Mathematical Problems in Engineering

Volume 2015, Article ID 425689, 14 pages

http://dx.doi.org/10.1155/2015/425689

## Parameter Determination of Milling Process Using a Novel Teaching-Learning-Based Optimization Algorithm

School of Mechanical and Instrument Engineering, Xi’an University of Technology, 5 South Jinhua Road, Xi’an, Shaanxi 710048, China

Received 24 July 2015; Revised 29 September 2015; Accepted 7 October 2015

Academic Editor: Anna Vila

Copyright © 2015 Zhibo Zhai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Cutting parameter optimization dramatically affects the production time, cost, profit rate, and the quality of the final products, in milling operations. Aiming to select the optimum machining parameters in multitool milling operations such as corner milling, face milling, pocket milling, and slot milling, this paper presents a novel version of TLBO, TLBO with dynamic assignment learning strategy (DATLBO), in which all the learners are divided into three categories based on their results in “Learner Phase”: good learners, moderate learners, and poor ones. Good learners are self-motivated and try to learn by themselves; each moderate learner uses a probabilistic approach to select one of good learners to learn; each poor learner also uses a probabilistic approach to select several moderate learners to learn. The CEC2005 contest benchmark problems are first used to illustrate the effectiveness of the proposed algorithm. Finally, the DATLBO algorithm is applied to a multitool milling process based on maximum profit rate criterion with five practical technological constraints. The unit time, unit cost, and profit rate from the Handbook (HB), Feasible Direction (FD) method, Genetic Algorithm (GA) method, five other TLBO variants, and DATLBO are compared, illustrating that the proposed approach is more effective than HB, FD, GA, and five other TLBO variants.

#### 1. Introduction

In modern manufacturing, determining optimal cutting parameters is of great importance to improve the quality of products, to reduce the machining costs, and to maximize the profit rate. The main cutting parameters in multitool milling operations include the feed per tooth, cutting velocity, and the radial and axial depths of cut. The conventional methods of selecting of cutting parameters mainly depend either on the operator experience or on machining data from handbooks. But it is a known fact that the cutting parameters obtained from these resources, in most cases, are extremely conservative. Consequently, it may not perform high productivity. So it is necessary to develop a new technique to investigate the cutting optimization problem.

There are many mathematical programming techniques to be used extensively for optimization of cutting parameter over the past few decades. In earlier studies, Gupta et al. [1] developed an integer programming for the determination of optimal subdivision of depth of cut in multipass turning with constraints. Subsequently, Wang et al. [2] used deterministic graphical programming to optimize machining parameters of cutting conditions for single pass turning operations. Shin and Joo [3] used a dynamic programming for the determination of optimum of machining conditions with practical constraints. Petropoulos [4] developed a geometric programming model for the selection optimal selection of machining rate variables.

Although these mathematical programming techniques have been applied to solve the cutting parameter optimization problem, these studies have not involved some important cutting constraints. Considering the number of constraints such as surface roughness, cutting force, cutting velocity, machining power, and tool life, cutting parameter optimization problem is very complicated. The additional variables due to number of passes make the solution procedure more complicated. These mathematical programming techniques incline to obtain local optima and may be only useful for a specific problem.

