Int J Adv Manuf Technol (2013) 68:1733–1744 DOI 10.1007/s00170-013-4971-4
ORIGINAL ARTICLE
Optimization of sequential subdivision of depth of cut in turning operations using dynamic programming Kaibo Lu & Minqing Jing & Xiaoli Zhang & Heng Liu
Received: 26 June 2012 / Accepted: 2 April 2013 / Published online: 20 April 2013 # Springer-Verlag London 2013
Abstract The cutting sequence in the optimization of multipass turning operations has not gained much attention in many previous studies. The objective of this paper is to present a novel method to determine the optimal sequence of cutting passes and machining parameters in turning operations with practical constraints. The optimization problem of minimizing the total production cost is solved in two phases. The first phase is to achieve the minimum production cost for each cutting pass for the predefined depths of cut. A hybrid solver which combines a genetic algorithm and sequential quadratic programming technique is employed to accomplish this step. In the second phase, a dynamic programming technique is introduced to obtain the optimal sequential subdivision of the total depth of cut. Examples interpret the proposed procedure in detail. The results have proved this proposed methodology effective and of generality in comparison with the prior works.
dmax, dmin
Keywords Optimization . Multipass turning . Cutting sequence . Dynamic programming
N p Pmax r
Nomenclature C Coefficient for Taylor’s life equation Cl Loading and unloading job cost per unit piece (in dollar per unit) CT Total production cost per unit piece (in dollar per unit) d Allowable depth of cut for a single machining pass (in millimeter) dt Total depth of cut (in millimeter) K. Lu : M. Jing (*) : H. Liu School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China e-mail:
[email protected] K. Lu : X. Zhang Division of Engineering and Physics, Wilkes University, Wilkes-Barre, PA 18766, USA
D0 D1 f fmax, fmin Fmax K i, j, k k0 kt L m, q, s
rt Rmax tc tl tm ts T Tmax, Tmin v vmax, vmin α, β
Maximum and minimum depth of cut (in millimeter) Initial workpiece diameter (in millimeter) Final workpiece diameter (in millimeter) Feedrate of the tool (in millimeter per revolution) Maximum and minimum feedrate of the tool (in millimeter per revolution) Maximum tangential cutting force (in kilogram force) Cutting force coefficient (in kilogram force per millimeterα+β) Index numbers Labor cost and overhead (in dollar per minute) Cost of a cutting edge (in dollar per edge) Length of machining (in millimeter) Exponents of speed, feed, and depth of cut for tool life equation Number of rough machining passes Production cost for a cutting pass (in minute) Maximum power (in kilowatt) Return or minimum production cost for a given total depth of cut (in minute) Nose radius of tool (in millimeter) Peak-to-valley height of surface roughness for finishing pass Tool replacement time (in minute) Loading and unloading job time per unit piece (in minute per unit) Actual machining time (in minute) Tool setting time per pass (in minute) Tool life (in minute) Maximum and minimum tool life (in minute) Cutting speed (in meter per minute) Maximum and minimum cutting speed (in meter per minute) Coefficients used in the empirical equation of cutting force
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η Δd Ω
Int J Adv Manuf Technol (2013) 68:1733–1744
Machine tool transmission efficiency Common difference of series of allowable depths of cut (in millimeter) Rotational speed of spindle (in revolution per minute)
Subscripts R Corresponding to the rough machining F Corresponding to the finish machining Superscript * Referring to the optimal value
1 Introduction Turning belongs to the class of single-point machining operations where a workpiece is held and rotated about its longitudinal axis on a lathe. The ultimate purpose of a machining operation is to produce parts that meet all the required specifications (dimensional tolerance, surface characteristics, physical property requirement, etc.) at the lowest possible cost or in the minimum possible time [1]. To fulfill such an objective, the optimum selection of operating conditions becomes a necessary consideration. Machining optimization problems have long been a topic of interest for many researchers. In the earlier works, the optimization of single-pass turning has been investigated. However, multipass turning operations comprising a number of roughing passes and a finishing pass are more widely conducted in industrial applications. Compared with the single-pass turning optimization, the multipass turning optimization problem has an extra decision variable viz. the number of cutting passes, to be considered. The introduction of this stochastic variability makes the problem much more complex. In the case of multipass turning, researchers have generally used traditional mathematical programming techniques [2–6], probabilistic or heuristic methods [7–14], and hybrid approaches [15–19] to optimize the machining conditions to satisfy an economic objective with practical machining constraints. References [20, 21] reviewed on the art of machining optimization techniques comprehensively. Recently, Chandrasekaran et al. [22] explored cloud computing techniques for the optimization of machining operations. Hinduja et al. [2] searched for the optimum cutting parameters for each pass from a confined feedrate versus depth of cut plane. This search is repeated until the sum of the optimum depths of cut equals or exceeds the total depth or stock to be machined. If the sum is greater than the given one, four correction algorithms could be chose to modify the optimum depths of cut. Shin and Joo [3] employed a Fibonacci search method to subdivide the total depth of cut to minimize the total production cost. The optimization of the multipass turning
