copyright 2004 Bruce A. McCarl and Thomas H. Spreen 1

Transcription

1 HAPTER III SOLUTION OF LP PROLEMS: A MATRI ALGERA APPROAH 3. Mtrix Formultion of the Liner Progrmming Problem 3. Solving LP's by Mtrix Algebr The Simplex Algorithm Exmple Solutions nd Their Interprettion Generl Solution Interprettion Exmples of Solution Interprettion Finding Limits of Interprettion Rnging Right-Hnd-Sides Rnging Obective Function oefficients Exmple hnges in the Technicl oefficient Mtrix 6 Illustrtive Exmple Finding the Solution Alterntive Optiml nd Degenerte ses Finding Shdow Prices for ounds on Vribles 3. Further Detils on the Simplex Method 3.. Updting the sis Inverse 3.. Advnced ses 3..3 Finding n Initil Fesible sis IG M Method Phse I/Phse II Method The Rel LP Solution Method 5 References 7 Tble 3.. GAMS Solution of Exmple Model 9 Tble 3.. Solution with ounds Imposed s onstrints nd s ounds 3 Tble 3.3. The Model s Redy for the ig M Method 3 Tble 3.. Solution to the ig M Problem 3 copyright ruce A. Mcrl nd Thoms H. Spreen

2 HAPTER III SOLUTION OF LP PROLEMS: A MATRI ALGERA APPROAH Liner progrmming solution hs been the subect of mny rticles nd books. omplete coverge of LP solution pproches is beyond the scope of this book nd is present in mny other books. However, n understnding of the bsic LP solution pproch nd the resulting properties re of fundmentl importnce. Thus, we cover LP solution principles from mtrix lgebr perspective demonstrting the simplex lgorithm nd the properties of optiml solutions. In ddition, we cover severl prcticl mtters. 3. Mtrix Formultion of the Liner Progrmming Problem The mtrix version of the bsic LP problem cn be expressed s in the equtions below. Mx s.t. A < b > Here the term is mximized where is n xn vector of profit contributions nd is n Nx vector of decision vribles. This mximiztion is subect to inequlity constrints involving M resources so tht A is n MxN mtrix giving resource use coefficients by the 's, nd b is n Mx vector of right hnd side or resource endowments. We further constrin to be non-negtive in ll elements. It is common to convert the LP inequlity system to equlities by dding slck vribles. These vribles ccount for the difference between the resource endowment (b) nd the use of resources by the vribles (A) t no cost to the obective function. Thus, define S = b - A s the vector of slck vribles. Ech slck vrible is restricted to be nonnegtive thereby insuring tht resource use is lwys less thn or equl to the resource endowment. One slck vrible is dded for ech constrint eqution. Rewriting the constrints gives A + IS = b, where I is n M M identity mtrix nd S is Mx vector. The slck vribles pper in the obective function with zero coefficients. Thus, we dd n xm vector of zero's to the obective function nd conditions constrining the slck vribles to be nonnegtive. The resultnt ugmented LP is MA + OS s.t. A + IS = b copyright ruce A. Mcrl nd Thoms H. Spreen

3 , S >. Throughout the rest of this section we redefine the vector to contin both the originl 's nd the slcks. Similrly, the new vector will contin the originl long with the zeros for the slcks, nd the new A mtrix will contin the originl A mtrix long with the identity mtrix for the slcks. The resultnt problem is 3. Solving LP's by Mtrix Algebr MA s.t. A = b > LP theory (Dntzig(963); zrr, et l.) revels tht solution to the LP problem will hve set of potentilly nonzero vribles equl in number to the number of constrints. Such solution is clled sic Solution nd the ssocited vribles re commonly clled sic Vribles. The other vribles re set to zero nd re clled the nonbsic vribles. Once the bsic vribles hve been chosen; the vector my be prtitioned into, denoting the vector of the bsic vribles, nd N, denoting the vector of the nonbsic vribles. Subsequently, the problem is prtitioned to become MA + N N s.t. + A N N = b, N. The mtrix is clled the sis Mtrix, contining the coefficients of the bsic vribles s they pper in the constrints. A N contins the coefficients of the nonbsic vribles. Similrly nd N re the obective function coefficients of the bsic nd nonbsic vribles. Now suppose we ddress the solution of this problem vi the simplex method. The simplex solution pproch relies on choosing n initil mtrix, nd then interctively mking improvements. Thus, we need to identify how the solution chnges when we chnge the mtrix. First, let us look t how the bsic solution vrible vlues chnge. If we rewrite the constrint eqution s = b - A N N. Setting the nonbsic vribles ( N ) to zero gives = b. This eqution my be solved by premultiplying both sides by the inverse of the bsis mtrix (ssuming non-singulrity) to obtin the solution for the bsic vribles, copyright ruce A. Mcrl nd Thoms H. Spreen 3

4 - = I = - b or = - b. We my lso exmine wht hppens when the nonbsic vribles re chnged from zero. Multiply both sides of the eqution including the nonbsic vribles by - giving = - b - - A N N. This expression gives the vlues of the bsic vribles in terms of the bsic solution nd the nonbsic vribles. This is one of the two fundmentl equtions of LP. Writing the second term of the eqution in summtion form yields = - b - N where N gives the set of nonbsic vribles nd the ssocited column vectors for the nonbsic vribles from the originl A mtrix. This eqution shows how the vlues of the bsic vribles re ltered s the vlue of nonbsic vribles chnge. Nmely, if ll but one ( ) of the nonbsic vribles re left equl to zero then this eqution becomes - x = - b - - This gives simultneous system of equtions showing how ll of the bsic vribles re ffected by chnges in the vlue of nonbsic vrible. Furthermore, since the bsic vribles must remin non-negtive the solution must stisfy i* = ( - b) i* - ( - ) i* = This eqution permits the derivtion of bound on the mximum mount the nonbsic vrible cn be chnged while the bsic vribles remin non-negtive. Nmely, my increse until one of the bsic vribles becomes zero. Suppose tht the first element of to become zero is i*. Solving for i* gives i* = ( - b) i* - ( - ) i* = where ( ) i denotes the i th element of the vector. Solving for yields = ( - b) i* /( - ) i*, where (b - ) i* This shows the vlue of which cuses the i *th bsic vrible to become zero. Now since must be nonnegtive then we need only consider cses in which bsic vrible is decresed by incresing the nonbsic vrible. This restricts ttention to cses where ( - ) i, is positive. Thus, to preserve non-negtivity of ll vribles, the mximum vlue of is copyright ruce A. Mcrl nd Thoms H. Spreen

