^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

This article attempts to develop a simultaneous optimization procedure of several response variables from incomplete multi-response experiments. In incomplete multi-response experiments all the responses (
*p*) are not recorded from all the experimental units (
*n*). Two situations of multi-response experiments considered are (i) on
units all the responses are recorded while on
units a subset of
responses is recorded and (ii) on
units all the responses (
*p*) are recorded, on
units a subset of
responses is recorded and on
units the remaining subset of
responses is recorded. The procedure of estimation of parameters from linear multi-response models for incomplete multi-response experiments has been developed for both the situations. It has been shown that the parameter estimates are consistent and asymptotically unbiased. Using these parameter estimates, simultaneous optimization of incomplete multi-response experiments is attempted following the generalized distance criterion [1]. For the implementation of these procedures, SAS codes have been developed for both complete (
*k* ≤ 5,
*p* = 5) and incomplete (
*k* ≤ 5,
*p*
_{1} = 2, 3 and
*p*
_{2} = 2, 3, where
*k* is the number of factors) multi-response experiments. The procedure developed is illustrated with the help of a real data set.

The experimental situations where more than one response is observed for a set of input combinations are known as multi-response experiments. In these experiments, experimenter is interested in determining the optimum combination of input factors. There are several experimental situations in which simultaneous optimization of several response variables is required. For example, in food processing experiments, a food technologist may be interested in determining the optimum combinations of various ingredients of a product or operability conditions for obtaining the product on the basis of acceptability, colour, flavour, nutritional value, economic and biochemical parameters, etc. Such experimental situations do exist in many other disciplines of science. For some more such experimental situations, a reference may be made to [

Multi-response optimization is the process of determining a combination of levels of input factors that yield optimal values of several response variables taken simultaneously. In such experimental situations, one should not obtain the optima for each response variable separately. When more than one response variables are under investigation simultaneously, the meaning of optimum combination becomes unclear since there is no unique way to order multivariate values of a multi-response function. Further the conditions that are optimal for one response variable may be far from optimal or may even be physically impractical for the other response variables. Therefore, the procedure of simultaneous optimization of several response variables is required. In the literature, different procedures for simultaneous optimization of several responses have been suggested (see e.g. [

[

There, however, do occur experimental situations particularly in osmotic dehydration studies where one wants to optimize the process keeping in view the maximum moisture loss and minimum solids gain. Such a situation has also been encountered by [

The above discussion relates to the experimental situations, where the data on all the response variables are collected from each design point. In some experimental situations, it may not be possible to observe data on all the response variables from all the experimental units. The data from only a subset of responses are collected from some experimental units and other subset of responses from other experimental units. Some of the responses may be common to two or more experimental units. For example, consider an experiment on modified atmospheric packaging (MAP) conducted to determine the optimum combinations of temperature, packaging material and time on the basis of different physical parameters like physiological loss of weight (PLW), moisture content (MC) and texture; biochemical parameters like total sugars, flavanoids and total soluble solids (TSS); microbiological parameters like, total count and caliform count and sensory parameters like, colour, flavour and overall acceptability. In such situations it may happen that after some days, the data on a subset of parameters such as sensory may not be available due to spoilage of the packaged food material for some of the treatment combinations. In this particular situation, the data on sensory characters may not be available at 25 degree centigrade after 11 days for the packaging material HDPE (High Density polyethylene). Further, it may be possible to record the data on all the response variables in some of the replications of the treatment combinations due to variability in the raw material and their differences in initial microbial count.

In response surface designs generally some of the design points are replicated for testing the lack of fit. Due to constraints on time or equipments, it may not be possible to collect data on all the response variables from each of the replicated design points. Therefore, the experimenter may divide the response variables in as many groups as the replications of a design point. Some of the response variables are common in each of the sets. Therefore, this is also an incomplete multi-response situation, where simultaneous optimization of several response variables is required.

Cold storage studies involving different temperatures may also encounter such situations. Field experiments where certain plots or treatment combinations are affected by pests or insects may also have situations as described above. For example, if a field crop is affected by insects/pests which affect only the leaf and damage it, we may be able to record data on other response variables but not on the response variables related to leaf or say if the grain/pod is affected and damaged, we may not be able to record data on yield but can record physical and other parameters like plant height etc.

