Multi-Piece Mold Design Based on Linear Mixed-Integer Program Toward Guaranteed Optimality

INTERNATIONAL CONFERENCE ON MANUFACTURING AUTOMATION (ICMA200) Multi-Piee Mold Design Based on Linear Mixed-Integer Program Toward Guaranteed Optimality Stephen Stoyan, Yong Chen* Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA 90089, USA *Corresponding author: yonghen@us.edu, (23) 740-7829 Abstrat Multi-piee molds are a type of molding tehnology, whih onsist of more than two mold piees and are assembled/dissembled lie a spae puzzle. Based on suh molds, omplex parts an be made for limited run prodution. Compared to traditional two-piee molds, parts with muh more omplex geometries an be made; however, this also brings hallenge in designing suh multi-piee molds. Previous wors to address the problem are all based on heuristis. In this paper, we present a multi-piee mold design framewor based on linear mixedinteger program. In our method, multi-piee mold design with guaranteed optimality on the number of mold piees an be generated for any given CAD model of a molded part. The formulation of multi-piee mold design as a linear mixed-integer program is presented. The related multi-piee mold design framewor is disussed. Some examples are provided whih illustrate the effetiveness and effiieny of our approah. Keywords Computer-aided design; deision support; rapid tooling; linear mixed-integer programming Bottom View Injetion Molded Part Top View. INTRODUCTION Multi-piee molds suh as spae puzzle modeling developed by Protoform GmbH (www.protoform.de), an have more than two mold piees. In the injetion molding proess, mold piees are first hand-loaded into a mold base mounted on the injetion molding mahine. During part injetion and ooling proess, the mold piees are aurately and seurely lamped into the holding devie. Finally eah mold piee is hand-removed from the mold base to release the injetion molded part. An example of multi-piee molds and related injetion molded part provided by Protoform GmbH is shown in Figure. As shown in the figure, eah mold piee an have its own parting diretion (PD), along whih, the mold piee an be separated from the part. () Compared to the traditional two-piee molds, multi-piee molding an produe limited run prodution parts with more omplex geometries. (2) Compared to rapid prototyping (RP) proesses suh as Stereolithography Apparatus (SLA), multi-piee molding an effiiently produe a small quantity of parts (e.g. 00); more importantly, the fabriated parts an be made in desired injetion molding materials that may not be available to RP proesses. Hene multi-piee molding has beome an important tooling tehnology in the era of mass ustomization, in whih limited run prodution is inreasingly beoming a ommon industrial pratie. Molded Piees Fig. : An example of multi-piee molds given by Protoform GmbH. Given the geometry of a part, depending on the seletion of mold design variables, a different number of mold piees may be required to form the part. It is desired to minimize the number of required mold piees beause fewer mold piees redues the tooling ost and simplifies the operation of the mold. Therefore, in this paper, we will onsider the automation of multi-piee mold design problem defined as: Problem MPMD: Multi-Piee Mold Design. Given a solid part and a mold base, design the minimum number of mold piees that an form the avity of the part in the material injetion proess, and an be disassembled properly in the part ejetion proess.. Related Wor The automation of mold design for injetion molding has been extensively studied before. Some representative wor an be found in [~6]. Most of the wor fouses on two-piee molds, inluding the determination of parting diretion, parting line, parting surfae, and underut deteting. Among them, the seletion of parting diretion has reeived muh

attention sine it is an important step in the automati mold design proess. Reently, new programmable graphis hardware aelerated algorithms have also been presented to test the moldability of parts and help in redesigning them [7]. Chen and Rosen [8, 9] first presented a multi-piee mold design method that allows three-dimensional mold deomposition. More reently, Gupta s group presented a set of geometri algorithms for automated design of multi-piee permanent molds [0], sarifiial multi-piee molds [], and multi-stage molding [2]. These wors provide an exellent groundwor on multi-piee mold design to improve upon. One ey problem of the urrent approahes is that the multi-piee molds are designed based on heuristis. Hene no optimality