quantitative platform selection in optimal design of...

Journal of Engineering DesignVol. 17, No. 5, October 2006, 429–446

Quantitative platform selection in optimal design of productfamilies, with application to automotive engine design

RYAN FELLINI*, MICHAEL KOKKOLARAS and PANOS Y. PAPALAMBROS

University of Michigan, Ann Arbor, Michigan USA

Product variants with similar architecture but different functional requirements may have commonparts. We define a product family to be a set of such products, and refer to the set of common parts as theproduct platform. Product platforms enable rapid adjustment to changing market needs while keepingdevelopment costs and time-cycles low. In many cases, however, the individual product requirementsare conflicting when designing a product family. The designer must balance the tradeoff betweenmaximizing commonality and minimizing individual product performance deviations. The designchallenge is to select the product platform that will generate family designs with minimum deviationfrom individual optima. We propose a methodology that combines two previous approaches developedfor making commonality decisions. In the first approach optimal values and sensitivity informationfrom the individually optimized variants are used to indicate components that are probable candidatesfor sharing. In the second approach a relaxed combinatorial problem is formulated to maximize sharingamong variants subject to bounds on performance reduction for the individually optimized values. Inthe combined methodology the first approach is used to identify an initial set of shared componentsand define the candidate platform to be considered by the second approach. The computational loadis reduced significantly and the platform-selection problem is solved in a more robust manner. Theproposed methodology is demonstrated on the design of an automotive engine family.

Keywords: Optimal design; Platform section; Product families; Automotive engines

1. Introduction

Product variants are defined as artefacts that have similar architecture but different functionalrequirements. Product variants may share a subset of the parts they consist of. A productfamily is a set of product variants, and the set of common parts in a family is referred to asa product platform. In many cases functional requirements of product variants are conflicting(Nelson et al. 1999). Platform-based family optimal designs will then be compromised relativeto individually optimized designs due to a tradeoff between maximizing commonality andminimizing individual performance deviations. The design challenge is to determine (select)the platform that will generate family designs with minimum deviation from individual (null-platform) optima.

Designing product families based on product platforms enables rapid adjustment to changingmarket needs while keeping development costs and time-cycles low (Meyer & Lehnerd 1997;

*Corresponding author. Email: [email protected]

Journal of Engineering DesignISSN 0954-4828 print/ISSN 1466-1837 online © 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/09544820500287797

430 R. Fellini et al.

Ericsson & Erixon 1999). Therefore, the primary objective is to share components that areparticularly costly and/or have low impact on design. Although the family design problemhas been studied extensively (Siddique et al. 1998, Simpson 1998, Conner et al. 1999, Nelsonet al. 1999, Simpson et al. 1999, Fellini et al. 2000, Messac et al. 2000, Kokkolaras et al. 2002),few mathematics-based methodologies have been proposed for determining which componentsshould be shared ahead of the family design step. One such method uses robust design princi-ples combined with an optimization problem to determine the platform for a family of scalableproducts (Chen et al. 1997, Nayak et al. 2000). Other methods focus on solving the combina-torial design problem for modular product families, since clustering parts to modules reducesthe problem size significantly. Examples are a process for simultaneous module design andcombination selection (Gonzalez-Zugasti et al. 1998, Gonzalez-Zugasti & Otto, 2000), andsimultaneous optimization of module combination and attributes (Fujita & Yoshida, 2001).The latter is based on previous work of the same research group (Fujita et al. 1998, 1999,Fujita 2000). In these methods a genetic algorithm (GA) is linked with sequential quadraticprogramming (SQP). The GA is used to choose a configuration of modules to share and SQPperforms product design optimization. Other researchers have also adopted GAs for solvingthe commonality selection and family design problems (D’Souza & Simpson 2002). The num-ber of possible combinations increases exponentially the number of products and/or variables.Combinatorial algorithms such as GAs may therefore be insufficient for solving even problemsof modest size.

