Sequence planning of sheet metal parts manufactured using progressive dies

Abstract

Progressive dies are widely used for the mass manufacture of sheet metal parts requiring several stamping operations. Sequencing these stamping operations is vital as it influences the design of the die, especially the number of stations required to make the part. This paper presents a method for automatically generating the sequence plan for parts manufactured using progressive dies. This method establishes a scheme that maps sheet metal features into their corresponding stamping operations, including shearing, bending and forming. The mapped operations are initially placed in one of two categories and then the operations in one category are clubbed together using heuristic rules. The essential characteristics of each group are analysed to establish fuzzy relationships between any selected group and the remaining groups. The final membership value vector obtained from the fuzzy set theory is used to sequence these groups. A program developed in C++ based on this method has been tested on representative industrial sheet metal parts. The methodology is validated by comparing the generated sequences, for two case studies, with those used in industry.

1 Introduction

Sheet metal industry plays an important role in engineering. Research on this topic has been conducted for over five to six decades and still attracts attention nowadays [1,2,3]. There are different tools to manufacture sheet metal parts, including (CNC) press brakes, laser cutting and dies. Some of the smaller sheet metal parts required in large quantities are usually made using progressive dies because the process is stable, has high productivity and is automated. A progressive die has several stations with one or more stamping operations performed at each station, thus making the design of such dies complicated.

Sequencing the operations is one of the most important tasks in the design of progressive dies as it influences part accuracy, the size (i.e. number of stations) of the die and hence its cost. Ever since the 1980s, researchers have tried to capture the knowledge of skilled engineers and embed it in computer-aided process planning (CAPP) systems to obtain feasible sequences for the stamping operations using progressive dies. These systems have used various methodologies, including knowledge-based [4, 5], fuzzy logic [6, 7], feature-based [8, 9], graph theory [10] and genetic algorithms [11, 12].

Although research on sequencing stamping operations in progressive dies has made vast progress, most of the existing methods focus on parts containing features requiring only shearing and/or bending operations, or in other words, they lack coverage of features involving multiple stamping and forming operations. In this paper, sheet metal features, be it a simple hole or a lance with a cut-out, are mapped into one or more stamping operations; these operations are then analysed to see if they can be grouped together. By analysing the common characteristics of the operations to be performed in a group, fuzzy set theory is used to sequence these groups.

The operations used in the fabrication of sheet metal parts include bending, shearing and forming operations and, over the years, researchers have developed systems to assist the user in sequencing these operations. Some researchers have focussed solely on bending operations and, needless to say, they have used different approaches to determine the optimum sequence, ranging from interactive intervention by the user to knowledge- and artificial intelligence-based systems. Li et al. classified bend features as “U”, “Z”, “L” or “V” types and interactively sequenced the bending operations using case-based reasoning algorithms [8]. Kannan and Shunmugam defined an optimal bending sequence as one which is collision-free, combines co-linear bends and minimises the number of tools, tool stations and part movement between flip-overs [13]. They determined the optimal sequence using a direct search for parts with few bends and a genetic algorithm for parts with many bends. When a bend is formed, the tolerance zones of the bends yet to be formed are attenuated and De Vin et al. were the first to consider this [14]. They considered a sequence to be feasible when the slack for the last bend is still positive. Faraz et al. predicted a bending sequence by considering not only the variations in the process but also the variations in the thickness of the sheet metal and its properties [15]. They achieved their goal by Monte Carlo simulations and finite element analysis. Duflou et al. formulated the bending sequence as a travelling salesman problem which they solved by using a branch-and-bound search method subject to a dynamic penalty system [16]. Ong et al. formalised the heuristic knowledge for the bending operations in the form of rules and then used fuzzy set theory to predict a feasible sequence [6]. In a more recent paper, Prasanth and Shunmugam use a two-stage algorithm to determine the bending operation sequence [3]. In the first stage, they identified, in the form of a matrix, feasible combinations of bends that could be performed without any collision. In the second stage, they demonstrated that the optimum sequence could be obtained using graph traversal or genetic algorithms. Whilst the above-mentioned researchers have sequenced the bending operations to be performed on a (CNC) press brake, Farsi and Arezoo used fuzzy logic theory for sequencing the bending operations intended for progressive dies [17]. Their rules grouped together operations which can be performed simultaneously, leading to a possible reduction in the number of tool stations.

