Geometry Inspired Algorithms for Linear Programming

Transcription

1 Geoetry Inpired Algorith or Linear Prograing Dhananjay P. Mehendale Sir Parahurabhau College, Tilak Road, Pune-4030, India Abtract In thi paper we dicu oe novel algorith or linear prograing inpired by geoetrical conideration and ue iple atheatic related to inding interection o line and plane. All thee algorith have a coon ai: they all try to approach cloer and cloer to centroid or oe centrally located interior point or peeding up the proce o reaching an optial olution! Iagine the line parallel to vector C, where C T denote the objective unction to be optiized, and urther uppoe that thi line i alo paing through the point repreenting optial olution. The new algorith that we propoe in thi paper eentially try to reach at oe eaible interior point which i in the cloe vicinity o thi line, in ucceive tep. When one will be able to arrive inally at a point belonging to all neighborhood o oe point on thi line then by oving ro thi point parallel to vector C one can reach to the point belonging to the uiciently all neighborhood o the point repreenting optial olution.. Introduction: There are two type o linear progra (linear prograing proble):. Maiize: C T Subject to: A b 0 Or. Miniize: C T Subject to: A b 0 Where i a colun vector o ize n o unknown. Where C i a colun vector o ize n o proit (or aiization proble) or cot (or iniization proble) T coeicient, and C i a row vector o ize n obtained by atri tranpoition o vector C. n. Where A i a atri o contraint coeicient o ize

2 Where b i a colun vector o contant o ize repreenting the boundarie o contraint. By introducing the appropriate lack variable (or aiization proble) and urplu variable (or iniization proble), the above entioned linear progra get converted into tandard or a: Maiize: C T Subject to: A + = b (.) 0, 0 Where i lack variable vector o ize. Or Miniize: C T Subject to: A = b (.) 0, 0 Where i urplu variable vector o ize. In the geoetrical language, the contraint deined by the inequalitie or the o called conve polyhedron bounded by the contraint plane, A = b, and coordinate plane, = 0, and it i traightorward to check that there eit at leat one verte, o thi polyhedron at which the optial olution or the proble i ituated when the proble at hand i well deined having at leat one olution and not unbounded or ineaible one. Further, we denote the plane deined by equation C T = d, or oe choen value, d, a the objective plane. Many a tie there eit be unique optial olution but oetie there ay eit any optial olution, e.g. when one o the contraint plane and the objective plane are parallel to each other then we can have a ultitude o optial olution. The ethod that i propoed in thi paper attept to utilize the geoetrical tructure o the linear prograing proble. A entioned above, in geoetrical language every linear prograing proble deine a conve polyhedron ored by interecting contraint plane and coordinate plane. The region inide o thi conve polyhedron i called the eaible region. It i ade up o eaible interior point lying inide thi conve polyhedron. Thee o called eaible interior point atiy all boundary contraint and are nonnegative. Solving the aiization (iniization) linear prograing proble in geoetrical ter eentially conit o puhing thi objective plane outward (inward) in the direction o C uch that thi objective plane will reach to the etree end o the polyhedron and will contain the etree verte. Thi etree verte i actually the optial olution o the proble at hand. In geoetrical

3 ter olving a linear prograing proble ean to arrive at the point repreenting etree verte, to deterine coordinate o thi point, and urther to ind value o the objective unction (the o called optial objective value) at thi point.. New Algorith or Linear Prograing: Becaue o the ar great practical iportance o the linear progra and other iilar proble in the operation reearch it i ot deired thing to have an algorith which work in a ingle tep, i not, in a ew tep a poible. No ethod ha o ar been ound which will yield an optial olution to a linear progra in a ingle tep ([], Page 9). We wih to ephaize in thi paper that when one i lucky enough to arrive at oe interior eaible point belonging to the line parallel to vector C which i alo paing through the point repreenting optial olution, where C T denote the objective unction to be optiized, then one can ind the point repreenting optial olution to the linear progra under conideration in a ingle tep by jut oving along thi line till one reache the deired etree verte on thi line at the boundary o the conve polyhedron!! We wih to develop oe NEW algorith or olving linear prograing proble. We now proceed with brie decription o thee algorith. We will try to capture the eence o the ethod with the help two geoetrical igure, FIG. and FIG. given below. We hope that thee igure will ake eplicit the geoetrical otivation that lie behind thee algorith. Coordinate Plane Starting interior point C line parallel to vector C through Optial Solution Contraint Plane Interior point near boundary C per New tarting interior point Objective Plane Optial olution Interior point near boundary FIG. 3