In the present work we combine two approaches developed in previous work to formulate animproved commonality strategy. The first method uses first-order information obtained fromindividual design optimizations to compute a metric for performance deviations attributed tocomponent sharing (Fellini et al. 2002a). This method has some limitation due to the heuristicinvolved when choosing a threshold that will determine what to share. The benefit, however,is that a large number of design variables can be filtered to identify an initial set of sharedcomponents using information readily available from the individual optimizations. In addition,it may be possible in practice to identify an appropriate threshold by means of knowledge-based design techniques. The second method uses an optimal problem formulation where thecombinatorial problem is relaxed and reformulated to maximize commonality among familymembers, while satisfying individual design constraints and observing a designer-specifiedbound on individual performance deviations (Fellini et al. 2002b). The designer can identifythe optimal platform and obtain optimal family designs for a number of scenarios based onthe willingness to sacrifice a certain amount of individual performance. This method has thebenefit of being rigorous and accurate. However, it is computationally more expensive thanthe first approach. Therefore, the use of surrogate models (e.g. artificial neural networks) isimperative. The main idea presented in this article is to combine these methodologies, usingthe first approach as a filter to reduce the platform selection problem size and the secondapproach to maximize commonality while minimizing individual performance deviations.

The article is organized as follows. The two approaches developed previously are reviewedfirst. The combined methodology is then formulated, and subsequently demonstrated on afamily of automotive engines. Results are discussed and conclusions are drawn.

2. Problem formulation

The following definitions will be used in reviewing the previous methodologies and describingthe combined strategy:

• P = {p1, p2, . . .}: set of m products.• xp is the column vector of design variables for the product p ∈ P .

Platform selection in optimal design of product families 431

Figure 1. Performance deviations due to commonality.

• Cp = {cp

1 , cp

2 , . . .} is the set of components that form a particular product p.• xcp

is the column vector of design variables for the component cp.• Spq is the set consisting of index pairs of elements that are shared between two products p

and q.• S = {Spq |p, q ∈ P ; p < q} is the set describing element sharing throughout the family.• S∗ is the set describing the ‘optimal’ product platform.• xp,o is the null-platform optimal design of product p, solution of equation (1).• f p,o is the null-platform optimal objective function value of product p.• xp,∗ is the platform-based optimal family design of product p , solution of equation (2).• f p,∗ is the platform-based optimal family objective function value of product p.

The individual optimal design problem for product variant p can be formulated as thegeneral optimization problem:

maxxp

f p(xp)

subject to gp(xp) ≤ 0

hp(xp) = 0.

(1)

The multiobjective family design problem is then formulated as

maxx=[xp1 ,xp2 ,...] {f p(xp)} ∀p, q ∈ P, (i, j) ∈ Spq, p < q

subject to gp(xp) ≤ 0

hp(xp) = 0

xp

i = xq

j ,

(2)

where the additional equality constraints represent commonality. In general, a platformselection methodology can be summarized as follows: quantify performance deviations byconsidering individual optimal designs and decide which components are to be shared(i.e. determine the platform) with minimal performance deviation (see figure 1); optimallydesign the product family around the chosen platform.

3. Platform selection strategy and optimal family design

In this section we will review the previously developed approaches for making commonalitydecisions and formulate a combined strategy for platform selection and optimal family design.


3.1 Platform selection using null-platform optimality information

Optimality and sensitivity information obtained from individual product optimization is usedto assess the potential deviation from the null-platform optimal design incurred by sharingparts with other product variants (Fellini et al. 2002a). Component sharing is represented bycommon design variables. The following assumptions are made:

1. Self-sharing (i.e. component sharing within the same variant) is not possible.2. Null-platform optimal designs lie ‘close enough’ to each other.3. The family optimum lies in the convex hull of the individual solutions (i.e. the null-platform

optima).4. Constraint inactivity remains unchanged between null-platform and family optimal designs.

We refer to the design solutions that satisfy these assumptions as ‘mild variants’. The for-mulation is derived based on a first-order Taylor series approximation. Therefore, the generalcondition is that the individual optimal designs lie relatively close to each other so that thelinear approximation is valid.

