A Computer Technique for Solving LP Problems with Bounded Variables


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1 Dhaka Unv. J. Sc. 60(2): , 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of Dhaka, Dhaka1000, Bangladesh Receved on Accepted for Publcaton on Abstract Lnear Programmng problem (LPP)s wth upper bounded varables can be solved usng the Bounded Smplex method (BSM), wthout the explct consderaton of the upper bounded constrants. The upper bounded constrants are consdered mplctly n ths method whch reduced the sze of the bass matrx sgnfcantly. In ths paper, we have developed MATHEMATICA codes for solvng such problems. A complete algorthm of the program wth the help of a numercal example has been provded. Fnally a comparson wth the bultn code has been made for showng the effcency of the developed code. Keywords: LPP, bounded smplex method, bounded varable, computer program, restrctons. I. Introducton Lnear Programmng problem (LPP) wth bounded varables has the form Maxmze, Z = {CX: (A, I) X = b; L X U} (1) One can solve such problems by regular smplex method by consderng the lower and upper bound constrants explctly whch s not computatonally effcent as the number of constrants as well as the number of varables become very large. Dantzg 1 developed the method for solvng lnear programmng wth upper bound restrctons on the varables. Wagner 2 developed the dual smplex method for LPP wth bounded varables, whch s further studed by Maros 3, 4. In 1972, Duguay 5 et al. studed lnear programmng wth relatve bounded varables. Y. Xa and J. Wang 6 used neural network for fndng an approxmate soluton of LPP wth bounded varables whch converges globally to the solutons to the LPPs. However, most of the real lfe problems deal wth large quanttes of data and parameters, whch can only reasonably be tackled by the calculatng power of a computer and researchers, are actvely engaged n developng programmng codes for solvng varous problems n Operatons Research e.g. Saha and Hasan 7, Hasan. 8, Morshed and Hasan 9. In ths paper, we wll dscuss the algorthm for solvng LPP wth Bounded Varables and develop a MATHEMATICA program for solvng such problems, whch s capable of dealng wth any number of varables. Fnally a numercal example obtaned by usng the program s provded and a comparson has been made wth the bultn MATHEMATICA code. * Correspondence author : Emal: II. Bounded Smplex Method 10 In LP models varables may have explct postve upper and lower bounds. An LPP may have n addton to the regular constrants, lower or upper bounds on some or all varables.e. constrants of the type L X U For the lower bounds.e. X L n (1), the followng substtuton X X = L, X 0 (2) can be used throughout and solve the problem n terms of X (whose lower bound now equals zero). The orgnal X s determned by back substtuton whch s legtmate because ts guarantees that X X = L wll reman nonnegatve for all X 0. However, for the upper boundng constrant, X U, the dea of drect substtuton.e. X + X = U, X 0 s not correct because back substtuton X + X = U, does not ensure that X wll reman nonnegatve. For smplcty we defne the upper bounded LP model as Maxmze, Z = {CX: (A, I) X = b, 0 X U} (3) The bounded algorthm uses only the constrants (A, I) X = b, X 0 explctly, whle accountng for X U mplctly through modfcaton of the smplex feasblty condton. Let X = B b be a current basc feasble soluton of (A, I) X = b, X 0 and suppose that accordng to the (regular) optmalty condton, P s the enterng vector. Then, gven that all the nonbasc varables are zero, the constrant equaton of the th basc varable can be wrtten as (X ) = (B b) B P x