4 Starting interior point Interior point near boundary Conve Polyhedron Centroid or Centrally located point New interior point near boundary C Farthet interior point near the boundary New arthet interior point near the boundary Conve Polygon FIG. A Brie Decription o Figure: In FIG. we have choen or the ake o iplicity a typical two dienional linear prograing proble, i.e. a proble which contain two variable, ay and y. In thi -dienional cae the objective plane, contraint plane, coordinate plane, are all actually line a hown in thi igure. Vector C which i perpendicular to objective plane (a line in thi cae) i hown. Alo, the (the ot iportant) line parallel to vector C and paing through the point repreenting optial olution (etree verte) i hown. A vector perpendicular to vector C, ay C per, i hown. So called tarting point, which i oe eaible interior point i hown. Starting at thi o called tarting interior point we ove in the direction o C till we reach at a arthet poible eaible interior point near boundary. In thi tep the value o objective unction iprove. Now, ro thi interior point near boundary we ove in the direction o C per till we reach at another interior point near boundary lying in thi direction. In thi tep the value o objective unction reain the ae, a thi new point i on the ae objective plane. Thi tep i carried out to locate the deired centrally located point. We take it a idpoint o line egent joining thee two interior point near the boundary and take thi idpoint a new tarting point to tart net iteration and to ove 4

5 in the direction o C till we again reach at a new interior point near boundary, and o on. In FIG. we have tried to depict higher dienional cae o a linear prograing proble, i.e. a proble containing at leat three variable (or ore: ay n variable). The contraint plane and coordinate plane give rie to a conve polyhedron (in n dienional pace) encloing eaible interior point a hown in the igure. The objective plane paing through oe eaible interior point interect with the polyhedron and the interection i a conve polygon a hown in the igure. Vector C which i alway perpendicular to (any) objective plane a hown. So called tarting point, which i oe interior point i hown. Starting at thi o called tarting interior point we ove in the direction o C till we reach at an interior point near boundary. We conider interection o the objective plane paing through thi interior point near boundary with polyhedron which i a polygon a hown. We deterine n point lying on the boundary (edge o conve polygon). We ue thee point (in place o vertice) and ind the centriod o thi polygon. We take thi centroid a new tarting interior point to tart net iteration and ove along the line parallel to vector C paing through thi centroid till we reach a new arthet poible eaible interior point near boundary, and o on. The other ethod ind a centrally located point a ean o interior point near boundary and arthet interior point near boundary a hown in igure. The arthet interior point i ound here uing ethod o calculu. For thi purpoe, we aiize the ditance unction. We conider line egent joining the point we arrived at near the boundary and the other interior point lying on the polygon at aued aiu ditance. We then take the ean o thee two point a centrally located point to be ued a new tarting point to tart new iteration i.e. to ove in the direction o C till again we reach at a new interior point near boundary, and o on. A Brie Decription o Firt Algorith: The irt algorith tart at oe eaible interior point. One then ove along the line paing through thi point and parallel to vector C in outward (inward) direction or aiization (iniization) proble till one will reach at an interior point very cloe to the boundary o the conve polyhedron deined by the proble. One then ind the value o objective unction (objective value) at thi newly obtained eaible interior point. One then conider objective plane paing through thi point and o having thi newly obtained objective value (or all it point). One then conider the interection o thi objective plane, 5