We consider two product variants A and B without loss of generality. Under assumption 3,the relation between the shared variables and the null platform can be rewritten as:(

x∗i − x

A,◦i

)= (1 − λi)

(x

B,◦i − x

A,◦i

),

with i ∈ S. The deviation of the objective f in one variant A due to sharing of the variablesxi, i ∈ S, is approximated by:

f A(x∗) − f A(xA,◦) ≈∑i∈S

∇ifA,◦

(x∗

i − xA,◦i

)

≈∑i∈S

(1 − λi)∇ifA,◦

(x

B,◦i − x

A,◦i

).

We can then estimate the upper bound on the total performance variation of product A by

�A ≤∑i∈S

(1 − λi)

∣∣∇if

A,◦∣∣ δi +∑j∈GA

max(∇igA,◦j δi, 0)

(3)

where δi = |xB,◦i − x

A,◦i | and GA is the set of indices of the active constraints at the null-

platform optimum of product A. A similar upper bound can be obtained for product B.We define next a performance deviation vector �, whose entries correspond to performancedeviations due to sharing, as follows:

�i = (1 − λi)

∣∣∇if

A,◦∣∣ δi +∑j∈GA

max(∇igA,◦j δi, 0)

+ λi

∣∣∇if

B,◦∣∣ δi +∑j∈GB

max(∇igB,◦j δi, 0)

. (4)

The l1 norm of the vector � provides an upper bound on the actual performance deviation� of the product family:

� = �A + �B ≤ ‖�‖1. (5)


Equations (4) and (5) can be adjusted straightforwardly for more than two products. The designvariables are arranged in order of increased performance deviation value and the number ofvariables to share is determined by a limit on acceptable design deviations. Figure 2 illustratesa typical use of this information in making commonality decisions (Fellini et al. 2002a). Froma total of 63 variables, the first 24 first variables are shared ‘naturally’; that is, they have thesame values at the null-platform optima. Next, choosing a threshold of 0.01, the designerdecides to share the first 54 variables. The information provided by the performance deviationvector allows the designer to choose in what order to begin sharing components. The methodis iterative, in that the designer probably have to increase or decrease the level of sharing untilthe desired level of product performance is achieved. The drawbacks are the heuristic mannerused to choose the components to share (according to some threshold) and the approximatenature of the method.

3.2 Platform selection by solving a relaxed combinatorial problem

This methodology integrates platform selection under bounds on performance deviation withoptimal product family design (Fellini et al. 2002b). The designer can choose what per-formance deviations are acceptable relative to null-platform optima. Component sharingis determined through the solution of a relaxed commonality maximization combinatorialproblem subject to these performance bounds.

The commonality decision problem is formulated as a mixed-discrete programmingproblem:

maxη, x=[xp1 ,xp2 ,...] {{f p(xp)}, ∑(i,j)pq η

pq

ij } ∀p, q ∈ P, (i, j) ∈ Spq, p < q

subject to gp(xp) ≤ 0hp(xp) = 0

ηpq

ij (xp

i − xq

j ) = 0

ηpq

ij ∈ {0, 1}.

(6)

Figure 2. Sorted performance deviation vector (reproduced from Fellini et al. 2002a).


The set Spq consists of component candidates for sharing between two products. The sharingdecision variables η

pq

ij multiplied by the commonality constraint can take a value of either 1 or0 depending on whether a component is shared or not. The multiobjective problem formulationreflects the tradeoff between individual product performance and commonality.

Equation (6) is reformulated to include bounds on individual performance deviations andreplace the sharing decision variables by a distance function D◦:

maxx=[xp1 ,xp2 ,...]

