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Engineering design: applications of goal programmingand multiple objective linear and geometricprogramming‡J. S. H. KORNBLUTH aa Jerusalem School of Business Administration , Hebrew University , Jerusalem, IsraelPublished online: 03 May 2007.

To cite this article: J. S. H. KORNBLUTH (1986) Engineering design: applications of goal programming and multipleobjective linear and geometric programming‡, International Journal of Production Research, 24:4, 945-953, DOI:10.1080/00207548608919779

To link to this article: http://dx.doi.org/10.1080/00207548608919779

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Engineering design: applications of goal programming and multiple objective linear and geometric programming$

In this paper we show how the standard goal programming approach to engineering design can be enhanced by the use of multiple objective linear and geometric programming formulations. For simple formulations a logarithmic transformation may be adequate to allow problems to be solved using linear programming and multiple objective linear programming methods. For some mow complex forms, geometric programming can be used.

Introduction Goal programming is an extension of linear programming (LP) which enables the

model builder t o indlude multiple (and conflicting) objectives in an optimizing model. It has been applied to a broad spectrum of management science, business and production problems. In a recent paper, Singh and Aganval (1983) applied linear goal programming to the optimum design of an extended ring.

I n this paper we will show how the.use of goal programming can be broadened by using multiple objective linear programming and geometric programming methods. The useof multiple objectivelinear programming methods allows thedecision maker to rwiew a wide range of alternative solutions to his problem, without having t o state nny a priori trade-offs between them.

Goal programming and multiple objective linear programming in engineering design l l a n y problems in engineering design can be formulated as non-linear goal

programming problenis of the form

where S are thc decision variables of the design problem; ji(S) are the various design or performance criteria (e.g. strength, rigidity, elasticity. weight), i= 1 , . . . , p ; gi are the goals set for the design or performance criteria, i = I , . . . p; G';, I,'+ are the multipliers required to equate f i (X) with g,, i = 1 , . . . , p; C; > 1 implies tha t the goal is under-attained. I..+ > I implies that the goal is over-attained; I;, 1: are subjective

Revision received Sovember 1984. t Jerusalem School of Business Administration, Hebrew University, Jerusalem, Israel. :This paper xas written whilst the author was on sabbatical leave a t the Department of

Management Science. Imperial College of Science and Technology, Exhibition Road, 1,ondon SW7 2BS

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w i g h t s i~pplied by t . 1 1 ~ tlc.cision-makw t o rinclw-attainmcmt or ovcr-att,;rinmc-nt. of his g ~ ~ i ~ l s : a t~ t l this func.t.ions f i are of thc* form

For an esample, s rc Singh and Agarwal (1983). Equation ( I ) is not the usual form of the (linear) goal programming prol)lem.

Csing the transformations

Equation (1) becomes

P min (I.: u: + 1.; u; )

i= l

q unconstrained 1 Equation (3) is a standard goal programming prohlem, .except tha t the non- negativity of x j has been relaxed. xj are unconstrained.

Standard LP packages normally assume non-negativity of all variables. I f x j are ass~inied positive in eqn. (3), then Sj are automatically constrained t o be greater than 1 in eqn. (1). The. transfoimation x j=r i -x j , with xi.r;BO can he used t o overcome this problem.

\Ye can use a similar transformation on any additional constraints in eqn. ( I ) of t 111. for111

\Yhereas it may he reletivelysimple tos ta te thegoalsofthedesign prol)lem ({gi)). it may Iw considerably more difficult t o define the appropriate values for {j.;), {I.:) wllich determine the relative importance of the non-attainment of goals. both intra- goal (over-nttainment versus under-attninment of a pnrliculur goal) and inter-goal (non-attainment of one goal versus non-attainment of any other). In (linear) goal programming two forms of j. are used, preemptive and weighted. Both could he :~pplirtl toeqn. (3). Singh and Aganval (1983) op t for the pre-emptire approach. The Ic~prritlirnic transformation clearly preserves the pre-emptire structure but may influence the weighted problem because of the induced sealing. An alternative approach to eqn. (3) ,might be t o solve the mukiple objective linear programming (.\IOI,P) problem