processes was decomposed into two separate optimization subproblems for rough and finish cutting operations. The optimization of the finishing operation was calculated prior to that of the roughing operation and an equal depth of cut strategy for per roughing pass was adopted. Based on Shin and Joo’s mathematical model, Gupta et al. [6] introduced an updated method to pursue the optimization, which could yield better results than those found by Shin and Joo’s method. First, the separate minimum production costs for the individual roughing and finishing passes were determined and tabulated for various fixed allowable values of depth of cut. Secondly, an integer programming technique was utilized to allocate the optimal depths of cut for multipass turning operations. Rather than using integer programming, Bhaskara et al. [7, 23] chose the genetic algorithm (GA) to acquire the optimal combination of depths of cut in multipass turning and milling processes. Satishkumar et al. [11] used nontraditional optimization techniques like genetic algorithms, simulated annealing, and ant colony optimization to optimize the multipass turning operations, respectively. It was proved that the nontraditional methods produced slightly more optimal objective values than that in [6]. Abburi and Dixit [15] proposed an optimization scheme of the combination of a real-parameter genetic algorithm and a sequential quadratic programming (SQP) to minimize the production cost, which gave better solutions by comparison with the results shown in [3, 6, 11]. Equal depths of cut for roughing passes were adopted as well in this method. Yildiz [16–19] developed several new hybrid algorithms for solving machining optimization problems. A significant amount of effort has been done on the optimization of turning processes. It should be noted that the job diameter before a cutting pass is associated with the objective function in multipass turning optimization problems and is dependent on the conducted depths of cut in the previous passes. However, a constant diameter was carried out in solving the optimization problem in many works. In addition, to the best of the authors’ knowledge, very little research work has highlighted the sequence issue of the subdivision of the depth of cut in turning processes. In certain cases the strategy of equal depth of cut for rough machining could not be reasonable. From the practical point of view the depth of cut has to be reduced during the last cutting passes to avoid the large deformation of the workpiece or chatter vibrations [2]. This phenomenon, especially, exists in multipass turning of long slender workpieces [24]. It is worth pointing out, however, that the cutting sequence in certain machining processes, such as in milling, generally has no effect on the economic objective values since the cutting mechanism of milling operations is different from turning operations [23, 25, 26]. In the multipass turning operations, the specific machining parameters should be configured for each machining pass in computer numerical control in advance. Dynamic programming (DP), developed by Richard Bellman in the early
Int J Adv Manuf Technol (2013) 68:1733–1744
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1950s, is a mathematical technique well suited for the optimization of multistage decision problems [27]. This technique is an implicit enumeration approach that can formulate a recursive procedure to improve the computational efficiency. Agapiou [4, 5] employed a DP technique to obtain the number of machining passes and the Nelder–Mead simplex method to determine the optimal machining parameters in turning processes. According to Agapiou’s method, Sonmez et al. [25] investigated the milling operation optimization problem. In this study, following the production cost model of Abburi and Dixit [15] we present a more general method for determining the optimal allocation of the total depth of material to be removed in turning operations, as well as the sequence of the machining passes. The minimum production cost per unit piece is obtained in two phases. In the first phase, the minimum production cost for each cutting pass is calculated and tabulated for various equally spaced fixed depths of cut. This step is accomplished by a hybrid solver of combining GA and SQP. In the second phase, a DP technique quite different from the one used in [5] is carried out to ascertain the optimal sequential depths of cut for cutting passes, the number of cutting passes, as well as the minimization of the production cost per unit piece. This paper is organized in six sections. In Section 2, the mathematical model for the optimization problem of turning processes is derived. Section 3 interprets the proposed methodology. In Section 4, numerical examples and the results are presented to evaluate the performance of this approach. The experimental results are discussed afterwards in Section 5. Lastly, Section 6 concludes this work.