∑pq |Spq | − ∑

(i,j)pq D◦(xp

i − xq

j ) ∀p, q ∈ P, (i, j) ∈ Spq, p < q

subject to gp(xp) ≤ 0hp(xp) = 0

f p(xp) ≥ (1 − Lp)f p,◦,

(7)

where

D◦(xp

i − xq

j ) ={

0 if xp

i = xq

j

1 otherwise.(8)

We approximate the binary function D◦ by the continuous function:

Dα(xp

i − xq

j ) = 1 − 1(xp

i − xq

j

α

)2 + 1

(9)

This function is constructed as a measure of the distance between designs, and approachesthe function D◦ as α goes to 0. Figure 3 shows Dα for α = 0.05. Since Dα is continuouslydifferentiable, gradient-based algorithms can be used to solve the approximate commonalityproblem.

Equation (7) can be solved in cases where components are described by a vector of designvariables by computing the l2 norm of the design difference:

maxx=[xp1 ,xp2 ,...]

∑(i,j)pq Dα(‖xc

p

i − xcq

j ‖2) ∀p, q ∈ P, (i, j) ∈ Spqc , p < q.

The solution of equation (7) will return just the ‘optimal’ platform, but not necessarily theoptimal design of the product family. The final step is therefore to design the product family

Figure 3. Continuous approximation of binary function (reproduced from Fellini et al. 2002b).


by solving equation (2). The design process is visualized in figure 4. This methodology canbe helpful in addressing the platform selection problem, which may be intractable in itsoriginal combinatorial form. However, as the number of products and/or shareable componentsincreases, the methodology becomes increasingly expensive.

3.3 Combining the approaches

The motivation for an integrated approach is to take advantage of the strengths of each of thetwo methods already described. The first approach will be used as a filtering step to reduce theproblem size, acting as an upper-bound computation in determining components consideredgood candidates for sharing. The second approach is then applied to the remaining candi-dates to complete the commonality selection. This reduced-size problem allows an efficientsolution of the relaxed combinatorial problem. The combined methodology is then linkedto an artificial neural network, due to computational expense, when using simulation-basedmodels.

The first method requires a minor modification so that the two approaches can be integratedefficiently, specifically to allow components described as a vector of design variables. This isdone by aggregating the performance deviation values of the design variables that define thecomponent into a single value, �c.

In the following demonstration study we will look at various ways to present the performancedeviation vector. Previously, the sorted performance deviation vector values were plotted aswas shown in figure 2. We will also look at the increasing cumulative value of the performancedeviation as more variables are shared.

The combined approach can be summarized in the following steps:

1. Determine the optimal null-platform design xp,◦ for each individual product p ∈ P bysolving the individual optimal design problem (equation (1)).

2. Identify the components that are good candidates for sharing among products (i.e. definethe candidate platform set).

3. Define the performance deviation Lp acceptable for each of the products.4. Compute the performance deviation vector using equation (4). Choose a subset of the

candidate platform set as components to share.

Figure 4. Product family design process (reproduced from Fellini et al. 2002b).


5. Solve the relaxed combinatorial problem (equation (7)) for the remaining components ofthe candidate platform set.

6. Based on the results obtained by solving the relaxed combinatorial problem, make a finalselection of components to be shared.

7. Solve the family optimal design problem (equation (2)).

In the following section we demonstrate the application of the combined strategy on a familyof automotive engines.

4. Automotive engine family design

This case study will examine designing a family of engines using the combined strategy alreadyformulated. Engine variants are defined based on different functional requirements.

4.1 Simulation tools and design variables

GT-Power by Gamma Technologies is used as the simulation tool (GTI 2001). A 24-valve2.5L V6 engine model, previously validated at various operating points, is used to generatethe family. Analysis is performed at a specified operating point given the engine speed andfuel rate, specifically at 5000 rpm and wide open throttle. The operating point characteristicsare as follows:

Ne = 5000 rpm (mean crank speed).tp = 90◦ (throttle angle).iv◦ = 331.0◦ (intake valve open with respect to the crank angle).ivc = −103.0◦ (intake valve closed with respect to the crank angle).ev◦ = 101.0◦ (exhaust value open with respect to the crank angle).evc = 397.0◦ (exhaust value closed with respect to the crank angle).

