Parameter values for lot size and quality level via CAE simulation,statistical method,and mathematical programming

Abstract

Because of the stochastic nature of production systems, it is necessary to first build an uncertainty model for subsequent real applications. Moreover, process parameter planning, quality design, and production inventory management are interdependent elements. In this research, a computer simulation model via computer-aided engineering (CAE) was developed to determine the optimal process parameters, lot size, and back order intervals for an integrated process design and inventory management system with simultaneous quality and cost considerations. Based on the estimated process time and costs obtained using CAE, the derived production rate and unit cost were then used for production inventory applications. In consideration of the uncertainty factor, the response surface method (RSM) was employed to analyze the output, namely the total costs incurred in employing the proposed approach, as well as the inputs, which include the cutting parameters, production quantity, and back order intervals. After the RSM was used to obtain the response functions, which represent the output of the collective interests, the mathematical programming (MP) was formulated based on the response functions to determine the optimal process parameters, process quality levels, production order quantities, and back order intervals. The total cost per set time unit was minimized by determining the required quality level, process parameter values, Economic Production Quantity (EPQ), and back order intervals. A cutting example was chosen to demonstrate the proposed approach. Two cases were used for comparison: the Integrated Case (the proposed approach herein) and the Disintegrated Case.

Keywords

Product and process value quality level production order quantity RSM CAE

Introduction

When a high quality product on the market has a low price, it is in a superior competitive state. The first step of developing a quality product is to understand customers’ needs. “Process design for quality” has become a popular practice in the business world, as it makes the product customer-driven and incorporates customers’ needs into in the production process. In the entire product process, production-inventory management, which involves a great deal of capital and influences the delivery time of goods, is often overlooked. As is known, quality levels and production costs are both directly related to product unit cost, which is further linked to the production quantity decision during inventory management. Paying additional attention to only process and quality design is not enough to ensure the economical attributes and quality of the product. Therefore, it is necessary to incorporate inventory management into the process and quality design.^1–4 For the purpose of demonstration in this research, machining operation with controllable process parameters, processing times, unit cost, and quality levels were taken to characterize the process and quality design. Economic Production Quantity (EPQ) with back order problems for inventory management was chosen as an example.

Well-designed process parameters can appreciably influence production costs and quality levels. Cutting operations, for example, have a major impact on the production of economically sustainable and quality goods in most industrial sectors. In order to minimize the cost of each produced workpiece, process parameter determination must be able to reduce production time and cost to the lowest-levels.^5–12 Most of these studies focus solely on production cost or process time minimization during the manufacturing process. However, production cost, process time, and quality level simultaneously influence product unit cost, successively impacting inventory carrying cost. These effects further determine production quantity, which then become turn-back influences on the determination of process parameters and quality levels related to process design. The turn-back influences mainly result from invalid process planning based on an unsound production quantity resulting from the lack of consideration of successive impacts. Under these successive or chain relationships, the cumulative influences become even greater under the combined impact of uncertain inventory cost and demand rate, which may arise in later production inventory management, with uncertain production cost and time estimation unconsidered in the earlier process design. Thus, the assumptions of invariable costs and time in traditional studies on the lot sizing problem are no longer valid under the chain influences.¹³ In this regard, the proposed approach takes these chain relationships into account by integrating product/process design and production inventory management under uncertain impacts.

Due to the uncertainties that arise during the product/process design and inventory management, robust modeling for the process design and production quantity ordering should be considered simultaneously during the application. The uncertainties involved in the inventory system result primarily from variations in unit cost, demand rate, production rate, inventory carrying rate, ordering cost, and backorder cost.^14–19 The variations in unit cost and production rate stem directly or indirectly from uncertainties that arise during the process, such as changes in the raw material cost, labor cost, overhead cost, set-up time, machine cost, replacement cost, product-process parameter determination, quality level establishment, and other production-related aspects. One of the robust methods used when confronting uncertainty is statistical analysis within the experimental design approach. With uncertainty as a consideration, this experimental design approach aims to build a functional connection between input and output, as well as to determine the factors that extensively affect the stability of output, such as quality and cost of a product/process, and production quantity of inventory management. The functions identified in the previous step are then used for further applications during system analysis and optimization. Using these common aspects, a few researchers have used regression analysis to form a functional relationship that can then be used to decide the best values of independent variables in the system of interest. Jacobs and Wagner²⁰ employ regression analysis for uncertain demand so that inventory costs can be lowered. Arreola-Risa et al.²¹ use a simulation technique to build a regression equation for stochastic production-inventory applications. The best values of decision variables are determined on the basis of regression equations. The present study adopted an experimental design approach called response surface methodology (RSM) to regress inputs and outputs under uncertain impacts.^22–24 RSM can use a factor design to obtain robust and effective information with less experiment cost and time. The interaction between factors is studied via analysis and adaptation patterns, and the extent to which multiple factors affect the response variables is discussed. At the same time, a mathematical model of the relationship between the simulated independent variables and the response value can be obtained, which significantly reduces the number of experiments and time. The inputs are the controllable variables, such as process parameters, production order quantity, and backorder periods, which all need to be determined. The outputs can be both the unit cost and total cost. The outputs are dependent on process parameters, material cost, setup cost, and tool cost. The total cost includes inventory cost, setup cost, and backorder cost. The uncertain impacts are considered as the noises occurring during the statistical analysis. The noises influencing tool replacement costs include: labor cost, overhead cost, set-up time, tool cost, and tool changing time. Other noise, such as raw material cost, also influences unit cost. The noises affecting production order quantity and backorder periods include: ordering cost, production rate, carrying rate, set-up cost, demand rate, and backorder cost.

