Design for agile assembly: an operational perspective

INT. J. PROD. RES., 1997, VOL. 35, NO. 1, 157-178

Design for agile assembly: an operational perspective

A. KUSIAKft and D. W. HEf

The concept of agile manufacturing is driven by the need to quickly respond to thechanging customer requirements. Agile manufacturing demands a manufacturingsystem to be able to produce efficiently a large variety of products and bereconfigurable to accommodate changes in the product mix and product designs.The manufacturing system reconfigurability and product variety are critical inagile manufacturing. The concept of agility has an impact on design of assemblies.To implement agile manufacturing, methodologies of design for agile manu-facturing are needed. Design for agile assembly is accomplished by consideringoperational issues of assembly systems at the eariy product design stage. In thispaf>er, three rules applicable to the design of products for agile assembly from anoperational perspective are proposed. These rules are intended to support thedesign of products to meet the requirements of agile manufacturing. Illustrativeexamples are provided to demonstrate the potential of the design rules. Proceduresand algorithms for implementing these rules are presented.

1. IntroductionAgile manufacturing is an emerging concept in industry that aims at achieving

flexibility and responsiveness to the changing market needs. The two importantcharacteristics of agile manufacturing are as follows (Nagel and Dove 1992, Sheridan1993, Kidd 1994, Noaker 1994):

(1) Greater product customization: manufacturing to order at a lower unit cost.(2) Dynamic reconfiguration of manufacturing systems: to accommodate swift

changes in product designs.

Agility is the ability of a company to produce a variety of products of high qualityat low cost. This demands that the manufacturing system be simple and flexible. Theagility concept has an impact on design of products and control of manufacturingsystems.

The design methodologies of incorporating hfe-cycle values of products at theearly design stages have been researched extensively. Besides functional requirements,other life-cycle concerns include manufacturability (Whitney 1988, Barkan et ai1990, Stoll 1991, Gupta et al. 1994), assemblability (Boothroyd and Dewhurst 1983,Boothroyd 1987, Andreasen et al. 1988), recyclability (Chen et al. 1994), quality(Sanchez et al. 1993), reliability (Birolini 1993), schedulability (Kusiak and He 1994),supply chain management (Mather 1987, Lee 1994), and so on. The concepts ofdesign-for-manufacturability (DFM) and design-for-assembly (DFA) are the mostmature. Different approacbes for implementing DFM have been developed. Bralla(1986) and Bolz (1977) proposed design rules based on the empirical knowledgeand experience. Suh (1990) developed a methodology for implementing DFM. The

Revision received March 1996.tintelligent System Laboratory, Department of Industrial Engineering, The University of

Iowa, Iowa City, Iowa 52242-1527, USA.JTo whom correspondence should be addressed.

0020-7543/96 SI2'00 © 1997 Taylor & Francis Ltd.

158 A. Kusiak and D. W. He

methodology is based on the relationships between design variables, processparameters, and functional requirements, expressed in the form of design matrices.The design for assembly method proposed by Boothroyd and Dewhurst (1983)essentially considers parts one by one to simplify them or combine some to reducethe part count, add features, and so on to enable the assembly process.

Most of the design-for-manufacturing literature does not consider constraintsrelated to operations of manufacturing systems, but rather deals with manufacturingprocesses (Kusiak 1994). Valuable as it is, considering manufacturability andassembly alone cannot meet the most fundamental requirements of agility as itdoes not take into account operations of manufacturing systems.

The success of agile assembly is largely determined by the design of products andthe system to manufacture them. Design for agile assembly must be accomplished byconsidering operational issues of assembly systems at the early product design stage.Consequently, design rules should be developed to support the design effort. In thispaper, three design rules for design of products in an agile assembly environmentfrom an operational perspective are proposed, The analytical insights ofthe benefitsof concurrent consideration of design and manufacturing issues are also provided.Algorithms and procedures for implementing these rules are proposed and illustratedwith examples.

2. Design rule 1 (Design for operational requirements): design a product to satisfy themanufacturing operations requirements

Products and components have been traditionally designed without consideringconstraints imposed by a manufacturing system. Design-for-manufacturing opera-tions offers a high potential for improvement of productivity in an agile assemblyenvironment. To reduce the cost of operating a manufacturing system, considerationshould be given to design of products and components.

An example of the printed circuit board (PCB) assembly is used to illustrate therule.