2 164 When the enterng varable x ncreases above zero level, (X ) wll ncrease or decrease dependng on whether B P s negatve or postve, respectvely. Thus, n determnng the value of the enterng varable x three condtons must be satsfed: Condton1: The basc varable (X ) remans nonnegatvethat s, (X ) 0. Ths s not volated f ( B 1 b) x θ mn : 1 B P ( B 1 P ) 0 (4) Condton2: The basc varable (X ) does not exceed ts upper bound.e., (X ) (U ), where U comprse the ordered elements of U correspondng to X. Ths s not volated f ( B 1 b) ( ) 1 x θ mn U B : B P 0 (5) ( B 1 P ) Condton3: The enterng varable x cannot assume a value larger than ts upper bound that s x u, where u s the th element of U. Combnng the three restrctons together, x enters the soluton at the level that satsfes all three condtons and the level s defned as x = mn {,, u } (6) The change of bass for the next teraton depends on whether x enters the soluton at level 1, 2 or u. Assumng that (X ) s the leavng varable, then we have the followng rules: Rule1: x = u. The basc vector X remans unchanged because x = u stops short of forcng any of the current basc varables to reach ts lower (=0) or upper bound. Ths means that x wll reman nonbasc but at upper bound. Followng the argument ust presented, the new teraton s generated by usng the substtuton x = u x. Rule2: x = ; (X ) leaves the basc soluton (becomes nonbasc) at level zero. The new teraton s generated n the normal smplex manner by usng x and (X ) as the enterng and the leavng varables, respectvely. S. M. Atqur Rahman Chowdhury and Sanwar Uddn Ahmad of, x =, wth one modfcaton that accounts for the fact that (X ) wll be nonbasc at upper bound. Because the values of and are developed under the assumpton that all nonbasc varables are at zero level. Ths s acheved by usng the substtuton (X ) = (U ) (X ), where (X ) 0. It s mmateral whether the substtuton s made before or after the new bass s computed. A te among,, and u may be taken arbtrarly. However, t s preferable to mplement the rule for x = u because t entals less computaton. The substtuton x = u x wll change the orgnal c, P and b to c = c, P = P and btob = u P. Ths means that f the revsed smplex method s used, all the computaton (e.g, B, X, and z c ) should be based on the updated values of C, A, and b at each teraton. III. MATHEMATICA codes For convenence we have presented the developed MATHEMATICA codes for solvng LPP wth bounded varables whle solvng a numercal example. LPP1 Maxmze, Z = 3x 1 + 5x 2 + 2x 3 Subect to, x 1 + 2x 2 + 2x x 1 + 4x 2 + 3x x 1 4, 0 x 2 3, 0 x 3 3 Introducng slack varables x 4 0, x 5 0 one to each constrant, we have Maxmze, Z = 3x 1 + 5x 2 + 2x 3 + 0x 4 + 0x 5 Subect to, x 1 + 2x 2 + 2x 3 + x 4 + 0x 5 = 10 2x 1 + 4x 2 + 3x 3 + 0x 4 + x 5 = 15 0 x 1 4, 0 x 2 3, 0 x 3 3, x 4 0, x 5 0 The followng codes are used to take the necessary nputs. Rule3: x = ; (X ), becomes nonbasc at ts upper bound. The new teratons s generated as n the case
3 A Computer Technque for Solvng LP Problems wth Bounded Varables 165 The nputs are made nto the followng nput box: Here, a, b are the 1 st and 2 nd row of the ntal table, K s the coeffcent matrx, U[[ ]] represents the upper bounds of x, CNT s the ntal basc feasble solutons (b ), B s the bass matrx and CJ[[ ]] represents the coeffcent of the proft vector (C ). Enter the desre functon Z: 3x 1 + 5x 2 + 2x 3 ; Then, n = 5; Enter Row 1: 1, 2, 2, 1, and 0 Enter Row 2: 2, 4, 3, 0, and 1 Enter coeff. of obectve functon: 3, 5, 2, 0, and 0 Enter upper bound: 4, 3, 3, 9999, and Enter coeff. of bass element: 0, 0 Enter constant: 10, 15 X = 0; X = 0; X = 0; X = 0; X = 0 The ntal smplex table s generated by the followng codes. Table. 1. Intal smplex table C Bass 0 0 x 4 x x 1 x 2 x 3 x 4 x Soluton(b ) C Z = 0
4 166 The followng codes are used for dentfyng the enterng nonbasc varable. S. M. Atqur Rahman Chowdhury and Sanwar Uddn Ahmad Table 3 s obtaned by rule 2 n secton 2 as, Table. 3. Basc x x x Soluton Z 1 5/2 3/2 39/2 x 0 1/21/2 5/2 x 2 3/2 1/2 3/2 The developed codes for Rule2 are, Here, l4 = C = C z = relatve proft vector, MP = Max{C : C > 0}, θ s the value of θ n (5), θ s the value of θ n (4) and θ represents the value of x n (6). By Rule1 n secton 2, we obtaned Table 2 as, Table. 2. Basc x x x Soluton Z x x For Rule3, developed codes are: The developed codes for Rule1 are,