It is normally difficult to determine precise unit cost in the early stage of the process and product design. However, owing to current developments in computer-aided design (CAD) software and in cooperation with robust design experiments for statistical analysis, design engineers are able to study unit cost-related process design problems in advance.²⁵ In this study, a cutting example was chosen to demonstrate the proposed approach, using computer-aided engineering (CAE) software such as CATIA (Computer Aided Three-dimensional Interactive Application) and CCS (Computer Cutting Data Service). Because of the convenience of unit cost estimation with CAE software, an extension with estimated unit cost, which includes the production-inventory system as an integrated system, is now possible. In this integrated system, input includes the cutting parameters from the process design system and production quantity, in addition to a backorder production period from the production-inventory system. Output of an integrated system refers to total cost, representing the measurement score. This functional relationship is used as an objective function in the mathematical programming (MP) for attaining optimization.^26,27 Consequently, with the aid received from CAE software, this study was able to identify the means of building a computer model that links systems, such as process parameter planning, quality level, design and production inventory management for robustness and optimization, with uncertainty as a factor of consideration.

The remainder of this paper is organized as follows. Section 2 presents the research background. Section 3 provides the proposed computer model for the present research. Section 4 offers an example to demonstrate the proposed approach. Section 5 concludes with a discussion and summary of the key findings.

Integrating process design and production inventory management

The unit cost and process time, which are required factors in production inventory management, cannot be determined until the completion of the process design for the required production effectiveness and quality level. Thus, like the conventional approach, which is called the Disintegrated Case in this research, the process parameter design occurs before production inventory management is carried out. However, quality level achievement, process costs, and process times are closely related to the determination of production order quantities. The example is the processing times for a job, which can be determined by controlling the setting of the process parameters, and adding or reducing manpower and resources as required. These decisions directly affect unit cost and process time, which advance the EPQ determination. For example, an effective production process with additional resources spent on manpower, facilities, and materials may not yield the desired benefits in the following EPQ determination due to the resulting extra inventory cost. An example of turn-back influence is that quality progress causing process modification usually takes place after inventory decisions have been completed. Thus, redesigning processes typically alters both unit cost and process time (production rate). Because of the chain relationship, this alteration further impacts inventory carrying cost, an important element in production-inventory management. This alteration then causes the planned production quantity and back order interval to deviate from the estimated decision values.

The proposed computer model

Unit cost is approximated as an RSM response function of the controllable design values determined during the product/process design stage. Processing times are the result of these controllable process parameters. Unit cost continues to affect inventory carrying costs and backorder costs, which in turn influence the economic production quantity (EPQ), as decided by the production inventory management system. Thus, integrating process design and production inventory management is necessary. The other elements that further prevent production quantity from achieving a truly optimal EPQ are uncertain factors, which include: material costs, tool change costs, carrying rates, backorder costs, stochastic demand, and setup time, incurred during the process design and production inventory management. Another difficulty occurring in the process design is that design functions are unknown in most situations. Fortunately, with the development of CAE, the process design can be performed before the production process is realized. In this regard, an integrated approach for process parameters and EPQ determinations for required quality levels via computer simulation in an uncertain environment needs to be established. One of the possible ways to deal with the preceding concerns, such as integration, unknown design functions, and uncertainties, is to use a statistical method such as RSM to cope with the process design requirements via CAE computer simulation and EPQ determinations through numerical simulation to form the response functions. MP is then formulated with the aforementioned RSM response functions to find the optimal solutions via the optimization software. The flow process of the integrated case is depicted in detail in Figure 1; the proposed approach is illustrated in the following steps.

Step 1: The parameters influencing machining time contain feed rate, cutting speed, depth of cut, and the percentage of tool diameter required for achieving a specified feature. Because there are multi-features in a part, the parameters are defined correspondingly: U_i₁ is the feed rate (mm/rev), U_i₂ is the cutting speed (m/min), U_i₃ is the depth of cut (mm), U_i₄ is the percentage of tool diameter, i is 1, 2, 3, …, and n denotes the operations for feature i. The levels of parameter values are arranged by following the experimental design matrix introduced in Appendices A and B. The parameters and the associated probabilistic distributions for material costs, labor costs, setup costs, process capability limits, demand rates, inventory-related costs, and setup times are provided for uncertain applications. Additionally, the required process quality levels are decided in advance. The quality level represented in the presented example involves surface roughness.

Step 2: The cutting time t_mi can be estimated via the CATIA software.²⁸ The tool failure costs which are functional of tool life may occur. Tool life is a function of the cutting parameters of U_i₁, U_i₂, and U_i₃. Tool life T_i is determined using CCS software. In order to effectively obtain cutting time and tool life, the levels associated with each controllable factor, as given in the example in Table 1, are combined as input by referring to the experimental design matrix shown in Appendices A and B, to obtain the output, cutting time t_mi and tool life T_i, respectively.

