic costs that are described as follows ( Rahim & Banerjee, 1993 ; Chih et al., 2011 ):

This paper presented a fast estimator of the solution space for a cost function model for the economic statistical design (ESD) of X-bar control charts. This estimator introduced the concept of 2D visualization of minimum cost scores (MCS) to illustrate the patterns of the cost function model for the ESD problem. It was observed that the minimum regions of the cost function for different reference data sets could be modelled with elliptical boundaries. Also, it was observed that sub-regions with the highest MCS were more likely to contain the optimal values for the chart’s parameters. Hence, this approach can be integrated within a heuristic method to perform an initial sampling of the solution space, making the search process more specific.

Nevertheless, more work must be performed to measure the suitability of this approach. The following points are considered for future work:

We are grateful for the financial support received, through PDA funds, from the Faculty of Management, University of New Brunswick. We gratefully acknowledge the valuable comments and guidance of reviewers as well as the editorial assistance of our GSRAs, Stebana Rose Mary George Micheal Rai and Edward Wayne Solomon.