||Constrained multiobjective experimental designs for the maximum ratio of utility and cost
||Department of Industrial and Information Management
Multiple objective experiments
The subject of this study is optimization of multiple objective experimental designs. It is very difficult to optimize multiple responses simultaneously through input variables because of the trade-off between responses. However, many industries face this problem nowadays. This study combines Pareto frontier, the desirability function and manufacturing costs to deal with multiple objective experimental designs. We consider the situation that the total number of experiments is limited by the resource. Therefore, we adopt the near D-optimal criterion for the multiple objective experimental designs and use an algorithm to generate a near D-optimal experimental design. Using this near D-optimal design to do the experiments, we can get the experimental results. Then, we find all the design points on the Pareto frontier and calculate the response predictive values of design points based on the experimental results. Next, we use these predictive values to update the Pareto frontier. Finally, we add extra information of the manufacturing costs to find the design point which has the maximum ratio of utility and cost. We use a case to illustrate the process and we discuss the influence of variance-covariance matrix on the Pareto frontier. At the end of the example, sensitivity analysis is performed to demonstrate the effect of the maximum value of the cost differences. This study expects to provide an instrument for decision-makers.
Key words: Multiple objective experiments, Pareto frontier, desirability function
Many quality characteristics are usually considered at the time when we measure the quality of products. These quality characteristics can be assessed by measuring the response variables. Response variables can be affected by one or several factors. When we deal with multiple response variables, we usually face the problem of trade-off. Trade-off means that one response variable can get the best result when the factors are set to specific levels, but the other response variables are not necessarily the best. That is, optimal factor settings for each response variable may be contradictory to each other. The experimental design that considers multiple responses is called multiple objective experimental designs. Ko et al. (2005), and Wu (2005) pointed out that the need for optimizing multiple objective experimental designs is increasing. Optimization of multiple objective experimental designs also has the trade-off problem. The common method is to convert response variables into a single value to access.
Generally, the more experiments are executed, the more accurate the result should be. In fact, we need to consider the cost when executing an experiment. The cost factors can be divided into two categories: the first category, the cost impacts on the total number of experiment. At this time, we need to find a feasible design under limited resource. The second category, the manufacturing costs of the different factor levels need to be considered. So, we need to consider the total manufacturing costs to produce one product.
In addition to the cost, limitation which exists between the factors also affects the design of experiment. So, we need to find an optimal experimental design. Berger and Wong (2005) pointed out that because of the existence of restrictions in the industrial, medical pharmaceutical, biomedical, epidemiology and other industries, the need to find the optimal experimental design problem is fairly common. Considering the total number of experiments is limited by cost and the restrictions between the factors. According to the description above, the purpose of this study is to find an optimal design of experiment under above limitation and then suggest a best design point when manufacturing costs are considered.
MATERIAL AND METHODS
In this study, we use a near D-optimal criterion to find a feasible design of experiment. The near D-optimal criterion does not depend on the variance-covariance matrix. In addition to the near D-optimal criterion, this study combines Pareto frontier, the desirability function and manufacturing costs to deal with the multiple objective experimental designs. The approach consists of the following steps: first, according to the response variables, we can get the Pareto frontier which is the set of non-dominated design points. We cannot tell which design points is better because they do not dominate each other. Second, we use the linear multiple responses model to predict the response variables. So, we can get the predictive values of non-executed experiments. Then, we update the Pareto frontier. Third, we set up an individual desirability function for each response variable according to the requirement of each response. The requirements can be divided into three categories: objective to maximize response variable, objective to minimize response variable and objective to be as close to the target as possible. After we calculate all individual desirability and the overall desirability of the design points on the Pareto frontier. Fourth, divide overall desirability by the total manufacturing cost of a design as the ratio of utility and cost. Finally, we choose the maximum ratio of utility and cost as the best design point.
RESULTS AND DISCUSSION
According to the case in this study, we find that the value of parameters is important when we calculate the overall desirability. The overall desirability is one of the important indices to choose the best design point. From the sensitivity analysis, we find the maximum value of cost differences has an impact on the best design point. If the factor with maximum value of the cost differences is set at high cost level, reducing its cost will not change the best design point. On the contrast, if the factor with maximum value of the cost differences is set at low cost level, reducing its cost may change the best design point.