Step 3: Process design-related conditions, including $C_{mat}$ , $C_{l}$ , $C_{0}$ , $t_{s}$ , $C_{ti}$ , and $t_{tci}$ , are considered as noises generated randomly from the probabilistic distributions introduced in the example in Table 2. These probabilistic distributions are assumed for convenient application.

Step 4: Substitute the preceding outputs $t_{mi}$ and T_i from Step 2 and the generated noises: $C_{mat}$ , $C_{l}$ , $C_{0}$ , $t_{s}$ , $C_{ti}$ , and $t_{tci}$ from Step 3 into equations (14) and (15) given in Appendix C, in order to find the unit cost C_U and the unit production time T_U. Based on T_U, the production rate P can then be obtained from equation (16) shown in Appendix C. Substitute the production inventory-related parameters and conditions obtained from Steps 1–4 into equations (17)–(20) to derive the total cost TC.

Step 5: Perform the RSM analysis using TC versus U₁₁, U₁₂, U₁₃, U₁₄, U₂₁, U₂₂, U₂₃, U₂₄, T_B₁, and Q to obtain the TC response function with SAS software.²⁹ SAS is capable of performing most statistical analyses. In addition, the job calculates the rank of one or more numerical variables in the observations of the SAS dataset, and then passes the rankings to the new SAS dataset. The controllable factors are ranked using ANOVA for possible improvement.

Step 6: Using the TC response function from Step 5, perform optimization through MP by GAMS software.³⁰ By inputting various quality levels, such as surface roughness R_a as a constraint Rosenthal of MP, sensitivity analysis can thereby be performed.

Step 7: If revision is required, the preceding steps can be redone by following Step 5.

Table 1.

Three levels of machining parameters for each factor.

Factors	Low level	Middle level	High level
U₁₁	0.056	0.103	0.15
U₁₂	100	125	150
U₁₃	1	3	5
U₁₄	25	50	75
U₂₁	0.056	0.103	0.15
U₂₂	100	125	150
U₂₃	1	3	5
U₂₄	25	50	75
T_B1	0.6	1.1	1.6
Q	225	275	325

Table 2.

Parameters conditions.

Parameters	Conditions
$C_{0}$	Uniform distribution, a = 1.25, b = 1.65
$C_{l}$	Uniform distribution, a = 0.30, b = 0.60
$C_{mat}$	Uniform distribution, a = 0.30, b = 0.70
$C_{t 1}$	Uniform distribution, a = 165, b = 195
$C_{t 2}$	Uniform distribution, a = 165, b = 195
$t_{s}$	Beta distribution, α 140, $β =$ 300
$t_{tc 1}$	Beta distribution, α 300, $β =$ 240
$t_{tc 2}$	Beta distribution, α 100, $β =$ 130
$r$	Uniform distribution, a = 0.01, b = 0.03
$S$	Uniform distribution, a = 13.5, b = 16.5
$D$	Normal distribution, $u = 13$ , $σ^{2} = 1$
$π$	Normal distribution, $u = 0.20 \times C_{U}$ , $σ^{2} = 1$

With the suggestions derived from the RSM experimental design matrix, the combined levels of controllable process parameters are input into the CAE simulation software to generate the simulated process machining time, tool life, and unit cost for multi-cutting operations. The generated parameter values of production inventory (Figure 1), the arranged levels of T_B₁, and Q (Figure 2) and the preceding multi-cutting unit cost are then collected for further application to account for the cost represented in the proposed approach (integrated case).

Figure 1.

Process design and production inventory management for integrated case.

Figure 2.

EPQ model with shortage.

Application

Numerical analysis

The example provided aims to deal with the minimization of machining costs of quality end-milling processes with solid parts, as shown in Figure 3. Production costs include machining operations costs and production inventory management costs.³¹ The choice of control factors depends on the experience of the engineer and can also be obtained by performing screen experiments. The different level values of machining parameters for each factor is set according to the range in which the practice can be operated. The machining operation involves removing excess material from the part body; the dimensions are 100 × 100 × 30 mm³. The tool geometry includes a normal diameter of 10 mm, an overall length of 100 mm, a measuring cutting length of 50 mm and a body diameter of 15 mm. Technical specifications of the tool are: six flutes, right hand rotation, rough quality machining, and a tooth material of High Speed Steel (HSS) coated with TiCN. The tool path style is inward helical. Because there are two features in the studied part, the parameters for the two features are defined correspondingly. U₁₁ and U₂₁ are the feed rates (mm/rev), U₁₂ and U₂₂ are the cutting speeds (m/min), U₁₃ and U₂₃ are the depths of cut (mm), and U₁₄ and U₂₄ are the percentages of tool diameter. As shown in Figure 2, the example for production inventory system is EPQ, with back order allowed. The conditions for overhead cost $C_{0}$ , material cost $C_{mat}$ , labor cost $C_{t}$ , set up time $t_{s}$ , tool change times $t_{tc 1}$ and $t_{tc 2}$ , tool costs $C_{t 1}$ and $C_{t 2}$ , production setup cost S, inventory carrying rate r, backorder cost $π$ , and demand rate D are listed in Table 2. As introduced in Step 1, based on the levels obtained fully or partially from Table 1 and the matrix presented in Appendices A and B, CATIA and CCS were used to obtain processing times $t_{m 1}$ and $t_{m 2}$ , and tool lives T₁ and T₂. The full usage from Table 1 and the application of Appendix A were applied for this integrated case. We selected a total of 10 experimental factors. Each factor has three levels. The Box-Behnken Design (BBD) can be fitted with three level response surfaces, and the first, second, and first order interaction polynomials can be estimated with a minimum number of experiments. Thus, according to the BBD method for experimental design, 170 experiments should be performed. Excluding controllable variables Q and T_B1, the partial usage of Table 1 and the application of Appendix B were applied for the Disintegrated Case. There were eight experimental factors. According to the BBD method, we carried out 120 experiments. The level of significance desired for $α$ was 0.05. Two cases are given in the following illustration.