From the above discussion we can see that there are two types of multi-response experiments viz. complete multi-response and incomplete multi-response experiments where interest is in simultaneous optimization of several response variables. As mentioned above, some procedures of simultaneous optimization of several response variables are available for complete multi-response situations. These procedures are involved quite algebraically, therefore, for the benefit of experimenters it is required to develop some computer based procedure for simultaneous optimization of several response variables.

For incomplete multi-response situations, it seems that no analytical procedure for simultaneous optimization of several response variables is available in the literature. The basic difference between simultaneous optimization from complete and incomplete multi-response experiments is in estimation of parameters in multi-response equations. As shown in [

In Section 2, the procedure of estimation of parameters from incomplete multi-response experiments is developed. Two situations of incomplete multi-response considered are (i) on _{1} = 2, 3 and p_{2} = 2, 3, where k is the number of factors) multi-response experiments. The codes are available with the first author and can be obtained by sending an E-mail to nandi_stat@yahoo.co.in.

In this section, we develop an estimation procedure of parameters of linear multi-response models for incomplete multi-response situations. We consider two cases for incomplete multi-response as described in the sequel.

Let there be p (= p_{1} + p_{2}) response variables which are measured for each design point of k input variables (factors)_{1} + n_{2}) experimental units. From n_{1} experimental units all the response variables are observed and from n_{2 }experimental units only p_{2} < p responses are observed. It is assumed that n_{1 }> m (the number of parameters to be estimated from the model). The pictorial representation in

For the first p_{1} response variables, model can be written as

where

y_{i}: n_{1} ´ 1 vector of observations on the i^{th} response variable,

X_{i}: n_{1} ´ m matrix of rank m of known functions of the settings of the coded variables,

b_{i}: m ´ 1 vector of unknown constant parameters, and e_{i}: n_{1} ´ 1 vector of random error associated with the i^{th} response variable

Model for each of the remaining p_{2} response variables is

y_{j}: (n_{1} + n_{2}) ´ 1 vector of observations on the j^{th} response variable,

X_{j}: (n_{1} + n_{2}) ´ m matrix of rank m of known functions of the settings of the coded variables,

b_{j}: m ´ 1 vector of unknown constant parameters, and e_{j}: (n_{1} + n_{2}) ´ 1 vector of random error associated with the j^{th} response variable

Now combining and rolling down the models (1) and (2) we have

where Y_{1} is (n_{1}p_{1} ´ 1) vector of observations on p_{1} response variables, _{1}p_{2} ´ 1) vector of observations on p_{2} response variables and _{2}p_{2} ´ 1) vector of observations on p_{2} response variables. So Y _{1} + p_{2}) response variables.

Here, X_{i} = X_{1} _{j} = X_{2}.

Using the above, the design matrix Z is

_{1} design points and

_{2} design points. Here _{1} experimental units. Therefore,

The vector of parameters is

where b_{1}(mp_{1} ´ 1) is the vector of parameters for first p_{1} response variables and b_{2}(mp_{2} ´ 1) is the vector of parameters for remaining p_{2} response variables. So b is the vector of parameters of order (mp_{1} + mp_{2}) ´ 1. Similarly, one can also write the expression for residual vectors as

where e_{1 }is (n_{1}p_{1} ´ 1) vector of residuals for first p_{1} response variables, _{1}p_{2} ´ 1) vector of residuals for remaining p_{2} response variables and _{2}p_{2} ´ 1) vector of residuals for p_{2} response variables.

From the structure of residual vector, the dispersion matrix of residual vector is

As is clear from (8) that the covariance between p_{1} and p_{2} can only be possible on n_{1} observations and for n_{2} observations covariance cannot be obtained (zero). In (8) _{1} response variables on n_{1} observations; _{1} and p_{2} response variables on first n_{1} observations and _{2} response variables on n_{2} observations.

