VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty barza_slvu@yahoo.com Abstract Ths paper wants to contnue the studes for a class of problems named constant sum nteger programmng ntroduced earler. Ths approach tres to see what s happenng when a varaton appears n rght h of constant sum condton. In ths paper I prove that lmted varaton nfluence no more than 3 model varables n model optmum there s a way to say whch these varables are. My consderaton could be mportant f we want to transform a constant sum condton n a stochastc form. Key-words: lnear programmng, combnatoral optmzaton, percentage programmng. AMS classfcaton: 90C05, 90C0, 90C27 Computng Revews classfcaton: G..6. Introducton In earler paper I had consdered constant sum nteger model wth non unform varables whch had explct form: (MSCDN) opt cx Ax b x = P x p { 0,,..., }, =,..., n constrants. In such a model X = P (or x + x 2 +... + xn = P) s constant sum 23
( 2 ) If c= c, c,..., cn has the property of monotony ( c c2... cn ) there exsts x R n, for whch Ax b X = P (x R n s a feasble soluton for (MSCDN)), then n a lexcographc sutable order the smallest feasble soluton x s optmal soluton for (MSCDN). In the same paper I gave an algorthm to generate optmal soluton for (MSCDN) In the above condton, the algorthm for (MSCDN) produced an optmal soluton of the form (,,...,,,0,...,0) x p p2 pk α =, () where for the value n poston k + I have 0 α < p k +. Also t was proved that ths result s a generalzaton for the case of 0- models wth monotone coeffcents n goal functon. 2. A famly of constant sum nteger model I wsh to contnue the study about constant sum nteger model. I consder a famly of constant sum nteger programs n whch I have a varaton for the value P of constant sun constrants. To smplfy the model I consder that monotony condton are fulfll for coeffcents. So, all the models wll have an optmal soluton of form (). The varaton consdered so far must be n maxmum length of p l. It s also assumed that n ths varaton the constant sum nteger model stll have optmal solutons. Startng wth (MSCDN) model I can buld the famly of models (MSCDN ): (MSCDN ) where opt cx Ax b x = x { 0,,..., p }, =,..., n [ P t P + t] N,, t mn p =, n 24
s chosen so that for any, y [ P t P + t] N optmal soluton., the (MSCDN ) model has an 3. Optmal soluton varaton for the famly of constant sum nteger model The mal goal of ths paper s to set a result about optmal soluton varaton for the famly presented n paragraph II. Frst I must observe that all members of (MSCDN ) model famly has an optmal soluton for whch there s a permutaton σ so that c σ () s a monotone strng ( () s of form (). Ths observaton allows me to gve the next result: Theorem. Let (MSCDN ) be a famly of models as t s defned n paragraph II. Then there exsts uvw,, {,2,..., n} so that wth excepton of postonsu, v w, optmal solutons are nvarant. In addton, f σ s a permutaton so that () consecutve ntegers. c σ are ordered, then σ ( u), σ ( v) ( w) σ are three Proof. The general case can be reduced to the one of ordered strng for the value of c, c2,..., cn Let σ be the permutaton whch ordered ntal values of coeffcent n obectve functon. Let x be the optmal value obtaned for value P as a rght h value n p constant sum constrant. Now, usng known propertes for model famly (MSCDN ), we have p (,,...,,,0,...,0) x = p p p α 2 ths soluton s gven by algorthm specfed n [2]. For an arbtrary [ t, t] N I have 3 cases. Case I. For α + p k + I use agan the algorthm gven n [2] to reach a soluton for P as a rght h of constant sum constrant, whch exsts because the famly (MSCDN ) has ths property. Then, there exsts an nteger k so that p+ p2 +... + pk P k 25