2.1 Objective function Consider a workpiece with the initial diameter D0 is to be machined to be a rod with the diameter D1 after multipass turning processes, as shown in Fig. 1. As mentioned earlier, the minimum production cost is explored as the optimized objective. The production cost per unit piece CT consists of three items: N X
pR ¼ k0 tm þ k0
tm tm tc þ kt þ k0 t s TR TR
ð2Þ
where the first term on the right side is the machining cost by the actual time in cut; the second term is the tool replacement cost; the third term is the tool cost; and the last one is the machine idle cost due to setting the tool or idling tool motion. The actual machining time tm can be represented by tm ¼
L Ωf
ð3Þ
where L is the length of machining, f is the feedrate. Since the rotational speed of the workpiece Ω is related to the cutting speed v by the equation Ω¼
1000v pD
ð4Þ
where D is the value of the workpiece diameter before a pass taking place. Obviously, this value is decreased progressively, as seen in Fig. 1. Substituting Eq. (4) into Eq. (3) yields tm ¼
pDL 1000vf
ð5Þ
Equation (2), therefore, can be expressed as pLD tc kt pR ¼ k0 þ k0 þ þ k0 ts 1000vf TR TR
2 Problem formulation
CT ¼
where N is the number of rough cutting passes, pR is the production cost for a single roughing pass, pF is the production cost for the finishing pass, and the last term Cl is the loading and unloading job cost. For an individual roughing pass the production cost pR can be represented by
pRi þpF þ Cl
ð1Þ
i¼1
ð6Þ
Similarly, the production cost for a finish machining pass pF can be expressed as pLD tc kt k0 þ k0 þ ð7Þ pF ¼ þ k0 t s 1000vf TF TF The loading and unloading job cost Cl is given by Cl ¼ k0 tl
ð8Þ
Fig. 1 Diagram of turning process
L
Workpiece
Chuck
D1
D Tool
f
D0
Live center
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And, the tool life is estimated by Taylor’s tool life equation as T¼
C vm f q d s
ð9Þ
where d is the depth of cut, and C, m, q, s are constants. 2.2 Constraints In most previous studies, four decision variables consisting of the depth of cut, cutting speed, feedrate, and number of cutting passes are only considered. In this formulated optimization problem, besides these four variables, the sequence of cutting passes is also investigated. These five variables are imposed by the following machining constraints, which are listed as bellow. Tool life constraint: Tmin T Tmax
ð10Þ
Cutting force constraints: KfRa dRb Fmax KfFa dFb Fmax
ð11Þ
Machine power constraint: KfRa dRb vR 6120η KfFa dFb vF 6120η
Pmax Pmax
ð12Þ
Geometric relation: N X
2dRi þ 2dF ¼ D0 D1
ð13Þ
i¼1
Cutting parameters bounds: vmin vR ; vF vmax fmin fR ; fF fmax dmin dR ; dF dmax
ð14Þ
Surface roughness constraint for finishing: fF2 Rmax 8rt
technique, based on the concept of suboptimization and Bellman’s principle of optimality, is suitable for the optimization of multistage decision problems. This technique decomposes the original multistage optimization problem into subproblems that also exhibit optimal properties and then solves these subproblems sequentially and recursively. The principle of optimality states that an optimal policy (or a set of decisions) has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the initial decision [27, 28]. It describes the phenomenon which enables problems to be viewed as a sequence of succession of decision problems, each one building on the last, until the problem is solved. As interpreted in Section 2, the objective function is related to the job diameter at the beginning of each pass, which is a descending variable. Hence, it is extremely complex to treat the decision variable of depth of cut as a continuous state variable in DP solving which often yields nonlinear partial differential equations to solve [27, 28]. Therefore, we define that the depth of cut is only allowed to be from a finite and discrete set of equally spaced values with a common difference Δd in this study, which differs from that the total depth was divided into equal sections in [5]. For this reason, this definition produces a rather distinguishing DP procedure from the one processed in [5], and may be more simplified and comprehensible. For the convenience of demonstration in following, the allowable depths of cut are supposed to be consecutive integers in this section. A path graph of dynamic configuration process of multipass turning operations is shown in Fig. 2, where point j corresponds to the jth allowable depth of cut, N is the total number of rough machining passes. Each path in Fig. 2 corresponds to a machining cost pj (i, J), which is the optimal production cost for the ith current cutting pass by using the jth allowable depth of cut to machine the rod which was machined with the Jth depth of cut in the previous pass. If the optimal solution to a total depth of cut n divides into N+1 steps j1, j2,…jN+1, such that n=j1 +j2 + … + p
1
ð15Þ p
1
2
ð16Þ
The meanings of the variables are as outlined in the Nomenclature section at the beginning of this paper.
Start
p
j
p
(2,1)
1
1
2
2
p
2
1
2 End
(1,0)
p
j
(2, j )
j
j
j p
j
3 Proposed methodology 0
(N+ 1,1)
(1,0)
j
To solve the formulated optimization problem, the cutting parameters should be specified for each cutting pass. The DP
1
(1,0)
Variable type: N is an integer:
1
1
2
N
( N+ 1, j )
N+1
Fig. 2 Dynamic configuration of multipass turning operations based on DP
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jN+1, then the minimization of the production cost or the return rn for the total depth n is
4 Examples and results
rn ¼ pj1 þ pj2 þ þ pjN þ1
The multipass turning example provided by Shin and Joo [3] is reconsidered for the purpose of illustration and comparison. Table 1 gives the data of the problem. As stated above, a finite set of depths of cut is defined for selecting. For both rough and finish machining, the allowable values of depth of cut distribute in the set with equal interval Δd equal to 0.5 mm as follows,
ð17Þ
Suppose that the optimal solution makes the first optimal cut to be depth k with a return of rk, then the optimal solution consists of an optimal solution to the remaining depth n−k with a return of rn−k, plus the first depth of cut k. Hence the optimal production cost rn can be obtained in terms of either the production cost of the entire depth n for only a cutting pass or the sum of two smaller depths of cut k and n−k. It is written in the following recursive equation,
dR ; dF 2 ½ 1; 1:5; 2; 2:5; 3 4.1 Phase 1
rn ¼ min ðpn ; rk þ rnk Þ 1k