6 which ha newly obtained objective value or it point, with the conve polyhedron deined by the proble. Thi interection will be a conve polygon. One then ind out the centroid o thi polygon. Thi centroid will be the new tarting point to ove along the line paing through thi centroid and parallel to vector C in outward (inward) direction or aiization (iniization) proble again till one will reach at an interior point very cloe to the boundary o the conve polyhedron deined by the proble. One then repeat the ae earlier tep o inding the value o objective unction (objective value) at thi newly obtained eaible interior point, conidering the objective plane paing through thi point and o having thi newly obtained objective value (or all it point), conidering the interection o the objective plane, having newly obtained objective value or it point, with the conve polyhedron deined by the proble, which will again a previou will be a polygon. One then ind out centroid o thi new polygon, again ove along the line paing through thi centroid and parallel to vector C in outward (inward) direction or aiization (iniization) proble again till one will reach at an interior point very cloe to the boundary o the conve polyhedron deined by the proble. The tep entioned above taken repeatedly will ake u arrive at dierent objective plane in ucceion which will have iproved value or the objective unction, and dierent polygon will be ored in ucceion by the interection o thee objective plane with the conve polyhedron deined by the proble. Thee polygon ored in ucceion will be having aller and aller ize (area). In each iterative tep one thu will be going in the direction parallel to vector C along the line paing through the centroid o thee polygon with reduced area generated through interection o objective plane arrived at in ucceive tep with the conve polyhedron. It i traightorward to ee that in ucceive tep o thi algorith one will clearly arrive inally either at a point on the line or at a point belonging to all neighborhood o oe point on the line which i parallel to vector C and alo paing through the point repreenting optial olution. By oving then ro thi point on the line or point belonging to all neighborhood o a point on the line parallel to vector C one can reach to the point repreenting optial olution or to the point belonging to the cloe neighborhood o the point repreenting optial olution. A Brie Decription o Second Algorith: The tep o the econd algorith are alot identical with that o the irt algorith. It alo tart at oe eaible interior point. One then ove in the direction o vector C till one reache at a eaible interior point lying jut inide 6

7 the boundary o the conve polyhedron deined by the proble in that direction. One then conider the objective plane paing through thi point which interect with the conve polyhedron and again a previou thi interection i a polygon. In the irt algorith we ound the centroid o thi polygon by inding and uing certain n point on the boundary o thi polygon. In thi econd algorith we dier in thi tep. Intead o inding centroid we ind oe other point which i alo centrally located. Starting with the eaible interior point jut obtained in the previou tep one ind the direction in which i one will proceed along oe line lying on the objective plane paing through jut obtained eaible interior point near the boundary till one reache at a eaible interior point on the other ide near the boundary o the conve polygon then thi eaible interior point will be at longet poible ditance. By thi choice or the direction to proceed ro obtained eaible interior point near the boundary o the polygon to oe other interior point on the other ide o the boundary uch that the length o thi line egent joining thee point i larget. By uing ethod o calculu we anage to aiize the ditance between the above entioned two boundary point o that we ake thi line egent a longet line egent that reide on the polygon. Thi obviouly ake thi line to proceed through central region o the polygon or enuring larget poible length or thi line egent. Thu, by oving in thi direction one will be oving in the cloe vicinity o the centroid! We ind idpoint o the line egent joining o thee two etree eaible interior point. Thi idpoint will autoatically be cloe to centroid a deired. One tart the new iteration here and again one ove in the direction parallel to vector C along the line paing through thi centrally located idpoint o the egent till one arrive at oe new point lying on the boundary o the conve polyhedron in the direction o vector C. Thu, alot all the tep o econd algorith are identical to irt algorith ecept the tep o inding centroid. Intead, here one ind a centrally located point (oething iilar to centroid) which alo work well to achieve ae reult a the irt algorith. A Brie Decription o Third Algorith: The tep o the third algorith are again alot identical with that o the earlier algorith. It dier in it ethod o locating centrally located point. It alo tart at oe eaible interior point. One then ove in the direction o vector C till one reache at a eaible interior point lying jut inide the boundary. Fro thi point near the boundary we draw perpendicular on each o the contraint plane (a well a the coordinate plane) and ind thee oot o perpendicular. Uing thee point which are oot 7