For future research, the continuous settings of factors can be considered. The value of cost differences are fixed in this study. In the future, cost differences can be a function of factors.
第一章 緒論 1
第二章 文獻回顧 4
2.2.3渴望函數(Desirability Function) 10
第三章 模型建構 19
第四章 實證分析 34
第五章 結論與未來研究建議 52
Alaeddini, A., Yang, K., Mao, H., Murat, A. and Ankenman, B. (2013), An adaptive
sequential experimentation methodology for expensive response surface
optimization – case study in traumatic brain injury. Modeling.Quality Reliability Engineering International.
Alaeddini, A., Yang, K. and Murat, A. (2013). ASRSM: A sequential experimental design
for response surface optimization. Quality Reliability Engineering International, 29(2), pp. 241-258
Allen, T. T., and Yu, L. (2002). Low-cost response surface methods from simulation
optimization. Quality Reliability Engineering International, 18(1), pp.5-17.
Arnouts, H., Goos, P., and Jones, B. (2010). Design and analysis of industrial strip-plot
experiments. Quality Reliability Engineering International, 26(2),pp. 127-136.
Athan, T. W., and Papalambros, P. Y. (1996). A note on weighted criteria methods for
compromise solutions in multi-objective optimization.Engineering Optimization,27(2), pp.155-176.
Berger, M. P. F., and Wong, W. K. (2005). Front Matter, in Applied Optimal Designs,
John Wiley and Sons, Ltd, Chichester, UK.
Box, G. E. P., and Wilson, K. B. (1951). On the experimental attainment of optimum
conditions.Journal of the Royal Statistics Society. Series B (Methodological),13(1), pp.1-45
Box, G. E. P., and Hunter, J. S. (1961).The fractional factorial designs PartI.
Technometrics, 3(3), pp. 311-351.
Box, G. E. P. (1999), Statistics as a catalyst to learning by scientific method Part II- A
Discussion.Journal of Quality Technology, 31, pp. 16-29.
Chackelson, C., Errasti, A., Ciprés, D., & Lahoz, F. (2013). Evaluating order picking
performance trade-offs by configuring main operating strategies in a retail
distributor: a design of experiments approach. International Journal of
Chang, S. (1997). An algorithm to generate near D-optimal designs for multiple responses
surface models. IIE transactions, 29(12), pp. 1073-1081.
Chankong, V., and Haimes, Y.Y.(1983). Multiobjective decision Making Theory and
Methodology. New York: Elsevier SciencePublishing.
Chen,W., Sahai, A., Messac, A. and Sundararaj, G.(2000).Explorationof the effectiveness
of physical programming in robust design. Journal of Mechanical Design, 122, pp.155-163.
Chernoff, H. (1953). Locally optimal designs for estimating parameters. The Annals of
Mathematical Statistics,24(4), pp.586-602.
de Aguiar, P. F., Bourguignon, B., Khots, M. S., Massart, D. L., and Phan-Than-Luu, R.
(1995). D-optimal designs. Chemometrics and Intelligent Laboratory Systems, 30(2),
Derringer, G., and Suich, R. (1980). Simultaneous optimization of several responses
variables. Journal of Quality Technology, 12(4), pp. 214-219.
Derringer, G.(1994). A balancing act: optimizing a product’s properties. Quality Progress,
26(6), pp. 51-58.
Fedorov, V. V. (1972). Theory of optimal experiments, Academic Press, New York.
de Aguiar, P. F., Bourguignon, B., Khots, M. S., Massart, D. L., and Phan-Than-Luu, R. (1995). D-optimal designs.Chemometrics and Intelligent Laboratory Systems,30(2), pp. 199-210.
Gadhe, A., Sonawane, S. S., and Varma, M. N. (2013). Optimization of conditions for
hydrogen production from complex dairy wastewater by anaerobic sludge using
desirability function approach. International Journal of Hydrogen Energy, 38(16),pp.6607-6617.
Harrington, E. C. (1965). The desirability function. Industrial Quality Control, 21(10), pp.