Another group of researchers has developed systems to sequence operations in a part requiring only piercing and blanking operations. Nee and Venkatesh were one of the earliest researchers who developed a nesting program to determine the optimal layout of parts requiring only blanking operations [4]. Similar work was done by Dequan et al. who developed a knowledge-based system for nesting irregular-shaped blanks and sequencing the shearing operations [18]. Lin and Sheu reduced the entire search space considerably by classifying the punches into one of five groups: punches that must be used first, last, simultaneously, exclusively and precede others [19]. Ghatrehnaby and Arezoo considered only the number of stations and torque acting on the die [20] and formulated the strip layout as a mathematical problem; they partitioned, using set theory, the punches into several subsets ensuring that user-defined precedence and intersectional constraints were satisfied.

Whilst the above-mentioned researchers have focussed solely on either bending or shearing operations, some other researchers have developed systems which address both types as well as forming operations. Most of these systems were developed for progressive dies. Nee and Foong’s knowledge-based system considered shearing and bending operations with heuristic rules embedded in the system using if–then statements [21]. Choi et al. used a similar approach for sequencing shearing and bending operations, with the sequence for shearing operations based on their total length [5]. The system developed by Vosniakos et al. was also knowledge-based but they assumed that the part is symmetric and contains a U-shaped feature [22]. Kim et al. developed a system which sequenced the operations and generated the strip layout automatically using fuzzy set theory [7] but their system requires the geometrical information for each plane has to be entered interactively; furthermore, the operations are not grouped together. Moghaddam et al. overcome this limitation using heuristic rules [23]. In Tang and Gao’s system, a feature is not only mapped into one or more stamping operations but also its stamping direction is considered [9]. Chang et al. used a genetic algorithm to determine the optimum sequence by minimising the moments exerted on the die [11], thus increasing the life of the die. Using graph theory, Chu et al. developed an algorithm to sequence shearing, bending and embossing operations [10]. They constructed a precedence graph, but user intervention was required to perform acyclic checks, which ensure that no cycle is present in the constructed graph. Kumar and Singh developed a production rule-based expert system for automating the determination of operation sequence, strip dimensions, number of tool stations and pilot holes in progressive dies [24]. In a methodology similar to that described in [19], Lin and Sheu determined and evaluated the feasible solutions for parts which required shearing and bending operations [25]. Their evaluation was based on four factors, i.e. feeding height, moments, number of stages and strip stability.

From the above literature review, it is clear that the methodologies for sequencing the operations made on CNC press brakes are quite involved [3, 14,15,16] because of the large number of bends and/or the tolerances imposed. These methodologies become less important for parts manufactured using progressive dies because there are fewer bends and there is less interaction amongst bends in such parts. Hence, as shown in the literature review above, sequencing the bends can be satisfactorily performed using knowledge-based or AI techniques. However, there are some research gaps that need to be addressed when sequencing the operations required for manufacturing these parts. They are:

• i. Researchers have mainly focussed on sequencing shearing and bending operations [5, 7, 8, 17,18,19, 21, 23, 25]. Only a few researchers [9, 10] have considered composite features (e.g. lance and jog), which require more than one stamping operation.

• ii. Although Tang and Gao [9] formalised an approach that maps features into their corresponding stamping operations, they did not discuss features commonly found in sheet metal parts.

• iii. Whilst a few researchers [7, 23] have considered embossing, other forming operations have been neglected.

• iv. Existing knowledge-based systems and some commercial sequencing systems such as 3DQuickStrip [26] require frequent user interactions from experienced engineers.

This paper presents a method that automatically generates sequence plans for sheet metal parts containing features ranging from simple holes to bridges, the input data for which is in STEP AP203 format, which overcomes the interoperability issue. The remaining sections of this paper are presented as follows. Section 2 maps sheet metal features into their corresponding manufacturing operations. These operations are subdivided into different types, which are further classified into two categories. Heuristic rules for grouping these operations and the criteria for establishing relationships amongst one group and other groups are discussed in Section 3. In Section 4, the developed system is tested for two sample parts, and the sequence plans generated are compared with near-optimal plans. Section 5 draws conclusions and discusses potential future work.

2 Mapping and classifying operations

Sheet metal parts manufactured using progressive dies result from a series of stamping operations. Since features are directly influenced by their corresponding manufacturing operations, it is necessary to establish the relationships between features and their manufacturing operations. This section first maps sheet metal features into their corresponding manufacturing operations, which are subsequently placed into different categories by considering their characteristics and manufacturing constraints.

2.1 Mapping feature onto operations

Sheet metal features require one or more operations to manufacture. Kannan and Shunmugam [27] and Yang et al. [28] suggested taxonomies that reflect the relationships between sheet metal features and their corresponding operations. In this paper, three different stamping operations, i.e. shearing (including blanking and piercing), bending and deform (forming discussed in Section 1) are discussed. Table 1 lists the sheet metal features considered in this paper, the category they belong to and the operations they are mapped into. Figure 1a shows a few of these features. Figure 1b shows two examples; the first is an emboss, the manufacture of which requires a deform operation. The second is a composite feature, i.e. a lance requiring piercing and deform operations.