It may be noted here that the form of Z in (5) with unequal number of observations for each set of responses (n_{1} for p_{1} and n_{1}+n_{2} for p_{2} response variables), requires additional effort to estimate

Asymptotically unbiased estimator of b is

The dispersion matrix of

Equations (10) and (11) require knowledge of

It is easy to prove that if we replace

Let E_{1} (n_{1} ´ p_{1}) and E_{2} ((n_{1} + n_{2}) ´ p_{2}) be the matrices of least-squares residuals for the p_{1} and p_{2 }response variables respectively. E_{2} can be partitioned analogously to X_{2} and Y_{2} as

Now we define the following quantities:

The consistent estimators of

Once a consistent estimator of

The dispersion matrix of

Following the proof of [

Let there be p (=p_{1} + p_{2}) response variables which are measured for each design point of k input variables (factors)_{1} + n_{2} + n_{3 }experimental units. All the p response variables are observed from n_{2} experimental units. Only p_{1} response variables are observed from first _{3} experimental units only p_{2} response variables are observed. It is assumed that

For the first p_{1} response variables, model can be written as

where

y_{i}: (n_{1} + n_{2}) ´ 1 vector of observations on the i^{th} response variable,

X_{i}: (n_{1} + n_{2}) ´ m matrix of rank m of known functions of the settings of the coded variables,

b_{i}: m ´ 1 vector of unknown constant parameters, and e_{i} : (n_{1} + n_{2}) ´ 1 vector of random error associated with

the i^{th} response variable

Model for each of the remaining p_{2} response variables is

where

y_{j}: (n_{2} + n_{3}) ´ 1 vector of observations on the j^{th} response variable,

X_{j}: (n_{2} + n_{3)} ´ m matrix of rank m of known functions of the settings of the coded variables,

b_{j}: m ´ 1 vector of unknown constant parameters, and e_{j}: (n_{2} + n_{3}) ´ 1 vector of random error associated with the j^{th} response variable

Now combining and rolling down the models (17) and (18) we have

where,

_{2} experimental units. Therefore,

where Y, b and e are column vectors with

We now regard the above model as a single equation regression model and making use of Aitken’s GLS technique asymptotically unbiased estimator of

In most of the practical situation

Let E_{1} ((n_{1} + n_{2}) ´ p_{1}) and E_{2} ((n_{2} + n_{3}) ´ p_{2}) be the matrices of least-squares residuals for the p_{1} and p_{2} response variables respectively. E_{i}’s can be partitioned analogously to Y_{i} and X_{i} in (19) and (20)

Now we define the following quantities:

The consistent estimators of

Once a consistent estimator of

As in Case I, following the proof of [

In the present study, simultaneous optimization of several responses based on minimization of generalized distance given by [_{1} = 2, 3 and p_{2} = 2, 3, where

For the sake of completeness, we describe the procedure in brief. Using the parameter estimates obtained in Section 2, the prediction equation for the i^{th} response at design point

where _{i} evaluated at the design point x, _{i}. It follows that

Combining all the response variables together we can get the prediction equation at design point

where

Let

and let

then an “ideal” optimum is said to be achieved. The problem of multi-response optimization is, therefore, obviously solved and no further work is needed. However, such an ideal optimum may rarely exist. In more general situations one might consider finding compromising conditions on the input factors that are somewhat favorable to all responses. Such a deviation of the compromising conditions from the ideal optimum is formulated by means of a distance function, which measures the distance of

where

The above distance function is termed as generalized distance by [

region W. If x_{0 }is the point in the experimental region at which

and if m_{0} is the value of this minimum, then one may describe the experimental conditions at x_{0} as being near optimal for each of the p response functions. The smaller the value of m_{0}, the closer these conditions are to representing an “ideal” optimum. To summarize, the steps for obtaining a simultaneous optima of a multi-response function consisting of p responses are:

1) Obtain the predicted equations using the complete/incomplete multi-response data and the method of generalized least squares.

2) Choose a distance measure

3) Optimize the predicted responses individually over the experimental region W to obtain the vector

4) The distance measure being a function of x alone is minimized over W.