5 A Computer Technque for Solvng LP Problems wth Bounded Varables 167 These codes are teratvely mplemented untl the optmum table s obtaned. Table. 4. Basc x x x Soluton Z 1/27/45/4 75/4 x 0 1/21/2 5/2 x 1/23/41/43/4 Table. 5. Optmum table Basc x x x Soluton Z 1/27/45/4 83/4 x 0 1/21/2 5/2 x 1/23/41/4 5/4 By back substtuton the optmal values of x, x, and x are x = u x = 4, x = u x = 4 7 and 83 x = 0. Fnally, the optmal value of Z s. 4 Output Table. 6. Comparson of used tmed for solvng LPP1 Machne Confguratons Intel[R] 4 CPU 2.00GHz,512M B of RAM Tme Used(Sec.) Bultn Code 10, 11 IV. Algorthm of BSM Developed Code In ths secton we present the algorthm of our developed code. Step 1. Calculate the net evaluaton C Z. For a maxmzaton problem f C Z 0 for the nonbasc varables at ther upper bound, optmum basc feasble soluton s attaned. It not, go to step2, revenue s true for a mnmzaton problem. Step 2. Select the most postve C Z. Step 3. Let x be a non basc varable at zero level whch s selected to enter the soluton. Compute the quanttes defned n (4) & (5) Step 4. x = mn (,, u ). Substep 1: If =. (X ) leaves the basc soluton (because nonbasc) at level zero and x enter by usng the regular row operaton of the smplex method. Substep 2: If =. (X ) leaves the basc soluton at level zero and x enters then (X ) beng nonbasc at ts upper bound must be substtuted out by usng where, (X ) = (U ) (X ) 0 (X ) (U ) Substep 3: If = u, x s substtuted at ts upper bound by x = u x whle remanng nonbasc. Flowchart: The flowchart of the developed code s presented n ths secton.
6 168 S. M. Atqur Rahman Chowdhury and Sanwar Uddn Ahmad Start Input Construct ntal Table Prnt C Z 0 No Stop Identfyng pvot column From (2)&(3), calculate θ & θ respectvely U = upperbound of x x = mn {,, u } x = x = x = u x enters the bass (X ) (X ) = (U ) (X ) x = u x V. Concluson Nowadays computer programmng plays an mportant role for solvng varous problems n Operatons Research and s a much needed tool for solvng large scale problems. In ths paper we have dscussed the algorthm for solvng LPP wth Bounded Varables, have developed a MATHEMATICA program for solvng such problems. It s evdent from table 6 that the developed program can reduce sgnfcantly the tme taken to provde optmum soluton. Despte the restrctons, ths program may be used more effcently for solvng LPP wth bounded varables Dantzg, G. B., 1955, Upper Bounds, Secondary Constrants and Trangularty n Lnear Programmng, Econometrca, 23, No Wagner, H. M., 1958, The Dual Smplex Algorthm for Bounded Varables, Nav. Res. Log. Quart. 5, Maros I, 2003a, A pecewse lnear dual phasei algorthm for the smplex method, Computatonal Optmzaton and Applcatons, 26, Maros I, 2003b, A generalzed dual phase2 smplex algorthm, European Journal of Operatonal Research, 149, C. Duguay, M. Todd and H.M. Wagner, 1972/73, Lnear programmng wth relatve bounded varables, Management Sc., 19, Youshen Xa and Jasong Wang, 1995, Neural Network for Solvng Lnear Programmng Problems transacton on neural networks, 6(2). wth Bounded Varables, IEEE 7. Ron Saha and M. Babul Hasan, 2010, A Computer Technque for Senstvty Analyss n Lnear Programs, Dhaka Unv.J.Sc. 58(2), Hasan, M.B. Hasan, 2008, Soluton Lnear Fractonal Programmng Problems through Computer Algebra, The Dhaka Unversty Journal of Scence, 57(1), Morshed, M.B. Hasan, 2005, Graphcal representaton of feasble regon of Lnear programmng problems usng Mathematca, The Dhaka Unversty Journal of Scence, 53(1), Taha H. A., 1999, Operaton Research An Introducton, PrentceHall of Inda Pvt. Ltd, New Delh. 11. Gupta, P.K. & Hra, D.S., 1998, Problems n Operatons Research, S. Chand & Company, New Delh.
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