Aled Mathematcal Scences, Vl. 2, 2008, n. 5, 241-248 A Nte n the Lnear Prgrammng Senstvty Analyss f Secfcatn Cnstrants n Blendng Prblems Umt Anc Callway Schl f Busness and Accuntancy Wae Frest Unversty, Wnstn-Salem NC, 27109, USA Anc@wfu.edu Abstract In blendng rblems there are tycally secfcatn cnstrants that lmt the cntent f varus rertes f the blend that t acqures frm the ngredents t certan maxmum r mnmum ercentages f the ttal blend. Fr sae f lnearty, these cnstrants are cmmnly ncluded n the rblem n a way that recludes drect senstvty analyss wth resect t changes n these ercentages. Ths nte shws that the senstvty analyss wth resect t changes n the target secfcatn ercentages can be derved frm the rdnary senstvty analyss wth mnmum addtnal effrt; and that cmmn LP cdes can be slghtly mdfed t facltate senstvty analyss f blendng ercentages.
242 Umt Anc Mathematcs Subect Classfcatn: 90C31 Keywrds: Lnear Prgrammng, Blendng Prblems, Senstvty Analyss. Thrugh ts hstry, ne f the mst frequently cted alcatn examles f lnear rgrammng has been the s-called blendng, rblem n whch varus ngredents (nuts) are mxed nt ne r mre blends (ututs) t satsfy sme bectve [1]. Blendng rblems, amng ther tyes f cnstrants, may nclude the secfcatn cnstrants. These cnstrants lmt certan rertes (such as msture) f the blend, whch t nherts frm the ngredents, t certan mnmum r maxmum ercentages f the ttal. Mst cmmn LP sftware utut des nt allw drect senstvty analyss f the target ercentages. The urse f ths nte s t shw hw smlex-based LP sftware may be slghtly mdfed t enable drect and full senstvty analyss f these target ercentages. A tycal such cnstrant, say the th cnstrant f the rblem, may be wrtten generally as: sx / x ( 1) where s s the ercentage f the rerty n questn cntaned n ngredent, and S s the subset f all ngredents whch may be used n blend. By selectng arrate zers and nes fr s, Eq (1) can als mdel a cmmn tye f
Lnear rgrammng senstvty analyss 243 secfcatn cnstrant ne whch lmts the ercentage f a certan subset f ngredents n the blend. Snce ths frm s nt lnear and thus unsutable t lnear rgrammng t s equvalently ncluded n the LP as: s x x 0 (2) Hwever ths transfrmatn recasts the senstvty wth resect t the RHS f Eq (1) nt the senstvty wth resect t several technlgcal cnstrants n Eq (2). Ths mre dffcult frm f senstvty has been studed extensvely as art f LP thery. See fr nstance Smnnard [2,. 145]. Hwever, the theretcal results cncernng the technlgcal cnstrants have nt been aled t the senstvty analyss f target ercentages; nr they have been mlemented n the mst cmmnly used LP sftware. Yet n many blendng and mxng rblems ths arameter, may be set as a matter f management lcy and thus effect f adtng dfferent lces n the tmal LP value may be qute a valuable gude. The tmal dual rce fr the mdfed cnstrant, say d, gves the rate f change n the bectve functn as the rght hand sde (RHS) f (2) s erturbed wthn the range n whch the tmal bass and thus d des nt change (allwable range). Let z / r dente ths rate, where z s the bectve functn value and r s the RHS f (2) (currently zer); and r and r dente the allwable ncrease