8 o perpendicular we ind centroid and ue it a new tarting point and ove in the direction parallel to vector C along the line paing through thi centroid till one arrive at oe new interior point lying on the boundary, and o on. Intead o dropping perpendicular on the contraint plane one can drop the on the edge o conve polygon that reult through the interection o objective plane paing through jut obtained eaible interior point near boundary interect with conve polyhedron deined by the proble under conideration. We ue thu obtained oot o perpendicular lying on the edge o the polygon to ind centroid and ue thi centroid a new tarting point to tart a new iteration and again ove in the direction parallel to vector C along the line paing through thi centroid till one arrive at oe new interior point lying on the boundary, and o on. 3. A Detailed Decription o Algorith: Firt Algorith ) We take oe interior point to tart with, ay = (,, L, n ), ) We then ove along the line paing through thi point,, and parallel to vector C in outward (inward) direction or aiization (iniization) proble till one will reach at an interior point very cloe to the boundary o the conve polyhedron deined by the i proble, ay. 3) We then ind the objective value (value o objective unction) at i, naely, T i i C = 4) We then or the objective plane deined by the equation, T C = Thi objective plane interect the conve polyhedron deined by the proble at hand and give rie to a polygon. We wih to ind centroid o thi polygon and or thi purpoe we proceed to ind the point, one on each edge o thi polygon, ay,, a decribed in the ollowing tep. Note that the atri equation, A = b, actually together repreent nuber o equation repreenting nuber o contraint d d i,, L, 8

9 plane. Thee contraint plane together with coordinate plane, = 0, give rie to conve polyhedron. 5) We olve together the equation deining irt contraint plane ay, A =, where A repreent irt row o atri A, and the b T i C = d and ind olution repreenting the objective plane irt point (verte), ay. We olve together the equation deining econd contraint plane ay, A = b, where T A repreent econd row o atri A and i C = d and ind the olution repreenting the econd point (verte), ay. We continue olving pair o equation till inally we will olve together the equation deining -th contraint plane ay, T A = b and -th point (verte), ay. 6) We now ind the centroid, naely, i C = d and ind olution repreenting the = i= R = i n Thi centroid will be ued a a new tarting point to proceed. We ove in the direction o vector C, along the line paing through thi centroid till again a previou we reach at an interior point very cloe to the boundary o the conve polyhedron deined by the proble, ay. We now treat thi i i i point a i (i.e. ) and go to tep 3) and begin net iteration o the algorith. 7) We continue iteration or uiciently any tie, which will produce polygon o aller and aller ize in ucceive iteration, and thi will inally take u to the point which repreent optial olution o the proble, or in very all neighborhood o uch a point which repreent optial olution o the proble. Second Algorith: Thi econd algorith i alot identical to irt algorith. All the tep ecept one are ae. The only dierence between thi algorith and the irt algorith i in the i 9

10 procedure that we ollow here to ind a centrally located point on the polygon. We thereore dicu only thi tep o inding centrally located point in order to avoid unneceary repetition. In the irt algorith we take centroid a uch a point. In thi algorith we don t ind and take centroid a uch a point but obtain the required centrally located point through dierent conideration. It i traightorward to ee ro geoetry that each tie (i.e. in each iterative tep) inding and chooing oe centrally located point a a tarting point to ove parallel to vector C till one reache very near to the boundary o polyhedron will take u near the point that repreent optial olution in a uch ater way cauing ubtantial iproveent in the value o the objective unction in each tep. Chooing oe other point on the polygon which i away ro a centrally located point to ove parallel to vector C till one reache very near to the boundary o polyhedron will how uch lower iproveent and o will deand large any tep to reach the point which either itel the optial point or lying in very all neighborhood o optial point. A i done in irt algorith we tart at oe interior point, ay and proceed along a line through thi point which i parallel to vector C, till we reach at an interior point near the boundary o polyhedron, ay. Thu, i the arthet interior point on the line through and parallel to vector C. One can write = + αc A b i atiied. To ind Where calar α i o choen that uch α we put in the irt contraint, i.e. we conider A b and ind condition on α like α l. We then put A b in the econd contraint, i.e. we conider and ind condition on α like α l. We continue in thi way and conider every contraint in ucceion and conider inally the -th contraint, i.e. we conider A b and ind condition on α like α l. We then et 0