2.1. Component location problem in FCB designThe speed and flexibility of PCB assembly are important in electronics manu-

facturing. A significant percentage of the final cost of a PCB attributed to theassembly process includes component placement/insertion. Due to the wide use ofsurface mount technology (SMT) in electronics assembly, the placement of surfacemount components is crucial. A high-flexibility placement machine is usually used forplacing components for products such as telecommunication equipment or com-puters. The robotic surface mount system works with fixed feeders and fixed boardlocations. In such a system, the robot head picks up components from differentfeeders, then moves and places them on the board. The system is capable of handlingvarious types of components and can be easily retooled and reprogrammed to handledifl'erent boards. This type of a system can be applied to a large percentage ofcommercial placement systems for SMT components.

The assembly operations performed by a tool head in a robotic surface mountsystem are of two types. The placement operation involves moving components fromfeeders to their positions on the board. The second type is the pick-up operation, i.e.the tool head moves from the components' positions to the feeders to pick upcomponents. Consequently, the total time required to finish a board includes theplacement time, i.e. the time required for performing the placement operation and the

Design for agile assembly: an operational perspective 159

pick-up time. i.e. the time required for performing the pick-up operation. The pick-uptime is minimized by solving a placement sequencing problem. The placementsequencing problem can be formulated as a linear assignment problem if the locationsof the feeders and the component's positions on the board are fixed (Ji et al. 1992).

The design of PCB involves four major steps (Byers 1991):

(1) Defining the physical size and shape of the PCB. The resulting board areadefines the maximum number of components that can be placed on the board.

(2) Assigning gates to components that contain multiple gates.(3) Locating components on the board.(4) Routeing the board.

The objective of step (3) in PCB design is to locate components on the board sothat the interconnections form the simplest pattern (Byers 1991). In addition, manyother constraints are considered in locating components, for example (Ginsberg1991):

• Circuit design constraints, e.g., critical signals must be short, bypass capacitorsmust be interspersed evenly between the DIPs {dual inline packages), andthermally critical components must be placed at specific locations.

• Some devices need to be placed near other devices, such as filter capacitors andresistors, which need to be near DIPs.

• Some devices require a certain clearance around them.

The location of components on a board has been performed with a specializedsoftware. Many automatic component location systems (algorithms) automaticallylocate the components on the PCB at strictly defined positions; other systems allowrandom positioning of components, restricting only component body interference.All systems attempt to minimize the length ofthe connection path (Ginsberg 1991).

In the current practice of PCB design the placement operations are not consideredin solving the component location problem. To produce a cost-effective PCB layout, adesigner should consider the constraints imposed by the placement operations in PCBassembly when the components are located on the board.

The surface mount technology further complicates the component locationproblem. SMT components can be placed on both sides of the board; they alsocome in a variety of shapes, Automatic assembly is mandatory for most SMTcomponents. Thus, the manufacturability issue must be taken into account (Ginsberg1991).

In this section, the component location problem in PCB design is solved from anassembly point of view. The formulation of the component location problem ispresented next.

2.2. Formulation ofthe component location problemAssume that there are A' components to be located on a board. Each location / has

coordinates (.vy,_Vj). There are M feeders required to deliver the components. Theposition of feeder /• is fixed with coordinates (x,,^,). Each feeder / delivers m^components. The components delivered by feeder / are grouped into componentgroup G,. The travelling time from feeder /• to a board location (.v,, Vy) is /,,. Theproblem is to assign A* locations to M feeders so that the placement time is minimized.

hti P = {p^.pi— ,P!^) be the set of locations and f ^ {/i,/;, • - . /w} the set offeeders. The component location problem can be described by a bipartite graph

160 A. Kusiak and D. W.He

Feeder

Figure 1. Bipartite graph G.

G — [N, A), with a set of vertices N = FU P and a set of arcs A —{{fhPj)\fi € f^Pi € ^} (see Fig. 1). Each arc {fi,Pj) has a weight /,y. The graph Gprovides a complete match of P to F such that every pi e P is the end point of onlyone arc, a n d / e i is the end point of m, arcs. The component location problem is tofind a complete match of F to F so that the total weight of the arcs is minimized.

The integer programming formulation of the component location problem ispresented next.

Define:

i = feeder or component group indexj = location index

N = number of components or number of locationsM = number of feedersm, = number of components to be delivered by feeder/Gj — component group /tji — time required to travel between feeder i and location j

{ 1, if location y is assigned to feeder (

0, otherwiseM N

Minimize Y^ Y^ tuXu (1)< = 1 j = l

M

Subjec t t o ^ Xy ^ 1, J =\,...,N (2)

y^;c = - 1 , . . . , M (3)

Xij = 0,1 (4)

The objective function (I) minimizes the total placement time. Constraint (2)ensures that only one location is assigned to a feeder. Constraint (3) ensures that thenumber of locations assigned to feeder / equals m,. Constraint (4) ensures integralityof the decision variable x,,.