an be guaranteed for a given arbitrary geometry. For example, in designing multi-piee permanent molds [0], a set of andidate parting diretions are seleted from () prinipal diretion, (2) planar fae normal, and (3) ylindrial/onial fae axis. Based on them, a greedy sheme is then used in identifying the minimum number of mold piees. In this paper, a multi-piee mold design method based on linear mixed-integer program is presented. In our method, a lower and upper bound on the number of mold piees are first identified for a given part geometry. Based on them, heuristis an then be used in further improving the generated design. Our design framewor, for the first time, provides a solid foundation in pursuing multi-piee mold design with guaranteed optimality..2 Problem Formulation Before disussing our approah, we first present a more aurate formulation of the aforementioned Problem MPMD as follows. Problem MPMD: Multi-Piee Mold Design. Given a polygonal model (P) and a mold base (Ψ), design a n-piee mold M = { m, m2,, m n } to: Minimize: number of mold piees (n). Subjet to: () Eah mi M is a onneted solid; (2) Eah mi M has a parting diretion d i suh that m i an be dissembled along d i ; (3) M n = m satisfies M = Ψ P. i The multi-piee mold design problem onsidered in this paper is essentially the same as the ones defined in [8] and [0]. 2. MULTI-PIECE MOLD DESIGN APPROACH For a part to be moldable, every faet on the part needs to be aessible from at least one diretion. Usually suh a diretion an be easily found for eah individual faet; however, the ore hallenge in mold design is to find the minimum number of diretions that are ommon to all the faets of the part. 2. Mold Design Based on Visibility of Faes Chen et al. [3] formulated demoldability as a visibility problem and presented a set of important omputational tehniques suh as visibility and Gaussian maps (V-Maps and G-Maps). For a planar surfae F, its V-map is a hemisphere entered on the unit outward normal. By alulating the intersetions of all V-maps of region faes, allowable draw ranges an be omputed. Several approahes and algorithms based on spherial polygons have been presented for different appliations [4-5]. However, the algorithms and related data strutures are rather ompliated. In addition, it taes onsiderable omputational time to ompute the exat V-map intersetions based on spherial surfaes. Instead, Chen and Rosen [8] presented an alternative approah based on linear program for evaluating the parting diretions of a set of surfaes. Suppose a onneted region R onsists of fae F i (with unit fae normal N i and area A i, i n). If a ommon diretion d(d x, d y, d z ) exists suh that every faet F i an be aessible from suh a diretion, d is a andidate parting diretion of the region. In addition, the ease of ejetion an be used as the riterion to hoose an optimal parting diretion from a feasible range. For a fae F i, its ease of ejetion an be determined by the draft angle and the area in shear ontat with the related part fae during the moldopening operation (i.e. Ai( Ni d) ). Hene an optimization problem for determining a parting diretion of region R an be formulated as follows. Maximize: n A( N d) i Subjet to: N d N d N d 0 i + + for fae F i xi x yi y zi z d + d + d = (sphere onstraint) 2 2 2 x y z A unit sphere related to the sphere onstraint an be approximated by a set of linear surfaes with aeptable errors. Suppose the equations of a planar surfae M are Mxldx + M yldy + Mzldz = μl with fae normal (M xl, M yl, M zl ) toward the inside. A linear problem an be formulated for evaluating a parting diretion of the region. Problem PDLP: Parting Diretion Linear Problem. n Maximize: Ai( Ni d) (2.) Subjet to: N d + N d + N d 0 for fae F i (2.2) xi x yi y zi z Mxldx + M yldy + Mzldz μl for fae M l. (2.3) A linear program problem in 3 dimensions an be solved in O(n) time (n is the number of onstraints) and on linear storage. Therefore, the running time to solve Problem PDLP is very fast, typially in milliseonds for thousands of faes based on a ommerial solver (e.g. LINGO system - www.lindo.om). More importantly, the omputation proess of finding a parting diretion for a set of faes beomes muh easier and more robust. 2