11 α = in {} l i.with thi α we get arthet poible interior point in the direction o vector C on the line paing through. A done in previou algorith we conider objective plane paing through and conider it interection with polyhedron which will be a conve polygon. In the irt algorith we obtained centroid o thi polygon to treat it a a new tarting point. Now, or thi algorith we wih to conider variou line paing through point and lying in the objective plane deined by equation T C = d having θ a the angle between any two neighboring w w,,, line. Let u uppoe that L are the eaible interior point lying on thee line through going in dierent direction and at the arthet poible ditance ro point, the aial nature o thee ditance i enured by oberving the ulillent o contraint aially. We then ai to earch out the line uch that or the obtained point, ay, aong the ditance w w i, i =,,L the ditance k i eay to check that uch line will pa cloely to centroid or centrally located point. A a ipliied verion or the above procedure let u conider ollowing dierent line lying on thi polygon and paing through the interior point. All thee line will belong to the T objective plane through, naely, C = d, where w w k i aiu. It T d = C and o will be perpendicular to vector C. Let u denote a vector belonging to objective plane through, by ybol C (or C per ), then clearly, C. C = 0. Let C = ( ω, ω, L, ωn ). Uing C. C = 0 we can eliinate one paraeter and in the epreion or C. Thu, we can have ω = φ ω, ω, L, ω ) and thereore we can write ( 3 n C = ( φ, ω, L, ω n ) where φ i unction o paraeter

12 ω, ω, 3, ω n L. Siilarly, we can have ω = φ ω, ω, L, ω ) and thereore we can write C ( 3 n = ( ω, φ, L, ω ) n, On continuing on thee line we can have variety o epreion or C,like 3 n C = ( ω, ω, φ, L, ωn ),..., C = ( ω, ω, L, φ ). Now in order to generate variou line paing through and T lying on the objective plane deined by equation C = d let u aign unit value to the paraeter in unction. We can build in thi way dierent vector perpendicular to vector C, like ( (,,,),, L C = φ L,), C = (, φ (,, L,), L,),, C n (,, L, n = φ (,, L,)). Now, ro the above obtained vector perpendicular to C we build ollowing vector like, g g g 3 3 = + µ C, = + µ C, = + µ 3C, g n, n = + µ nc. For each vector we ind the larget poible value or µ i uch that all the contraint reain valid and the vector g i g i repreent eaible interior point. Aong thee vector we chooe the one or which the Euclidean g k ditance between the choen vector ay, and vector i aiu. It i eay to check that or uch vector joining the point repreented by vector g k and g k the line will pa nearet to a centrally located point aong thee vector. We now chooe a new tarting point to ove in the direction o n g = + k. It i eay to check that all vector C, naely, ( ) thee tep will inally take u to the point which repreent optial olution o the proble, or in very all neighborhood o uch a point which repreent optial olution o the proble. In the ethod jut dicued to locate a centrally located point we have uggeted to tet and ind ro everal g i