Here the controlled random search procedure given in [

The results of ridge analysis for optimization of individual responses can be obtained using PROC RSREG in SAS [_{1} = 2, 3 and p_{2} = 2, 3). The SAS code is available with the first author and can be obtained by sending an E-mail to nandi_stat@yahoo.co.in. The method of ridge analysis for determining optimum conditions given in [

In this section, we illustrate the procedure developed in Sections 2 and 3 for simultaneous optimization of several response variables through the data from a quantitative factorial experiment.

Example 4.1: An experiment was conducted on osmotic dehydration of banana to determine the optimum combinations of power levels, temperature and air velocity at Division of Agricultural Engineering, ICAR- Indian Agricultural Research Institute, New Delhi. The levels of the 3 factors tried are shown in

The data were collected on energy use efficiency (%)_{1}), temperature (X_{2}) and air velocity (X_{3}) are coded, using the following expression given in [

where x_{i} is the coded variable,

The coded levels of the factors and data on 5 response variables in given in

As mentioned in Section 1 that in case of complete multi-response situations, the parameter estimates obtained by taking all the response variables together is same as obtained individually for each of the response variables. A second order response surface model was fitted to each of the 5 response variables separately to obtain the parameter estimates. The regression coefficient estimates, their standard errors, and the coefficients of determination

It was observed that stationary points are outside the experimental region for some of the response variables, we have performed ridge analysis to obtain individual maxima. The individual maxima for the 5 response

Factors | Levels |
---|---|

Microwave power (W) | 140, 210 and 280 |

Temperature (˚C) | 25, 45, 55 and 65 |

Air velocity (m/s) | 0.5, 1.5 and 2.5 |

Microwave power | Temperature | Air velocity | |||
---|---|---|---|---|---|

Actual | Coded | Actual | Coded | Actual | Coded |

X_{1} | x_{1} | X_{2} | x_{2} | X_{3} | x_{3} |

140 W | −1 | 25˚C | −1 | 0.5 m/s | −1 |

210 W | 0 | 45˚C | 0 | 1.5 m/s | 0 |

280 W | 1 | 55˚C | 0.5 | 2.5 m/s | 1 |

65˚C | 1 |

Serial number | Power level (W) | Temperature (˚C) | Air velocity (m/s) | Energy use efficiency (%) | Rehydration Ratio | TSS (˚Brix) | Total Sugars | Total Carbohydrates (mg/g dry matter) |
---|---|---|---|---|---|---|---|---|