p + p +... + pk + p k + > P 2 α = P p p2... pk By addng to the frst two relatons wrtten above I obtan p+ p2 +... + pk + P+ p + p + + p + p + > P+. 2... k k+ In our case condton, α + p k + t follows that p + p +... + pk + pk P+ 2 + p + p + + p + p + p > P+. 2... k k+ k+ 2 If I consder β = pk + α, than the soluton for P+ as a rght h of constant sum constrant s x = p, p,..., p, p, β,0,...,0 p+ 2 k k+ 2 so, new values appear only n postons k + k +. Case II. For α + < 0 By smlar consderaton, new soluton s x = p, p,..., p, β,0,...,0, so, new values appear n postons p+ 2 k k k +. Case III. For 0 α + < pk + Ths s a smlar case to case I only value or rang k + s modfed. The concluson for these three cases, for any values [ t, t] N, no more than three values are modfed n soluton vector, the values of rang k, k + k + 2. I must remember that obectve functon coeffcents are ordered. Comng back to general stuaton, optmal soluton s reached by applyng σ to optmal soluton n ordered coeffcent case. So we have only 3 non nvarant component n general soluton x, placed n postonu, v w whch s ` k, k σ k + 2. And so I fnsh the theorem demonstraton. σ σ ( + ) 26
4. Optmal soluton varaton subspace Now, because I know that varaton for rght h of constant sum constrant produce modfcaton for no more than three values n optmal soluton, t s possble to make a proecton of soluton space n 0 f = = { 0,,..., } S S p nto 3-dmenson space Sf = { 0,,..., pu} { 0,,..., pv} { 0,,..., pw }, where u, v w are modfed value rang n optmal soluton so that u<v<w. If proecton space s S 0 S f, the goal for ths secton s to determne proecton space form. If I follow the demonstraton for the above theorem, for ordered coeffcents model I observe that f modfcaton produces varaton only n x k + then the value for ths element s between 0 p σ. So I have that ( k+ ) ( p, u y,0 ) S0, pu = p σ, y { 0,,..., p } ( k ) v wth pv = p σ ( k + ) P + α, pv α P t, P + t [ ] [ ] N represent rght h n constant sum constrant. In a smlar way, f modfed values n ordered coeffcents are of rang k k+ then ( z,0,0) S0, z { pu t+ α,..., pu}. Also, f rang k+ k+2 values are modfed, then ( pu, pv, s) S0, s { 0,,..., P+ t p v}. Now I can gve the followng result. Lemma. If (MSCDN ) s a famly of models defned n secton II wth optmal soluton space n 0 f = = { 0,,..., } S S p non nvarant optmal soluton values of rang u, v, w so that u<v<w then pr S = x,0,0 x p t+ α,..., p U p, y,0 y 0,,..., p U 0 { { u u} } ( u ) { v} ( p, p, z) z { 0,,..., P+ t p } S f { u v v } { } U 27
5. Conclusons In the real world, t s the best model formulaton for a real problem n the feld of stochastc model. Tll now, all studes about percentage programmng are n respect wth determnstc models. Ths paper shows that a sgnfcant part of optmal soluton for constant sum nteger model s nvarant under the above assumpton. I consder that ths result s a prelmnary one t prepares the future studes n whch to have a constant sum stochastc nteger programmng. Such a model can have a value for constant sum whch s a dscrete rom varable wth some specfc probablstc dstrbuton. Another stochastc model n ths area can have an estmated value for constant sum. Both subects wll be consdered later. REFERENCES. S. Bârză, Programare procentuală: formulăr ş propretăţ, Revsta de Informatcă, nr., Inforec, Bucureşt, 2004. 2. S. Bârză, Programare procentuală: propretăţ ale soluţlor, Revsta de Informatcă, nr. 2, Inforec, Bucureşt, 2004. 3. S. Bârză, Modele întreg cu sumă constantă ş varable cu domenu unform, Analele Unverstăţ Spru Haret, Sera Matematcă-Informatcă, nr.2, Bucureşt, 2006. 4. S. Bârză, Constant Sum Integer Programmng wth Non Unform Defned Varables, Proceedngs of Internatonal Conference on Operatonal Research for Developments, Inda, 2005. 5. Goemans M.X., Semdefnte Programmng Combnatoral Optmzaton, Doc. Math. Extra Volume ICM: 657-666, 998. 6. Goemans M.X., Rendl F., Semdefnte Programmng n Combnatoral Optmzaton, November, 999. 7. Hoffman K.L., Combnatoral Optmzaton: Current Successes Drectons for the Future, Journal of Computatonal Appled Mathematcs, 24, pp. 34, 2000. 8. Nemhauser G.L., Wolsey L.A., Integer Combnatoral Optmzaton, John Wley & Sons Inc, New York, 999. 28