(C) 998 Gerald B Sheblé, all rghts reserved Lnear Prograng Introducton Contents I. What s LP? II. LP Theor III. The Splex Method IV. Refneents to the Splex Method What s LP? LP s an optzaton technque that allocates lted resources aong copetng actvtes n the best possble anner. A. Possble applcatons of LP Mnzng total producton costs n a power sste Maxzng the profts of a anufacturng process B. LP Background LP was developed for ateral oveent durng WWII. C. LP Theor a. Feasble Soluton The sste of constrant equatons n atrx for: A x <= b x s an n vector, b s an vector, and A s an x n atrx. A vector satsfng ths set of constrants s sad to be a feasble soluton. Graphcal Representaton Exaple Fgure b. Basc Soluton Separate A nto sub atrces B and D. B conssts of lnearl ndependent coluns; a bass for the space E. x = B x b + D x nb Obtan a soluton for A x = b b solvng B x b =b. Ths B soluton s known as a basc soluton. Ths equaton wll alwas have at least one basc soluton. A vector that s feasble (satsfes A x = b) and s also basc s sad to be a basc feasble soluton (BFS). c. Connectng the Graphcal and the Matheatcal The Fundaental Theore of LP - t s onl necessar to consder BFS when seekng an optal soluton to an LP proble because the optal value s alwas acheved at such a soluton. Equvalence Theore - A vector x s an extree pont of the convex set f and onl f x s a BFS to A x = b. 6/03/2003 of 9 2:9 PM
(C) 998 Gerald B Sheblé, all rghts reserved Solve the LP proble b traversng around the convex set of extree ponts n such a wa as to contnuall decrease the value of a gven obectve functon untl a nu s reached. d. Splex Method of Solvng LP. Steps of the Splex Method. Start at a corner-pont feasble soluton. 2. Move to a better adacent corner-pont feasble soluton. Repeat ths step as often as needed. 3. Stop when the current corner-pont feasble soluton s better than all ts adacent corner-pont feasble solutons. Ths s an optal soluton. A graphcal plot of the constrants and obectve can show the optal soluton. If there are ore than three decson varables then I cannot use standard graphcal ethods.. Splex Standard For The splex tableau ethod s capable of solvng LP probles wth an nuber of varables. Maxze: Z = su c x for all Subect to: a x >= b for all x > 0 for all The frst equaton s known as the obectve functon snce t represents the obectve of the proble. The rest of the equatons are collectvel called constrants and act to lt or constran the obectve functon. An econoc nterpretaton of the varables of the standard for s useful n showng what each varable represents: x - the level of actvt of a process c - unt cost of actvt Z - total cost fro all actvtes b - aount of resource avalable a - aount of resource consued b each unt of actvt. Splex Tableau Method Frst, convert lnear algebrac equatons to tableau. 6/03/2003 2 of 9 2:9 PM
(C) 998 Gerald B Sheblé, all rghts reserved Consder the proble below: axze 3x +5x 2 subect to: x +x = 4 3 2x +x = 2 2 4 3x +2x +x = 8 2 5 x > 0 The orgnal tableau wll be as follows: x x2 x3 x4 x5! b ------------------------------------------! 0 0 0! 4! 0 2 0 0! 2! 3 2 0 0! 8!----------------------------------------- rcc! -3-5 0 0 0! 0 Note that x3, x4 and x5 are basc varables. (Wh?) Note that the value for the obectve functon s zero (0). Exane the tableau. Note that the obectve functon has been changed to standard for. Should the unt atrx be stored? (No.) Appl the Splex procedure (4 an steps): A. Choose the enterng varable - ost negatve to decrease Z. Deterne non-basc varable for r< 0. If none then stop. Else, select varable wth ost negatve rcc. B. Choose the departng varable - deterne the nu rato. Mn RHS/a where a s strctl postve. Ths fnds sallest step b deternng whch varable goes to zero frst. C. Pvot on the (enterng, departng) eleent New row = Old row (pvot colun coeffcent) new pvot row Update tableau as Pvot on th eleent. D. STOP when all coponents of r are > 0. v. Exaple Soluton Splex Tableau Soluton Procedure 6/03/2003 3 of 9 2:9 PM
(C) 998 Gerald B Sheblé, all rghts reserved Exaple contnued Orgnal tableau: x x2 x3 x4 x5! b ------------------------------------------! 0 0 0! 4! 0 2 0 0! 2! 3 2 0 0! 8!----------------------------------------- rcc! -3-5 0 0 0! 0 Deterne non-basc varable for r< 0. If none then stop. Select x2. Fnd sallest step b deternng whch varable goes to zero frst. 2/2 = 6 selected snce nu 8/2 = 9 not selected (ht vertex later) Pvot on row 2, colun 2. Next Tableau x x2 x3 x4 x5 b -------------------------------------------! 0 0 0! 4! 0 0 /2 0! 6! 3 0 0 -! 6!------------------------------------------ rcc! 3 0 0 5/2 0! 30 Deterne non-basc varable for r< 0. If none then stop. Select x. Fnd sallest step b deternng whch varable goes to zero frst. 6/0 not evaluated (no connecton) 6/3 = 2 selected (onl one) Next Tableau x x2 x3 x4 x5 b -------------------------------------------! 0 0 /3 -/3! 2! 0 0 /2 0! 6! 0 0 -/3 /3! 2!------------------------------------------ rcc! 0 0 0 3/2! 36 Deterne non-basc varable for r< 0. If none then stop. So stop. Soluton X = 2; x2 = 6; x3 = 2; x4 = 0; x5 = 0 Z = 36 6/03/2003 4 of 9 2:9 PM