5 = {( - b) i /( - ) i } for ll i where ( - ) i > The procedure is clled the minimum rtio rule of liner progrmming. Given the identifiction of nonbsic vrible, this rule gives the mximum vlue the entering vrible cn tke on. We lso know tht if i * is the row where the minimum is ttined then the bsic vrible in tht row will become zero. onsequently, tht vrible cn leve the bsis with inserted in its plce. Note, if the minimum rtio rule revels tie, (i.e., the sme minimum rtio occurs in more thn one row), then more thn one bsic vrible reches zero t the sme time. In turn, one of the rows where the tie exists is rbitrrily chosen s i * nd the new solution hs t lest one zero bsic vrible nd is degenerte. Also, note tht if ll the coefficients of re zero or negtive ( - ) i -- for ll i -- then this would indicte n unbounded solution, if incresing the vlue of the nonbsic vrible increses the obective function, since the vrible does not decrese the vlue of ny bsic vribles. Another question is which nonbsic vrible should be incresed? Resolution of this question requires considertion of the obective function. The obective function, prtitioned between the bsic nd nonbsic vribles, is given by Substituting the eqution (3.) yields Z = + N N or Z = ( - b - A N N ) + N N or Z = - b - - A N N + N N Z = - b - ( - A N - N ) N This is the second fundmentl eqution of liner progrmming. Expressing the second term in summtion nottion yields Z - b c N This expression gives both the current vlue of the obective function for the bsic solution ( - b since ll nonbsic equl zero) nd how the obective function chnges given chnge in the vlue of nonbsic vribles. Nmely, when chnging Z - b c A degenerte solution is defined to be one where t lest one bsic vrible equls zero. copyright ruce A. Mcrl nd Thoms H. Spreen 5

6 Since the first term of the eqution is equl to the vlue of the current obective function (Z), then it cn be rewritten s Z Z - For mximiztion problems, the obective vlue will increse for ny entering nonbsic vrible if its term, - - c, is negtive. Thus the criterion tht is most commonly used to determine which vrible to enter is: select the nonbsic vrible tht increses the vlue of obective function the most per unit of the vrible entered. Nmely, we choose the vrible to enter s tht vrible such tht the vlue of - - c, is most negtive. This is the simplex criterion rule of liner progrmming nd the term - - c, is clled the reduced cost. copyright ruce A. Mcrl nd Thoms H. Spreen 6 If there re no vribles with negtive vlues of - - c, then the solution cnnot be improved on nd is optiml. However, if vrible is identified by this rule then it should be entered into the bsis. Since the bsis lwys hs number of vribles equl to the number of constrints, then to put in new vrible one of the old bsic vribles must be removed. The vrible to remove is tht bsic vrible which becomes zero first s determined by the minimum rtio rule. This criteri gurntees the non-negtivity condition is mintined providing the initil bsis is non-negtive. These results give the fundmentl equtions behind the most populr method for solving LP problems which is the simplex lgorithm. (Krmrkr presents n lterntive method which is ust coming into use.) 3.. The Simplex Algorithm Formlly, the mtrix lgebr version of the simplex lgorithm (ssuming tht n initil fesible invertible bsis hs been estblished) for mximiztion problem follows the steps: ) Select n initil fesible bsis ; commonly this is composed of ll slck vribles nd is the identity mtrix. ) lculte the sis inverse ( - ). 3) lculte - - c for the nonbsic vribles nd identify the entering vrible s the vrible which yields the most negtive vlue of tht clcultion; denote tht vrible s if there re none, go to step 6. ) lculte the minimum rtio rule Min bi / where i i i Denote the row where the minimum rtio occurs s row I*; if there re no rows with ( - )i > then go to step 7. 5) Remove the vrible tht is bsic in row I* by replcing the vrible in the *th column of the bsis mtrix with column nd reclculte the bsis inverse. Go to step 3. 6) The solution is optiml. The optiml vrible vlues equl -b for the bsic vribles nd zero for the nonbsic vribles. The optiml reduced costs re - - c

7 7) - - c (lso commonly clled Z - c ). The optiml vlue of the obective function is - b. Terminte. 8) The problem is unbounded. Terminte. 3.. Exmple Suppose we solve Joe's vn conversion problem from hpter II. After dding slcks tht problem becomes Mximize Z = fncy + 7 fine s.t. fncy + fine 5 fncy + fine 8 fncy, fine Now suppose we choose s nd s to be in the initil bsis. Thus, initilly A S S 5 N 7 fncy fine b 8 - A N - N A N N 5 Now using criterion for selecting the entering vribles ( - N - c N ): Tking the vrible ssocited with the most negtive vlue (-) from this clcultion indictes the first nonbsic vrible fncy, should enter. omputtion of the minimum rtio rule requires the ssocited - nd - b 5 5 b. 8 copyright ruce A. Mcrl nd Thoms H. Spreen 7