Figure 3.

Geometrical solid model of the part.

Integrated case: Figure 4, which is extended from Figure 1, represents the integrated case in detail for the introduced example. The TC is obtained from Steps 1–4. Table 3, containing 170 experimental trials, provides an example that summarizes the tasks conducted through Steps 1–4. RSM is performed for TC in Step 5. The results of the RSM analysis are given in Table 4. R² value refers to the percentage that can be explained in the total variation of the regression model. A large value indicates that the fit of the regression model is better. In general, a R² value greater than 0.7 is considered strong effect size.³² Based on the ANOVA table in Table 4, the F-Ratio and important ranking for total cost TC are shown in Table 5. The order is U₁₁, U₂₁, Q, U₂₂, U₁₂, U₁₃, U₂₄, U₁₄, T_B₁, and U₂₃, sequentially. This order can be used as the priority reference for sorting the potential candidates for design improvement. The TC response functions are listed in Table 6.

Table 3.

RSM experiment design for total cost TC (Integrated Case).

No.	U₁₁	U₁₂	U₁₃	U₁₄	U₂₁	U₂₂	U₂₃	U₂₄	T_B1	Q	TC
1	0.103	100	3	50	0.103	100	1	50	1.1	225	306.9389
2	0.103	100	3	50	0.103	100	1	50	1.1	325	188.6197
3	0.103	100	3	50	0.103	100	5	50	1.1	225	262.4657
4	0.103	100	3	50	0.103	100	5	50	1.1	325	288.8596
5	0.103	100	3	50	0.103	150	1	50	1.1	225	316.3369
						⋮
100	0.056	125	3	25	0.103	125	5	75	1.1	275	268.6025
101	0.056	125	3	75	0.103	125	1	25	1.1	275	203.6499
102	0.056	125	3	75	0.103	125	1	75	1.1	275	162.4612
103	0.056	125	3	75	0.103	125	5	25	1.1	275	173.79
104	0.056	125	3	75	0.103	125	5	75	1.1	275	172.3497
						⋮
						⋮
170	0.103	125	3	50	0.103	125	3	50	1.1	275	326.4788

Table 4.

Response surface analysis for the total cost TC

Response surface for variable time
Response mean					278.947276
Root MSE					42.005534
R-square					0.7297
Coefficient of variation					15.0586
Regression	Degrees of freedom	Type I sum of squares	R-square	F value	p > F
Linear	10	306604	0.4517	17.38	<0.05
Quadratic	10	50932	0.075	2.89	<0.05
Cross-product	45	137767	0.203	1.74	<0.05
Total regress	65	495303	0.7297	4.32	<0.05
Factors	Degrees of freedom	Sum of squares	Mean square	F value	p > F
U₁₁	11	102,855	9350.413	5.3	<0.05
U₁₂	11	73,081	6643.761	3.77	<0.05
U₁₃	11	59,033	5366.65	3.04	<0.05
U₁₄	11	28,211	2564.62	1.45	0.1607
U₂₁	11	101,584	9234.915	5.23	<0.05
U₂₂	11	75,255	6841.397	3.88	<0.05
U₂₃	11	18,110	1646.339	0.93	0.5121
U₂₄	11	48,320	4392.77	2.49	<0.05
T_B1	11	23,062	2096.517	1.19	0.3043
Q	11	82,185	7471.327	4.23	<0.05

Table 5.

The F-ratio and important ranking for total cost TC.

Factor	TC
	F-ratio	Important ranking
U₁₁	5.30	1
U₁₂	3.77	5
U₁₃	3.04	6
U₁₄	1.45	8
U₂₁	5.23	2
U₂₂	3.88	4
U₂₃	0.93	10
U₂₄	2.49	7
T_B1	1.19	9
Q	4.23	3

Table 6.

Response functions of TC, C_U, and T_U.