1 | −1 | −1 | −1 | 19.54 | 1.94 | 60.98 | 47.73 | 555 |

2 | −1 | −1 | 0 | 13.01 | 1.9 | 61.74 | 47.85 | 558.09 |

3 | −1 | −1 | 1 | 12.53 | 1.904 | 62.5 | 48.04 | 561.53 |

4 | −1 | 0 | −1 | 22.45 | 1.805 | 59.82 | 48.69 | 569.69 |

5 | −1 | 0 | 0 | 13.22 | 1.733 | 60.87 | 49 | 575.45 |

6 | −1 | 0 | 1 | 8.57 | 1.71 | 60.43 | 49.89 | 584.31 |

7 | −1 | 0.5 | −1 | 20.51 | 1.68 | 58.71 | 49.36 | 581.36 |

8 | −1 | 0.5 | 0 | 12.82 | 1.783 | 60.06 | 50.54 | 594.09 |

9 | −1 | 0.5 | 1 | 8.21 | 1.702 | 60.31 | 50.54 | 595.32 |

10 | −1 | 1 | −1 | 24.71 | 1.603 | 57.47 | 49.95 | 590.59 |

11 | −1 | 1 | 0 | 11.96 | 1.77 | 58.86 | 50.28 | 597.52 |

12 | −1 | 1 | 1 | 7.49 | 1.78 | 59.3 | 50.37 | 598.15 |

13 | 0 | −1 | −1 | 35.76 | 2.125 | 64.31 | 50.17 | 583.99 |

14 | 0 | −1 | 0 | 32.8 | 2.093 | 65.08 | 50.9 | 590.8 |

15 | 0 | −1 | 1 | 16.66 | 2.212 | 67.32 | 51.7 | 603.29 |

16 | 0 | 0 | −1 | 31.68 | 1.819 | 63.78 | 50.95 | 596.09 |

17 | 0 | 0 | 0 | 21.69 | 1.993 | 64.39 | 51.18 | 601.88 |

18 | 0 | 0 | 1 | 13.98 | 2.018 | 66.71 | 52.02 | 608.66 |

19 | 0 | 0.5 | −1 | 36.76 | 1.775 | 63.18 | 52.52 | 617.34 |

20 | 0 | 0.5 | 0 | 22.83 | 1.671 | 64.49 | 51.9 | 615.73 |

21 | 0 | 0.5 | 1 | 17.53 | 1.906 | 66.73 | 52.69 | 622.7 |

22 | 0 | 1 | −1 | 34.29 | 1.691 | 63.51 | 53.37 | 630.47 |

23 | 0 | 1 | 0 | 21 | 1.684 | 63.76 | 51.55 | 612.53 |

24 | 0 | 1 | 1 | 14.7 | 1.88 | 64.46 | 52.74 | 626.69 |

25 | 1 | −1 | −1 | 52.09 | 2.237 | 71.41 | 52.54 | 614.24 |

26 | 1 | −1 | 0 | 44.65 | 2.307 | 71.04 | 52.7 | 623.26 |

27 | 1 | −1 | 1 | 39.1 | 2.403 | 70.98 | 53.96 | 629.86 |

28 | 1 | 0 | −1 | 49.29 | 2.353 | 72.94 | 53.44 | 628.08 |

29 | 1 | 0 | 0 | 32.96 | 2.433 | 72.53 | 54.7 | 641.83 |

30 | 1 | 0 | 1 | 25.3 | 2.505 | 70.46 | 54.34 | 639.48 |

31 | 1 | 0.5 | −1 | 46.25 | 2.793 | 71.13 | 54.36 | 640.54 |

32 | 1 | 0.5 | 0 | 33.67 | 2.579 | 71.86 | 55.39 | 651.29 |

33 | 1 | 0.5 | 1 | 26.47 | 2.609 | 71.51 | 55.23 | 651.4 |

34 | 1 | 1 | −1 | 53.2 | 2.821 | 70.69 | 54.75 | 647.96 |

35 | 1 | 1 | 0 | 32.57 | 2.64 | 70.04 | 54.85 | 647.34 |

36 | 1 | 1 | 1 | 24.69 | 2.696 | 70.81 | 54.71 | 649.41 |

Regression coefficients | |||||
---|---|---|---|---|---|

Model term | Y_{1} | Y_{2} | Y_{3} | Y_{4} | Y_{5} |

Intercept | 22.2847 (1.1205) | 1.8608 (0.0589) | 65.2523 (0.3336) | 51.7167 (0.2112) | 607.1644 (2.1368) |

x_{1} | 12.1479 (0.5381) | 0.3580 (0.0283) | 5.5159 (0.1602) | 2.4656 (0.1014) | 29.6592 (1.0261) |

x_{2} | −2.1499 (0.5899) | −0.0285 (0.0310) | −0.8924 (0.1756) | 1.0097 (0.1112) | 16.4039 (1.1250) |

x_{3} | −8.5617 (0.5381) | 0.0292 (0.0283) | 0.5675 (0.1602) | 0.3829 (0.1014) | 5.1491 (1.0261) |

x_{1}^{*}x_{1} | 1.4958 (0.9189) | 0.2480 (0.0483) | 0.8754 (0.2736) | −0.0071 (0.1732) | 0.2271 (1.7524) |

x_{2}^{*}x_{1} | −2.1095 (0.7174) | 0.1582 (0.0377) | 0.6565 (0.2136) | −0.1479 (0.1352) | −2.7435 (1.3681) |

x_{2}^{*}x_{2} | 2.5217 (0.9774) | 0.0466 (0.0514) | −0.5885 (0.2910) | −0.1804 (0.1842) | −0.383 (1.8639) |

x_{3}^{*}x_{1} | −2.1788 (0.6498) | −0.0037 (0.0342) | −0.4981 (0.1935) | 0.0025 (0.1225) | −0.2087 (1.2391) |

x_{3}^{*}x_{2} | −1.9400 (0.7174) | −0.0059 (0.0377) | 0.0004 (0.2736) | −0.2629 (0.1352) | −2.7094 (1.3681) |

x_{3}^{*}x_{3} | 2.3083 (0.9189) | 0.0331 (0.0483) | −0.0102 (0.2136) | 0.0992 (0.1732) | 0.2721 (1.7524) |