8 Using the criterion for leving vrible Min i b / i / i Min. i * 8 / 5. In this cse, the minimum rtio occurs in row. Thus, we replce the second bsic vrible, s, with fncy. At this point, the new bsic nd nonbsic items become S fncy N S fine 7 N 5 A N nd the new bsis inverse is - -/5 /5 Recomputing the reduced costs for the nonbsic vribles fine, nd s gives - A N - N -/5 / Observe tht the procedure implies fine should enter this bsis. The coefficients for the minimum rtio rule re - -/5 /5 /5, /5 - b.8. The minimum rtio rule computtion yields.8/(/5) Min i./ /5 * copyright ruce A. Mcrl nd Thoms H. Spreen 8

9 In the current bsis, s is the bsic vrible ssocited with row. Thus, replce s with fine. The new bsis vector is [ fine fncy ] nd the bsic mtrix is now 5 In turn the bsis inverse becomes /5 /5 The resultnt reduced costs re - A N - N 5 - -/5 / Since ll of these re greter thn zero, this solution is optiml. In this optiml solution fine - 5 -/5 b fncy - /5 Z = = c = 5 6 N 8 8 This method my be expnded to hndle difficulties with finding the initil nonnegtive bsis using either the Phse I/Phse II or IG M methods discussed below. 3.3 Solutions nd Their Interprettion LP solutions rise nd re composed of number of elements. In this section we discuss generl solution interprettion, common solver solution formt nd contents, specil solution cses nd sensitivity nlysis Generl Solution Interprettion The two fundmentl equtions developed in section 3. my be utilized to interpret the LP solution informtion. The first (3.) shows how the bsic vribles chnge s nonbsic vribles re chnged, - b - N - x nd the second (3.) give the ssocited chnge in the obective function when nonbsic vrible copyright ruce A. Mcrl nd Thoms H. Spreen 9

10 is chnged - Z b c x, N Suppose we ssume tht n optiml bsic solution hs been found nd tht nd - re the ssocited bsis nd bsis inverse. Now suppose we consider chnging the constrint right hnd sides. The implictions of such chnge for the solution informtion my be explored using clculus. Differentiting the bove equtions with respect to the right hnd side b yields b Z b These results indicte tht - is the expected rte of chnge in the obective function when the right hnd sides re chnged. The vlues - re clled the shdow prices nd give estimtes of the mrginl vlues of the resources (lter they will lso be clled the Dul Vribles or Dul Solution). Similrly, - gives the expected rte of chnge in the bsic vribles when resources re chnged. Thus when the first right hnd side is chnged, the bsic vribles chnge t the rte given by the first column within the bsis inverse; i.e., the first vrible chnges t rte ( - ), the second t ( - ) nd so on. Other results my be derived regrding chnges in nonbsic vribles. Prtilly differentiting the obective function eqution with respect to nonbsic vrible yields Z x - ( - - c ) N This shows tht the expected mrginl cost of incresing nonbsic vrible equls the negtive of - - c, consequence the - - c term is usully clled reduced cost. The mrginl effect of chnges in the nonbsic vribles on the bsic vribles is obtined by differentiting. This yields x - - N which shows tht the mrginl effect of the nonbsic vribles on the bsic vrible is minus -. The - constitutes trnsltion from the originl resource use spce (i.e., ) into the bsic vribles spce nd tells us how mny units of ech bsic vrible re removed with mrginl chnge in the nonbsic vrible. We cn lso use these results to further interpret the Z x eqution. The mrginl revenue due to incresing non bsic vrible is equl to its direct revenue (c the obective function coefficient) less the vlue of the bsic vribles ( ) times the mount of the copyright ruce A. Mcrl nd Thoms H. Spreen

11 bsic vribles diverted ( - ). Thus, this eqution tkes into ccount both the direct effect from incresing plus the substitution effect for the bsic vribles Exmples of Solution Interprettion This set of generl interprettions my be pplied to the Joe's Vn exmple bove. The pproprite mthemticl expressions for ech of the four items re s follows. Z b b Z N N ( N A N N ) The first expression, which gives the prtil of Z with respect to b, tells how the obective function chnges when the right hnd sides chnge. Thus, if the cpcity limit were chnged upwrd from, one would expect the obective function to increse $5 per unit. Similrly if the second right hnd side or the lbor limit were incresed upwrds from 8 then one would expect return of $6 per hour. The second expression indictes the nticipted chnge in the vlues of the bsic vribles when the right hnd sides re chnged; the bsic vribles in the model re rrnged with fine being first nd fncy being second. The first column of the bsis inverse corresponds to wht hppens if the vn cpcity right hnd side is chnged; wheres, the second column corresponds to wht hppens if the lbor right hnd side is chnged. Thus, if cpcity were expnded to 3, one would expect to produce 5 more fine vns nd less fncy vns. Similrly, if lbor ws expnded, the number of fine vns would decrese by /5 per unit nd the number of fncy vns would increse by /5. The prticulr signs of these trdeoffs re cused by the originl dt. Fncy vns use more lbor then fine vns. Thus, when cpcity is expnded, more fine vns re mde since they use lbor more intensively while, if lbor is incresed, one mkes more fncy vns. Now let us exmine the effects of chnges on the obective function when the nonbsic vribles re ltered. In this problem we hve two nonbsic vribles which re the slck vribles for the two resources. The effect of incresing the nonbsics is $5 decrese if we increse slck cpcity, nd $6 decrese if we increse slck lbor. This is exctly the opposite of the resource vlues discussed bove, since the consequence of incresing the slcks is the sme s tht of decresing the resource endowments. copyright ruce A. Mcrl nd Thoms H. Spreen

