مدل‌سازي مسايل متغير‌هاي چندپاسخه با توزيع نامعين غير نرمال با استفاده از برنامه‌ريزي ژنتيك

Modeling of Multi-Response Problems with Non-Deterministic Non-Normal Distributed Responses using Genetic Programming

در طراحي و تحليل آزمايش‌ها، پس از تعيين متغير‌هاي موثر بر متغير پاسخ، كشف رابطه‌ي بين آنها و ارايه‌ي مدل پيش‌بيني مد نظر است. در روش‌هاي كلاسيك لازم است مفروضات اوليه‌يي براي شناسايي رابطه‌ي بين متغيرهاي پاسخ و متغيرهاي كنترلي بررسي و تاييد ‌شوند كه در دنياي واقعي اغلب متغيرهاي پاسخ چنين شرايطي را ندارند. برنامه‌ريزي ژنتيك ازجمله روش‌هاي نوين براي پي بردن به رابطه‌ي بين دسته‌يي از متغيرهاست و از مزيت‌هاي آن مي‌توان به عدم وابستگي آن به نوع توزيع باقي‌مانده‌ها اشاره ‌كرد. اين روش برخلاف الگوريتم ژنتيك به‌دنبال كشف رابطه بين متغيرهاي اثرگذار است. در اين پژوهش، روش برنامه‌ريزي ژنتيك براي كشف رابطه بين متغير‌هاي وروديِ يك طرح آزمايش كه چند متغير پاسخ دارد پيشنهاد شده و در ادامه از الگوريتم ژنتيك به‌منظور بهينه‌سازي استفاده مي‌شود.

In most experiments, the experimenter is interested in identifying effective controllable factors to model their relationship function. The classic approaches of response surface methodology and experimental design need to meet some requirements such as residual normality. However, in many real world applications, the assumptions may be violated. In such cases, data transformation methods can be an alternative. However, the mentioned method may increase total error in multiple response analyses. Genetic programming is a meta-heuristic approach in determination of effective controllable variables, and has been previously applied to many areas. One of the major differences between GP and the GA (Genetic Algorithm) is in the representation of the solution. In addition, GP is used to identify a suitable relationship function between variables, while the GA is used to optimize an objective function and find the near optimal values of decision variables. Therefore, each solution in GP represents one equation of the relationship function between variables. In this paper, genetic programming is applied for determination of the relation function between the response variables and controllable factors for non-deterministic, non-normal distributed responses. In other words, three steps are considered in the proposed method. In the first step, a relation function is estimated for each response according to the GP. Then, all estimated response functions are aggregated to a single response by the desirability function. In the last step, a GA is used for optimization of the extracted integrated function. Moreover, three examples are used to illustrate applications of the proposed method. In the first example, the efficiency of the proposed method in a single response problem is considered. The second example is used to compare the performance of the proposed method with the result of the regression method, while residuals have non-normal distribution. In the last example, the proposed method is applied to a multi-response problem in a real case study from the literature. Finally, the computational results of simulated data and previous studies confirm that the proposed method has a proper performance in determination of a suitable level of controllable factors.