Recently, nontraditional optimization approaches recently have been developed to solve the cutting parameter optimization problem. Shunmugam et al. [5] used a Genetic Algorithm (GA) to optimize cutting parameters in multipass milling operations, exploiting total production cost as the objective function. Li et al. [6] developed a two-phase GA to optimize the spindle speed and feed and select the tools for drilling blind holes in parallel drilling operations to obtain the minimum completion time. Krimpenis and Vosniakos [7] used a GA to optimize rough milling for parts with sculptured surfaces and select process parameters such as federate, cutting speed, width of cut, raster pattern angle, spindle speed, and number of machining slices of variable thickness. Although GA has some advantages over traditional techniques, the successful application of GA depends on the population size and the diversity of individual solutions in the search space. If its diversity cannot be maintained before the global optimum is reached, it may prematurely converge to a local optimum. Liu and Wang [8] proposed a modified GA to optimize milling parameter selection. The operating domain is defined and changed to be around the optimal point in its evolutionary processes so that the convergence speed and accuracy are improved. Wang et al. [9] presented a parallel genetic simulated annealing to select optimal machining parameters for multipass milling operations. The Taguchi method was initially used to predict cutting parameter performance measures, and then the GA was utilized to optimize the cutting conditions. Subsequently, Öktem [10] discussed the utilization of Artificial Neural Network (ANN) and GA for predicting the best combinations of cutting parameters to provide the best surface roughness. Li et al. [11] suggested combining the ANN and GA to minimize the make-span in production scheduling problems. António et al. [12] used a GA based on an elitist strategy to minimize manufacturing costs of multipass cutting parameters in face milling operations. Zhou et al. [13] applied fuzzy particle swarm optimization algorithm (PSO) to select the machining parameters for milling operations. Zarei et al. [14] proposed a Harmony Search (HS) algorithm to define the optimum cutting parameters for a multipass face milling operation. Mahdavinejad et al. [15] developed a new hybrid optimization approach by combining the immune algorithm with ANN to predict the effect of milling parameters on the final surface roughness of Ti-6Al-4V work pieces. Briceno et al. [16] selected an ANN for modeling and simulating the milling process. Orthogonal design and specifically equally spaced dimensioning showed that ANN is a good method to define process parameters. Venkata Rao and Pawar [17] used an Artificial Bee Colony (ABC) algorithm to minimize production time of a multipass milling process to determine the optimal process parameters such as the number of passes, depth of cut for each pass, cutting velocity, and feed. Onwubolu [18] used a new optimization technique based on tribes to select the optimum machining parameters in multipass milling operations such as plain milling and face milling by simultaneously considering multipass rough machining and finish machining.

Although some improvements in optimizing machining parameters in milling operations have been made, these nontraditional optimization approaches require a lot of specific controlling parameters except for the common parameters such as number of generation and population size. For instance, the GA involves crossover and mutation probability. Similarly, the HS algorithm includes harmony memory considering rate, bandwidth rate, and a random select rate. These specific controlling parameters affect significantly the performance of the above mentioned algorithms. Improper parameters of algorithms either raise total complexity of consumption time or fall into the local optimum. So there remains a need for efficient and effective optimization algorithms for the cutting parameters determination.

Very recently, Rao et al. proposed a Teaching-Learning-Based Optimization (TLBO) [19] algorithm. This algorithm does not need specific controlling parameters except for the common parameters such as number of generation and population size. As a stochastic search strategy, it is a new algorithm based on swarm intelligence having the characteristics of rapid convergence, simple computation, and no specific controlling parameters except for the common parameters such as number of generation and population size. However, it has some undesirable dynamical properties that degrade its searching ability [20]. One of the most important issues is that the population tends to be trapped in the local optima solution because of diversity loss. To improve the performance of the original TLBO, a few modified or improved algorithms are proposed in recent years, such as teaching-learning-based optimization with dynamic group strategy (DGSTLBO) [20], teaching-learning-based optimization with neighborhood search (NSTLBO) [21], an elitist teaching-learning-based optimization algorithm (ETLBO) [22], and a variant of teaching-learning-based optimization algorithm with differential learning (DLTLBO) [23]. These modified TLBOs have better performance than the original TLBO on classical benchmark functions. Although the abovementioned variants TLBO have some improvements, they never focus on correct assignment problem. That is to say, each learner should be assured correct assignment of learning objects in the “Learner Phase.” To this aim, we present a novel version of TLBO, TLBO with dynamic assignment learning strategy (DATLBO), in which all the learners are divided into three categories in the “Learner Phase”: good learners, moderate learners, and poor ones. Good learners are self-motivated and try to learn by themselves; each moderate learner uses a probabilistic approach to select one of good learners to learn; each poor learner also uses a probabilistic approach to select several moderate learners to learn. The modification tries to both enable the diversity of the population to be preserved in order to discourage premature convergence and achieve balance between the explorative and exploitative tendencies of achieving better solution. A case study in multitool milling operations is used to verify DATLBO. The results are compared with results from GA [24], the feasible direction method [25], handbook recommendations [26], and five other TLBO variants.

The paper is organized as follows. Section 2 gives a short introduction to modeling of milling operations. Original TLBO algorithm and the proposed algorithm, DATLBO, are described in Section 3. This case study of multitool milling parameter optimization is presented in Section 4, and the summary and conclusions are given in Section 5. The last section provides the nomenclature.

#### 2. Modeling of Milling Operations

Milling is a machining process which uses rotary multiple tooth cutters to remove material from a work piece. Figure 1 displays two kinds of milling operations: end milling and face milling. As the cutter rotates, each tooth removes a small amount of material from the advancing work piece during each spindle revolution.