Factor	R-square	Response functions of TC, C_U, and T_U
TC	0.7297	1384.635801 − 705.859364U₁₁− 2.142367U₁₂− 37.457215U₁₃− 2.351487U₁₄− 3803.178069U₂₁− 9.803158U₂₂− 38.088918U₂₃− 3.117492U₂₄+ 326.865653T₁− 1.297061Q + 8948.861088U₁₁²+ 4.691438U₁₂U₁₁+ 0.005036U₁₂²+ 11.893966U₁₃U₁₁− 0.003152U₁₃U₁₂+ 0.272978U₁₃²− 2.436111U₁₄U₁₁+ 0.025145U₁₄U₁₂− 0.501584U₁₄U₁₃+ 0.001198U₁₄²+ 12660U₂₁U₁₁+ 34.019569U₂₁U₁₂+ 111.53223U₂₁U₁₃− 1.679416U₂₁U₁₄+ 20.843921U₂₁²− 2.594161U₂₂U₁₁− 0.005765U₂₂U₁₂+ 0.458585U₂₂U₁₃+ 0.019892U₂₂U₁₄+ 5.586321U₂₂U₂₁+ 0.015219U₂₂²+ 126.244639U₂₃U₁₁+ 0.275519U₂₃U₁₂− 0.238509U₂₃U₁₃+ 0.018545U₂₃U₁₄− 69.907318U₂₃U₂₁+ 0.053853U₂₃U₂₂− 3.040233U₂₃²+ 2.262849U₂₄U₁₁− 0.008186U₂₄U₁₂− 0.021276U₂₄U₁₃+ 0.002427U₂₄U₁₄+ 3.53877U₂₄U₂₁+ 0.016996U₂₄U₂₂+ 0.0234632U₂₄U₂₃− 0.028944U₂₄²− 418.030309T₁U₁₁− 0.974626T₁U₁₂− 7.372452T₁U₁₃+ 0.400358T₁U₁₄+ 325.8718T₁U₂₁+ 0.576023T₁U₂₂+ 1.061034T₁U₂₃+ 0.387099T₁U₂₄− 32.016769T₁²− 7.169353QU₁₁− 0.007641QU₁₂+ 0.046597QU₁₃− 0.009048QU₁₄− 9.023153QU₂₁+ 0.012804QU₂₂− 0.007409QU₂₃+ 0.008178QU₂₄− 0.722948QT₁+ 0.004631Q²
C_U	0.8804	467.6513 − 936.795U₁₁− 1.21994U₁₂− 79.2093U₁₃− 2.71748U₁₄− 1072.47U₂₁− 0.080598U₂₂+ 44.45378U₂₃− 1.19353U₂₄+ 3926.342U₁₁²+ 1.163936U₁₂U₁₁+ 0.006099U₁₂²+ 248.3242U₁₃U₁₁− 0.0798U₁₃U₁₂+ 7.966298U₁₃²+ 0.796779U₁₄U₁₁− 0.00341U₁₄U₁₂− 0.38887U₁₄U₁₃+ 0.028269U₁₄²+ 70.69904U₂₁U₁₁+ 0.438323U₂₁U₁₂− 26.9579U₂₁U₁₃+ 4.994659U₂₁U₁₄+ 4883.771U₂₁²− 6.21247U₂₂U₁₁− 0.00318U₂₂U₁₂+ 0.127193U₂₂U₁₃+ 0.022552U₂₂U₁₄+ 2.266377U₂₂U₂₁+ 0.003468U₂₂²− 54.9799U₂₃U₁₁+ 0.05509U₂₃U₁₂− 1.6479U₂₃U₁₃+ 0.101691U₂₃U₁₄− 50.9727U₂₃U₂₁− 0.25067U₂₃U₂₂− 0.37688U₂₃²− 3.83272U₂₄U₁₁+ 0.011903U₂₄U₁₂+ 0.135273U₂₄U₁₃− 0.01918U₂₄U₁₄− 14.806U₂₄U₂₁+ 0.009948U₂₄U₂₂− 0.08584U₂₄U₂₃+ 0.022224U₂₄²
T_U	0.9099	127.9681 − 451.349U₁₁− 0.34273U₁₂− 15.7478U₁₃+ 0.210216U₁₄− 321.807U₂₁− 0.39171U₂₂+ 5.579496U₂₃+ 0.287428U₂₄+ 874.879U₁₁²+ 1.485201U₁₂U₁₁+ 0.000569U₁₂²+ 53.63086U₁₃U₁₁+ 0.017152U₁₃U₁₂+ 1.196435U₁₃²− 1.26354U₁₄U₁₁− 0.00193U₁₄U₁₂− 0.04983U₁₄U₁₃+ 0.003597U₁₄²− 202.49U₂₁U₁₁− 0.82579U₂₁U₁₂− 10.9301U₂₁U₁₃+ 0.40222U₂₁U₁₄+ 975.4017U₂₁²− 0.77397U₂₂U₁₁+ 0.000858U₂₂U₁₂+ 0.01403U₂₂U₁₃+ 0.000894U₂₂U₁₄+ 1.949167U₂₂U₂₁+ 0.000592U₂₂²− 1.2655U₂₃U₁₁− 0.00126U₂₃U₁₂− 0.25049U₂₃U₁₃+ 0.021462U₂₃U₁₄+ 2.788558U₂₃U₂₁− 0.04167U₂₃U₂₂+ 0.015197U₂₃²+ 0.364846U₂₄U₁₁+ 0.000266U₂₄U₁₂+ 0.008801U₂₄U₁₃− 0.00182U₂₄U₁₄− 2.08667U₂₄U₂₁− 0.00118U₂₄U₂₂− 0.01591U₂₄U₂₃+ 0.002558U₂₄²

Figure 4.

Computer model via computer software, statistical methods, and optimization techniques.