Motivated by suh a linear program approah, we further develop a linear mixed-integer program for determining a minimum number of parting diretions for all the fae of a given part. The optimization formulation is disussed in Setion 3. A simple example (Test in Setion 6) is shown in Figure 2 to illustrate the objetive in Problem PDLP, whih is also used in the linear mixed-integer program. For a simple ube, there is an infinitely number of parting diretions that an be used in its mold design. However, onsidering the ease of ejetion, the two diretions, d and d 2 as shown in the figure, are the most desired. In suh diretions, the projetion area n Ai( Ni d + Ni d2 ) is also the maximum. Fig. 2: A simple example to illustrate the optimization objetive in PDLP. 2.2 Basi Elements for Multi-Piee Mold Design In Problem PDLP, a single fae F i is used as the basi element in evaluating a feasible parting diretion. However, suh a fae is not a good element in identifying a minimum number of parting diretions in multi-piee mold design. This is due to the aessible diretion to a fae may be bloed by its neighboring faes (e.g. two neighboring faes between a onave edge); however, suh bloage is hard to be inorporated if faes are direted used as the basi element. As shown in Figure 2, any pair of diretions an be used in mold design if the given part ontains only onvex edges. Based on suh an idea, Chen et al. [3] omputed a set of poets based on the onvex hull of an objet to apture all the non-onvex regions. Aordingly, the visibility map of eah poet an be omputed and the parting diretions to maximize the number of ompletely visible poets an be determined. Top View Bottom View Fig. 3: An example of poets for a test part. An example of poets for a test part (Test 3 in Setion 6) is given in Figure 3. The four poets of the part are shown in different olors. It an be seen that a poet is a molding feature whih ontains multiple onave and onvex edges. As disussed in [8], poets give us less design freedom in trying different ombinations of elements; hene they are also not good elements for identifying a minimum number of parting diretions in multi-piee mold design. As disussed in [8], onave edges (i.e. the dihedral angle of two neighboring faes is bigger than 80 o ) an apture the bloage of aessible diretions between neighboring faes. Based on suh edge lassifiation, two types of elements an be defined as follows. Definition 2.. A fae F is a onvex fae if all edges of F are onvex edges. Definition 2.2. A Conave Region is a set of onneted faes suh that: () all internal edges of the region are onave edges; and (2) all boundary edges of the region are onvex or flat edges. As an example, the onave regions for the test part in Figure 3 are shown in Figure 4. There are a total of 2 onave regions whih are shown in different olors; in addition, there are 68 onvex faes whih are drawn in wireframe in the figure. Hene, ompared to poets, the design freedom based on them for identifying a minimum number of parting diretions has been signifiantly inreased. Top View Bottom View Fig. 4: An example of onave regions and onvex faes for a test part. Similar to [8], onave regions and onvex faes are used as the basi elements in our multi-piee mold design method. A similar strategy has also been adopted in their generation. However, different from [8] that is based on a 3D geometri modeler (ACIS from Spatial www.spatial.om), our urrent method is based on polygonal meshes (i.e. the input model is defined in a STL file). Hene the approah in omputing onave regions and onvex faes is aordingly modified as follows by adding a onept of Convex_Edge_Vertex. () Classify all edges as onave, onvex and flat. (2) Assign two faes to the same region if they share a onave or flat edge. (3) Identify all the internal onvex edges of eah region and mar the verties of suh edges as Convex_Edge_Vertex. Notie we do not want a region to have internal onvex edges (i.e. all the internal edges should be onave as shown in Definition 2.2). (4) Regenerate regions by assigning two faes to the same onave region if they share a onave edge, or a flat edge if suh a flat edge has no vertex that is mared as Convex_Edge_Vertex. 3

2.3 Approah Overview Based on the identified basi elements (a set of onave regions and onvex faes), the essene of multi-piee mold design is to try a different ombination of them, and aordingly identify a design with the best performane (i.e. the minimum number of mold piees and the easiness of ejetion if the same number of mold piees is ahieved). There have been various approahes to solve suh ombination problem between elements (e.g. the well-nown napsa problem). General solution methods involve: (i) searhing all possible ombinations of variables, (ii) using heuristis suh as greedy heuristi, or (iii) optimization methods suh as utting planes or the branh and bound method. Eah solution method has advantages