12 The interprettion of the bsis inverse lso llows us to get further informtion bout the interprettion of the chnge in the obective function when the right hnd sides hve chnged. Nmely, if chnging cpcity cuses five more fine vns to be produced (ech worth $7, leding to $85 increse but, four less fncy vns worth $8) the net effect then is $5 increse equling the shdow price. Similrly, the lbor chnge cuses $ more worth of fncy vns to be produced but $3 less fine vns for net vlue of $6. Overll, this shows n importnt property of liner progrmming. The optiml solution informtion contins informtion bout how items substitute. This substitution informtion is driven by the reltive uses of the constrint resources by ech of the lterntive ctivities. This is true in more complex liner progrmming solutions Finding Limits of Interprettion The bove interprettions only hold when the bsis remins fesible nd optiml. Rnging nlysis is the most widely utilized tool for nlyzing how much liner progrm cn be ltered without chnging the interprettion of the solution. Rnging nlysis dels with the question: wht is the rnge of vlues for prticulr prmeter for which the current solution remins optiml? Rnging nlysis of right-hnd-side (b i ) nd obective function coefficients (c ) is common; mny computer progrms vilble to solve LP problems hve options to conduct rnging nlyses lthough GAMS does not esily support such fetures (See chpter 9 for detils) Rnging Right-Hnd-Sides Let us study wht hppens if we lter the right hnd side (RHS). To do this let us write the new RHS in terms of the old RHS, the size of the chnge nd direction of chnge, b new = b old + r where b new is the new RHS, b old is the old RHS, is sclr giving the mgnitude of the chnge nd r is the direction of chnge. Given n r vector, the resultnt vlues for the bsic vribles nd obective function re - b new b r b r old old while - c is unchnged. The net effect is tht the new solution levels re equl to the old solution levels plus - r. Similrly the new obective function is the old one plus - r. For the bsis to still hold the bsic vribles must remin nonnegtive s must the reduced costs ( - - c ). However, since the reduced costs re unltered we must only worry bout the bsic vribles. Thus the condition for cn be written with non-negtivity imposed - - b b new old - r copyright ruce A. Mcrl nd Thoms H. Spreen

13 nd merits further exmintion in terms of the dmissible vlue of The bove expression gives simultneous set of conditions for ech bsic vrible for which one cn solve those conditions. Two cses which rise cross the set of conditions depending on the sign of individul elements in - r. i - r - b old -, where - i r i nd i - r - b old -, where - i r i much s in the row minimum rule where positive vlues of - r limit how negtive cn be nd negtive numbers limit how positive cn become. This result shows the rnge over which cn be ltered with the bsis remining optiml. Exmple Suppose in the Joe's vn fctory exmple we wished to chnge the first right hnd side. Ordinrily, if one wishes to chnge the i th RHS, then r will be vector with ll zeros except for one in the i th position, s illustrted below... th r i element... Thus when we chnge row in our two row problem b new 8 8 copyright ruce A. Mcrl nd Thoms H. Spreen 3

14 The resultnt vlues of becomes new /5 /5 5 - which implies Therefore the first right hnd side cn be chnged up by or down by.8 without the bsis chnging. Note tht during this ltertion the solution ( - b) does chnge, but - does not. Furthermore, this gives rnge of vlues for b for which the mrginl vlue of the resource ( - ) remins the sme. This pproch lso encompsses generliztion of the RHS rnging problem. Nmely, suppose we wish to lter severl RHS's t the sme time. In this cse, the chnge vector (r) does not hve one entry but rther severl. For exmple, suppose in Joe's vn tht Joe will dd both cpcity nd n employee. In tht cse the chnge vector would look like the following: b new Rnging Obective Function oefficients The nlysis of rnging obective function coefficients is conceptully similr to RHS rnging. We seek to nswer the question: wht is the rnge of vlues for prticulr obective function coefficient for which the current bsis is optiml? To exmine this question, write the new obective function s the old obective function plus, which is chnge mgnitude, times T which is direction of chnge vector. new old T Simultneously, one hs to write n expression for the obective function coefficients of the bsic vribles new old T copyright ruce A. Mcrl nd Thoms H. Spreen

15 copyright ruce A. Mcrl nd Thoms H. Spreen 5 where T gives the wy the 's re ltered. Subsequently, one cn reexpress the restriction tht the reduced cost vlues must be nonnegtive s new new new c c which reduces to old new T T c c In turn, we discover for nonbsic vribles c while for bsic vribles,, - old T where T T T c T where T T T c Exmple Suppose in our exmple problem we wnt to lter the obective function on fncy so it equls, +. The setup then is 7 new nd T so new for the nonbsics equls / /5 - -/ which implies -3 5 or tht the bsis is optiml for ny obective function vlue for fncy between 5 nd 7. This shows rnge of prices for fncy for which its optiml level is constnt. A - - new

16 hnges in the Technicl oefficient Mtrix The bove nlysis exmined chnges in the obective function coefficients nd right hnd sides. It is possible tht the technicl coefficients of severl decision vribles my be simultneously vried. This cn be done simply if ll the vribles re nonbsic. Here we exmine incrementl chnges in the constrint mtrix. For exmple, frmer might purchse new piece of equipment which lters the lbor requirements over severl crop enterprises which use tht equipment. In this section, procedures which llow nlysis of simultneous incrementl chnges in the constrint mtrix re presented. onsider the liner progrmming problem Mx s.t. Z A b where the mtrix of the technicl coefficients is to be ltered s follows A A M where A, A, nd M re ssumed to be mxn mtrices. Suppose the mtrix M indictes set simultneous chnges to be mde in A nd tht the problem solution is nondegenerte, possessing n unique optiml solution. Then the expected chnge in the optiml vlue of the obective function given M is Z new Z old U * M * where * nd U * re the optiml decision vrible vlues ( - b) nd shdow prices ( - ) for the unltered originl problem. The eqution is n pproximtion which is exct when the bsis does not chnge. See Freund(985) for its derivtion nd further discussion. Intuitively the eqution cn be explined s follows: since M gives the per unit chnge in the resource use by the vribles, then M * gives the chnge in the resources used nd U * M * then gives n pproximtion of the vlue of this chnge. Further, if M is positive, then more resources re used nd the Z vlue should go down so minus is used. Mcrl, et l.(99) investigted the predictive power of this eqution nd conclude it is good pproximtion for the cse they exmined. Illustrtive Exmple To illustrte the procedure outlined in the preceding section, consider the Joe's vn shop copyright ruce A. Mcrl nd Thoms H. Spreen 6