244 Umt Anc and decrease n the RHS resectvely. Althugh d gves sme useful senstvty nfrmatn ertanng t the current tmal slutn, t des nt drectly answer the mre legtmate questn f the behavr f the tmal slutn fr changes n the arameter, the RHS f (1). The lnear frm (2) wth a RHS f r s s x x r. It can be re-wrtten as s x / x ( r / x) S whch, n turn, means that changng the RHS f Eq (2) by r s equvalent t changng the RHS f (1) by r / x. Therefre we have: S = r / x S () 3 = r / x and S = r / x (4) S z / = ( z / r) x. S (5) Suse that the tmal slutn t the blendng rblem, wth Eq. (2) and ts senstvty analyss nfrmatn s avalable frm a standard LP rgram. Althugh z / r = d s cnstant n the allwable nterval; z / may nt be cnstant, that s z() may be nn-lnear n. The behavr f the tmal slutn vectr and the bectve functn value, as changes can be estmated by smly evaluatng the quantty d x at the current tmal slutn x. Hwever, ths
Lnear rgrammng senstvty analyss 245 wuld nly be arxmate, because as the RHS f Eq (2) changes wthn the allwable range, whle the tmal bass and d stay cnstant, the tmal x and thus z / may nt. The qualty f ths arxmatn deends n the magntude f the change n the quantty x. Whle n sme cases ths change mght be S small and can be gnred, n thers t mght be cnsderable enugh t sgnfcantly dstrt the senstvty results wth resect t arameter,. The standard LP slutn wth Eq (2) hwever, can be used t erfrm a full and exact senstvty analyss f the slutn fr changes n the arameter,. Let x ( r) be the vectr f slutn values as a functn f the change n the RHS f (2), vectr b the rgnal rght hand sdes f the LP wth Eq (2), and B the current tmal bass. We have: 1 x ( r) = B ( b u r ) = x B 1 u r, (6) where u s the th unt vectr. In (6), the term, B 1 u traces ut the th clumn f the tmal bass nverse. Let us dente the elements f ths clumn by β. Therefre, we have: x ( ) = r x r β, where S s the subset f S S S crresndng t the currently basc vectrs. Wth ths nfrmatn the exact lmts f, and are btaned as: = r /( x r β ) and = r /( r ). ( 7 x β ) S
246 Umt Anc Als the exact rate f change n the bectve functn value, as the RHS f Eq. (1) changes, may be wrtten as: z / = z / r ( x r β ). T exress ths rate n terms f rather than r, we can slve r = ( x r β ) fr r and substtute n abve t gve: z / = d x β x 1 β, (8) whch s the average rate f change n z, as changes by an amunt. The tmal value f the LP as changes by wthn the allwable range, can be btaned by multlyng (8) thrugh by and smlfyng z( ) = z d x 1 β. (9) In Eq (9), when β = 1, z( ) becmes undefned. Hwever ths trublesme stuatn des nt ccur, because1 β > 0. T rve ths, assume that β < 0. As changes n the negatve drectn frm 0, t reaches ( 0) befre t becmes / β, that s t say >1/ β. Ths s 1 easly shwn by substtutng r /( x r β ) fr and rearrangng terms t r β get < 1. x β r Ths s true whenever x > 0, snce bth β and r
Lnear rgrammng senstvty analyss 247 are nn-stve. Furthermre, snce 1 / β < we have 1 0 β >. A smlar argument hlds fr the case β > 0. It s als straghtfrward t examne the functnal behavr f z( ) changes n the allwable range. The frst dervatve f z( ) as s d x 1 β ( ) 2 and thus has the same sgn as d. Thus z( ) s nn-decreasng f d 0; nn-ncreasng therwse. The secnd dervatve s 2d β x (1 β ) 3 whch mles that f β = 0, then z( ) s lnear n the allwable nterval; als snce x and 1 β are bth nn-negatve, z( ) s cnvex f d and β have the same sgn, cncave therwse. The abve suggests that standard LP sftware acages can be mdfed slghtly t enable users t cnduct exact senstvty analyses f secfcatn-tye cnstrants. All that s requred, ssbly as a user selectable tn, s t rert the β quanttes crresndng t thse cnstrants that the user has cded as secfcatn-tye durng nut. REFERENCES [1] Gerge Dantzg, Lnear Prgrammng and extensns. Prncetn Unversty Press, Prncetn, N.J., 1963
248 Umt Anc [2] Mchel Smmnard, Lnear Prgrammng. Prentce Hall, Englewd Clffs, N.J., 1966. Receved: June 5, 2007