and disadvantages. Searhing all ombinations, for example, has osts with respet to solution time, espeially with large-sale problems where validating a solution may tae years. Heuristis typially generate fast solutions, but they do not guarantee optimality. Optimization methods an guarantee optimality and use tehniques that onverge faster than searhing the whole solution spae; however, they generally perform slower than heuristis. When models involve integer variables (suh as the one in the next setion), the problems with the various solution methods desribed above get even worse. Although optimization methods have their trade-offs, depending on the omplexity of the problem, urrent state-ofthe-art solvers suh as CPLEX from ILOG (www.ilog.om) an solve large-sale problems in reasonable time. As disussed in Setion, the previous multi-piee mold design methods are based on heuristis. They are effetive but do not provide optimality and also may fail for geometries that have not been onsidered when generating suh heuristis. In this paper, we formulate the Problem MPMD into a linear mixed-integer program and solve it based on optimization methods. As shown in Setion 6, by using CPLEX, suh a problem an be solved within 0 seonds for optimal solutions of the four test problems. An overview of our method based on an example (Test 2 in Setion 6) is shown in Figure 5. () A given part to be injetion molded is shown in Figure 5.a. The part has 64 triangles. They an be lassified into 3 onave regions (drawn in different olors) and 40 onvex faes (drawn in wireframe) as shown in Figure 5.b. (2) Based on suh elements, a linear mixed-integer program is formulated and programmed in CPLEX. Aordingly two parting diretions, d = (0, 0, ) and d 2 = (0, -, 0), are identified by the optimization solver. (3) The optimization solution provided us a lower bound on the solution (i.e. n=2). Based on them, all the onave regions and onvex faes an be ombined into two mold piee regions (shown in two different olors in Figure 5.). Sine multiple solutions may exist, a best one may be identified based on molding design nowledge. (4) Based on the generated mold piee regions and related parting diretions, parting lines and parting surfaes an be identified. Aordingly two mold piees, M and M 2, an be onstruted (refer to Figure 5.d). Notie there is a one to one orrespondene between the mold piees and the mold piee regions. Also the two mold piees an be assembled and disassembled properly in the related parting diretions. Additional loing features an be added in the mold piees. In this paper, we will mainly fous on the proess of generating mold piee regions from onave regions and onvex faes. Aordingly, the remainder of the paper is organized as follows. In Setion 3, we present the optimization formulation for Problem MPMD. In Setion 4, we disuss the fae onnetivity of mold piee regions based on the identified part diretions. The further improvements of the mold design are disussed in Setion 5. The test results are disussed in Setion 6. Finally, we give onlusions in Setion 7. (a) () (d) M d d 2 (b) Fig. 5: An overview of our method. M 2 3. OPTIMIZATION FORMULATION FOR COMPUTING PARTING DIRECTIONS Before getting into the intriate details of the problem, we first outline the deision variables and parameters assoiated with the model. To begin we define: n: the total number of parting diretions of mold piees; 4

p: the total number of onave regions; m: the total number of approximated spherial surfaes; and their respetive sets as: φ :={i: i [,n]}: the set of parting diretions; ψ := {: [,p]}: the set of onave regions; θ := { : [,m]}: the set of spherial surfaes. In addition to the number of onave regions, there also exists a number of surfaes assoiated with eah region, whih do not neessarily have the same length. Thus, we define the set of surfaes assoiated with eah onave region and introdue the set of onvex surfaes as: β : the set of surfaes in eah onave region =,...,p; β : the set of onvex surfaes. The deision variables involved in the optimization problem are as follows: d i =[d i x,d i y,d i z ]: the vetor of parting diretions for mold piees,...,n; g i (): the binary variable used to enfore onstraints related to parting diretions,...,n for =,...,p onave regions; g i( β ): the binary variable used to enfore onstraints related to onvex surfaes for parting diretion,...,n; z i : a ontinuous variable used to satisfy the absolute value funtion in the objetive. Finally, the parameters involved with the deision variables in the design are defined as: N(, β )=[N x (, β ),N y (, β ),N z (, β )]: the matrix with β surfaes related to onave regions =,...,p; N ( β )=[N x( β ),N y( β ),N z( β )]: the matrix of onvex faes with β surfaes; M =[ M x, M y, M z ]: the vetor defining approximated spherial surfaes =,...,m; A(, β ): the area of onave regions with β surfaes for =,...,p; A ( β ): the area of onvex regions with β surfaes; u : a salar value related to the approximated spherial surfaes =,...,m; L: the lower bound of parting diretion for mold piees; U: the upper bound of parting diretion for mold piees. The design desribed in the earlier setions requires the solution to two optimization problems. Given a set of onstraints that define the mold we want to reate, the first problem (I) entails finding the minimum number of vetor parting diretions d i neessary to design the mold. After we now the minimum number of parting diretions, the seond problem (II) involves maximizing the surfae area overed in forming the mold piees. The first optimization problem (I) is the following: After solving the problem above, we let n be the number of d i 0 in the solution to (3.)-(3.). Then, the seond optimization problem (II) is: The onstraints are the same for both problems sine they define the boundaries of the mold to be reated. Constraints (3.2)-(3.3) and (3.4)-(3.5) ensure that at least one parting diretion vetor d i satisfies the desired inequality and the rest an be turned ``off" via the binary variables g i () and g i( β ); respetively. The problem with both (3.)-(3.) and (3.2)- (3.3) is the objetive funtions and onstraints are nonlinear and also involve binary variables g i () and g i( β ). However, the NonLinear Mixed-Integer Program (NLMIP) of (3.)-(3.) and (3.2)-(3.3) an be onverted to an equivalent linear program with the introdution of a deomposition strategy and a few additional variables and onstraints. The linear transformation allows the problem to be muh more tratable and easier to solve. Fig. 6: Problem (I) subproblem deompositions for variables d i. 5

We now desribe the linear transformations and deomposition strategy used to mae the problem more tratable. In the objetive of problem (I), the NLMIP minimizes the number of variables d i uses in the solution. Sine we are looing for the lowest number i that satisfies (3.2)-(3.), we deompose problem (I) into subproblems that inrease in size by one variable d i. As shown in Figure 6, problem (3.)-(3.) is solved for the number of variables orresponding to eah subproblem until a solution is generated, in whih ase we stop. Given the nature of the problem, we now that at least two variables will be needed to generate a solution, thus subproblem () begins with d and d 2. Then we stop after the first instane when subproblem (i n) obtains an optimal solution. The objetive of problem (II) aims to maximize the surfae area assoiated with the absolute value of the diretion vetor and the orresponding surfae region. This has a linearly equivalent set of equations by introduing the following: This equates to solving the following linear Mixed-Integer Program (MIP): This is also done for N ( β )d i where z i is used, whih is shown in equations (3.22), (3.25) and (3.26) below. The nonlinear onstraints in the problem an also be addressed in a similar fashion, whih mae the problem more tratable and less omplex. The nonlinear onstraints in (3.2) and (3.4) an be onverted into linear onstraints by using the following equation: where Λ is a large value. Here, when g i () is equal to zero then this onstraint is essentially turned ``off" sine the large Λ value will satisfy the inequality for any d i. When g i () is equal to one then N(, β )d i 0 must be satisfied, whih is the desired funtion of onstraint (3.2). This is again repeated for g i( β )N ( β )d i, as is shown below in equation (3.29). Finally, the upper and lower bound onstraints of (3.7) and + (3.8) an be linearly expressed by defining d i =d i - d - i, d i+ 0, + d i- 0. Hene, d i and d - i are simply the positive and negative omponents of d i ; respetively. Then, satisfying the absolute value of the onstraints is provided by the following equations: where δ i i φ. Thus, the original problem of solving problem (I) and (II) separately an now be done in one equivalent linear problem. where (3.22)-(3.43) is solved using an inreasing number of parting diretions d i until the first instane that generates a solution, as desribed above and shown in Figure 6. 