Note that the formulation (l)-{4) assigns groups of components rather than eachindividual component to locations. This allows one to link the operational issueswith design constraints. The location problem solution obtained from the formula-tion (l)-(4) can serve as an initial location of components on the board. The initial


1

4

4 1

2

5

3

6

iFeeder f Feeder f-

Figure 2. The board aod feeder locations.

i

11i111

j

123456

/,-, (sec)

20304051525

222222

; •

123456

tij (sec)

60402045255

Table 1. Travel time from feeder i to location^.

location of components can be fine-tuned by moving the components to satisfy somespecific design requirements.

The formulation (l)-(4) provides a basis for economic design justification. Atrade-off analysis can be performed by comparing the savings in the total placementtime with the cost increase due to the changes in board design.

Example I: Consider locating six components on a PCB. The six board locations andtwo feeder locations are shown in Fig. 2.

The components are grouped in two component groups: Gi = (c|,C2) andGi — (c3,C4,C5,f'6) and delivered by two feeders /[ and / j ^ respectively. The timerequired to travel between the feeders and locations is given in Table 1.

Solving the formulation {l)-(4), component group G[ is assigned to locations 1and 4 and component group G^ is assigned locations 2. 3. 5, and 6.

Since groups of components rather than each individual component are assignedto the locations, the assignment of each component of a component group to thelocations can be made based on design constraints. For example, if component C] is aDIP and component CT, is either a filter capacitor or resistor then cy can be placed inlocation 2 or 5. Note that this decision does not have any impact on the solution of theformulation (I)-(4).

The location solution obtained by solving the formulation (l)-(4) also providesflexibility in the routeing process. It allows one to swap components to enhance theinterconnectivity without affecting the solution of the fonnulation {l)-(4). Forexample, suppose that components Cj, c^, and C4 are assigned to locations 1, 2, and3. respectively. The interconnection between components C], CT,, and c^ is shown inFig. 3(a). After checking the interconnection between these components, componentsCi and C4 are swapped. The resulting interconnection is shown in Fig. 3{b). Swappingcomponents cj, and Q reduces the number of cross connections without affecting thesolution of the formulation (l)-(4).


(a)

(b)

0

o

0

a^

1 1

: ^

1 1

0 ^

•o o0 OO 0

(a)

Figure 3. Component swapping: (a) before swapping, (b) after swapping.

(b)

(c) (d)

1 t

(e)

" ^ ^ "

Assembly station

Direction of product flow

Figure 4.

3.

Five types of product flows; (a) repeat operation, (b) serial flow, (c) by-pass flow,(d) backtracking flow, (e) branch/merge flow.

Design rule 2 (Simplify the flow of products): design products to simplify the flow ofproducts in a multi-product assembly lineMost research on design for assembly has focused a single product rather than

multiple products (see Gairola 1986. Yokota and Brough 1992, Kroll et al. 1993). Noresearch has studied the collective impact of product designs on the product flow in amulti-product assembly system. In an agile assembly environment, a large variety ofproducts are produced. The production of a large variety of products createsdifficulties in design and control of agile assembly systems, e.g., line balancing and


Figure 5. Three products and their superimposed assembly graph: (a) assembly structures ofthree products, (b) superimposed assembly graph.

flow control. To reduce the cost of designing, operating, and control of assemblysystems, all these issues should be considered at the product design stage. It isdesired that an analytical tool be available to design products for manufacturingperformance.

In design of a multi-product assembly line, the flow of products is an importantfactor to be considered. Ho et al. (1993) discussed four different product flows: repeatoperation, serial flow, by-pass flow, and backtracking flow (see Fig. 4a-d). Inaddition, the branch/merge flow can be observed (see Fig. 4e).

Of these five flows, the serial flow is the most desirable because it eases control ofthe manufacturing process and material handling. Backtracking is the least desirableflow since it complicates the flow.

In design of multi-product assembly lines, the assembly structures of individualproducts can be combined into a superimposed assembly graph.

An assembly structure of product i is represented by a graph G t with a set oi Nj^nodes and a set of ^ ^ arcs, where:

The nodes represent operations and the arcs precedence relations. Thus an arcfrom node (operation) i to node (operation) j indicates that operation / must becompleted before starting operation 7. It is assumed that G is an acyclic graph. Theassembly structures of N products can be combined into a superimposed assemblygraph G with A' nodes and A arcs where:

A^ = A' , U ^ 2 U • • • U iV;v

A= AiUA2Li---LiA^

An arc (i, j) inG is redundant if in addition to the arc itself there exists a chainfrom node / to n o d e / The redundant arcs can be omitted. The example in Fig. 5shows the assembly structures of three products and the superimposed assemblygraph.