17 model nd suppose we wish to consider the effect of n equl chnge in the lbor coefficients. For chnge equl, the problem becomes Mx s. t. Z 5 fncy fncy fncy fncy, fine fine fine fine, S S, S S For no chnge ( = ), the optiml solution to this problem is * - b 8 * - U 5 6 with the optiml vlue of the obective function equl to,8, our chnge mtrix in this cse is M. Thus, the chnge in the vlue of the obective function is given by Z new - Z old - U - * M * Suppose the lbor requirement is reduced by hour for both vns so tht = -, then the nticipted increse in the obective function tht would result from using the new mchine is Z = -U * M * = 7 Solution of the revised problem shows the obective function chnges by Finding the Solution As shown bove, the liner progrmming solution contins lot of informtion reltive to the wys the obective function nd bsic vribles chnge given chnges in prmeters. However, not ll this informtion is included in n optiml solution s reported by modeling systems such s GAMS. onsider the following problem copyright ruce A. Mcrl nd Thoms H. Spreen 7

18 Mx Z 3x x x x, x x x x.5x x ZROW ONSTRAIN ONSTRAIN The GAMS solution informtion for this problem ppers in Tble 3.. The optiml obective function vlue equls 6.5. Then GAMS gives informtion on the equtions. For this problem, there re 3 equtions s nmed in the prentheticl sttements bove. For ech eqution informtion is given on the lower limit (lbeled LOWER), vlue of A * (lbeled LEVEL), upper limit (lbeled UPPER), nd shdow price - (lbeled MARGINAL). The obective function row (ZROW) does not contin interesting informtion. The constrint equtions show there is ) no lower bound on the first eqution (there would be if it were ) b) the left hnd side equls (A * ) nd c) the right hnd side is (UPPER) while the shdow price is.5 (MARGINAL). Similr informtion is present for the second eqution. Turning to the vribles, the solution tble gives the vrible nme, lower bound (LOWER), optiml level (LEVEL), upper bound (UPPER) nd reduced cost (MARGINAL). The solution shows = 6.5 nd = 3.5 while the cost of forcing 3 into the solution is estimted to be $. per unit. We lso see the obective function vrible (Z) equls 6.5. The solution informtion lso indictes if n unbounded or infesible solution rises. GAMS output does not provide ccess to the - or - mtrices. This is mixed blessing. A row model would hve quite lrge - nd - mtrices, but there re cses where it would be nice to hve some of the informtion. None of the GAMS solvers provide ccess to this dt Alterntive Optiml nd Degenerte ses Liner progrmming does not lwys yield unique priml solution or resource vlution. Non-unique solutions occur when the model solution exhibits degenercy nd/or n lterntive optiml. Degenercy occurs when one or more of the bsic vribles re equl to zero. Degenercy is consequence of constrints which re redundnt in terms of their coefficients for the bsic vribles. Mthemticlly, given problem with M rows nd N originl vribles, nd M slcks, degenercy occurs when there re more thn N originl vribles plus slcks tht equl zero with less thn M of the originl vribles nd slcks being non-zero. Most of the discussion in LP texts regrding degenercy involves whether or not degenercy mkes the problem hrder to solve nd most texts conclude it does not. Degenercy lso hs importnt implictions for resource vlution. onsider for exmple the following problem: copyright ruce A. Mcrl nd Thoms H. Spreen 8

19 Mx The solution to this problem is degenerte becuse the third constrint is redundnt to the first two. Upon ppliction of the simplex lgorithm, one finds in the second itertion tht the vrible cn be entered in plce of the slck vrible from either the second or third rows. If is brought into the bsis in plce of the second slck, the shdow prices determined re (u, u, u 3 ) = (, 75, ). If is brought into the bsis in plce of the third slck, the vlue of the shdow prices re (u, u, u 3 ) = (5,, 75). These differ depending on whether the second or third slck vrible is in the bsis t vlue of zero. Thus, the solution is degenerte, since vrible in the bsis (one of the slcks) is equl to zero (given three constrints there would be three non-zero vribles in non-degenerte solution). The lterntive sets of resource vlues my cuse difficulty in the solution interprettion process. For exmple, under the first cse, one would interpret the vlue of the resource in the second constrint s $75, wheres in the second cse it would interpret nominlly s $. Here the shdow prices hve direction nd mgnitude s elborted in Mcrl (977) (this hs been shown numerous times, see Drynn or Gl, Kruse, nd Zornig.). Note tht decresing the RHS of the first constrint from to 99 would result in chnge in the obective function of s predicted by the first shdow price set, wheres incresing it from to would result in $5 increse s predicted by the first shdow price set. Thus, both sets of shdow prices re vlid. The degenerte solutions imply multiple sets of resource vlution informtion with ny one set potentilly misleding. oth Mcrl (977) nd Pris discuss pproches which my be undertken in such cse. The underlying problem is tht some of the right hnd side rnges re zero, thus the bsis will chnge for ny right hnd side ltertions in one direction. Another possibility in the simplex lgorithm is the cse of lternte optiml. An lterntive optiml occurs when t lest one of the nonbsic vribles hs zero reduced cost; i.e., - - c for some N equl to zero. Thus, one could pivot, or bring tht prticulr vrible in the solution replcing one of the bsic vribles without chnging the optiml obective function vlue. Alterntive optimls lso imply tht the reduced cost of more thn M vribles in problem with M constrints re equl to zero. onsider the following problem: Mx 5 5 In this problem the optiml solution my consist of either = or = 5 with equl obective function vlues one or the other of these vribles will hve zero reduced cost t optimlity. Alterntive optimls my cuse difficulty to the pplied modeler s there is more thn one nswer which is optiml for the problem. Pris (98, 99); Mcrl et l. (977); Mcrl nd Nelson, nd urton et l., discuss this issue further. copyright ruce A. Mcrl nd Thoms H. Spreen 9