Table 1 Feature category, feature and the mapped operations

Fig. 1

Mapping sheet metal features into stamping operations

2.2 Operation classification

Once features have been mapped into operations, these operations are classified into one of two categories, an approach which has been previously used by other researchers [7, 17, 23]. The underlying principle is that all operations in category I are performed before those in the second category. In our work, an improved method that considers and classifies shearing, bending and deform operations is presented (see Fig. 2). Shearing operations associated with holes and slots are classified into category I and all bending operations into category II. Since operations in category I are performed before those in category II, it is understandable to classify hole- and slot-related operations into category I. If they were placed into category II, it would mean that a side wall is bent first and the holes and slots punched later, a sequence of operations which is unadvisable.

Fig. 2

Classification of stamping operations

Whilst the classification of shearing and bending operations is straightforward, deform operations are more involved since there are three different types. Deform operations which can form a complete feature such as an emboss and bead are said to be of type I (Fig. 3a). Deform operations which require a pre-operation to generate the feature and have no axis along which the feature shape is formed are said to be of type II. Figure 3b shows a flanged hole and a louvre, and neither of these two features has an axis about which the material is bent. Deform operations of type III also require a pre-operation, but there is one or more axis about which the material is bent to form the feature completely. Figure 3c shows a lance and a jog; the lance is formed by bending the material about one axis whereas in the jog, the material is bent about two axes.

Fig. 3

Examples for different types of deform operation: (a) type I, (b) type II, (c) type III

Next, as mentioned above, these operations are placed in one of two categories which are referred to “shearing and deform” and bending and deform” categories. The first category contains the following types of operations.

• i. Shearing operations that result in features such as a hole and slot.

• ii. Shearing operations that make the cut part of a composite feature such as the contour of a lance, jog and bridge. Figure 4a shows an example of this situation.

• iii. Deform operations of type I, type II and some of type III.

Fig. 4

Example of a special case in classifying deform operations

The second category includes:

• i. bending operations that result in bent walls, and

• ii. some deform operations of type III such as those that result in the deformed part of a lance, jog, bridge, etc. (see Fig. 4a).

The reason for having deform operations of type III is that in some cases they can be performed simultaneously with another bending operation in category II. A deform operation of type III is placed in category II if it is located on the mother plane and there is a side wall which can be formed simultaneously with it. Otherwise, it is placed in category I. For example, the deform operation associated with jog 1 shown in Fig. 4b is placed in category I since it is not located on the mother plane. On the other hand, the deform operation associated with jog 2 is placed in category II because it lies on the mother plane, and it can be performed simultaneously with the side wall, assuming of course that no collisions take place between the tools for bending the side wall and jog.

3 Sequencing the operations

To sequence the operations for a given sheet metal part, empirical knowledge is used to club together related operations into groups, and then fuzzy set theory is applied to prioritise the groups. Empirical knowledge is one important source in optimising design and manufacturing procedures. For the manufacture of sheet metal parts using progressive dies, operations that share common manufacturing constraints can be grouped together and performed simultaneously at one station, thus optimising the sequence plan and reducing the number of stations in the die. Farsi and Arezoo [17] and Moghaddam et al. [23] discuss this grouping strategy for operations for simple cutting and bending features. The work described here extends this grouping strategy to include more complex features. Extension of this strategy is based on knowledge elicited from experienced engineers at the collaborating company.

Fuzzy set theory [29] is used to prioritise these groups by quantifying their final membership values. The groups in one specific category are prioritised according to some of their characteristics, such as their shear length and location. Considering the characteristics that influence sequencing, several criteria are proposed to establish sequencing relationships amongst one selected group and the remaining groups. A high value for one group in one relationship indicates that the group operations should be performed early on. These values form the elements of a fuzzy matrix, and the final sequence of each group of operations can be obtained by deriving these values in the matrix. This procedure is performed twice, once each for the group of operations in a category.

3.1 Shearing and deform category

3.1.1 Pre-definition

Generally, sheet metal designers are creative, and the external contour of a sheet metal part can take different shapes. In our work, it is assumed that the external contour is made by one or more blanking operations and the contour is created at one station.

3.1.2 Grouping rules

The rules for grouping together operations in this category are discussed below.