13 g i i = + µ C the bet one. We now proceed to ee a vector i arter ethod to ind a centrally located point. In thi ethod we ake ue o the ethod o calculu a ollow: We conider a g general point repreented by vector on the objective plane paing through point repreented by vector, thu g = + µ C, whoe coordinate are n paraeter. We correlate and deterine thee paraeter uing aial nature o g ditance between point repreented by point and. g When thee point and are aially eparated we can g epect thee point and to be ituated diaetrically oppoite to each other with repect to a circle encloing the polygon. So, urther we can aely epect that the line joining g point repreented by repectively and will pa cloely to centrally located point (o the polygon) i the Euclidean ditance g between point repreented by repectively and will be g aiu, i.e. = µ C i aiu. We take ( C ) C = ( φ, ω, L, ω n ), etup equation = 0 or all ( ωi ) i =,3, L, n and ind interrelation between paraeter ω i. A a g conequence, we inally get the equation, = + µ C, where C = ( β, β, L, β n ) and thee β i are certain contant, and we have to deterine larget poible value or µ again uing contraint (a it wa done above to ind larget poible value or g α ) o that will be arthet interior point ro. A i done early, we now ind a new tarting point to ove in the n g direction o vector C, naely, = ( + ) It i eay to check that all thee tep will inally take u to the point which repreent optial olution o the proble, or in very all neighborhood o uch a point which repreent optial olution o the proble. We have thu dicued only that portion o econd 3

14 algorith that dier and or the entire ret portion both algorith proceed in identical way! Third Algorith: Thi third algorith i alot identical to irt two algorith. All the tep ecept one are ae. The only dierence between thi algorith and the irt two algorith i in the procedure that we ollow here to ind a centrally located point. We thereore dicu only thi tep o inding centrally located point in order to avoid unneceary repetition. A i done in previou algorith we tart at oe interior point, ay and proceed along a line through thi point which i parallel to vector C, till we reach at an interior point near the boundary o polyhedron, ay. Thu, i the interior point near boundary on the line parallel to vector C through. Fro thi point we draw perpendicular on contraint plane, A = or all i =,,..,. and ind each oot o perpendicular, i b i u u,,, u L. Again a previou we ind the centroid, naely, u i i R = = = n Thi centroid will be ued a a new tarting point to proceed along the line paing through it in the direction parallel to vector C, and o on. A an alternative, we draw perpendicular on edge o conve polygon, i.e. on line ored by interection o each contraint plane, i i T, with the objective plane, C = d, where T A = b or all i =,,.., d = C and ind each uch oot o perpendicular, v v,,, v L. Again a previou we ind the centroid, naely, v i i R = = = n Thi centroid will be ued a a new tarting point to proceed along the line paing through it in the direction parallel to vector C, and o on. 4

15 Eaple : Maiize: + y Subject to: + y 4 + y 4 + y, y 0 Solution: It i eay to check that (, ) i a eaible interior point. The value o objective unction at thi point i + y = + =. So, we tart by taking (, ) a tarting interior point. We now have to proceed in the direction o vector C till we reach near boundary and thu reach at an interior point near boundary. Now, C = (, ) o our new interior point will be (, ) + α C = (, ) + α (, ) = (α +, α +) Now, we deterine α uch that we get interior point near boundary, i.e. a point near boundary uch that all the contraint will be atiied. Subtituting the point in the irt contraint α + + α + 4, i.e. α /3 Subtituting the point in the econd contraint -α - + α +, i.e. 0 Subtituting the point in the third contraint 4α α +, i.e. α Thu, we take α = in {α } = /3. Uing thi α =/3 we get the deired point near boundary, naely, (, ) + α C = (, ) + α (, ) = (α +, α +) = (4/3, 4/3) The value o objective unction at thi point i + y = 4/3 + 4/3 = 8/3 =.66 Thu, the value o objective unction got iproved a epected. We have C = (, ), thereore, C per = C = (-, ). Starting ro the jut obtained interior point near boundary we now ove in the direction o C till we reach the (arthet) interior point on the (other ide o the) boundary. Thu, we ind (4/3, 4/3) + β C = (4/3, 4/3) + β (-, ) = (4/3 - β, 4/3 + β ) Subtituting thi point in the contraint equation we ind β and uing β = we get (4/3, 4/3) + β C = (0/3,-/3), and we then have the centroid a /[(4/3, 4/3) + (0/3, -/3)] = (7/3, /3). Subtituting point (7/3, /3) + α C = (7/3, /3) + α (, ) in the contraint we ind α =/3. Thi yield new point near boundary a (7/3, /3) + α (, ) = (7/3, /3) + (/3, /3) = (8/3, /3). The value o objective unction at thi point i 5