A cycle in a superimposed assembly graph represents a possible backtracking flowin a multi-product assembly line. Consider the products and their superimposed


(a) (b)

(j)(3)Station 1 Station 2 Station 1 Station 2

(c)

Station 1 Station 2

Figure 6. Three possible designs of a two-stage serial line.

(a)

PI:

Figure 7. The redesigned assembly structure: (a) new assembly structures of three products,(b) new superimposed assembly graph.

assembly graph in Fig. 5. Three possible designs of a two-stage serial line are shown inFig. 6.

One can see from Fig. 6 that except for design (a), the two designs (b) and (c) resultin a backtracking flow. This example illustrates that designing products withoutconsidering operational and control issues may lead to poor designs. Product designsforming a superimposed assembly graph with cycles are said to have a backtrackingflow. An agile assembly system should be designed without a backtracking flow.

A pair of operations {i,j) is defined as a cycle pair if both operations i and jbelong to a cycle and there is an arc from operation / toy. For example, in Fig. 5, thepair (4.2) is a cycle pair while the pair (1,2) is not. A cycle can be eliminated bycombining operations in the cycle pairs. Consider the example in Fig. 5. If the pair(4.2) of product P3 can be combined into operation S1. the cycle in the superimposedassembly graph is eliminated. The resulting assembly structures of three products andthe superimposed assembly graph are shown in Fig. 7.

Not combination of operations in every cycle pair eliminates a cycle in a super-imposed assembly graph. For example, combination of operations in the cycle pair(2.3) does not eliminate the cycle. Figure 8 shows the new assembly structures of threeproducts and the superimposed assembly graph after the operations of cycle pair (2,3)is combined into operation S2.


(a) (b)

PI:

P2:

P3:

Figure 8. The redesigned assembly structure: (a) new assembly structures of three products,(b) new superimposed assembly graph.

A cycle pair is critical if the combination of two operations in the cycle paireliminates the cycle. For example, in Fig. 5(b), the cycle pair (4,2) is critical while thecycle pair (2,3) is not.

Combining two operations of a critical pair can eliminate a cycle in the super-imposed assembly graph. A designer can decide whether operations of critical pairsshould be combined into subassemblies through redesigning the products or assignedas combined operations to the assembly stations through balancing the assemblysystem.

Yokota and Brough (1992) categorized the previous studies examiningsubassembly structures at the product design stage as follows:

(1) Subassemblies are explicitly declared, when the assembly is represented at thedesign stage, and are treated as given thereafter. Some related work hasbeen conducted by Leibermann and Wesley (1977), Wesley et al. (1980), andKitajima and Yoshikawa (1984).

(2) Particular types of subassembly are defined, based on the characteristics ofparts or patterns and distribution of parts, and algorithms are devised todetect these types in the assembly. Some examples of research are included inSekiguchi et al. (1983) and Boneschanschcr and Heemskerk (1990).

(3) Rules or algorithms are applied to assembly information and a group of partsthat satisfy certain criteria is recognized as a subassembly. One such exampleis presented by Ko and Lee (1987).

As indicated by Yokota and Brough (1992), the third approach in comparisonto the first two approaches, provides a designer with the responsibility and flexibilityin identifying appropriate subassemblies. However, the criteria in selecting asubassembly must be meaningful.

Research on identifying subassemblies based on their impact on the design ofassembly systems has not been condticted. In this section, an algorithm for identifica-tion of critical pairs of operations is developed. The algorithm can be used as a designadvisory tool. It can advise a designer how to improve the product design in terms ofsimplifying the product flow in a multi-product assembly system.

Define an adjacency matrix Aj^ — [a,y] of product k, with an entry

1, if there is an arc directed from operation / to operation j in Gf^

0, otherwise

Note that each operation can be considered as having an arc directed from and toitself. Therefore, a'lj ^ 1 for all / in the adjacency matrix A^.

Let V be the superimposed adjacency matrix (corresponding to the superimposed


assembly graph). An entry Vjj in V is defined as follows:

0, otherwise

For the convenience of presentation, only non-zero entries are shown in thematrices.

Theorem \. In the triangular matrix, a non-empty entry in the lower-triangularmatrix means there exists a cycle in the corresponding superimposed assembly graph.

This theorem is proven as Theorem 5 in Kusiak et al. (1993). It is known fromTheorem I that if the superimposed matrix V for a set of products can be transformedinto an upper-triangular matrix then no cycle exists in the superimposed assemblygraph.

