As far as I know, quality can be treated in two ways in econometric models. One way is to use dummy variables. For instance if you want to capture sex, you will code 1 for male and 0 for female, or the reverse.

Another way is to create a quality index. Say you want to model the comfort of cars. Then you can use an index that say add the head room to the leg room. As far as optimization goes, you will have to fit indifference curves kind of relationship among the variables. A good managerial economics textbook can help you on this.

I am attaching an article of mine in which I used quality indices in prices car. Hope it helps.

ISSUE: Cost optimization or how to model minimum cost for implementing quality system?

MATHEMATICAL OPTIMIZATION: The issue appears to fall into mathematical optimization analysis. Mathematical optimization deals with the selection of minimization cost and maximization of benefits. Here, “optimizing cost” would mean minimizing cost and maximizing benefits.

Let there be a function f for which a data set call A exists and that set A is comprised of real number R, then x0 is an element of set A such that minimization and maximization may be defined as:

(1)   f(x0) = f(x) for maximization.

The function f(x) is called objective function, loss function or cost function. The next step is to look for the model for f(x).

COST & BENEFIT MODEL: The implementation of quality system ISO 9001: 2008 will result in cost and benefits. The objective then is to determine an appropriate model that would provide minimum cost and maximum benefit. The means to reach this goal is mathematical optimization in arguments (1) and (2). The cost and benefit analysis model is given by:

(3)   NPV = sum (B – C) / (1 – r)t

… where NPV = present value; B = benefits; C = costs; r = discount rate; t = year; and n = analytic horizon in years.

The objective is to maximize NPV, which is the as saying to find the minimum costs. The quality under the ISO 9001: 2008 may run for a period for more than on year. Thus the “sum” in equation (3) allows us to accommodate these multiple years, i.e. NPV0, NPV1, …, NPVn from time t0 to tn. This series of data of NPV may be tested for how well does the observed NPV is from the expected NPV. This best fit analysis may be accomplished through chi-square test. The general chi-square test is given by:

(4)   X2 = sum(O – E)2 / E

… where O = observed value; and E = expected value (value = NPV).

Among NPV0, NPV1, …, NPVn, find the highest value of NPV. That maximum NPV was produced by the lowest cost or optimum cost. Now use that NPV as the observed value and compare it to NPV(E). If this maxNPV > NPV* then the cost produced that NPV is the optimal cost.

The “goodness-of-fit” under the chi-square test tells us whether cost is optimized. For instance, at 0.95 confidence interval for NPV values at time t0, t1, …, tn at degree of freedom 4 (if the observation period is 5 years), in order to confirm minimize cost or cost optimization, the critical value for the observed chi-square must be equal to or greater than 9.50. Optimal cost (minimum cost is found at the observed point that is higher than 9.5 (statistically significant).

REFERENCE:

(1) Bradley, S.; Hax, A.; Magnanti, T. (1977). Applied mathematical programming. Addison Wesley.

(2) Rardin, Ronald L. (1997). Optimization in operations research. Prentice Hall. p. 919.ISBN 0-02-398415-5. "copyright: 1998

(3) Magnanti, Thomas L. (1989). "Twenty years of mathematical programming". In Cornet, Bernard; Tulkens, Henry. Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987). Cambridge, MA: MIT Press. pp. 163–227. ISBN 0-262-03149-3.

I recommend Cobb-Douglas production function which can model the relation between independent and dependent variables. you can find good information by seeking on the web.

 There are classic designs - see J.M. Juran's model and modern designs. If you enter the library of theses ASE Bucharest, see find work,, Contributions to analyze the efficiency of implementation of quality management in the Romanian Industry ", which treats these econometric models.

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