Coefficient of determination (R^{2}) | 0.9698 | 0.8987 | 0.9807 | 0.9643 | 0.9766 |

^{*}The number in the parentheses is the standard error.

variables obtained by the method of ridge analysis are given in

simultaneous maxima for the 5 response variables is obtained by minimizing

2.48, 71.63, 54.48, 643.65) and the results are given in

Now, to illustrate the procedure for incomplete multi-response situations, we have deleted the data on some of the response variables on some of the design points in Example 4.1. Example 4.2 relates to Case I of incomplete multi-response situation as discussed in Section 2.1.

Example 4.2: For the purpose of illustration, consider the experiment in Example 4.1 for complete multi-response experiments. Let us consider that the data on Energy use efficiency (%), Rehydration ratio and TSS for the design points (1, 1, 0) and (1, 1, 1) at serial number 35 and 36 in

In case of incomplete multi-response experiments the parameter estimates obtained by taking each response variable individually is not same as the parameter estimates obtained taking all the response together. In this case to estimate the parameters of the second order response surface are obtained as per procedure of Case I in Section 2.1. After obtaining individual estimates based on the available data, residuals are used to obtain variance-covariance matrix. Using the variance-covariance matrix, we have obtained the GLS estimate of the parameters of the second order response surface model. Comparing

It was observed that the stationary points are outside the experimental region for some of the response variables, we have performed ridge analysis to obtain individual maxima.

simultaneous maxima for the 5 response variables is obtained by minimizing

2.50, 71.72, 54.49, 643.68)' and the results are given in

The location at which the linear incomplete multi-response function attains its simultaneous optima is [1.000 (280 W), 0.610 (57.2˚C), −0.470 (1.03 m/s)] and the corresponding maxima (40.70, 2.57, 71.42, 54.55, 643.76) for the response variables are given in

Location of maxima | ||||
---|---|---|---|---|

Response | Individual maxima | x_{1} (X_{1}) | x_{2} (X_{2}) | x_{3} (X_{3}) |

40.92 | 0.99 (279.8) | −0.16 (41.6) | −0.61 (0.88) | |

2.48 | 0.98 (279.0) | 0.14 (47.8) | 0.09 (1.59) | |

71.63 | 0.99 (279.7) | −0.02 (44.4) | −0.002 (1.49) | |

54.48 | 0.99 (279.8) | 0.35 (52.1) | 0.05 (1.55) | |

643.65 | 0.99 (279.9) | 0.45 (54.1) | 0.14 (1.64) |

Distance measure | Generalized distance | |
---|---|---|

Minimum distance | 3.81 | |

41.02 | ||

2.55 | ||

71.22 | ||

54.51 | ||

643.16 | ||

Location of Minima^{*} | X_{1} | 0.995 (278.95 W) |

X_{2} | 0.59 (46.55˚C) | |

X_{3} | −0.52 (0.98 m/s) |

^{*}Actual value of optimum point is given in parenthesis.

Regression coefficients | |||||
---|---|---|---|---|---|

Model term | Y_{1} | Y_{2} | Y_{3} | Y_{4} | Y_{5} |

Intercept | 22.3701 (0.9848) | 1.8627 (0.0027) | 65.2435 (0.0836) | 51.7166 (0.0322) | 607.1644 (3.2975) |

x_{1} | 12.1903 (0.2399) | 0.3603 (0.0006) | 5.5555 (0.0203) | 2.4655 (0.0074) | 29.6591 (0.7603) |

x_{2} | −2.0867 (0.3120) | −0.0249 (0.0008) | −0.8334 (0.0263) | 1.0097 (0.0089) | 16.4038 (0.9139) |