Rule 1: Related operations

Operations that give rise to features of the same type and have similar key parameters are said to be related. It is logical to perform these operations together at one station because they are very likely to have similar manufacturing requirements, e.g. the same tolerances. For the part shown in Fig. 5a, holes A and B, both of which have the same diameter, are said to be related and can be grouped together. Features C and D are the cut part of two identical jogs, and therefore they are related. Similarly, features E and F are related as they are the deformed part of two identical flanged holes.

Fig. 5

Examples for grouping rules 1 and 2

Rule 2: United operations

Sometimes it may be possible to combine two similar operations into one even though the shapes generated by the two operations are different. For example, features G and H shown in Fig. 5b are both cut-outs and would normally be performed at different stations. However, if the distance between the two features is sufficient to permit the use of two different punches or if the distance is within a pre-defined value, it would be possible to use an integrated tool. In both these cases, the two features are created simultaneously. Also, use of an integrated punch merges two punches into one thus increasing the reliability of the tool. The union of such operations may lead to a reduced number of die stations.

Rule 3: Piloting operations

One method to achieve positional accuracy of the die is to use pilot holes which are located either outside the part or are internal features. The pilot holes, for obvious reasons, are made at the first station. Figure 6a shows two holes of the part being used as pilot holes [18] whereas in Fig. 6b, the pilot holes are external to the shape of the part [23].

Fig. 6

Internal and external pilots: (a) part with internal pilots [18], (b) part with external pilots [23]

3.1.3 Handling criteria and Fuzzy set theory

Several handling criteria are proposed to establish the fuzzy relationships amongst one selected group and the remaining groups. These criteria consider several common characteristics of operations in this category and are discussed below.

Criterion 1: Operation location

This criterion indicates where the feature occurs in the blank of the part. As suggested in [7], three different types of relations that characterise the location are suggested. They are:

• (i) is-on for deform operation of type I (see Fig. 7a);

• (ii) is-in for shearing operations and deform operations of type II (see Fig. 7b), and

• (iii) is-along for blanking operations that make the contour (see Fig. 7c).

Fig. 7

Operation location: (a) is-on, (b) is-in, (c) is-along

Criterion 2: Processing length

The processing length of an operation reflects the size of the feature. The longer the length, the greater will be the effect caused by the operation on the other features present in the part. To decrease this influence, a feature with a long processing length is given preference. The processing length of a feature is given by the perimeter of the cut shape, or the contour of the deformed shape and the processing length of a group is the sum of all the lengths of the features in one group.

Criterion 3: Number of operations in a group

It is assumed that the number of operations in a group influences the quality of the part. The bigger this number is, the greater will be the influence. A group containing a small number of operations is given preference over groups containing several operations.

Criterion 4: Operations for composite features

The manufacture of composite features requires more than one operation and these operations have to be performed in a certain sequence. The shearing operation is performed prior to the deform operation as the material has to be relieved before it can be deformed.

The fuzzy membership functions for these four criteria are shown in Fig. 8a–d.

Fig. 8

Fuzzy membership functions: (a) operation location, (b) processing length, (c) number of operations in a group, (d) operations for composite features

Let ${\mathrm G}_1$ = {${\mathrm g}_{1\mathrm i}$ | $\mathrm i=1,2\dots\mathrm n$} represent the set of operation groups that are present in the first category and where ${\mathrm g}_{1\mathrm i}$ represents a particular group.

Let ${\mathrm C}_1$ = {${\mathrm c}_{1\mathrm j}$ | $\mathrm j=1,2,3,4$} represent the set of defined criteria mentioned above, where ${\mathrm c}_{1\mathrm j}$ represents the fuzzy set for criterion $\mathrm j$.

Elements in sets ${\mathrm G}_1$ and ${\mathrm C}_1$ have a one-to-one mapping fuzzy relationship $f$ and their mapped result is defined as a combined grade, ${\mathrm R}_{\mathrm i,\mathrm j}$.

$${R}_{i,j}= f({g}_{1i},{c}_{1j})$$

(1)

The fuzzy matrix ${R}_{1}$ is of size $n\times 4$, where n is the number of groups in category I. There are four columns and the values in each column reflect the importance of a specific criterion. This matrix is populated by repeating the above operation for all the groups and the resulting structure of this matrix is shown in Table 2.

