VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES

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1 VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty 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 xn = P) s constant sum 23

2 ( 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

3 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+ p pk P k 25

4 p + p pk + p k + > P 2 α = P p p2... pk By addng to the frst two relatons wrtten above I obtan p+ p pk + P+ p + p + + p + p + > P k k+ In our case condton, α + p k + t follows that p + p pk + pk P+ 2 + p + p + + p + p + p > P 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

5 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

6 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, S. Bârză, Programare procentuală: propretăţ ale soluţlor, Revsta de Informatcă, nr. 2, Inforec, Bucureşt, S. Bârză, Modele întreg cu sumă constantă ş varable cu domenu unform, Analele Unverstăţ Spru Haret, Sera Matematcă-Informatcă, nr.2, Bucureşt, S. Bârză, Constant Sum Integer Programmng wth Non Unform Defned Varables, Proceedngs of Internatonal Conference on Operatonal Research for Developments, Inda, Goemans M.X., Semdefnte Programmng Combnatoral Optmzaton, Doc. Math. Extra Volume ICM: , Goemans M.X., Rendl F., Semdefnte Programmng n Combnatoral Optmzaton, November, Hoffman K.L., Combnatoral Optmzaton: Current Successes Drectons for the Future, Journal of Computatonal Appled Mathematcs, 24, pp. 34, Nemhauser G.L., Wolsey L.A., Integer Combnatoral Optmzaton, John Wley & Sons Inc, New York,