subject t o the constraints of eqn. (3). I \\'here - eff represents the search for the set of all optimal (efficient) solutions (in

a minimizing sense). For definitions and examples see Evans and Steuer (1973) or

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Zclr~ly (I!)i-t). In this case tlw (q',q;) pairs rcprescnt the intra-goal weighting of over-attainment versus under-attainment. So intcr-goal weighting is needctl sincc! ;dl possil)le inter-goal weights are providctl by the X1OI.P algorithm.

Since in many cases the problems will not involve more than say 10-15 design goals, and maybe 30-40 design parameters, the use of eqn. (4) is quite reaaonahle- even for a micro-computer. Using eqn. (4) only requires tha t the decision-maker determines the relative values of q' and q;, i.e. of over-attainment and under- attainment of goal i, a relatively easy task, and does not require any inter-goal comparisons which may be difficult t o make a priori. The set of solutions to eqn. (4) will include all possible solutions for different sets of inter-goal weighting for given (v: 1.

An example Consider the esample presented by Singh and Agarwal (1983). The original

problem can be formulated as

0.7r U; s.t. -.-= (sensitivity goal)

Ebt2 L': I Ebt3 6'; -.-= lo6 (rigidity goal)

r3 U :

r - = 1.25 (minimum radius) v :

b - 5.0 (minimum width) r;:

( r . / I . t ;in. I I W r i~tl ir~r. width and th ichess , respectively of an octagonal ring.) The goal progr;unn~ing form of this problem is

4

min 2 (2:~: +;.;I[;) i = l

-.rl - 2 , r 2 + , r 3 + ~ / ; --I(: = 0.47771

- J , -3.r2+3x3 +U;--U; = 032222

2 3 - u: = 0.0969 1 2' - U: = 0.69897

xi unconstrained, u', u; > 0

The nlultiple objective linear programming form is

effzi=q:u'+q;u;, i=1, ..., 4 - subject to the constraints of eqn. (6).

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Ifwegivvequal w igh t too\-t*r-a1t:tinment atid I I I~IT-a t ta inment with r ~ s p w t to each goal, there are only two efficient (non-dominated) solutions to eqn. (7). These arr shown in Tahlc 1.

In solving eqn. ((i). Singh and Agarwal use eight different objectivc functions correspond.ing to eight different pre-emptive weight structures, only one of which (Z,) has different emphasis on over-attainment and under-attainment of any goal. Their solutions (corrected for an arithmetical error in the first three calculations of the under-achievement of the rigidity goal) are shown in Tahle 2.

Solutions S, and P2 (the efficient non-dominated solutions) correspond with Z6 and Z,, respectively. As might be expected,. these solutions are 'better' than any others in the set. Solution S, (=Z6) dominates Z,, Z2 and Z,. S,, Z,. Z, and Z, all attain the sensitivity goal. but S , comes closest to simultaneously attaining thc rigidity goal. Similnrly S, (=Z,) dominates Z,, Z, and 2,. S, attains the rigidity goal (as do both 2, and Z,) but it is nearer to attaining the sensitivity goal than either Z, or 2,. Z, attains less of the sensitivity goal than S2 whilst deviating from the rigidity goal (which S2 attains). The change of emphasis in Z , has made no practical difference.

From the decision-making viewpoint, all the relevant decision-making inform- ation is encapsulated in the set {S,,S,}. Using the multiple objective sensitivity analysis approach suggested by Kornhluth (1974). we can infer that there arc two 'domains' of preference for the decision-maker. He will 'prefer' S, over S2 if

where(; : 5; represents the relative importance ofdeviating from goal 2 from above. as opposed to deviating from goal I from helow. If all intra-goal weights are equal. 5; : (; \vould be the relative importance of deviating from goal 2 as opposed from deviating from goal 1. Applying parametric analysis to rows of the 1.P used for efficiency tests (Evans and Steuer 1978, p. 57, problem (P)), it is theoretically possible to identify the ranges for which intra-goal weightings leave the efficient set unalterrd. In practivr one \vould carry out the analysis hy interactively changing thc intra-goal weights.

