Journal of Applied Sciences

Year: 2008 | Volume: 8 | Issue: 14 | Page No.: 2606-2612
DOI: 10.3923/jas.2008.2606.2612

Computerized Analysis of Discrete MRR Optimization for Constrained Due-Date

Tian- Syung Lan

Abstract: This study not only applies Material Removal Rate (MRR) into the objective function mathematically, but also implements calculus of variations to resolve the dynamic machining control problem comprehensively. In addition, the optimal solution of the Machining Project Control (MPC), model is proposed and the decision criteria to determine the optimal solution are also recommended. Moreover, the computerized analyses with a numerical simulation to compare with the traditional machining model are fully prepared. This study definitely contributes the applicable approach to dynamic control of the material removal rate and provides the efficient tool to concretely optimize the cost of a machining project for operation research engineers in today`s machining industry with profound insight.

Keywords: material removal rate, calculus of variation, operations research and Optimal control

INTRODUCTION

Machining conditions of a cutting tool have been the most critical variables in machining operation. The cutting speed, feed rate and depth of cut were considered as three factors of input cutting parameters (Lan et al., 2002, 2008). To calculate the optimum cutting conditions is the objective for production (Meng et al., 2000). Koren et al. (1991) have described several methods to be used under stepwise constant variation in feed, speed, or depth of cut, but none is practically applicable when two or more cutting conditions are changed. Therefore, the method of controlling cutting conditions with fixed material removal rate has been introduced (Balazinski and Ennajimi, 1984; Choudhury and Appa Rao, 1999). In most studies on this viewpoint, the material removal rate is fixed because of the expensive observation of the control. Nevertheless, through the computer-integrated interface to program the machining feed rate on modern computer numerical controlled (CNC) machines with fixed cutting speed and depth of cut, the material removal rate is capable of being dynamically controlled (Balazinski and Songmene, 1995; Lan et al., 2008).

In addition, the tool life is a critical parameter of the machining process (Davim and Conceicao Antonio, 2001; Lan et al., 2008). Novak and Wilkund (1996) proposed a suitable implementation to predict tool life and Lee et al. (1992) proposed a method of optimal control to ensure maximum tool life. Meng et al. (2000) also provided a modified Taylor tool life equation to minimize tool cost. As a matter of fact, the maximum tool life or the minimum tool cost will not guarantee the minimal cost of a machining operation. Besides, the various tool checking periods for a tool change from different operators will decrease the productivity and increase the cost during the machining project significantly. In order to manage the consumption of tools well, a fixed tool life is practically considered into the machining project. However, the production period is certainly related to the order quantity of a project. For the convenience of project scheduling, the production period is also proposed to be determinable and then introduced as fixed into the study.

Moreover, the cost to machine each part is a function of the machining time (Jung and Ahluwalia, 1995). As the marginal cost of production is a linear increasing function of production rate (Kamien and Schwartz, 1991; Lan et al., 2008) the marginal cost of machining operation is also considered to be a linear function of the material removal rate in this study. It is that the higher machining rate results higher operational cost such as machine maintenance, loading-unloading and machine depreciation costs.

Although, several time series modeling on the control of machining process are mentioned (Kim et al., 1996; Yeh and Lan, 2002), none is capable to achieve minimum cost. They are all emphasizing on the maximal tool usage or minimal tool cost. Actually, the production cost and the production period of a machining project are mostly concerned problems confronting the manufacturing industry. Besides, the need of operating CNC machines efficiently to obtain the required payback is even greater in the case of rough machining, since a greater amount of material is removed thus increasing possible savings (Meng et al., 2000). With the reasons above, there is an economic need to control the material removal rate of rough machining operation for a machining project. Hence, the necessity of finding the optimal solution to reach the minimum cost of a machining project with fixed tool life and production period is absolutely arising.

Before formulating the problem, several assumptions and notations are to be made. They are described as follows:

Comparing to the productivity of traditional machining model (fixed MRR), it is necessary for MPC Model to competitively satisfy L <=x( )<=B . In addition, for the machining quality of all products, the lower limit of material removal rate is applied to determine the number of tools required for the production project. Therefore, with A = aQ/L and T = A, the tools required for production and the production period for project scheduling is then achieved and proposed as given for the model.

In this study,

and

denote the machining cost during for a fully consumed tool and during for a partially consumed tool, respectively. Thus, the objective function for the machining project is constructed as below

Set c = h₁ + h₂/a as the overall holding cost per unit chip per unit time. Therefore, the MPC Model and its constraints are formulated and described as below.