Table 2 Fuzzy matrix ${R}_{1}$

Because there are several criteria, their importance from a sequence viewpoint is weighted. The weighting factors for the first three criteria are 1.5, 0.6 and 0.6 respectively [7, 23] whereas the fourth weighting factor was obtained by trial and error as 0.5, making the weighting vector, ${{W}_{1}}^{T}=\{1.5, 0.6, 0.6, 0.5\}$. Preference is given to the operation location criterion whereas the processing length and number of operations are given a much lower, but equal, weighting. The fuzzy matrix ${R}_{1}$ is then weighted by ${W}_{1}$ to give ${FV}_{1}$, a column vector containing the final values of which indicate the priority of the groups in this category.

$${FV}_{1} = \left[\begin{array}{c}{fv}_{1}\\ {fv}_{2}\\ \begin{array}{c}\vdots \\ {fv}_{n}\end{array}\end{array}\right]= {R}_{1}\times {W}_{1}$$

(2)

3.2 Bending and deform category

Operations discussed in this category include bending operations and deform operations of type III. Before discussing the rules for grouping the operations, several pre-definitions are required.

3.2.1 Pre-definitions

Feed direction: The progressive die is fed with raw material by a coil feeder. The direction in which the raw material moves is the feed direction, which is specified by the user.

Bend axis: The bending operations require an axis about which material is bent. The direction of this axis subdivides the operations in this category into three different types.

• i. Feed_parallel. The bend axis is parallel to the feed direction, as is the case with bends 1 and 2 in Fig. 9.

• ii. Feed_perpendicular. The bend axis is perpendicular to the feed direction, as is the case with bends 3, 4 and 5 in Fig. 9.

• iii. Feed_inclined. The bend axis makes an angle which is neither 0 nor 90 degrees with the feed direction as is the case with bend 6 in Fig. 9.

Fig. 9

Different types of operations in the bend and deform category

3.2.2 Grouping rules

Three heuristic rules are used to check if operations can be clubbed together to form a group and they are discussed below.

Rule 4: Co-axial bends

This rule groups together bending operations whose axes are co-axial. For example, bends 1 and 2 in Fig. 9 form a group as they have a common bending axis.

Rule 5: Parallel bends

Operations to create bends whose axes are parallel but not co-axial can be grouped together provided: (i) the bends are located on opposite sides of the mother plane [7]; (ii) the bends are in the same direction; and (iii) bends have the same number of planar walls between them and the mother plane. In Fig. 9, bends 3 and 4 satisfy these conditions and they can be grouped together.

Rule 6: Related operations

If deform operations associated with composite features have the same characteristics as external bends, then they can be grouped together with these external bends. For example, lances 1 and 2 in Fig. 9 can be grouped together with bends 1 and 2 as they are bent in the same direction as the bends, and their bending axes are parallel to those of bends 1 and 2. For the same reasons, the jog in Fig. 9 can be clubbed together with bends 3 and 4.

3.2.3 Handling criteria and Fuzzy set theory

The handling criteria for establishing the relationships amongst one selected group and the remaining groups are summarised as follows.

Criterion 5: Bend distance

This criterion considers the location of a bend by quantifying its distance in terms of the number of planar faces between it and the mother plane. The further the bend is, the earlier it is performed. In Fig. 9, bends 1–6 are at zero distance as they are directly connected to the mother plane. In Fig. 10, bends 1 and 2 are also at zero distance as they are directly connected to the mother plane, whereas bends 3 and 4 are at a unit distance away from the mother plane.

Fig. 10

A sheet metal part with four bend features

Criterion 6: The number of operations in one group

The more the number of operations in one group, the greater is likely to be its influence on the quality of the part. For example, since this category contains bend and deform operations, any bent/deformed material will decrease the space available for the movement of the tool. Hence, the group which contains more operations is given a lower priority.

Criterion 7: Bend angle

If the bend angle of a feature is greater than 90°, it is bent in two operations. Therefore, bend angles smaller than 90° should be given preference over bends with angles greater than 90°.

Criterion 8: Bend orientation

Bending operations require sufficient space for the tooling in the upper and lower die to move. According to [17], a bend parallel to the feed direction requires a larger space than a bend perpendicular to the feed direction. Therefore, bends which are perpendicular to the feed direction are given priority.

The fuzzy membership functions of these criteria are shown in Fig. 11a–d.

Fig. 11

Fuzzy membership functions: (a) Bend distance, (b) Number of operations in one group, (c) Bend angle, (d) Bend orientation

The weighting factors for the criteria 5 to 8 are 1.2, 0.8, 0.6 and 0.2, respectively [6, 7, 17], making the weighting vector.

$${{W}_{2}}^{T}= \{1.2, 0.8, 0.6, 0.2\}$$

(3)

By weighting ${R}_{2}$ with ${W}_{2}$, the vector ${FV}_{2}$ is obtained the elements of which indicate the priority of the groups in this category.

$${FV}_{2} = {R}_{2}\times {W}_{2}$$

(4)

Note that the size of ${R}_{2}$ is ${n}_{2}\times 4$ where ${n}_{2}$ is the number of groups in category II.