Izraelevitz, A. M., Anderson-Cook, C. M., & Hamada, M. S. (2011). Illustrating the use of
statistical experimental design and analysis for multiresponse prediction and optimization. Quality Engineering, 23(3), 265-277.
Islam, M. A., Alam, M. R., and Hannan, M. O. (2012). Multiresponse optimization based
on statistical response surface methodology and desirability function for the production of particleboard. Composites Part B: Engineering, 43(3), pp.861-868.
Khuri, A.I., andCornell, L.A. (1987). Response surfaces – designs and analyses. Marcel
Dekker, New York.
Ko,Y. H., Kim, K. J., and Jun, C. H. (2005). A new loss function-based method for
multiresponse optimization. Journal of Quality Technology, 37(1), pp. 50-59
Lu, L., Anderson-Cook, C. M., and Robinson, T. J. (2012). Optimization of designed
experiments based on multiple criteria utilizing a Pareto frontier. Technometrics, 53(4), pp. 353-365.
Lu, L., and Anderson‐Cook, C. M. (2012). Balancing multiple criteria incorporating cost
using Pareto front optimization for split‐plot designed experiments. Quality and
Reliability Engineering International, 30(1), 37-55.
Mahalanobis, P. C.(1936).On the generalized distance in statistics. Proceedings National
Institute id Science of india. pp. 49-55
Mao, M. and Danzart, M. (2008). How to select the best subset of factors maximizing the
quality of multi-response optimization. Quality Engineering, 20(1), pp. 63-74
Marler, R. T., and Arora, J. S. (2004).Survey of multi-objective optimization methods for
engineering. Structural and Multidisciplinary Optimization, 26(6), pp. 369-395.
Montgomery, D. C. (2012). Design and Analysis of Experiments. 8thed. New York: Wiley.
Muyanja, A. W.,Atichat. T. and Porter, J. D.(2013) An experimental study on the effect
of pattern eecognition parameters on the accuracy of wireless-based task time
estimation. International Journal of Production Economics.
Ngatchou, P., Zarei, A., and El-Sharkawi, M. A. (2005).Pareto multi objective
optimization. Proceedings of IEEE International Conference on Intelligent Systems Application to Power Systems, Arlington, VA, pp.84-91.
Pal, S. and Gauri, Susanta Kumar. (2010). Multi-response optimization using multiple
regression-based weighted signal-to-noise ratio(MRWSN).Quality Engineering, 22(4), pp. 336-350.
Smith, K. (1918). On the standard deviations of adjusted and interpolated values of an
observed polynomial function and its constants and the guidance they give towards a proper choice of the distribution of observations.Biometrika,12(1/2), pp. 1-85.
Student.(1908).The probable error of mean.Biometrika, 6(1), pp.1-25.
Taguchi, G., and Rajesh, J. (2000). New trends in multivariate diagnosis. Sankhyā: The
Indian Journal of Statistics, Series B, pp. 233-248.
Taguchi, G., and Wu, Y. (1980). Introduction to off-line quality control. Central Japan
Quality Control Assoc.
Taguchi, G. (1987).Introduction to Quality Engineering. Tokyo: Asian Productivity
Wald, A. (1943). On the efficient design of statistical investigations. The Annals of
Mathematical Statistics,14(2), pp. 134-140.
Walter, M., Sommer-Dittrich, T., & Zimmermann, J. (2011). Evaluating volume
flexibility instruments by design-of-experiments methods. International Journal
of Production Research. 49(6), 1731-1752.
Wijesinha, M.e. (1984).Design of expenments for multiresponse models.Ph. D. thesis.
Department of Statistics.University ofFlonda, Gainesville, FL.
Wu, F. C. (2005). Optimization of correlated multiple quality characteristics using
desirability function.Quality Engineering,17(1), pp. 119-126.
Yu, P. L., and Leitmann, G. (1974). Compromise solutions, domination structures, and
Salukvadze's solution.Journal of Optimization Theory and Applications,13(3), pp. 362-378.
Zeleny, M.(1982).Multiple Criteria Decision Making. NewYork: McGraw Hill.