Let (x*,y*) be the optimal solution of MPC Model and (t_x*, t_y*) be the optimal queuing time for a fully and a partially consumed tools, respectively. Assume that the time interval are the maximal subintervals of [0, ] and [0Z ], respectively to satisfy Euler Equation (Kamien and Schwartz, 1991; Chiang, 1992).

There are two possible situations to be discussed in this study.

Situation 1: x*´(t) (y*´(t)) will not touch B before (Z )

The optimal solution for Situation 1 is shown as follows:

The detail is described in Appendix A.

Here, one PROPERTY is proposed and shown as follow:

PROPERTY: If the line Y = x*´(t) (Y = y*´(t)) touches the line Y = B, two lines should overlap to be Y = B from the touch point

The proof of PROPERTY is discussed in Appendix B.

Situation 2: x*´(t) (y*´(t)) touches B at time before

The optimal solution for Situation 2 is shown as follows:

The detail is described in Appendix C.

Decision criteria: From Eq. 1 and 3, the maximum values of x*´(t) and x*(t) are found at t = when t_x* = 0, the following criteria are made.

If aQ/A <= B /2, x*´(t), will not reach the upper limit B before .

If aQ/A > B /2, x*´(t), will reach the upper limit B before .

In addition, from Eq. 2 and 4, the maximum value of y*´(t) and y*(t) are found at t = Z when t_y* = 0 and the criteria are then obtained as below.

If aQ/A <= B /2, y*´(t), will not reach the upper limit B before Z .

If aQ/A > B /2, y*´(t), will reach the upper limit B before Z .

It is noticed that the criteria for x*´(t) and y*´(t) are exactly identical. This denotes that x*´(t) and y*´(t) will both either never reach B before (Z ), or touch B before (Z ). Therefore, when aQ/A<=B /2, both x*´(t) and y*´(t) will not reach the upper limit B; the optimal solution is Situation 1. When aQ/A > B /2, both x*´(t) and y*´(t) will reach the upper limit B; the optimal solution is Situation 2.

For a specific turning operation, there are ranges for cutting conditions suggested in the machining handbook. Therefore, there must exist a maximum material removal rate U and a minimum material removal rate L derived from the maximum and the minimum cutting conditions respectively. Each material removal rate between U and L can feasibly be selected as the upper limit B. From the well-known Taylor`s expression of the tool life (DeGarmo et al., 1997), it is then modified to be Bx ⁿ = k when the cutting speed and depth of cut are selected fixed. Therefore, the fixed tool life for each feasible upper limit is then obtained. However, for every selected feasible upper limit, there exist a lowest cost for the MPC Model and the traditional machining model with a fixed MRR between U and L is also possible to be optimal. To find the minimum cost solution among all the possibilities, a computer program written in MATLAB to analyze the problem is then developed. The concept of the flow chart is described as follows:

Q, a, b, c, U, L, n, k, c₁ and c_s should be given before the following algorithm

Initialize B = L

Step 1: Compute , A, Z, T for MPC,
then compute A_m = aQ/B , Z_m, T_m for traditional.

Step 2: Compute

Step 3: Plot and P(B,T_m), then go to Step 4.

Step 4: If aQ/A>B /2, go to Step 6; otherwise go to Step 5.

Step 5: Compute t_x, t_y; then compute

Go to Step 7.

Step 6: Compute; then compute

Go to Step 7.

Step 7: Plot , then go to Step 8.

Step 8: If B=>U, stop the program;

otherwise, set B = B+0.025, as initialized, return to Step 1.

The example referred to a rough turning operation of specific shafts from a machining company in Taipei is studied. The machining process is assigned to a CNC lathe with FANUC controller. All data provided are converted and listed as follows:

Q = 1000 parts, a = 3.375 in³, b = 0.150 dollars-min/in³, c = 0.100 dollars/in³, U = 10.0 in³/min, L = 6.0 in³/min, n = 0.2, k = 12.0, c₁ = 0.350 dollars/min and c_s = 12.0 dollars.

From Fig. 1, it is observable that the MPC Model is superior and less costly than the traditional machining model for the whole allowable MRR range. In addition, as the selected upper limit B increases, the production cost will decrease with the MPC Model, while it increases for the traditional model. Therefore, the MPC Model is the optimal solution for production and the maximum MRR generated from machining handbook is the optimal upper limit B for the minimum cost. Moreover, with the three different production quantities in Fig. 1, it is noted that the production cost per unit product of the MPC Model will slightly decrease for each feasible machining speed selected, while the cost per product for the traditional model stays. Thus, when the production quantity increases, the MPC Model is much more competitive in minimizing the production cost.