4 Case studies

The sequence planning logic described in the previous sections has been developed as a module in our sheet metal system, which has been developed in C++ and a Windows environment. The input data to this system is a boundary representation model of the sheet metal part in neutral format STEP AP203. The feature recognition module, described in [28], extracts the features and their characteristics, and they serve as input to the sequence planning module. Herein, two industrial parts (see Figs. 12 and 14) are analysed and the sequence plans generated by our logic are validated by comparing the sequence plan for the first part with that described in [30] and the sequence plan for the second part with that currently used by our industrial collaborator.

Fig. 12

Case study I

Case study I

Figure 12a shows the stamped sheet metal part analysed by Tor et al. [30] and its unfolded shape in Fig. 12b. The feed direction of this part is also shown in Fig. 12b.

This part has sixteen features which are mapped into nineteen operations, and these operations are then placed into eight groups, each of which is subsequently placed in either category I or II. The features, operations, groups and categories are discussed below.

• Group 1. Features: pilot holes 1 and 2. Since the part does not have holes which can be used as pilot holes, two pilot holes 1 and 2 are pierced but they are located outside the part in the strip sheet. These pilot holes require piercing operations P₁ and P₂; these operations are related and form a group, which is placed in category I.

• Group 2. Feature: external contour. For this part is assumed the contour of the part is made by two blanking operations C₁ and C₂; they form a group and are placed in category I.

• Group 3. Features: holes 1, 2 and 3, and flanged holes 1 and 2. The holes are mapped into piercing operations H₁, H₂ and H₃ which are then grouped together. The manufacture of a flanged hole requires two operations, i.e. a piercing operation (FP) and a deform operation (FD). The two piercing operations FP₁ and FP₂ are related not only to each other but also to operations H₁ to H₃. Hence, this group consists of five operations, and it is placed in category I.

• Group 4. Feature: slot 1. This feature also requires only one operation S₁; since it is not related to any other operation, this one-operation group is placed in category I.

• Group 5. Features: embosses 1, 2 and 3. Each emboss requires only one operation. The three operations, E₁, E₂ and E₃ are related and form one group which is place in category I.

• Group 6. Features: flanged holes 1 and 2. The two deform operations FD₁ and FD₂ required for these flanged holes are related and they form one group which is placed in category I.

• Group 7. Features: bends 1 and 3. These two bends are on opposite sides of the mother plane and since both point upwards, the two operations required to form these features, i.e. B₁ and B₃, form a group which is placed in category II.

• Group 8. Features: bends 2 and 4. These two bends are also on either side of the mother face and they point upwards. The corresponding bending operations, B₂ and B₄ form a group which is placed in category II.

Once the groups are formed, the coefficients of fuzzy matrices are calculated by determining the four membership grades for each of the groups. The resulting fuzzy matrices ${R}_{1}$ and ${R}_{2}$ are shown in Tables 3 and 4 respectively. Referring to the coefficients in the first column of ${R}_{1}$, which indicate the importance of feature location, the three operations in group 5 are given a value of 1 because the embosses are located on (i.e. “is-on”) the surface of the part whereas the piercing operations in groups 1, 3, 4 and 6 are given a value of 0.8 because the holes are “is-in” the part.

Table 3 Fuzzy matrix ${R}_{1}$

Table 4 Fuzzy matrix ${R}_{2}$

The second column of ${R}_{1}$ indicates the importance of the total processing length of the operations in the group. Since group 2 has the longest processing length of 997.92 mm, it is given a value of unity, whereas group 1, with the shortest length, is given a value of zero. The values for the other groups are obtained by linear interpolation. Table 5 gives the total processing length for each of the six groups.

Table 5 Total processing length of operations in each group

The coefficients in the third column of ${R}_{1}$ indicate the importance of the number of operations in a group. Group 3 has the highest number of operations and is therefore given a value of zero, whereas group 4, with one operation, has a unit value. Groups 1, 2 and 6 with two operations each are given a value of 0.75 and group 5 a value of 0.5.

For criterion 4, all the groups, except group 6, have a membership grade of one since all the operations in these groups are of the shearing type. Group 6 is given a value of zero as it consists of two deform operations for the flanged holes.

The determination of the coefficients for fuzzy matrix ${R}_{2}$ is relatively straightforward as there are only two groups. The bends in both groups are bent by 90° (criterion 7), are directly connected to the mother plane (criterion 5) and each group requires two bend operations (criterion 6). The only difference between the groups is that the bends in group 7 are in the feed direction whereas those in group 8 are in the perpendicular direction (criterion 8).