Extensions to geometric programming The primal problem in geometric programming has heen defined by Duffin et 01.

(1967) as follows. Find the minimum value of a function yo subject to the constraints

and

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Tlw r s ~ ~ m ( v ~ I s I I ~ ~ ~ ' Y ' :trI)itri~l:\- r ( d r l ~ ~ n ~ l w r s 1 ~ 1 t tlw cvwttic.icwtsri arc. asstr~nt:tl t o lw ~wsitivc-. ~ ~ ( 1 ) in ( ~ I I . ( 1 0 ) i ~ r v k n f n v ~ ~ as p o s y ~ ~ o t ~ ~ i i ~ l s . (For d(*tinitions set: Ih t l in el nl. I!Ni.)

Pastwal and I+n-Israel (1971) have shown how the standard geometric 1wogr11niming mc.1 I~ods can Iw used t o s o l w prol~lems of rector minimization, i.e. thc idwlitiration of etfiricnt non-dominant minimizing solutions. For design prol)lems a h e r r both objective criteria and constraint functions are posynomials and conform t o the pr in~al equations stated at)orc their met.hod is immediately available.

Geometric programming can also he extended t o goal-seeking situations, with some litnitations. Given the posynomial fi(x) antl a goal g,. the goal constraint fi(r) should Iw less than gi can he modelled Ijy the geometric programming constraints

and

with the associated minimiaation of [I:. ( In geometric programming form. eqn. ( 1 3) \voultl be written a s (C:)-'$1.) S o t e that the left-hand side of eqn. (12) is a pnsynoniial. A goal constraint fi(x) should he greater than gi cnnnol he modelled in a similar way since the reciprocal [ fi(z)]-I need not be a posynomial. This is only the case if fJr) is of the form

'She goal ~wogr;mtning form of geometric* programming can therefore be used as follows i n Tal)le 3.

Table 3.

Tlw esatnple presented by Pascual and Ikn-Israel (1971) is essentially type ( h ) . i~ncl is in fact similar t o problem ( I ) . Imt us consitlrr an extension of the esamplc in I'uswsl and Ikn-Israel (1971) to show the effect of the posynon~ial form. The ~ r o l ~ l e l u is that of designing a tensile rod of circular cross-section. using w i g h t antl ~(1st criteria. \Ye have enlarged the original prol,lern I y adding a surfact. area const~xint antl a weight : cost relationship.

An example of geometric programming in engineering design Let

d diameter of the rod (maximum value dm,,) L length of the rod (minimum value Lmi,)

11- weight of the rod

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C.' cost I I ~ Lhe rod S, yield strrngth of th r rod A' safety factor of the rod c unit. weight cost

11- s p ~ r i f i r wpight

The r~rol)lem formulation is

minimize W = (n/4)wd 2L, and

minimize C=(n/4)cwd2L.

s.t. 4PN/xS,d C 1 (performance) (14)

kl(n/2)d + klxdL$l (surface area restriction) (15)

d m < 1 (maximum diameter) (16)

L i / L 1 (minimum length) (17)

k, / (cd) < 1 (weight: cost relationship) (18)

c,w, L , d a O

P, iV, n, S,, k,, k,, 1, d ,,,, L,,, all constants. The set of efficient solutions for this problem can be identified by solving either

min {a,(nwd Ll4) + a2(ncwd 'L/4)} (19)

or

min (xwd 2L/4)"'(ncwd 'L/4)"'= (xwd 2L/4)c'2 (20)

subject to eqns. (14)-(18), for varying (a,,a,), a , +a,= 1, a i 2 0 . Xote tha t the surface area restriction (eqn. (15)) is a posynomial constraint,

preventing the problem from being solved using the logarithmic transformation. Using duality theory described in Duffin et al. (1967). problem (19) can be solved

via the dual problem

max v(a, 6 ) = (aln/461)d'(a2n/4d2)d1(4PS/nS,)d~(kln/26,)d4

(kln/65)d'(l /d,,,)d6(~,ii)a6(k2)d~(64 + 65)d4+d5

s . t . 6, +62 = 1

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Problem (22) is easier t o solve than (21 ) because i t involves only 2 degrees of freedom in the cvnstraints as opposed to 3 degrees in eqn. (21).