4. CONNECTIVITY OF MULTI-PIECE MOLD DESIGN As disussed in Setion 2.3, the formulated linear Mixed- Integer Program an be programmed in a CPLEX optimization solver. Advaned MIP algorithms based on methods suh as branh and bound an be used in finding an optimized solution. One benefit we gain based on our approah is that we an easily inorporate different draft angle requirements in our problem. That is, for a part to be injetion molded, its surfaes that are parallel to the paring diretion must be drafted at least an angle γ in order to ease the ejetion of the part and redue the damaging possibility of the part and molds. The minimal draft angle mainly depends on the molding proess and material. For some molding proesses suh as urethane rubber molding, a zero or even slightly negatively draft angle are aeptable. 6

Fig. 7: Different draft angle requirements. For a minimal draft angle γ, we an ompute τ = sin( γ ) and use it to replae 0 in Equations (2.2) and (3.27). Hene our optimization formulation an be hanged as: N d + N d + N d τ for fae F i, and xi x yi y zi z Λ( gi( )) + N(, β) di τ, i φ, ϕ. Aordingly, the omputed optimization solution, if any, will satisfy the given draft angle requirements. In addition, any non-drafted or under-drafted surfaes an be deteted if no solution is found for them. Notie, however, in our MIP formulation, only the demoldability requirement has been inorporated. We had diffiulties in onverting the fae onnetivity of a mold piee region into omputable formulation; hene suh onnetivity has not been inorporated. Hene, the optimal solution given by solving the MIP is atually a lower bound on the multi-piee mold design. That is, if without onsidering the onnetivity of mold piee regions, the minimum number of mold piees is found to be n (e.g. n= 5), it is impossible to find a better solution (i.e. n < 5) after the onnetivity of n mold piee regions has been onsidered. An example of suh onnetivity problem is shown in Figure 8. For a test part as shown in Figure 4, a minimum of three parting diretions has been identified after solving the MIP (i.e. d, d 2, and d 3 as shown in Figure 8). Aordingly three mold piee regions (m, m 2, and m 3 ) are required for them respetively. However, for a onave region (CVR ), its solution is d sine it has the biggest projetion areas in suh a diretion; nonetheless, CVR is not onneted to other regions that use the same parting diretion. 5. REFINING MULTI-PIECE MOLD DESIGN Based on the omputed parting diretions, we an eep on refining the generated mold piee regions suh that a mold design an be ahieved that is loser to the lower bound (n) instead of the upper bound (n+m). This an be done based on various heuristis. For example, we an ompute the projetion areas of eah onave region for the given parting β diretions (i.e. ). An example of suh results is Ai( Ni d) shown in Table for the test ase in Figure 8. In the table, a mar is assigned to a region if a related parting diretion annot satisfy the demoldability of the region. Parting TABLE THE PROJECTION AREA OF NINE CONCAVE REGIONS FOR TEST 3 Conave Region # (9 out of 2) Dirs. 0 2 3 4 5 6 7 8 d (0,0,-) 0.3 4 5 5 4 d 2 (,0,0) d 3 (-,0,0) 0 0 0 0.5 0 2.5 0 0 0 0.5 0 2.5 Hene, for CVR that is identified as isolated region in diretion d, we an he the other diretions d 2 and d 3. Sine they are both demoldable and onneted, we an reassign CVR to another mold piee region instead of adding a new one. Notie, as shown in Table, there are several onave regions that have a unique assignment to a related parting diretion. For example, CVR 3, CVR 4, and CVR 8 an only be assigned to d 3, d 2, and d respetively. We all suh onave regions as the ore onave regions of the related mold piee regions. Their assignments will not be hanged during the refining proess while other onave regions and onvex faes may be. In addition, we an also ompute a onnetivity table of all the hangeable onave regions and onvex faes based on the ore onave regions. In addition to demoldability and fae onnetivity, a wealth of mold design and manufaturing nowledge has been haraterized into a set of heuristis [~2]. These heuristis an also be onsidered in the proess of refining mold piee regions. For example, it is desired to have a smooth parting line. Hene for a mold design result generated by our system for test 3, whih is shown in Figure 9, we an further refine it by hanging the assignment of some faes for ahieving a smoother parting line (refer to red lines as shown in the figure). Fig. 8: An example of onnetivity of mold piee regions. Based on a set of given parting diretions, it is trivial to he the fae onnetivity between onave regions and identify all the isolated regions (suppose m of them are identified). Aordingly we have an upper bound on the multipiee mold design. That is, a solution must be able to found by using n+m mold piees after onsidering both demoldability and fae onnetivity requirements. Fig. 9: An example of inorporating mold design heuristis. 6. EXAMPLES Four test examples are given to illustrate our method. Test, 2, and 3 are shown in Figure 2, 5 and 3 respetively. Test 4 7