Next, the fuzzy matrices ${R}_{1}$ and ${R}_{2}$ are weighted by ${W}_{1}$ and ${W}_{2}$ to give ${FV}_{1}$ and ${FV}_{2}$, respectively (see Table 6). The higher the value of ${fv}_{i}$, the earlier the group operations should be performed. Since ${fv}_{5}$ has the highest value, operations in group 5 should be performed first. But according to rule 3, the pilot holes should also be stamped in the first station; therefore, operations in groups 1 and 5 should be performed at the first station. The slot in group 4 should be performed next, followed by the five holes in group 3. The deform operations in group 6 are performed at the fourth station, and the external contour is formed at the fifth. The bending operations B₂ and B₄ in group 8 have a higher final value because of their orientation and are performed at station 6, whilst operations B₁ and B₂ are done at the seventh station. Finally, the part is cut off at the eighth station.

Table 6 Final sequence plan for case study I

Figure 13a shows how the shape of the part has evolved, and this part evolution is compared with that of Tor et al. in Fig. 13b [30]. With respect to the number of stations, both results compare favourably, with our logic suggesting one more station. In our solution, the piercing operation to create the slot is the only operation performed at station 2, whereas in Tor et al.’s solution, the slot-making operation is combined with the operations that make the deformed part of the flanged holes.

Fig. 13

Strip layouts for case study I: (a) heuristic method, (b) plan proposed by Tor et al. [30]

Another difference between the solutions is with the operations that form the contour of the part. In our solution, the contour is formed using two operations at one station, whereas Tor et al. use five operations at two different stations. These differences aside, there are similarities as well. In both solutions, the group with pilot holes and embosses, and the group containing five hole-making operations are identical. Also, the groups containing bending operations are identical, although their sequences are slightly different. This plan may need modification to satisfy other manufacturing constraints such as tool collisions and the available tonnage. As it stands, it provides the manufacturing engineer with an initial sequence plan which could be improved upon.

Case study II

The second example is a slightly modified version of a part currently manufactured by our industrial collaborator. Figure 14a shows this part which is used in the electronic industry. The unfolded part and the user-defined feed direction are shown in Fig. 14b. This part has thirteen features which are mapped into twenty operations. The different grouping of these operations and the category they are placed in are discussed below.

• Group 1. Features: holes 1 and 2. This part contains two holes which can be used as pilot holes, and they require two operations H₁ and H₂ which are related and hence grouped together. This group of operations is placed in category I.

• Group 2. Features: jogs 1, 2, 3 and 4. There are four jogs, and each jog requires a piercing operation (denoted as JP) and a deform operation (JD). Operations JP₁ to JP₄ are related and therefore grouped together. This group of operations is placed in category I.

• Group 3. Features: jogs 1, 2, 3 and 4. Operations JD₁ to JD₄ are required for the deformed part of the jogs; these operations are related and therefore form a group. This group of operations is placed in category I.

• Group 4. Feature: external contour. The presence of a bridge-carrier divides the material to be removed between the blank and the part into four pieces. The fuzzy set theory solution is unaware of the facts that the part is symmetric, and that integrated tooling can be used. Assuming that one piece of material is removed by one blanking operation, this group contains four blanking operations, C₁ to C₄. This four-operation group is placed in category I.

• Groups 5–9. Features: bends 1–6. The part has six bends, and each bend requires one bending operation (denoted as B₁–B₆), but only two of them, i.e. B₃ and B₄ are related. These two bending operations are on opposite sides of the planar wall forming the base of the U-shaped channel; both these operations can be grouped together (group 7) because they are equidistant from the mother plane. B₁, B₂, B₅ and B₆ are not related to each other and therefore four groups (groups 5, 6, 8 and 9) are formed, each containing one bending operation. These five groups are placed in category II.

Fig. 14

Case study II

The total processing lengths for the groups are given in Table 7 and the fuzzy matrices ${R}_{1}$ and ${R}_{2}$ for this case study are shown in Tables 8 and 9 respectively. Whilst determination of most of the membership values is self-explanatory, those in column 1 of ${R}_{2}$ need some explanation. Bend 6 which is furthest away (3 planar faces) from the mother plane has a fuzzy value of 1, whereas bends 5, 2 and 1 (in groups 8, 6 and 5 respectively) are 2, 1 and 0 planar faces away from the mother plane. This results in the operations in these groups being given fuzzy values of 0.67, 0.33 and 0 respectively.

Table 7 Total processing length of operations

Table 8 Fuzzy matrix ${R}_{1}$ for case study II

Table 9 Fuzzy matrix ${R}_{2}$ for case study II

The final membership values of these groups are shown in Table 10, from which it is clear that the pilot holes are pierced first, followed by the piercing operations to create the shape of the four jogs. The bent parts of the four jogs are formed at station 3 and the external contour of the part at station 4. The bending operation in group 9 is performed next.