20 3.3.6 Finding Shdow Prices for ounds on Vribles Liner progrmming codes impose upper nd lower bounds on individul vribles in specil wy. Mny modelers do not understnd where upper or lower bound relted shdow prices pper. An exmple of problem with upper nd lower bounds is given below. Mx s.t. 3-5 The second constrint imposes n upper bound on, i.e., <, while the third constrint, >, is lower bound on. Most LP lgorithms llow one to specify these prticulr restrictions s either constrints or bounds. Solutions from LP codes under both re shown in Tble 3.. In the first solution the model hs three constrints, but in the second solution the model hs only one constrint with the individul constrints on nd imposed s bounds. Note tht in the first solution there re shdow prices ssocited with constrints two nd three. However, this informtion does not pper in the eqution section of the second solution tble. A closer exmintion indictes tht while nd re non-zero in the optiml solution, they lso hve reduced costs. Vribles hving both non-zero vlue nd non-zero reduced cost re seemingly not in ccordnce with the bsic/nonbsic vrible distinction. However, the bounds hve been treted implicitly. Vribles re trnsformed so tht inside the lgorithm they re replced by differences from their bounds nd thus nonbsic zero vlue cn indicte the vrible equls its bound. Thus, in generl, the shdow prices on the bounds re contined within the reduced cost section of the column solution. In the exmple bove the reduced costs show the shdow price on the lower bound of is nd the shdow price on the upper bound of is -3. Notice these re equl to the negtive of the shdow prices from the solution when the bounds re treted s constrints. One bsic dvntge of considering the upper nd lower limits on vribles s bounds rther thn constrints is the smller number of rows which re required. 3. Further Detils on the Simplex Method The simplex method s presented bove is rther idelistic voiding number of difficulties. Here we present dditionl detils regrding the bsis in use, finding n initil nonnegtive bsis nd some comments on the rel LP solution method. 3.. Updting the sis Inverse In step 5 of the mtrix simplex method the bsis inverse needs to be chnged to reflect the copyright ruce A. Mcrl nd Thoms H. Spreen

21 replcement of one column in the bsis with nother. This cn be done intertively using the soclled product form of the inverse (Hdley(96)). In using this procedure the revised bsis inverse ( - ) is the old bsis inverse ( - ) times n elementry pivot mtrix (P), i.e., - new P - old This pivot mtrix is formed by replcing the I *th (where one is pivoting in row I * ) column of n identity mtrix with elements derived from the column ssocited with the entering vrible. Nmely, suppose the entering vrible column updted by the current bsis inverse hs elements * -. then the elements of the elementry pivot mtrix re P P * * i i * ki / * k * i / * i, k i * Suppose we updte the inverse in the Joe's vn exmple problem using product form of the inverse. In the first pivot, fter fncy hs been identified to enter the problem in row, then we replce the second column in n identity mtrix with column with one over the pivot element (the element in the second row of - divided by the pivot element elsewhere. Since - equls, the pivot 5 mtrix P is P -/5 /5 nd the new - is - P - -/5 /5 -/5 /5 P - / /5 5 /5/ / 5 Similrly in the second pivot we find the minimum in the first row nd hve - / 5 =, so tht / 5 in forming P, the first column of n identity mtrix ws replced since 5 will enter s the first element of the bsis vector. Multipliction of - by P gives copyright ruce A. Mcrl nd Thoms H. Spreen

22 - P - 5 -/5 / /5 /5 which equls the bsis inverse computed bove. 3.. Advnced ses The process of solving LP is hunt for the optiml bsis mtrix. Experience with LP revels tht the simplex method usully requires two or three times s mny itertions s the number of constrints to find n optiml bsis. This implies tht when solving series of relted problems (i.e., chnging price of n input), it my be worthwhile to try to sve the bsis from one problem nd begin the next problem from tht prticulr bsis. This is commonly supported in LP solution lgorithms nd is quite importnt in pplied LP involving sizble models. In recent study, it took more thn thirty-five hours of computer time to obtin n initil bsis, but from n dvnced bsis, series of relted problems with few chnges in prmeters could be solved in two hours. Dillon (97) discusses wys of suggesting bsis for problems tht hve not previously been solved. Modeling systems like GAMS do not redily tke n dvnced bsis lthough one cn be ttempted by choice of initil levels for vribles (GAMSAS (Mcrl (996)) permits this). However, once n initil model solution hs been found, then ny dditionl solutions re computed from n initil bsis. Furthermore, n dvnced bsis cn be employed by restrting from stored file Finding n Initil Fesible sis When n LP problem includes only less-thn constrints with non-negtive right hnd sides, it is strightforwrd to obtin n initil fesible bsis with ll non-negtive vrible vlues. In tht cse the slcks form the initil bsis nd ll decision vribles re nonbsic, equling zero, with ech slck vrible set equl to the RHS (s i = b i ). The initil bsis mtrix is n identity mtrix. In turn, the simplex lgorithm is initited. However, if one or more: ) negtive right hnd sides, b) equlity constrints, nd/or c) greter thn or equl to constrints re included, it is typiclly more difficult to identify n initil fesible bsis. Two procedures hve evolved to del with this sitution: the ig M method nd the Phse I/Phse II method. onceptully, these two procedures re similr, both imply n inclusion of new, rtificil vribles, which rtificilly enlrge the fesible region so n initil fesible bsis is present. The mechnics of rtificil vribles, of the ig M method nd the Phse I/ Phse II problem re presented in this section. Models which contin negtive right hnd sides, equlity nd or greter thn constrints do not yield n initil fesible solution when ll 's re set to zero. Suppose we hve the following copyright ruce A. Mcrl nd Thoms H. Spreen