Since a superimposed assembly graph represents a number of acyclic assemblystructures, there may be more than one critical pair of operations in a cycle of thesuperimposed assembly graph. To select a critical pair of operations, some design andmanufacturing constraints must be considered. From a manufacturing point of view,the combination of two operations that share similar components and auxiliaryequipment is preferred. Therefore, a similarity index of components and auxiliaryequipment between two operations can be used to select a pair of operations.

An assembly plan for a product specifies the type and number of components, thesequence of the operations, and the requirement for auxiliary equipment such astools, fixtures, grippers and so on. The requirement for auxiliary equipment by eachproduct / can be represented by the following column vector:

where:

1, if auxiliary equipment q is required by product''' 10, otherwise

The similarity index between any two distinct products can be expressed by theweighted Hamming distance (Kusiak 1990). For any two products Pj and Pj, definesimilarity index as the weighted Hamming distance

where:

^" ' \ 0, otherwise

and Wg is the weight of the auxiliary equipment.The value of weight n' can be assigned to an auxiliary equipment q depending on

its importance in the assembly process. A way to determine the value of weight w^ forall q is to set it proportional to the estimated setup time caused by the auxiliaryequipment q.

Next, a graph theory based algorithm for identification of critical pairs ofoperations in a superimposed assembly graph is proposed.


Before the algorithm is presented, the following terminology used in Kusiak et a!.(1993) is defined. An operation is called origin operation (OO) if there is no otheroperation proceeding it. In an adjacency matrix an operation i is an OO if the /thcolumn of the adjacency matrix has only one non-empty entry (a diagonal entry). Anoperation is called destination operation (DO) if there is no other operation followingi t. I n an adjacency matrix an operation / is a DO if the i ih row of adjacency matrix hasonly one non-empty entry (a diagonal entry).

Let S and Q be the sequences of pairs of operations. W the set of critical pairsof operations identified, and J the total number of operations. The algorithm foridentification of critical pairs of operations is presented next.

3.1. Identification algorithmStep 0. Construct a superimposed adjacency matrix V with an arbitrary sequence of

the operations {1,2,3, . . . ,7 ) .Set W = 0.

Step 1. End the algorithm if V is empty.Identify an operation which is an OO or a DO.Go to step 3 if neither an OO nor a DO is found.

Step 2. Delete the row and column associated with the operation found in step 1 fromthe matrix V, and go to step 1.

Step 3. Find a cycle.Step 4. Apply procedure I (see below) to the cycle found in step 3.Step 5. Update matrix V and go to step 1.

In step 4 of the identification algorithm, procedure 1 is called to eliminate thecycles detected in step 3 of the identification algorithm by combining operations ofcritical pairs. A pair of cycle operations with the highest value of similarity index isselected. Procedure 1 is presented next.

Procedure 1Step 0. Construct a superimposed adjacency matrix U for each cycle.Step 1. Rank the non-diagonal entries in the increasing order of similarity index.

Denote the sequence of entries by S.

Step 2. Start with the first entry Ujj in S.Delete H,, from S.Combine operations i and,/ of the pair (/,_/) into an operation and update thesuperimposed adjacency matrix U.Add {ij) to W.

Step 3. Apply procedure 2 (see below) to the adjacency matrix constructed in step 2.If the adjacency matrix is an upper-triangular matrix, stop; otherwise go tostep 2.

In step 3 of procedure 1, procedure 2 is called to organize an adjacency matrix intoan upper-triangular matrix. Procedure 2 presented next is basically a simplifiedversion of the triangularization algorithm in Kusiak et al. (1993).

Procedure 2Step 0. Construct an adjacency matrix with any arbitrary sequence of operations

( )


(a)

PI:

(b)

Figure 9. Six products and their superimposed assembly grapii; (a) assembly structures of sixproducts, (b) superimposed assembly graph.

P3:

(b)

Figure 10. Six products and their superimposed assembly graph after redesign: (a) assemblystructures of six products, (b) superimposed assembly graph.

Step 1. Identify an operation which is an OO or a DO.Go to step 5 if neither an OO or DO is found.

Step 2. Apply the SORTING RULE to the operation identified in step 1.Step 3. Underline the operation identified in step 1.Step 4. Delete the row and column associated with the operation in step I from the

adjacency matrix, and go to step 1.Step 5. Build an adjacency matrix with the sequence of operations in Q.

The SORTING RULE used in step 2 of procedure 2 is directly from Kusiak et al.(1993) and is defined as follows.

SORTING RULE: If an operation is an OO move it to the most left position inthe sequence of the operations that are not underlined in Q. If an operation is a DO,move it to the most right position in the sequence of the operations that are notunderlined in Q.