Table 10 Final sequence plan for case study II

Figure 15a and b show the sequence plans obtained from the proposed method and the plan provided by our industrial partner, respectively. There are several differences between the two sequences. In our solution, the three operations which result in the pilot holes, the cut part of the four jogs and the external contour of the part are performed at the first, second and fourth stations, whereas in the industrial solution shown in Fig. 15b, these three operations are performed at the very first station. This difference is because, in our solution, the size of the tools has not been considered and it was assumed that if only one group of operations is performed at a station, the problem of tool collisions will not arise. Furthermore, we assumed the contour of the part was made by one group of four operations, whereas in the industrial solution advantage is taken of part symmetry by having only one group of two operations. This difference leads to a lower membership value for group 4 (which forms the contour) than the group making the cut part of the four jogs. If our logic had considered part symmetry and used two operations to make the contour, operation groups 1, 2 and 4 would have been performed at the first station, thus making it identical to the industrial solution.

Fig. 15

Strip layouts for case study II: (a) heuristic method, (b) industrial solution

The second difference is in the formation of the external contour. In both solutions, the width of the strip is equal to that of the part, requiring only the top and bottom edges of the part to be formed. In the industrial solution, advantage is taken of the fact that the part is symmetric, and both the top and bottom edges are stamped to form the two lugs in one operation between the first and second stations. In our fuzzy set system, it is not possible to detect and take advantage of geometric symmetry.

The third difference is the sequence in which the bending operations are performed. The industrial solution again takes advantage of the fact that, when two parts are placed next to each other, bend 4 of one part is very close to bend 3 of the other part, thus making it possible to combine operations B₃ and B₄ by using one integrated tool. This combined operation is performed between stations 2 and 3. But in our solution, B₃ and B₄ are two distinct operations, and they are performed at station 8. This causes difficulty in manufacturing these two bends since they are connected to a planar wall that has been already bent vertically up.

Another difference is the sequence in which the bending operations B₁, B₂, B₅ and B₆ are performed. In the heuristic solution, the bend furthest away from the mother plane is given priority which means that the sequence is B₆, B₅, B₂ and B₁. Although the industrial solution performed B₆ before the other three bending operations, it has a different sequence compared to our solution. Since the width of the U-shaped channel is small and there are jogs, it prevents the insertion of any tool to form the channel in one operation. The industrial solution forms bend 2 completely but only half of bend 1 at station 5. The reason for bending it by only 45° is to have enough space for another tool to complete bend 1. It is only after this that bend 5 is formed.

The above case studies have demonstrated that the plans generated by the heuristic method bear a good degree of similarity with the plans suggested in [30] and our industrial collaborator. It can be assumed that the latter plans are near-optimal. The use of features allows not only a feature to be mapped into one or more operations but also facilitates the grouping together of related features. It is this grouping of the features and their corresponding operations which makes the heuristic sequence plans similar to the near-optimal plans. For example, the three embosses in case study I are recognised as being related, and the operations to form them are performed at one station, which is the same as that suggested by Tor et al. [30]. Furthermore, there is fairly good agreement between the two sets of plans regarding the number of stations required, with the heuristic method always suggesting one/two more stations. These additional stations could be dispensed with if the heuristic method is combined with a geometric processor, which would detect if the part were symmetric and there are no tool collisions. If, for example, the geometric processor was able to establish that the part is symmetric and that there are no tool collisions, then the operations performed at stations 1, 2 and 4 in case study II could be combined together, thus reducing the number of stations by two.

The advantage of the heuristic method is that it is able to generate a sequence plan without any user intervention in a few minutes, a task which would take even an experienced engineer a few days to complete. The only difference is that the latter is likely to be optimal, whereas the former may require minor modifications for it to become near-optimal.

5 Conclusion and future work

This paper presents a heuristic method that automatically generates sequence plans for sheet metal parts manufactured using progressive dies. From the case studies discussed in this work, the following conclusions can be drawn:

• i. The heuristic method tends to suggest more workstations than the near-optimal solution.

• ii. Whilst the sequence plans generated by the heuristic method are not as good as the near-optimal solutions, they suggest a similar grouping of operations.

• iii. To generate near-optimal sequence plans, the heuristic method needs to be coupled with a geometric processor to detect part symmetry and tool collisions.

Since the heuristic sequence plans are quite similar to the near-optimal solutions, an alternative to (iii) would be for an experienced engineer to use the heuristic sequence plan as a starting point and achieve a near-optimal solution by making minor modifications.