As shown in Tables 4 and 5, the significant factors are feed rates U₁₁ and U₂₁, economic production quantity Q, and cutting speeds U₁₂ and U₂₂. The insignificant factors are backorder period T_B₁, depth of cut U₂₃, and percentage of tool diameter U₁₄. To balance the ordering cost and inventory carrying cost in a production inventory management system, the most important decision variable is production quantity Q. Hence, Q is considered the most significant factor of response function TC formation. In cutting process planning, cutting speed and feed rate together determine the material removal rate, which is the production rate that further affects production quantity Q. Thus, these two factors appear significant influence in building TC response function. On the other hand, rest of the factors are relatively less significant. The backorder period T_B₁ is the least significant among all factor because a high backorder cost, as compared with inventory carrying cost, results in a small backorder period T_B₁, which has even less significant influence on the total cost TC response function.

As indicated in Step 6 in the preceding discussion, the TC response function is used as an objective function to form the MP. The decision variables are: U₁₁, U₁₂, U₁₃, U₁₄, U₂₁, U₂₂, U₂₃, and U₂₄. They are restricted by the upper and lower limits, S_Ui and S_Li, as defined in Table 1. These restrictions are expressed in equations (2)–(11).

The problem for the integrated case is formulated as follows:

Minimize TC

(1)

0.056 ⩽ U_{11} ⩽ 0.15

(2)

100 ⩽ U_{12} ⩽ 150

(3)

1 ⩽ U_{13} ⩽ 5

(4)

25 ⩽ U_{14} ⩽ 75

(5)

0.056 ⩽ U_{21} ⩽ 0.15

(6)

100 ⩽ U_{22} ⩽ 150

(7)

1 ⩽ U_{23} ⩽ 5

(8)

25 ⩽ U_{24} ⩽ 75

(9)

0.6 ⩽ T_{B 1} ⩽ 1.6

(10)

225 ⩽ Q ⩽ 325

(11)

The optimal solutions via the above MP²⁹ are given in Table 7.

Table 7.

The optimal solutions for integration case.

Controllable variables	Optimal values
U₁₁	0.056
U₁₂	100.000
U₁₃	5.000
U₁₄	75.000
U₂₁	0.125
U₂₂	100.000
U₂₃	5.000
U₂₄	25.000
T_B1	0.600
Q	325.000
Total cost	63.3807

Note. R_max = 0.00010.

The following equation ensures that an acceptable quality level restricted below the maximum surface roughness, R_max.

R_{a} ⩽ R_{\max}

(12)

R_{a} \approx 0.032 \frac{f^{2}}{R}

(13)

where R_a is the ideal arithmetic average surface roughness, f is the tool feed, and R is the tool tip radius.^33–35

The results of sensitivity analysis under different quality levels R_a are shown in Table 8.

Table 8.

Comparison between integration and disintegration among various surface roughness.

R_max	0.00007	0.00008	0.00009	0.00010
Integration case
C_U	142.737	141.968	141.632	141.673
T_U	33.6203	33.5027	33.2296	32.935
Q	309.775	314.24	320.087	325
Q_B	0.766	0.796	0.867	0.944
I	13.099	13.769	15.154	16.620
I_B	0.010	0.011	0.012	0.013
T	23.829	24.172	24.622	25
T_B1	0.600	0.600	0.600	0.600
TC	96.284	85.1291	74.1411	63.3807
Disintegration case
C_U	79.839	76.299	73.598	71.594
T_U	11.032	10.669	10.474	10.359
Q	225	225	225	225
Q_B	18.306	19.194	19.697	20.002
I	61.643	61.950	62.094	62.171
I_B	1.062	1.151	1.204	1.236
T	17.308	17.308	17.308	17.308
T_B1	0.600	0.600	0.600	0.600
TC	116.254	112.971	109.982	107.583

Note. I and I_B are average inventory level and back order average inventory level respectively. Q_B is back order quantity. T is reorder interval. Q_B and T are defined in Figure 2.

Disintegrated case: After completing the analysis of the Integrated Case, the analysis of the Disintegrated Case was performed for comparison. Without considering the variables of the production inventory system, such as backorder period T_B₁ and production quantity Q, the controllable variables for the process design in the first stage are the cutting parameters: U₁₁, U₁₂, U₁₃, U₁₄, U₂₁, U₂₂, U₂₃, and U₂₄. We used the experimental design matrix from Appendix B to run the computer simulations through CATIA and CCS, and acquired the cutting times, t_m₁ and t_m₂, and tool lives T₁ and T₂. Steps 1 and 2 were then used the same way as in the Integrated Case; instead of TC for the Integrated Case, the obtained unit cost C_U and unit producing time T_U were used as the response values for the RSM analysis, in order to derive the response functions of C_U and T_U for the Disintegrated Case. The response functions of C_U and T_U are stated in Table 6.

Similarly, with the constraints from equations (2)–(9) and equation (12), instead of TC as the response function used in the Integrated Case, the obtained C_U response function was considered as an objective function in the formulation of MP for the Disintegrated Case. C_U was optimized in advance before the lot sizing was carried out in the second stage of the Disintegrated Case. Then, T_U was obtained from the optimal cutting parameters determined via the above MP formulation. The production rate P was identified through equation (A.3). TC was found from the above C_U and P via equations (A.4)–(A.7). TC became a function of T_B1 and Q. MP was formed with the TC response function as an objective function, with equations (8) and (9) as constraints. Similar to the Integrated Case, a sensitivity analysis was performed with the various R_a, as given in equation (12). The results are shown in Table 8. These results can be further used for comparison.