From Fig. 2, it is also observed that the production period for the MPC Model is longer than the traditional model for the whole range. Because that the fixed tool life is derived from the maximum cutting conditions suggested in the handbook and the number of tools required is generated from the minimum cutting conditions suggested in the handbook; the production period for the MPC Model is then less competitive with this aspect. However, when there is a need for the production period to be shorter than proposed in the model, the production period for the MPC Model can always be possibly considered within aQ/L<=T<=aQ/B. Thus, the required tools for production will become reduced and the tool cost for production will also be minimized. Besides, this will neither change the optimal solution nor the competition of the MPC Model, but fortunately increase the flexibility in production period for the MPC Model.

The tool life, tool cost, operational cost, holding cost, production period, production quantity, average material removal per unit part machined and upper limit are considered simultaneously to determine the optimal control of material removal rate and queuing time for the machining project. This is an extremely hard-solving and complicated issue. However, the problem becomes concrete and solvable through the MPC Model.

In addition, the three characteristics from the optimal solution of MPC Model are illustrated as follows: First, from the optimal solution of material removal rates, x*´(t) and y*´(t) are strictly increasing linear functions of t before reaching upper limit B. Second, by PROPERTY described before, if the material removal rate, x*´(t) or y*´(t), touches the upper speed limit B; the optimal material removal rate will stay to be upper limit B. Third, with the maximum values of x*´(t), y*´(t), x*(t) and y*(t); it is found that x*´(t) and y*´(t) will both either never reach B before (Z ), or touch B before (Z ).

Moreover, from the computerized analyses and numerical simulation with the MATLAB program, the three remarks are then provided. First, the MPC Model is the optimal solution for a machining project and the maximum allowable MRR from the handbook is the optimal upper limit B for the MPC Model. Second, when the production quantity increases, the MPC Model is much more competitive in minimizing the production cost. Third, when it is acquired for the production period to be shorter than proposed, the production period for the MPC Model is always possible to be scheduled within aQ/L<=T<=aQ/B. With these remarks above, the application flexibility for the MPC Model is significantly extended. Thus, the production planning, production cost estimating and even the contract negotiation can be further approached with this study.

The material removal rate is an important control factor of a machining project and the control of machining rate is also critical for production planners. This study not only delivers the idea of controlling the material removal rate to the machining technology, but also leads a machining project towards to achieve minimal cost. Future researches with the modeling of dynamic optimization on multi-tool machining process and multi-project production control are absolutely encouraged. In sum, this study surely generates a reliable and applicable concept of machining control to the techniques and also provides a better and practical solution to this field.

Appendix A: The optimal solution for Situation 1

Suppose that the material removal rate x*´(t) (y*´(t)) will never reach the upper limit B before tool life . From Euler Equation (Kamien and Schwartz, 1991; Chiang, 1992), it is derived that

There exists k₁ and to satisfy

Integrating Eq. A1 and A2 with t, it is obtained that

With the transversality conditions for free t_x and t_y (Kamien and Schwartz, 1991; Chiang, 1992), then

Using Eq. A5, A6 and the boundary conditions, x(t_x) = 0 and y(t_y) = 0, it is derived that

From Eq. A1 and A2, it is then found

Using Eq. A3, A4, A7 and A8, x(t_x*) and y(t_y*) = 0; we have

Applying Eq. A7, A8, A9 and A10 into Eq. A1, A2, A3 and A4; x*´(t), y*´(t), x*(t) and y*(t) are then obtained.

With the boundary conditions, x( ) = aQ/A and y(Z ) = Z aQ/A, hence t_x* and t_y* are derived.

Appendix B: The proof of PROPERTY.

Proof: From Eq. 1, x*´(t) is a strictly increasing linear function of t. And it holds for any subinterval during [0, ] satisfying 0<=x*´(t)<=B. Therefore, x*´(t) in the time interval cannot exist because it contradicts to be a decreasing linear function of t, the PROPERTY of x*´(t) is verified. In addition, the PROPERTY of y*´(t) can also be derived with the same proof.

Appendix C: The optimal solution for Situation 2.

Before touching the upper limit, Eq. 1, 2, 3 and 4 are satisfied for this situation either.

Using the transversality condition for free end point (Kamien and Schwartz, 1991; Chiang, 1992), it is derived that

With Eq. 3, 4, C1, C2 and PROPERTY; we have

In addition, from boundary conditions,

and PROPERTY, it is found that

By Eq. C3, C4, C5 and C6, t_x*, t_y*, can be determined.

From Eq. 3, 4, PROPERTY, x( ) = aQ/A and y(Z ) = Z aQ/A; x*(t) and y*(t) are then obtained.

REFERENCES