The geometry of components from the intake manifold through the exhaust system aremodelled in the simulation. The design variables of particular interest in this study are asfollows:

x1 is the bore (b).x2 is the stroke (s).x3 is the connecting rod length (l).x4 is the compression ratio (cr).x5 is the intake valve diameter (di).x6 is the intake cam-timing angle (icta).x7 is the intake angle multiplier (iam).x8 is the exhaust valve diameter (de).x9 is the exhaust cam-timing angle (ecta).x10 is the exhaust angle multiplier (eam).i1 is the number of cylinders (nc).


The two exhaust valves are modelled as a single valve by using the equivalent area:

(input diameter)2 = 2 (valve diameter)2. (10)

In addition, the theoretical height of the combustion chamber is computed as a function of thestroke and compression ratio:

hc = s/(cr − 1). (11)

The responses to be computed are as follows.

R1 is the break power (performance).R2 is the break torque (performance).R3 is the brake-specific fuel consumption (efficiency).R4 is the NOx (emissions).R5 = dP mx/DCA (NVH, knock).R6 = Pmax (stress/durability).

The brake power and brake torque are measures of the product dynamic performance. Thebrake specific fuel consumption is a measure of efficiency, and measured emissions are throughthe Nox produced. Finally, dP mx/DCA, the mean pressure rise with respect to the crank angle,and Pmax , the maximum cylinder pressure, contribute to various measures such as NVH, knock,stress, and engine durability.

The components we focus on are the following: exhaust cam(s), intake cam(s), exhaustvalve(s), intake valve(s), cylinder head, piston(s), connecting rod, and engine block (seefigure 5). Finally, we map the design variables to their respective components:

xc1 = [x10].xc2 = [x7].xc3 = [x8].xc4 = [x5].xc5 = [x1, hc, i1].xc6 = [x1].xc7 = [x3].xc8 = [x1, x2, x3, i1].

Note that the number of cylinders, i1, will be fixed at 6 for all engines.Simulation is computationally quite expensive. We therefore developed a surrogate model

by training a radial basis function artificial neural network. This model was constructed usinga 2500-point data sample from a Latin-hypercube design of experiments. The upper and lowerbounds of the design variables were set as follows:

A 500-point data sample of random design points was used to validate the model. Thecomputed average errors µ and standard deviations σ are presented in table 1. All reportedresults have been obtained using the artificial neural network in place of the GT-power modelsimulation.

4.2 Optimal design model

The first step in the design process is to define the optimal design model. Various engine designrules of thumb on the bore to stroke ratio, the connecting rod to stroke ratio, and so on, are


Figure 5. Engine components of interest.

x1,l , x1,u = 70.0, 95.0 (mm).x2,l , x2,u = 70.0, 95.0 (mm).x3,l , x3,u = 105.0, 237.5 (mm).x4,l , x4,u = 8.0, 10.0.x5,l , x5,u = 22.0, 35.0 (mm).x6,l , x6,u = −10.0 ◦, 10.0◦.x7,l , x7,u = 0.9, 1.1.x8,l , x8,u = 30.0, 43.0 (mm).x9,l , x9,u = −10.0◦, 10.0◦.x10,l , x10,u = 0.9, 1.1.

available in the literature (Heywood 1988). In addition to geometric constraints, limits areplaced on pressure gradients, in-cylinder pressures, and mean piston velocities to maintain thereliability of the engine. The following inequality constraints must be satisfied by all familyproducts:

gp

1 , gp

2 : 0.8 ≤ b/s ≤ 1.2.

gp

3 , gp

4 : 350 ≤ πb2s/4 × 10−3 ≤ 650 (cm3).

gp

5 , gp

6 : 1.5 ≤ l/s ≤ 2.5.

gp

7 : di ≤ 0.37b (mm).

gp

8 : de ≤ 0.45b (mm). (12)

gp

9 : (2sNe)/(60 · 1000) ≤ 15.0 (m s−1).