16 + y = 8/3 + /3 = 0/3 = Thu, the value o objective unction got iproved a epected. Actually, one can eaily check by carrying out one ore iteration that we have already reached the optial olution! Eaple : Maiize: Subject to: ,, 3 0 Solution : We have C = (0, 6, 4). Let u take a tarting interior point, = (,, ). Value o objective unction at = 0. We now ove in the direction o vector C till we reach near boundary and thu reach at an interior point near boundary. So our new interior point will be (,, ) + α C = (,, ) + α (0, 6, 4) = (0α +, 6α +, 4α +) Now, we deterine α uch that we get interior point near boundary, i.e. a point near boundary uch that all the contraint will be aially atiied. Subtituting the point ucceively in the contraint we get α = in {4.85, 4.034, 5.7} = 4. Thu, we get the point near boundary a (4, 5, 7). Value o objective unction at thi point i ( ) = 68. Thu, the value o objective unction got iproved a epected. We now ind point on the interection o objective plane = 68 eparately with each contraint plane and ind the interection 3 point a ollow: = (38, 0, 6), = (54, 4, 0), = (49.38, 0, 33.5), thereore, centroid = (47., , 3.84). Moving in the direction o vector C through centroid we reach to (47., , 3.84) (0, 6, 4). Value o objective unction at thi point i = Again, we now ind point on the interection o objective plane = eparately with each contraint plane and ind the interection point a 3 ollow: = (4.8, 0, 57.7), = (-00, 76, 0), = (67.73, 0, -5), thereore, centroid = (3.336, 9, 7.56). Moving in the direction o vector C through centroid we reach to (3.336, 9, 7.56) + 0.(0, 6, 4). Value o objective unction at thi point i = We thu ee that the value o objective unction i iproving in the ucceive tep and by continuing thee tep we can ind out the deired optial olution. 6

17 Solution : We have C = (0, 6, 4). Let C = ( λ, β, γ ). Uing 6 4 C. C = 0, we have C = ( β γ, β, γ ). We aiize ditance, i.e. F = ( β γ ) + β + γ i aiu. By 0 0 F F equating partial derivative to zero i.e. etting = 0, = 0 we β γ get γ =. 3β or γ = 0. 03β. Let u take a tarting interior point, = (,, ). Value o objective unction at = 0. We now ove in the direction o vector C till we reach near boundary and thu reach at an interior point near boundary. So our new interior point will be (,, ) + α C = (,, ) + α (0, 6, 4) = (0α +, 6α +, 4α +) Now, we deterine α uch that we get interior point near boundary, i.e. a point near boundary uch that all the contraint will be aially atiied. Subtituting the point ucceively in the contraint we get α = in {4.85, 4.034, 5.7} = 4. Thu, we get the point near boundary a (4, 5, 7). Value o objective unction at thi point i ( ) = 68. Uing relation γ =. 3β or aiization o ditance we have other point (4, 5, 7) + µ ( 3.9,,.33), where uing contraint we get µ =.35 and o, we have other point = (35.708, 3.65, 3.9). Thereore, the centrally located point = (38.354, 4.35, 4.645). Thu, we conider (38.354, 4.35, 4.645) + α (0, 6, 4) and ind α uing contraint a α = 0.64 giving rie to point (44.75, 8.65, 7.05). Thu, value o objective unction at thi point = = A arked iproveent!! We thu ee that the value o objective unction i iproving in the ucceive tep and by continuing thee tep we can ind out the deired optial olution. Acknowledgeent The author i thankul to Dr. M. R. Modak or oe ueul dicuion. Reerence. Hadley G., Linear Prograing, Naroa Publihing Houe, New Delhi 0 07, Eighth Reprint,