In MP, the problem is formulated into a mathematical model, which involves an objective function subjective to constraints. MP is used for obtaining the most optimal solution for a problem with given constraints. All the constraints intersect a region giving a feasible region for optimization. The feasible region explains all of the values that a mathematical model can provide. The optimal solutions of decision variables are the best values of objective function in the feasible region. In other words, the optimal solutions are dependent on the given equations of objective functions and constraints. Table 4 lists the important levels of controllable variables (factors) that influence the variability of response values of response functions. However, the presented MP is an optimization technique for finding the best mean values of problem model. Due to the insignificant consideration in Table 4, the elimination of partial controllable variables from response functions sometime may lead to biased optimal solutions. Hence, all significant and insignificant controllable variables, such as TC, CU, and TU, should be included to form response functions.

Results discussion

The results summarized for both cases in Table 8 indicate that C_U, T_U, and TC decrease as R_max increases. The magnitudes Q, Q_B, I, I_B, and T increase as R_max increases, since poorer quality levels with greater R_max require less production time and money spent on producing a unit. The shorter production time and cheaper unit cost result in a greater production quantity and inventory level because of the higher production rate and lower carrying cost. Although C_U and T_U in the Integrated Case are greater than those in the Disintegrated Case, the Integrated Case has a lower average inventory level I and TC because the Integrated Case assimilates process design and production inventory management under the consideration of balance. In further comparing these two cases, the Integrated Case shows greater production quantity Q and periodic reordering interval T than the Disintegrated Case does. These phenomena reflect that greater production time T_U in the Integrated Case indicates a slower production rate and a higher unit cost resulting from a poor process design, as compared to a lower production time with higher production rate and lower unit cost from an effective process design in the Disintegrated Case.

An additional application with the proposed approach is to find the best parameters for product development. The best combination of product parameters is the obtained parameters associated with an alternative that holds the minimum response TC. The corresponding TCs are found with varying product design parameters. Consequently, the optimal parameters for simultaneous product and process design can thus be determined.

In contrast, the Disintegrated Case aimed to minimize the manufacturing cost of the process design in the first stage and then the production cost of inventory management in the second stage, sequentially and independently. As in the presented example for the Disintegrated Case, the act of cost reduction in the first stage solely with effective process design, namely, a smaller production time T_U, or unit cost C_U, may not prove beneficial in a further attempt at cost saving in the second stage. Consequently, greater collective TC results for the Disintegrated Case, since the higher production rate is decided solely in the first stage (process design), caused overproduction in the second stage (inventory management system). Overproduction results in a higher inventory level and carrying cost in the second stage. Thus, the reason that subsystem optimization sequentially and independently cannot obtain the global optimization for an entire system is that the local optimization for the predecessor subsystem may exert a negative impact on the successor subsystems.

Thus, the reason that subsystem optimization sequentially and independently cannot obtain the global optimization for an entire system is that the local optimization for the predecessor subsystem may exert a negative impact on the successor subsystems.

Computer simulations via CAE can replace physical experiments, decrease the cost of experimentation, and accelerate the process design and production development. As shown in Figure 4, the noise impacts from Table 2 are considered in the proposed approach, which makes the integrated system more robust during the applications, especially in uncertain environments. The proposed approach contains statistical analysis for importance ranking, which assists designers to identify the reasons for changes and to receive instantaneous advice for design progress. The proposed approach with CAE, RSM, and MP optimization for an integrated system can proceed equally well on personal computers.

Summary

The proposed approach in this study was used to develop a computer model to obtain the minimal cost with acceptable quality levels for simultaneous process design and production inventory management under uncertainty. Computer software CATIA and CCS were adopted to simulate the relevant data. Then, the statistical RSM method and the MP optimization technique were used to derive the optimal process parameters and production quantities. An example was introduced to demonstrate the proposed approach. The results show that integrating machining operations and inventory management could achieve lower costs and higher quality. Consequently, economically viable and high quality product/process design and production planning can be completed simultaneously in the early stages of a product’s life cycle.

In addition, the integrated process and quality design of whole product and process life cycle could be proposed as multi-objective programming, which can be solved with newly developed algorithms, such as the ant colony algorithm, genetic algorithm, simulated annealing algorithm, etc. Future work will explore whether it is possible to find a more accurate and optimal value.

Footnotes

Appendix A: Matrix of Box-Behnken design for integrated case

Factors = 10, runs = 170

\begin{matrix} U_{11} & U_{12} & U_{13} & U_{14} & U_{21} & U_{22} & U_{23} & U_{24} & T_{B 1} & Q \end{matrix}

[\begin{matrix} 0 & \pm 1 & 0 & 0 & 0 & \pm 1 & \pm 1 & 0 & 0 & \pm 1 \\ \pm 1 & \pm 1 & 0 & 0 & \pm 1 & 0 & 0 & 0 & 0 & \pm 1 \\ 0 & \pm 1 & \pm 1 & 0 & 0 & 0 & \pm 1 & \pm 1 & 0 & 0 \\ 0 & \pm 1 & 0 & \pm 1 & 0 & \pm 1 & 0 & 0 & \pm 1 & 0 \\ \pm 1 & 0 & 0 & 0 & 0 & 0 & 0 & \pm 1 & \pm 1 & \pm 1 \\ 0 & 0 & \pm 1 & \pm 1 & \pm 1 & 0 & 0 & 0 & 0 & \pm 1 \\ \pm 1 & 0 & 0 & \pm 1 & 0 & 0 & \pm 1 & \pm 1 & 0 & 0 \\ 0 & 0 & \pm 1 & 0 & \pm 1 & 0 & \pm 1 & 0 & \pm 1 & 0 \\ \pm 1 & 0 & \pm 1 & 0 & 0 & \pm 1 & 0 & 0 & \pm 1 & 0 \\ 0 & 0 & 0 & \pm 1 & \pm 1 & \pm 1 & 0 & \pm 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