(:oi11s such as 1I.G II', and C G C , van be moclelled a s

suhjctd to eqns. (14)-(18) and solved via the suitably rcformulatrd dual. Sensitivity analysis of the optimal solution can be carried ou t using the methods

of N o l ~ a n and Agarwal (1683) and Dembo (1080).

Conclusions I:or many forms of engineering design problen, the logarithmic transformntion

may Iw sufficient to allow the designer to obtain the set of all feasible efficient solntions to his or her design problem. For some more complex cases. geometric propxmming can be successfully applied.

Acknowledgments Tllc author gratrfully ac:knowlcdgrs the verx helpful comments of the referees

u.hic.11 h a w been inc.oq)orated in this paper.

Ihns rrt artic.1~. noun dtmontrons comment la fac;on standard tl'nborder la programmation de la conception en inghierie peut Btre aniCliorbe grice B I'utilisation de forn~ulationn c k programmation gGombtriques et li11Caires multi- ples. Pourde simples formulations. une transformation lo~arithmique peut sufiirr pour rtsoudre les prob1i.mt.s graw I'utilisation clrs 1nCthodt.s dr proprammation lintaires rt a objectifs lintaires multiples. Pour des formes pllw wmplenes. on p u t utiliser la programmation gbombtrique.

In dieser Abhandlung zeigen wir, wie sich der standardmahge Tisunps\veg der Zielprogrammierung fur die Konstruktionsplanung durch den Einsatz objektiver linearer oder geometrischer Programmierformulierungen verbessern IiiBt. Fiir einfache Formulierungen ksnn eine logarithmische Transformation zur Problemliisung mit den Methnden der Linearprogrammierung oder der

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~lrlir~;~~~l~zirl-I.ine;~r~~ro~ra~n~r~irr~~n~ ausreirhrn. Fiir knmplizierterr For- mulirrut~gt-n kann die geometriwhe I 'r~~grammi~rrung eingwetzt werden.

References DRM\(RO. R. S.. 1980. The sensitivity of optimal engineering designs using geometric.

~~rogranimitig. Engineering Oplimi:olion, 5, 27-40. DVFPIS. R. J.. PETEHSOS. E. L.. and ZENER. C.. 19fi7, Geometric Programming-Theory and

Application (Sew York: John N'illey 6 Sons Inr) . E v ~ s s . . l . P.. and STECER. R. E.. 1973. A revised simples method for linear multiple o h j ~ c t i v r

programs. Nolhenmlicol l'rogramnring, 5, KO. 1 . 54-72. KORS~I.L-TII. J. S. H.. 1974, Lluality indifferenceandsensitivity analysis in multipleobjectivr

linear programming. Operalioml Research Quarterly. 25, S o . 4. 599-614. J l o l t ~ s . C., and AOARWAL. \'.. 1983. A computational algorithm for the sensitivity analysis of

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P ~ s c c ~ l . . L. D.. and BES-ISRAEL, A,, 197 1. Vector-valued criteria in geometric programming. Operational Research. 19, S o . 1 . 98-104.

SISGII . S., and A ~ A R ~ A L . S. K.. 1983, Optimum design ofan extended octagonal ring by goal programming. Inlernalioaal Journnl of Production Research. 21. S o . 6 , 891-898.

ZELESY. M., 1974. Idinear Mulliobjectice I'rcgramming. Lecture Sotes in Economics and Jlathematical Systems (Berlin: Springer-Verlag).

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