x_{3} | −8.5909 (0.2264) | 0.0287 (0.0006) | 0.5756 (0.0192) | 0.3828 (0.0074) | 5.1490 (0.7603) |

x_{1}^{*}x_{1} | 1.5489 (0.6679) | 0.2509 (0.0018) | 0.9250 (0.0567) | −0.0070 (0.0216) | 0.2270 (2.2177) |

x_{2}^{*}x_{1} | −2.0244 (0.4788) | 0.1628 (0.0013) | 0.7358 (0.0404) | −0.1479 (0.0132) | −2.7435 (1.3517) |

x_{2}^{*}x_{2} | 2.6119 (0.8199) | 0.0516 (0.0022) | −0.5043 (0.0694) | −0.1804 (0.0245) | −0.3830 (2.5089) |

x_{3}^{*}x_{1} | −2.2336 (0.3507) | −0.0046 (0.0009) | −0.4829 (0.0297) | 0.0025 (0.0108) | −0.2087 (1.1088) |

x_{3}^{*}x_{2} | −1.9985 (0.4247) | −0.0068 (0.0011) | 0.0059 (0.0360) | −0.2628 (0.0132) | −2.7094 (1.3517) |

x_{3}^{*}x_{3} | 2.0922 (0.6828) | 0.0253 (0.0019) | −0.0684 (0.0579) | 0.0991 (0.0216) | 0.2720 (2.2177) |

^{*}The number in the parentheses is the standard error.

Location of maxima | ||||
---|---|---|---|---|

Response | Individual maxima | x_{1} (X_{1}) | x_{2} (X_{2}) | x_{3} (X_{3}) |

Y_{1} | 40.61 | 1.00 (280.0) | −0.16 (41.8) | −0.56 (0.93) |

Y_{2} | 2.50 | 0.99 (279.6) | 0.15 (48.02) | 0.09 (1.59) |

Y_{3} | 71.72 | 1.00 (280.0) | −0.01 (44.76) | 0.01 (1.51) |

Y_{4} | 54.49 | 1.00 (280.0) | 0.35 (52.12) | 0.05 (1.55) |

Y_{5} | 643.68 | 1.00 (280.0) | 0.45 (54.12) | 0.12 (1.62) |

Distance measure | Generalized distance | |
---|---|---|

Minimum distance | 3.36 | |

Y_{1} | 40.70 | |

Y_{2} | 2.57 | |

Y_{3} | 71.42 | |

Y_{4} | 54.55 | |

Y_{5} | 643.76 | |

Location of Optima^{*} | X_{1} | 1.000 (280 W) |

X_{2} | 0.610 (57.2˚C) | |

X_{3} | −0.470 (1.03 m/s) |

^{*}Actual value of optimum point is given in parenthesis.

function obtained is 3.36 which is lower (3.81) than that obtained from complete multi-response situation as given in

On the similar lines, one can obtain the simultaneous optima for Case II of incomplete multi-response situations.

In the present investigation, a methodology was developed for estimation of parameters and optimization of simultaneous optimization from multi-response experiments where some responses could not be obtained on some design points. The procedure of obtaining a consistent estimator of unknown dispersion matrix in case of incomplete multi-responses is also suggested. The simultaneous optimization procedure is implemented through SAS codes for specified number of input factors (k ≤ 5) and response variables (p_{1} = 2, 3 and p_{2} = 2, 3). The procedures developed have been illustrated through an example. Further efforts are required to obtain a general procedure for any number of factors and any number of response variables in incomplete multi-response situations.

Authors are thankful to Dr Abhijit Kar for sharing the data that has been used for illustration in Section 4. The authors are also thankful to reviewer and the editor for their valuable suggestions that led to improvements in the presentation of the results. The present article is part of Ph.D. work of first author.

Pradip KumarNandi,RajenderParsad,Vinod KumarGupta, (2015) Simultaneous Optimization of Incomplete Multi-Response Experiments. Open Journal of Statistics,05,430-444. doi: 10.4236/ojs.2015.55045