Example 2: Consider six products in Fig. 9.Apply the identification algorithm for products in Fig. 9(b). The two critical pairs

identified by the identification algorithm are: W = {(3,1), {2,4)} (see the Appendix).Assume that operations of the critical pairs are combined into subassemblies, i.e.operations of the pairs (3,1) and (2,4) are combined into subassemblies SAl andSA2,respectively. The final product designs and the corresponding superimposed assemblygraph are shown in Fig. 10.

There are two cases in which the identification algorithm can be used. In the firstcase, the products are designed by different design teams, or products are designed indifferent time periods. In such a case, the identification algorithm can be used tosuggest some possible redesigns to simply the production flow. In the second case,new products are designed to fit the assembly structures of the existing products. Inthis case, the new products are designed using the identification algorithm to avoidpoor designs. In any of the two cases, the identification algorithm can be used to aidthe designer in design and redesign of products for manufacturing performance.

To perform the design trade-off analysis, the total flow distance for differentdesigns can be computed (Aneke and Carrie 1986, Ho et al. 1993). By comparing thesavings in total flow distance with the cost of product redesigns that simplify theproduction flow, the designer can make an economic design decision along with otherfactors i.e. marketing and manufacturing.

4. Design rule 3 (compatible design): design new products compatible with the existingproduction facilities and product mix

In an agile manufacturing environment, production facilities should be able toproduce a wide range of products. New products are often introduced to themanufacturing system. It is expected that the introduction of a new product shouldnot have a negative impact on the manufacturing system performance.

In this section, a design rule for compatible design is proposed. The design rule isillustrated with an example of a just-in-time (JIT) assembly system. It is shown thatsignificant performance improvement can be made when introducing a product thathas been designed for an existing assembly line.

Before the rule is demonstrated, the scheduling of a JIT assembly system isdiscussed.

4.1. Scheduling a JIT assembly systemThe JIT assembly system considered for scheduhng is in the form of an assembly

line without buffers between stations. The objective is to minimize the meancompletion time. The JIT assembly line is not balanced with respect to a cycletime, rather, the products follow a no-delay schedule. Given a schedule of products Z,the mean completion time of Z is derived next (He et al. 1996).

Define:

M = number of stationsZ — {1,2, . . .} , product scheduletjk — processing time of product / at station kD{i, (• + 1) — the idle time between the /th product and the / + 1th product on the

first stationIk{i,i+ 1) = the idle time between the ith product and the / + 1th product on

station k


C/ ^ completion time of /th product in the schedule

The completion time of the /th product in Z can be represented as follows:

M - l I ( - 1

c = (/ > 2) - E '>^+E '"'w + E ^^C''''+1) (5)k=\ h=l A-1

where:

iMih,h+1) = /)(/i,/i+1) + y ^ h-i-i t - y ^ tht (6)

ZJ(A,A+l)^ max ^ V / / , . - y / f i + i . , 0 l (7)

Replace the third component in (5) with (6), then Q can be represented asfollows:

M-\ i l-\ i-\ fM-\ M \

*=1 A=l A=l h=\ \k=\

M 1-1 (-1

+ 1) (8)

Let C{Z) be the completion time of schedule Z. With (8), the mean completiontime C{Z) can be represented as follows:

NC{Z) ^N

c - l (=2E

h=\ h=\

N M N (-1 A' (-1

EE'.* + EE''H+EE( = 1 k=\ 1=2 A=l (=2 /i=l

4.2. Compatible design of a new productLet Z' be the new product schedule by adding the new product to Z. Then the

impact of introducing a new product on the performance of the existing system ismeasured by the change in mean completion time, AC — C{Z') - C{Z). It is desiredthat AC be as small as possible when a new product is introduced into the assemblysystem. The basic idea of compatible design is to design a new product so that AC isminimized.

The compatible design rule can be implemented using a simple algorithm. IfAC < 0, i.e. adding a new product to the existing system results in a betterperformance, the algorithm is stopped. If AC > 0, attempt to redesign the productso as to reduce AC.


©Station 1 Station 2 Station 3

Figure 11. The layout of the assembly line.

Product I

1234

Table 7

Schedule Z,

Zi = {P4, P3,Zj = {P3, P4,Zj = {P3,P2,2^ = {P3,P2,

P2P2P4PI

Processing time

tn tn

3 13 13 13 0

^ 3

5302

Processing time data.

C(Zi)

! P 1 } 10-75,P1} 1100,P4} 1150

'C(Zi)

180105130180

Table 3. The computation of CiZ^ and AC(Z,).