is shown in Figure 0. A summary of the omputed results is given in Table 2. The results were generated by solving the model desribed in Setion 3. The running time is based on a 3GHz Intel Xeon CPU using CPLEX 9.0. Test # Total Tri # TABLE 2 EXPERIMENTAL RESULTS OF FOUR TESTS Conave Region # Convex Fae # 2 0 2 2 64 3 40 3 08 2 68 4 528 86 250 Resulted Parting Dir. # d, d 2 (0, 0, ), (0, 0, -) d, d 2 (0, 0, ), (0, -, 0) d,d 2,d 3 (0, 0, -), (, 0, 0), (-, 0, 0) d,d 2,d 3, d 4 (0,, 0), (0, 0, -), (-, 0, 0), (, 0, 0) Running Time (se) 0.0 0.05 0.3 8.5 The largest problem we solved is test 4 whih has 86 onave regions and 250 onvex faes. CPLEX solved the linear MIP to optimality with a CPU time of 8.5 seonds inluding the omputation of both (d, d 2 ), (d, d 2, d 3 ) and (d, d 2, d 3, d 4 ). (a) (b) () Top View 86 onave regions d 4 Fig. 0: Sreen aptures of omputed results of test 4. d 3 4 mold piee regions 7. CONCLUSION For the multi-piee mold design of an arbitrary part model, we presented a novel approah by formulating a linear mixedinteger program based on a set of basi elements, onave regions and onvex faes. By using a state-of-the-art optimization solver, suh a problem an be solved for optimal d 2 Bottom View d solutions in reasonable time. Hene the optimality of multipiee mold design an be provided by identifying a lower and upper bound on the number of mold piees. The multiplepiee mold design an be further improved based on heuristis. Four examples were given and the test results have demonstrated the effetiveness and effiieny of our method. ACKNOWLEDGMENT We anowledge Prof. Satyandra K. Gupta at University of Maryland for providing us the part models of test 3 and 4. REFERENCES [] K. Hui, Geometri Aspets of the Mouldability of Parts, Computer-aided Design, 29(3), pp. 97 208, 996. [2] T. Wong, S. T. Tan, and W. S. Sze. Parting line formation by sliing a 3D CAD model. Engineering with Computers, 4(4), pp. 330-343, 998. [3] M. W. Fu, J. Y. H. Fuh, A. Y. C. Nee, Underut Feature Reognition in an Injetion Mould Design System, Computeraided Design, 3, pp. 777 790, 999. [4] Z. Yin, H. Ding, Y. Xiong. Virtual prototyping of mold design: Geometri mouldability analysis for near-net-shape manufatured parts by feature reognition and geometri reasoning. Computer Aided Design, 33(2), pp. 37-54, 200. [5] X. G. Ye, J. Y. H. Fuh, K. S. Lee. Automati Underut Feature Reognition for Side Core Design of Injetion Molds. Journal of Mehanial Design, 26, pp. 59-526, 2004. [6] S. MMains, X. Chen. Finding underut-free parting diretions for polygons with urved edges. Journal of Computing and Information Siene in Engineering, 6(), pp. 60-68, 2006. [7] R. Kharderar, G. Burton, and S. MMains. Finding feasible mold parting diretions using graphis hardware. Computer Aided Design, 38(4), pp. 327-34, 2006. [8] Y. Chen, D. W. Rosen. A region based method to automated design of multi-piee molds with appliation to rapid tooling, Journal of Computing and Information Siene in Engineering, 2(2), pp. 86-97, 2002. [9] Y. Chen, D. W. Rosen. A reverse glue approah to automated onstrution of multi-piee molds, Journal of Computing and Information Siene in Engineering, 3(3), pp. 29-230, 2003. [0] A. Priyadarshi, S. K. Gupta. Geometri algorithms for automated design of multi-piee permanent molds. Computeraided Design, 36(3), pp. 24--260, 2004. [] J. Huang, S. K. Gupta, K. Stoppel. Generating sarifiial multipiee molds using aessibility driven spatial partitioning. Computer-Aided Design, 35(3), pp. 47--60, 2003. [2] A. K. Priyadarshi, S. K. Gupta. Algorithms for generating multistage molding plans for artiulated assemblies. Robotis and Computer Integrated Manufaturing, 32(3/4), pp.350-365, 2009. [3] L. L. Chen, S. Y. Chou, T. C. Woo. Parting diretions for mould and die design Computer-Aided Design, 25, pp. 762 768, 993. [4] T. C. Woo, Visibility maps and spherial algorithms, Computer- Aided Design, 26(), pp. 6 6, 994. [5] S. Dhaliwal, S. K. Gupta, J. Huang, A. Priyadarshi. Algorithms for omputing global aessibility ones. Journal of Computing and Information Siene in Engineering,3(3), pp.200-209, 2003. 8