23 Mx s.t. R D F H b - e g p generl problem where b, e, nd p re positive. onversion of this problem to the equlity form requires the ddition of slck, surplus nd rtificil vribles. The slck vribles (S nd S ) re dded to the first nd second rows (note tht while we cover this topic here, most solvers do this utomticlly). Surplus vribles re needed in the lst constrint type nd give the mount tht left hnd side (H) exceeds the right hnd side limit (p). Thus, the surplus vribles (W) equl p - H nd the constrint becomes H - W = p. The resultnt equlity form becomes Mx O S O S R D F H, I S S, I S S, O I W W W b - e g p Where the I's re ppropritely sized identity mtrices nd the O's re ppropritely sized vectors of zeros. Note tht when =, the slcks nd surplus vribles do not constitute n initil fesible bsis. Nmely, if S nd W re put in the bsis; then ssuming e nd p re positive, the initil solution for these vribles re negtive violting the non-negtivity constrint S = -e nd W = -p. Furthermore, there is no pprent initil bsis to specify for the third set of constrints (F = g). This sitution requires the use of rtificil vribles. These re vribles entered into the problem which permit n initil fesible bsis, but need to be removed from the solution before the solution is finlized. Artificil vribles re entered into ech constrint which is not stisfied when = nd does not hve n esily identified bsic vrible. In this exmple, three sets of rtificil vribles re required. copyright ruce A. Mcrl nd Thoms H. Spreen 3

24 Mx R D F H, O S I S S, O I S S S, O I W W W, - I A A, I 3 A A 3 3, I A A b - e g p Here A, A 3, nd A re the rtificil vribles which permit n initil fesible nonnegtive bsis but which must be removed before "true fesible solution" is present. Note tht S, A, A 3, nd A cn be put into the initil bsis. However, if elements of A 3 re nonzero in the finl solution, then the originl F = g constrints re not stisfied. Similr observtions re pproprite for A nd A. onsequently, the formultion is not yet complete. The obective function must be mnipulted to cuse the rtificil vribles to be removed from the solution. The two lterntive pproches reported in the literture re the IG M method nd the Phse I/Phse II method IG M Method The IG M method involves dding lrge penlty costs to the obective function for ech rtificil vrible. Nmely, the obective function of the bove problem is written s Mx + O S + O S + O W - M A - M 3 A 3 - M A where M, M 3, nd M re conformble sized vectors of lrge numbers tht will cuse the model to drive A, A 3, nd A out of the optiml solution. An exmple of this procedure involves the problem Mx 3x x x x x x, x x x x x x 3 nd the model s prepred for the ig M method is in Tble 3.3. The optiml solution to this problem is in Tble 3.. This solution is fesible since A, A 3, nd A hve been removed from the solution. On the other hnd, if the right hnd side on the second constrint is chnged to -, then A cnnot be forced from the solution nd the problem is infesible. This, with the IG M method one should note tht copyright ruce A. Mcrl nd Thoms H. Spreen

25 the rtificil vribles must be driven from the solution for the problem to be fesible so M must be set lrge enough to insure this hppens (if possible) Phse I/Phse II Method The Phse I/Phse II method is implemented in lmost ll computer codes. The procedure involves the solution of two problems. First, (Phse I) the problem is solved with the obective function replced with n lterntive obective function which minimizes the sum of the rtificil vribles, i.e., Min L A + L 3 A 3 +L A where L i re conformbly sized row vectors of ones. If the Phse I problem hs nonzero obective function (i.e., not ll of the rtificils re zero when their sum hs been minimized), then the problem does not hve fesible solution. Note this mens the reduced cost informtion in n infesible problem cn correspond to this modified obective function. Otherwise, drop the rtificil vribles from the problem nd return to solve the rel problem (Phse II) using the Phse I optiml bsis s strting bsis nd solve using the norml simplex procedure. The ddition of the slck, surplus nd rtificil vribles is performed utomticlly in lmost ll solvers including ll tht re ssocited with GAMS. 3.. The Rel LP Solution Method The bove mteril does not fully describe how LP solution lgorithm works. However, the lgorithm implemented in modern computer codes, while conceptully similr to tht bove is opertionlly quite different. Tody some codes use interior point lgorithms combined with the simplex method (for instnce, OSL, Singhl et l.). odes lso del with mny other things such s compct dt storge, bsis reinversion, efficient pricing, nd round-off error control (e.g., see Orchrd-Hys or Murtgh). In terms of dt storge, lgorithms do not store the LP mtrix s complete MN mtrix. Rther, they exploit the fct tht LP problems often be sprse, hving smll number of non-zero coefficients reltive to the totl possible number, by only storing non-zero coefficients long with their column nd row ddresses. Further, some codes exploit pcking of multiple ddresses into single word nd economize on the storge of repeted numericl vlues (for in-depth discussion of dt storge topics see Orchrd-Hys or Murtgh). Perhps the most complex prt of most modern dy LP solvers involves inversion. As indicted bove, the - ssocited with the optiml solution is needed, but in forming - the code usully performs more itertions thn the number of constrints. Thus, the codes periodiclly construct the bsis inverse from the originl dt. This is done using product form of the inverse; copyright ruce A. Mcrl nd Thoms H. Spreen 5