Station 1

Station 2

Station 3

P3 P4 P2 PI12

[Pl]3 4 6

.-P310 12 13

P2 PI

4 6 8 10 13 18Figure 12. Gantt chart of the product schedule {P3,P4,P2,PI}.

4.3. The compatible design algorithmStep L Determine a new product schedule Z ' when the new product is introduced.Step 2. If AC < 0, stop; else go to step 3.Step 3. Check if it is possible to reduce AC by redesigning the product.

If yes, redesign the product and go to step 1; else stop.

Next, an example is used to demonstrate the compatible design algorithm.

Example 3: Consider a JIT assembly line as shown in Fig. 11.The processing time data for products PI, P2, P3 and new product P4 are given in

Table 2.Assume that the optimal mean completion time schedule of the original product

mixis{P3,P2,PI}.Apply the compatible design algorithm for new product P4.


Station 1

Station 2

P3 P2

3

3

P3 P4

4 fP3

1 M

PI

S 9

P2

7 9 10

P2 PIStation 34 7 10 15

Figure 13. Gantt chart of the schedule Z ' .

Step 1. Determine a new product schedule Z ' .The optimal mean completion time of Z is C{Z) ^ (4 -I- 10 + 15)/3 - 9-7.Let Z, be the /th product schedule generated by inserting the new product inZ and AC(Z,) be the corresponding change of the mean completion time.The eomputation of C(Z,) and AC(Z,) for / - 1, . . . ,4 is shown in Table 3.The schedule Z2 with the smallest increase of the mean completion time isselected. The Gantt chart corresponding to the minimum mean completiontime schedule Z ' is shown in Fig. 12.

Step 2. Since AC > 0, go to step 3.Step 3. Check if it is possible to reduce AC by redesigning the product.

From Fig. 12 one can see that schedule Z' results in idle times at stations 2and 3. Assume that the new product P4 can be processed at any station. Inorder to take advantage of the low utilization of stations 2 and 3 in theexisting assembly system the new product P4 is redesigned. The processingtime data for the redesigned product is: /41 ^ 0, 42 ^ 2,743 ^ 3. Go to step 1.

Step 1. With the redesigned product, a new product schedule Z ' is determined as{P3,P4, P2,P1}. Its minimum mean completioti time and AC are computed:

C{Z') = {4 + l+\0+ l5)/4 = 9, AC(Z') = -0-7.

Step 2. Since AC < 0, the algorithm terminates.

The Gantt chart ofthe schedule Z ' is shown in Fig. 13. One can see that designinga new product for an existing assembly line and the product mix improves theperformance of the assembly system.

The computation of change in the mean completion time AC provides an input toeconomic design trade-off analysis. The savings in mean completion time due to thecompatible design of a new product should be compared with the cost of generating anew design.

5. SummaryThe concept of agile manufacturing is driven by the need to quickly respond to

changing customer requirements. Agile manufacturing demands a manufacturingsystem to be able to produce efficiently a large variety of products and be recon-figurable to accommodate changes in the product mix and product designs. Themanufacturing system reconfigurability and product variety are critical in agilemanufacturing. To implement agile manufacturing, methodologies of design for


Rule Parameters, variables, andno. objectives Implementation procedures

Parameters and variables:/,-,—travelling time from feeder/ tolocation yG,—component group delivered by/jObjective:Minimize the placement time

Parameters and variables:Ai^—adjacency matrix of product k[vjj]—superimposed adjacencymatrixObjective:Eliminate cycles in a superimposedassembly grapfi

Parameters and variables:^/t^-pfocessing time of product / atstation kZ—schedule of existing product mixZ—schedule of new product mixC(*)—mean completion time ofschedule *Objective:Minimize AC = CiZ') - C(Z)

Assign a component of group G,corresponding to the smallest ;,-,•to location j

(1) Construct A^ for all products(2) Construct [ti,-J(3) Check cycles in [u,- ](4) Eliminate cycles by

redesigning products

(1) Check the availability ofidlecapacity on each station k

(2) Determine /j^ for a newproduct to fit the idlecapacity

Table 4. Summary of design rules.

agile manufacturing are needed. Design for agile assembly must be accomplished byconsidering the operational issues in assembly systems at the early product designstages.

In this paper, three design rules for the design of products for agile assembly froman operational perspective are proposed. The first rule is to design a product to satisfythe manufacturing operations requirement. An PCB design example was used toillustrate this rule. The second rule is to simplify through the design of products theirflow in a multi-product assembly system. This rule suggests that in order to avoidbacktracking in a multi-product assembly line, products should be designed so thatcycles in a superimposed assembly graph are eliminated. The third rule is to design anew product for a compatible production schedule. Illustrative examples wereprovided to demonstrate analytical insights and the benefits from applying thedesign rules. Procedures and algorithms for implementing these design rules werepresented. The trade-off analysis associated with each design rule was discussed.Table 4 summarizes the implementation of these rules.