Table 1. Artificial neural network modelaverage errors and standard deviations.

Response µ σ

R1 6.5521 5.6571R2 6.5520 5.6570R3 1.2758 1.3417R4 1.7756 3.4038R5 4.3606 4.3559R6 3.5425 3.8723


gp

10: s/(cr − 1) ≥ 5.0 (mm).

gp

11: l + s + s/(cr − 1) + 0.5b ≤ 350.0 (mm).

gp

12: dP mx/DCA ≤ 3.0 (bar per degree).

gp

13: Pmax ≤ 110 (bar).

where

gp

1 , gp

2 are the bounds on the bore-to-stroke ratio.g

p

3 , gp

4 are the bounds on the displacement of the cylinder.g

p

5 , gp

6 are the bounds on the connecting rod length-to-stroke ratio.g

p

7 is the upper bound on the intake valve diameter with respect to bore.g

p

8 is the upper bound on the exhaust value diameter with respect to bore.g

p

9 is the upper bound on the mean piston speed.g

p

10 is the upper bound on the clearance height above the piston crown.g

p

11 is the upper bound on the overall engine height.g

p

12 is the upper bound on the pressure rise rate.g

p

13 is the upper bound on the cylinder pressure.

We define three variants by means of three functional requirements. The first engine variantis designed to maximize power, the second to minimize fuel consumption, and the third tominimize emissions. The optimal design problems are formulated as the following:

maxxp

f p = Power (kW)

subject to gp

1 , gp

2 , . . . , gp

13

gp

14: NOx ≤ NOx,max (p.p.m.)

gp

15: Power · BSFC ≤ 30,000 (g h−1)

(13)

minxp

f p = Power · BSFC (g h−1)

subject to gp

1 , gp

2 , . . . , gp

13

gp

14: NOx ≤ NOx,max (p.p.m.)

gp

15: Power ≥ 80 (kW)

(14)

minxp

f p = NOx(p.p.m.)

subject to gp

1 , gp

2 , . . . , gp

13

gp

14: Power ≥ 80 (kW)

gp

15: Power · BSFC ≤ 30,000 (g h−1)

(15)

The value of NOx,max is based on the baseline value of 25,546 p.p.m. multiplied by 110%;namely, we do not want to produce more than 10% additional emissions with respect to thebaseline model.

5. Results and discussion

Following the steps of the combined methodology we first solve equation (1) to obtain thenull-platform optima for each of the three products independently. The results are presented


in table 2. For the engine designed for maximizing power we find that three of the inequalityconstraints are active. These are the upper bounds on the mean piston speed, engine height,and NOx emissions. For the engine designed for maximizing fuel efficiency, four inequalityconstraints are active. These are the lower bounds on cylinder displacement and power, andthe upper bounds on connecting rod to stroke ratio and intake valve diameter with respectto bore. For the engine designed for minimizing emissions, three inequality constraints areactive. These are the upper bounds on the connecting rod to stroke ratio and intake valvediameter with respect to the bore, along with the lower bound on power. For the maximumpower and fuel efficiency engines the compression ratio is maximized, which in turn increasescombustion efficiency. For the low emissions engine the compression ratio is minimized, whichcorresponds to the strong correlation of NOx , production with the increased heat due to highercompression. There is relatively little natural commonality between the three individuallyoptimized engines. The only common component is the exhaust valve between the enginedesigned for fuel efficiency and low emissions.

We computed the performance deviation vector for the product family. All functions arenormalized and design variables are scaled to be in the range [0, 1]. Figure 6 depicts thesorted performance deviation vector in terms of both individual �i and cumulative values.Figure 7 illustrates the sorted performance deviation vector with respect to engine components(aggregating deviations that correspond to component variables) in terms of both individual�c

i and cumulative values. The performance deviation vector values sorted with respect to thedesign variable illustrate which variables we could share if variable sharing were the objective.Our focus is on the deviations sorted with respect to components. From the performancedeviation vector values sorted with respect to components, we find that the �c

i values arerelatively low for the connecting rod, intake and exhaust valves, and the intake cam. Wealso observe on the cumulative deviation plot that after sharing the first four components(components 7, 4, 2, and 3), the performance deviation increases significantly. Therefore,these first four components will be shared and the relaxed combinatorial problem will besolved for the remaining components.

