A dynamic machining model to optimize the control of material removal rate (MRR) for a cutting tool undergoing the considerations of fixed tool life and maximum machining rate is established in this paper. This study not only applies material removal rate mathematically into the objective function, but also implements Calculus of Variations to comprehensively optimize the control of material removal rate. In addition, the optimal solution for the dynamic machining model to gain the maximum profit is provided, and the decision criteria for selecting the optimal solution of the dynamic machining model are then recommended. Moreover, the computerized analyses to simulate both the dynamic and traditional machining models for a numerical example are also promoted. This study definitely contributes an applicable approach to the dynamic function of material removal rate and provides the efficient tool to concretely optimize the profit of a cutting tool for operation planning and control in modern machining industry with profound insight.

The cutting conditions of a cutting tool have been the most critical variables in the machining process. Cutting speed, feed rate, and depth of cut were considered as three factors of input cutting parameters. Liew and Ding [

In addition, Cámara et al. [

Moreover, the cost to machine each part is a function of the machining time, and the marginal cost of production is a linear function of production rate. Therefore, the marginal cost of operation for the machining process is also proposed to be a linear function of the material removal rate in this study. This explicates that more machining rate causes more operational cost such as machine maintenance, loading-unloading, and machine depreciation costs.

According to Zong et al. [

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

The cutting process is a continuous rough turning operation with one type of tool.

Each tool performs the same fixed length of cutting time (tool life).

The upper limit of material removal rate is generated from the maximum cutting conditions suggested in the handbook, and the fixed tool life is derived from the Taylor’s tool life equation [

The marginal cost of operation is a linear function of the material removal rate [

There is no chattering or scrapping of parts occuring during the machining process.

All chips from cutting and finished parts are held in the machine until a tool change.

All machined parts are paid at a given price immediately at the tool change.

One has the following:

One has the following

When considering the machining profit of an individual cutting tool under fixed tool life, the mathematical model representing the machining profit under dynamic

Since the marginal operation cost of production is a linear increasing function of production rate [

Whilst the marginal operation cost of production is a linear increasing function of production rate and the operational cost is directly proportional to the square of the production rate [

As all chips from cutting and finished products are usually held and stored at the machine until a tool replacement,

Let

Let

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

The optimal solution for

The detail is developed in Appendix

From (

This denotes that the overall holding cost of unit chip for the period

Applying the constraint

This represents that twice of the marginal cost at upper limit

Before solving the optimal solution for

If the line

The proof of the Property is illustrated in Appendix

The optimal solution for

The detail is developed in Appendix

From (

When

When

For a specific turning operation, there are bounds for cutting conditions suggested in the machining handbook. Therefore, there must exist a maximum material removal rate

Compute

Go to

Plot

If

Otherwise, go to Step

Compute the profit

Go to Step

Compute

Go to Step

Plot

If

Otherwise, set

With a machining project from AirTAC Corporation in Taipei, Taiwan, the numerical example referenced to a rough turning operation of specific cylindrical parts with step diameters is studied. The machining operation is assigned to a

The simulated results of the MATLAB program are presented as the profit analysis and production quantity analysis in Figures

From Figure

With the analyses of the dynamic model above, when there is a need for the cutting tool to acquire other productivity than the maximum profit solution, it is always feasible to slightly compromise on the profit for the required productivity. This will fractionally change the profit, but satisfactorily increases the productivity for the dynamic machining model.

The tool life, operational cost, holding costs, contribution per unit part machined, average volume of material machined per unit part, and upper speed limit are considered simultaneously to dynamically optimize the control of material removal rate. It is an extremely hard-solving and complicated issue. However, through the dynamic machining model, the problem becomes concrete and solvable.

In addition, four characteristics of the dynamic machining model are illustrated as follows. First, the optimal material removal rate

Moreover, from the computerized analyses and numerical simulation with the MATLAB program, the four remarks are then provided. First, the dynamic model receives much more balanced profit than the traditional model within the whole range of allowable machining rate. Second, the dynamic machining model is much more profitable than the traditional machining model for the higher range of machining rate. Third, when the machining rate increases, the difference of production quantity between two models will decrease. Fourth, when there is a need to obtain higher productivity than the optimal solution, it is always feasible for the dynamic machining model to slightly compromise on the profit in achieving the productivity. With these remarks above, the applicability and flexibility of the dynamic machining model are significantly expended.

The material removal rate is an important control factor of machining operation, and the control of machining rate is also critical for production planning. This study not only delivers the idea of controlling the material removal rate to the machining technology, but also leads a cutting tool towards to achieve maximum profit. This study focuses on the modeling and optimization of the

The optimal solution for

Suppose that the material removal rate

From the Euler Equation [

There is a constant

With the transversality condition of salvage value for free end value

Substituting (

The proof of Property.

From (

Possible condition of lines

Profit analysis of a cutting tool with different models.

Production quantity analysis of a cutting tool with different models.

The optimal solution for

It is assumed that the material removal rate

The objective function of the dynamic machining model is then modified as

With the transversality condition of salvage value for free end point

Thus, with

The author would like to thank the anonymous referees who kindly provided the suggestions and comments to improve this paper.