(C) 998 Gerald B Sheblé, all rghts reserved e. Refneents to the Splex Tableau. Upper Boundng Theore Strealnes the Splex Method when there are bounds gven on the varables. The bounds are handled outsde of the tableau. Ths reduces the sze of the tableau atrx. Upper Boundng Procedure. Deterne non-basc varable for rcc < 0. If none stop, else contnue. 2. Evaluate 3 possbltes: a. h 0 b. n ----for > 0 - h 0 c. n -------- for <0 where h s the upper bound assocated wth the th basc varable. 3. Fnd sallest of the three, then update tableau as follows. For a when x goes to opposte bound. RHS = RHS - (h col) col = -col No pvot requred.. For b wth as nzng ndex, then pvot on th eleent.. For c, where s nzng ndex n c above, then ove th basc var to opposte bound. RHS = RHS - h = - col = -col Pvot on th eleent. 6/03/2003 5 of 9 2:9 PM
(C) 998 Gerald B Sheblé, all rghts reserved 4. Return to step. Upper Boundng Exaple Intal Tableau x x2 x3 x4 x5! b!!! 0-2! 5! 0 2 2! 9!! T! r 0 0 - -2 5! -9 sgn + + + + + Calculate step 2 above. a. 5 b. 9/2 = 4.5 c. 2 Case c apples New Tableau x x2 x3 x4 x5! b!!! - 0-2! -2! 0 2 2! 9! T! r 0 0 - -2 5! -9 - + + + + Pvot x x2 x3 x4 x5! b!! 0 - -2! 2! -2 4 0 5! 5!! T! r 2 0-3 0! -5 - + + + + a. b. 5/4 =.25 c. 3 Case a apples and no pvot s requred x x2 x3 x4 x5! b!!! 0-2! 3! -2-4 0 5!!! T! r 2 0 3 0! -2 - + - + + 6/03/2003 6 of 9 2:9 PM
(C) 998 Gerald B Sheblé, all rghts reserved Snce all rcc coeffcents are nonnegatve, ths s an optal soluton. Soluton: D. Senstvt Analss x = 7 (ax); x2 = ; x3 = (ax); x4 = 3; x5 = 0; Z=2 Paraeters are usuall educated guesses. Senstvt Analss allows the nvestgaton of the effects of changng these paraeters and lookng at the behavour of the obectve functon. We wll be concerned prarl wth two tpes of changes n the rght hand sde (b) paraeters and changes n the cost coeffcents (c) paraeters Assue the followng for of the Fnal Tableau at the optu: x... xn xn+... xn+ b --------------------------------------------! a... a s... s b n n!....!..........!.! a... a s... s b n n!! r... r...! n 0! Note that these entres a be wrtten as the followng n ters of ust the slack varable coeffcents and the odel paraeters: o = b = b k = b sk for k =, 2,,, = z c = c + a for =, 2,, n, k a = a = s k = Consder the followng changes: b b + b, for =,..., c c + c, for =,...,n a a + a Norall onl one of the paraeters wll change. 6/03/2003 7 of 9 2:9 PM
(C) 998 Gerald B Sheblé, all rghts reserved Frst change the fnal splex tableau to reflect changes. These changes nclude: o k = = = b k b = b s, for k =, 2,, ( z c ) = c + a, for =, 2,, n a k = = a s k = Now the soluton a not be feasble or optal. Both cases ust be corrected. Consder the followng procedure. Step. Calculate the changes gven the above forulas and add the (when nonzero) to the correspondng entres n the fnal splex tableau. Step 2. Convert resultng tableau to proper for of dentfng and evaluatng the present basc soluton b pvotng (as necessar) for the splex ethod. Step 3. Test soluton for feasblt b checkng all basc varable values n the colun for volatng the nonnegatve assupton. Step 4. Test soluton for optalt b checkng all reduced coeffcent varable values n the row for volatng the postve assupton. Step 5. If ether test fals, then reappl the splex rules to fnd new optal soluton. Consder the followng cases when onl those paraeters change. Case. Changes n Rght Hand Sde (b) Step. Calculate: o k = = = b k b = b s, for k =, 2,, And add to respectve entres n rght hand sde colun of fnal splex tableau. Step 2. Not Applcable. Step 3. Test as stated. Step 4. Test as stated f soluton s not feasble. Step 5. Solve for optal value f step 3 fals usng new tableau. 6/03/2003 8 of 9 2:9 PM
(C) 998 Gerald B Sheblé, all rghts reserved Case 2 Changes n Nonbasc varable coeffcents Step. Calculate: ( z c ) = c + a k = k = a = a s, for k =, 2,, Add to respectve entres n x colun of fnal splex tableau. Step 2. Not Applcable. Step 3. Not Applcable. Step 4. Test as stated. New coeffcent of x n row 0 of tableau ust stll be nonnegatve. Step 5. Solve for optal value f step 4 fals usng new tableau. Case 3 Changes n Basc Varable coeffcents (assue lnear ndependence stll holds) Step. Calculate: ( z c ) = c + a k = k = a = a s, for k =, 2,, And add to respectve entres n x colun of fnal splex tableau. Step 2. Pvot as stated to for unt atrx under basc varables. Step 3. Test as stated. Step 4. Test as stated. Step 5. Solve for optal value f ether step fals usng new tableau. Case 4 Changes n Nuber of Varables (added) Student should do. Case 5 Changes n nuber of Constrants (added) Student should do. E. Possble Applcatons In Power Econoc Dspatch and Interchange Capablt are two possble applcatons. The extenson to ED s ver straghtforward f a pece-wse lnear approxaton of the Input/Output Curve s used. Then t s necessar onl to use the approprate segent to fnd the optal soluton. 6/03/2003 9 of 9 2:9 PM