Table 2. Null-platform optima.

Engine Power (A) Fuel usage (B) Emissions (C)

b (mm) 84.43 75.39 95.00s (mm) 79.95 78.40 90.00l (mm) 199.87 196.00 201.02cr 10.00 10.00 8.84di (mm) 31.24 27.90 27.55icta (degree) −10.00 10.00 −10.00iam 0.99 1.08 1.03de (mm) 35.57 30.00 30.00ecta (degree) −10.00 10.00 −10.00eam 1.10 1.00 0.90hc 8.88 8.71 11.48disp. (cm3) 2686 2100 3828Power (kW) 114.29 80.00 80.00Torque (Nm) 218.27 152.79 152.79BSFC (gh−1 kW−1) 263.50 265.82 331.17NOx (p.p.m) 26089.02 25717.78 17853.09dP mx/DCA (bar/degree) 1.40 1.28 0.88Pmax (bar) 52.36 47.97 35.43Fuel usage (gh−1) 30000.00 21265.32 26493.90

BSFC, brake-specific fuel consumption.


Figure 6. Sorted performance deviation vector with respect to design variables (the plot on the top shows the sortedvalues of �i , while the plot on the bottom shows cumulative values of the deviation vector).

In the second step we solved the relaxed combinatorial optimization problem for a deviationfactor of 5%, 6%, 7%, 8%, 9%, and 10%. The most interesting results were obtained for 6% and7%, and are presented in table 3, where 0 and 1 denote no sharing and sharing in product pairs,respectively. Note that if a component is shared in all product pairs, then it is shared amongall products. The results indicate that we can share the intake and exhaust valves along withthe connecting rod and intake cam across all engine variants for both performance deviationbounds. Depending on the allowable deviations from the optimal designs, it is also possible to


Figure 7. Performance deviation vector with respect to engine components (the plot on the top shows the sortedvalues of �c

i , while the plot on the bottom shows cumulative values of the deviation vector).

share the exhaust cam across the family. The piston is consistently shareable only between thehigh-power and low-emissions engines. With slightly more performance deviation it is alsopossible to share the entire engine block between these two engines. Note that the cylinderhead, piston, and engine block are ‘modules’ that happen to be shared in two cases (the pistonand engine block between products A and C). Exchanging these components produces thevariety in this engine family.


Table 3. Relaxed combinatorial optimization problem results (0 and 1 denote nosharing and sharing, respectively).

Deviation = 6% Deviation = 7%

Shared between: A & B A & C B & C A & B A & C B & C

Exhaust cam 0 0 1 1 1 1Intake cam 1 1 1 1 1 1Exhaust value 1 1 1 1 1 1Intake valve 1 1 1 1 1 1Cylinder head 0 0 0 0 0 0Piston 0 1 0 0 1 0Connecting rod 1 1 1 1 1 1Engine block 0 0 0 0 1 0

The final step is to design the product family. We design each of the engines to minimizethe performance deviation from the null-platform optima. Therefore, the objective function:

minx=[xp1 ,xp2 ,...]{((f

p,◦ − f p(xp))/f p,◦)2}

is used when solving equation (2). The optimal family designs for the 6% and 7% deviationbound cases are presented in tables 4 and 5, respectively. Performance optima and associ-ated deviations are summarized in table 6; they demonstrate that the bounds on performancedeviation due to commonality are not violated. The results have been validated by solving theentire problem using the relaxed-problem formulation. The optimization results are consistentto the combined approach. Additionally, the sharing order that is computed by the first-ordermethod is confirmed.