Appendix B: Matrix of Box-Behnken design for disintegrated case

Factors = 8, runs = 120

\begin{matrix} U_{11} & U_{12} & U_{13} & U_{14} & U_{21} & U_{22} & U_{23} & U_{24} \end{matrix}

[\begin{matrix} \pm 1 & \pm 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \pm 1 & 0 & \pm 1 & 0 & 0 & 0 & 0 & 0 \\ \pm 1 & 0 & 0 & \pm 1 & 0 & 0 & 0 & 0 \\ \pm 1 & 0 & 0 & 0 & \pm 1 & 0 & 0 & 0 \\ \pm 1 & 0 & 0 & 0 & 0 & \pm 1 & 0 & 0 \\ \pm 1 & 0 & 0 & 0 & 0 & 0 & \pm 1 & 0 \\ \pm 1 & 0 & 0 & 0 & 0 & 0 & 0 & \pm 1 \\ 0 & \pm 1 & \pm 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & \pm 1 & 0 & \pm 1 & 0 & 0 & 0 & 0 \\ 0 & \pm 1 & 0 & 0 & \pm 1 & 0 & 0 & 0 \\ 0 & \pm 1 & 0 & 0 & 0 & \pm 1 & 0 & 0 \\ 0 & \pm 1 & 0 & 0 & 0 & 0 & \pm 1 & 0 \\ 0 & \pm 1 & 0 & 0 & 0 & 0 & 0 & \pm 1 \\ 0 & 0 & \pm 1 & \pm 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & \pm 1 & 0 & \pm 1 & 0 & 0 & 0 \\ 0 & 0 & \pm 1 & 0 & 0 & \pm 1 & 0 & 0 \\ 0 & 0 & \pm 1 & 0 & 0 & 0 & \pm 1 & 0 \\ 0 & 0 & \pm 1 & 0 & 0 & 0 & 0 & \pm 1 \\ 0 & 0 & 0 & \pm 1 & \pm 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \pm 1 & 0 & \pm 1 & 0 & 0 \\ 0 & 0 & 0 & \pm 1 & 0 & 0 & \pm 1 & 0 \\ 0 & 0 & 0 & \pm 1 & 0 & 0 & 0 & \pm 1 \\ 0 & 0 & 0 & 0 & \pm 1 & \pm 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & \pm 1 & 0 & \pm 1 & 0 \\ 0 & 0 & 0 & 0 & \pm 1 & 0 & 0 & \pm 1 \\ 0 & 0 & 0 & 0 & 0 & \pm 1 & \pm 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & \pm 1 & 0 & \pm 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & \pm 1 & \pm 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

Appendix C: Equations represent for machining operations and EPQ production inventory systems

The unit cost is³⁰:

(A.1)

\begin{matrix} C_{U} = C_{mat} + (C_{l} + C_{0}) t_{s} + \sum_{i = 1}^{m} (C_{l} + C_{0}) t_{mi} \\ + \sum_{i = 1}^{m} C_{ti} (\frac{t_{mi}}{T_{i}}) + \sum_{i = 1}^{m} (C_{l} + C_{0}) t_{tci} \end{matrix}

where

m: number of machining operations required to produce a unit of product

$C_{U}$ : unit cost ($)

$C_{mat}$ : cost of raw material per part

$C_{l}$ : labor cost ($/min)

$C_{0}$ : overhead cost ($/min)

$C_{ti}$ : ith tool cost ($)

$t_{s}$ : set-up time for machining process

$t_{tci}$ : ith tool changing time (min)

$t_{mi}$ : ith machining time (min)

T_i: ith tool life for feature i

The production time for each unit, T_U:

(A.2)

T_{U} = t_{s} + \sum_{i = 1}^{m} t_{mi} + \sum_{i = 1}^{m} t_{tci}

The production rate, P:

(A.3)

P = \frac{1}{T_{U}}

The average setup cost per time:

(A.4)

A_{S} = S . \frac{D}{Q}

where ordering cost S and demand rate D are known

Q is unknown production quantity needed to be determined

The average inventory carrying cost per time:

(A.5)

A_{I} = \frac{r C_{U} (P - D) {(T_{P} - T_{B 1})}^{2}}{2 T_{P}}

where T_P and T_B₁ are defined in Figure 2

T_B ₁ is unknown backorder period needed to be determined

T_P is unknown production cycle which is $\frac{Q}{P}$

r is inventory carrying rate

The average backorder cost per time:

(A.6)

A_{B} = \frac{π T_{B 1}^{2} (P - D)}{2 T_{P}}

where $π$ is inventory backorder cost

The total cost:

(A.7)

TC = A_{S} + A_{I} + A_{B}

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Angus Jeang

Chien-Ping Chung

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