26 but this lso involves such diverse topics s LU decomposition, reduction of mtrix into lower tringulr form nd mtrix fctoriztion. For discussion in these topics see Murtgh. LP codes often cll the formtion of reduced costs the pricing pss nd number of different pproches hve been developed for more efficient computtion of pricing (see Murtgh for discussion). Finlly, LP codes try to void numericl error. In computtionl LP, one worries bout whether numbers re relly non-zero or whether rounding error hs cused frctions to compound giving flse non-zeros. Solver implementtions usully mke extensive use of tolernces nd bsis reinversion schemes to control such errors. Murtgh nd Orchrd-Hys discuss these. The purpose of the bove discussion is not to communicte the intriccies of modern LP solvers, but rther to indicte tht they re fr more complicted thn the stndrd implementtion of the simplex lgorithm s presented in the first prt of the chpter. copyright ruce A. Mcrl nd Thoms H. Spreen 6

27 References zrr, M.S. nd J.J. Jrvis, nd H.D. Sherli. Liner Progrmming nd Network Flows. New York: John Wiley & Sons, 99. urton, R.O., J.S. Gidley,.S. ker nd K.J. Red-Wilson. "Nerly Optiml Liner Progrmming Solutions: Some onceptul Issues nd Frm Mngement Appliction." Americn Journl of Agriculturl Economics. 69(987): Dntzig, G.. Liner Progrmming nd Extension. Princeton, New Jersey: Princeton University Press, 963. Dillon, M. "Heuristic Selection of Advnced ses for lss of Liner Progrmming Models." Opertion Reserch. 8(97):9-. Drynn, R.G. "Forum on Resolving Multiple Optim in Liner Progrmming." Review of Mrketing nd Agriculturl Economics. 5(986):3-35. Freund, R.M. "Postoptiml Anlysis of Liner Progrm Under Simultneous hnges in Mtrix oefficients." Mthemticl Progrmming Studies. (985):-3. Gl, T., H.J. Kruse, nd P. Zornig. "Survey of Solved nd Open Problems in the Degenercy Phenomenon." Mthemticl Progrmming. (988):5-33. Hdley, G. Liner Progrmming. Reding, Mss.: Addison-Wesley Publishing o., 96. Krmrkr, N. "A New Polynomil-Time Algorithm for Liner Progrmming." ombintori. (98): Mcrl,.A. "Degenercy, Dulity nd Shdow Prices in Liner Progrmming." ndin Journl of Agriculturl Economics. 5(977):7-73. Mcrl,.A., W.V. ndler, D.H. Doster, nd P. Robbins. "Experiences with Frmer Oriented Liner Progrmming for rop Plnning." ndin Journl of Agriculturl Economics. 5(977):7-3. Mcrl,.A., D.E. Kline, nd D.A. ender. "Improving on Shdow Price Informtion for Identifying riticl Frm Mchinery." Americn Journl of Agriculturl Economics. 7(99): Mcrl,.A. nd.h. Nelson. "Multiple Optiml Solutions in Liner Progrmming Models: omment." Americn Journl of Agriculturl Economics. 65(983):8-83. Murtgh,.A. Advnced Liner Progrmming. McGrw-Hill ook o., New York, 98. copyright ruce A. Mcrl nd Thoms H. Spreen 7

28 Orchrd-Hys, W. Advnced Liner- Progrmming omputing Techniques. McGrw-Hill, Inc: New York, 968. Optimiztion Subroutine Librry (OSL). IM Guide nd Reference, Relese. Ed. J.A. George nd J.W.H. Liu. IM orportion, Pris, Q. "Multiple Optiml Solutions in Liner Progrmming Models." Americn Journl of Agriculturl Economics. 63(98):7-7. Pris, Q. An Economic Interprettion of Liner Progrmming. Iow Stte University Press: Ames, Iow, 99. Singhl, J.A., R.E. Mrsten, nd T.L. Morin. "Fixed Order rnch-nd-ound Methods for Mixed- Integer Progrmming: The ZOOM System." Deprtment of Mngement Informtion Systems, University of Arizon, Tucson, AZ, 857, December, 987. copyright ruce A. Mcrl nd Thoms H. Spreen 8

29 Tble 3.. GAMS Solution of Exmple Model SOLVE SUMMARY MODEL PROLEM OJETIVE Z TYPE LP DIRETION MAIMIZE SOLVER MINOS5 FROM LINE 37 **** SOLVER STATUS NORMAL OMPLETION **** MODEL STATUS OPTIMAL **** OJETIVE VALUE 6.5 EIT -- OPTIMAL SOLUTION FOUND LOWER LEVEL UPPER MARGINAL ---- EQU ZROW EQU ONSTRAIN -INF EQU ONSTRAIN -INF ZROW ONSTRAIN ONSTRAIN OJETIVE FUNTION FIRST ONSTRAINT SEOND ONSTRAINT LOWER LEVEL UPPER MARGINAL ---- VAR INF VAR INF VAR 3.. +INF VAR Z -INF 6.5 +INF. 3 Z FIRST VARIALE SEOND VARIALE THIRD VARIALE OJETIVE FUNTION copyright ruce A. Mcrl nd Thoms H. Spreen 9

30 Tble 3.. Solution with ounds Imposed s onstrints nd s ounds Solved with onstrints Vrible Vlue Reduced ost Sttus Eqution Level Shdow Price Sttus sic sic sic 3 Solved with ounds 3 - Vrible Vlue Reduced ost Sttus Eqution Level Shdow Price Sttus -3 (U) sic (L) Tble 3.3. The Model s Redy for the ig M Method Mx 3x + x + S + S + W - 99A - 99A 3-99 x + x + S = x - x + S - A = - -x + x + A 3 = 3 x + x - W + A = x, x, S, S, W, A, A 3, A Tble 3.. Solution to the ig M Problem Vrible Vlue Reduced ost Eqution Shdow Price x x.333 S S W.667 A -99 A A -99 copyright ruce A. Mcrl nd Thoms H. Spreen 3