Although the design rules discussed this paper were developed independently, theycan be combined to improve performance of manufacturing systems. For example,design rule 1 can be applied to PCBs in a multi-PCB assembly system to reduce theplacement time in the system, design rule 2 can be applied to check the design of PCBsto simplify the production flow in the system, and design rule 3 can be applied toreduce the production time when a new PCB is introduced into the system. Thedevelopment of design rules discussed in this paper is by no means complete. Futureresearch will lend to the discovery of new design principles dealing closer operationalissues with the design of products.


AcknowledgmentThis research has been partially supported by the National Science Foundation

(grant No. DDM-9215259), the US Army (contract No. DAAE07-93-C-R080), andRockwell International Corporation.

Appendix: solution procedure for example 2Step 0. Construct a superimposed assembly matrix V as follows:

1 2 3 4 5 6 7

1234567

1

11 1

11

11

1

1

11

I1

1

Set W ^ 0 .

Step 1. Since neither an OO nor a DO is found, go to step 3.Step 3. Start with operation 1, a cycle Cl = (1,5,3,1) is found.Step 4. Apply procedure I to cycle Cl .Procedure 1:Step 0. Construct a superimposed adjacency matrix V for cycle Cl as follows:

1 5 3

1 1 1( 7 - 5 1 1

3 1 1

Step 1. Suppose that the non-diagonal entries have been arranged in an increasingorder of similarity index as: S = («3i,U53,«i5).

Step 2. Select the first entry W31 in the sequence S found in step 1 and combineoperations of the pair (3,1) into operation Si.Delete M3, from 5 , 5 = (u53,«i5). Add (3,1) to W,W = {(3,1)}.Update the superimposed adjacency matrix U as follows:

1 3 5 SI

1 1

SI 1 1

Step 3. Applying procedure 2 to the matrix found in step 2, the following upper-triangular matrix is obtained:

1 3 SI 5

1 13 1

SI5

1

1 11


Since the upper-triangular matrix is found, procedure 1 terminates. Return to theidentification algorithm.Step 5. Update the matrix V as follows:

1 2 3 4 5 6 7 SI

1 I234

^567

SI

1

I

1

1

1

1

1

1

I

1

1

1

1

11

Notice that the entry v^ is eliminated as a consequence of combining operations ofthe pair (3,1) into operation SI. Go to step 1.

Step 1. Operation I is an OO.Step 2. Delete the row and column associated with operation 1 from matrix V.

Go to step I.Step 1. SI is an OO.Step 2. Delete the row and column associated with SI from matrix V.

Go to step 1.Step 1. Operation 5 is an OO.Step 2. Delete the row and column associated with operation 5 from matrix V.



Go to step 1.Step 1. Since neither an OO nor a DO is found, go to step 3.Step 3. Start with operation 2, a cycle C2 = (2,4,6,2) is found in matrix V.Step 4. Apply procedure 1 to cycle C2.

Procedure 1:

Step 0. Construct the superimposed adjacency matrix V for cycle C2 as follows:

2I

1

411

6

1I

Step 1. Suppose that the non-diagonal entries have been arranged in an increasingorder of similarity index as: S = («34, W46.«62)-

Step 2. Select the first entry U24 in the sequence S found in step 1 and combineoperations of the pair (2,4) into operation S2.


Delete U24 from S,S = (M46,U62). Add (2,4) to W,W = {(3,1), (2,4)}.The updated superimposed adjacency matrix U is as follows:

2 4 6 S2

2 1^ ^ 4 1 1

6 1 1S2 1 1

Step 3. Applying procedure 2 to the matrix found in step 3, the following upper-triangular matrix is obtained:

4 S2 6 2

4 1S2

62

1111 1

1

Since the upper-triangular matrix is found. Procedure 1 terminates. Return to theidentification algorithm.Step 5. Update the matrix V as follows:

V

2_ 4

6S2

21

1

4

1

6

1

11

S2

1

Step 1. Operation 4 is an OO.Step 2. Delete the row and column associated with operation 4 from matrix V.

Go to step 1.Step 1. S2 i sanOO.Step 2. Delete the row and column associated with S2 from matrix V.



Go to step 1.Step 1. Since matrix V is empty, stop.

The two critical pairs identified by the identification algorithm are:(^ = {(3,1), (2,4)}.

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Design for agile assembly: an operational perspective

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