Table 4. Product family designs for 6% deviation.


b (mm) 95.00 75.03 95.00s (mm) 79.17 79.17 90.00l (mm) 197.92 197.92 197.92cr 10.00 10.00 8.69di (mm) 26.65 26.65 26.65icta (degree) −5.32 10.00 −10.00iam 1.05 1.05 1.05de (mm) 30.00 30.00 30.00ecta (mm) 10.00 10.00 −10.00eam 0.99 0.90 0.90hc 8.80 8.80 11.71disp. (cm2) 3367 2100 3828Power (kW) 109.62 80.54 80.00Torque (Nm) 209.36 153.81 152.79BSFC (gh−1 kW−1) 273.67 269.50 330.72NOx (p.p.m.) 25795.82 25252.87 18110.20dP mx/DCA (bar/degree) 1.29 1.26 0.87Pmax (bar) 48.86 47.29 35.22Fuel usage (gh−1) 30000.00 21704.10 26457.97


Table 5. Product family designs for 7% deviation.


b (mm) 91.02 74.13 91.02s (mm) 90.00 81.09 90.00l (mm) 202.72 202.72 202.72cr 10.00 10.00 8.64di (mm) 27.43 27.43 27.43icta (degree) 3.42 10.00 −10.00iam 1.02 1.02 1.02de (mm) 30.00 30.00 30.00ecta (degree) 10.00 10.00 −10.00eam 0.91 0.91 0.91hc 10.00 9.01 11.78disp. (cm3) 3513 2100 3513Power (kW) 106.29 84.39 80.28Torque (Nm) 203.00 161.16 153.31BSFC (gh−1 kW−1) 282.25 268.83 324.71NOx (p.p.m.) 24638.78 25237.59 18883.23dP mx/DCA (bar/degree) 1.22 1.28 0.89Pmax (bar) 46.20 48.15 35.86Fuel usage (gh−1) 30000.00 22685.43 26066.62

Table 6. Optimal product family design results and associatedperformance deviations.

Variant A (kW) B (g h−1) C (p.p.m.)

Null platform 114.29 21,265.32 17,853.09Platform with Lp = 6% 109.62 21,704.10 18,110.20

Performance loss (%) 4.09 2.06 1.44Platform with Lp = 7% 106.29 22,685.43 18,883.23

Performance loss (%) 7.00 6.68 5.77

6. Conclusions

A methodology was proposed for making commonality decisions and designing optimal pro-duct families, by combining two previously developed approaches. The methodology wasapplied successfully to the design of an automotive engine family consisting of three variantswith different functional requirements modelled by means of artificial neural networks. Theadvantage of the new method is improved computational efficiency. This allows for commonal-ity decisions on larger product sets. The limitations include the heuristic nature still inherent tothe first-order method. However, the consequence of this limitation has been reduced throughthe combination of the strategies. Future work for the methodology could include addingsearch heuristics in an attempt to further increase the efficiency of the utilized algorithms. Themixed-discrete case could be explored by implementing evolutionary algorithms together withthe continuous algorithm. Future work for the engine application includes looking at enginevariants defined by different power specifications and using the number of cylinders as a designvariable. An important next step is to integrate the engine model into a full vehicle model toobtain a more comprehensive operating cycle. From a methodology viewpoint, the next stepis to link the engineering decisions with financial analysis models (Georgiopoulos et al. 2002)so that product families can be examined in the context of a combined business-engineeringproduct development and assessment process.


Acknowledgements

The authors would like to thank Nestor Michelena and Alexis Perez-Duarte for their numer-ous helpful suggestions and discussions regarding the commonality strategy, and GeorgeDelagrammatikas, Terry Wagner, and Guangquan Wu for their assistance with the engine sim-ulations and design model. This research was partially supported by the Automotive ResearchCenter, a US Army Center of Excellence in Modeling and Simulation of Ground Vehicles, by aUSArmy Dual-Use Science and Technology Project, and by the General Motors CollaborativeResearch Laboratory at the University of Michigan. This support is gratefully acknowledged.

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