Determine the Optimal Solution for Linear Programming with Interval Coefficients

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1 IOP Coferece Series: Materials Sciece ad Egieerig PAPER OPEN ACCESS Determie the Optimal Solutio for Liear Programmig with Iterval Coefficiets To cite this article: E R Wula et al 2018 IOP Cof. Ser.: Mater. Sci. Eg View the article olie for updates ad ehacemets. This cotet was dowloaded from IP address o 11/01/2019 at 14:41

2 The 2d Aual Applied Sciece ad Egieerig Coferece (AASEC 2017) IOP Publishig IOP Cof. Series: Materials Sciece ad Egieerig (2018) doi: / x/288/1/ Determie the Optimal Solutio for Liear Programmig with Iterval Coefficiets E R Wula 1*, M A Ramdhai 2 ad Idriai 3 1,3 Mathematics Departmet, Sciece ad Techology Faculty, UIN Sua Guug Djati, Jl. AH. Nasutio 105 Badug, West Java, Idoesia 2 Iformatics Egieerig Departmet, Sciece ad Techology Faculty, UIN Sua Guug Djati, Jl. AH. Nasutio 105 Badug, West Java, Idoesia *elis_rata_wula@uisgd.ac.id Abstract. The covetioal liear programmig model requires the parameters which are kow as costats. I the real world, however, the parameters are seldom kow exactly ad have to be estimated. Liear programmig with iterval coefficiets is oe of the tools to tackle ucertaity i mathematical programmig models. This paper presets a problem solvig liear programmig with iterval coefficiets. The problem will be solved by the algorithm geeral method to solve liear programmig with iterval coefficiets. Thus, the best ad the worst optimum solutios ca be obtaied, the upper bouds ad lower bouds for the optimum value of the mai problems ca be determied. 1. Itroductio I the liear programmig problem, sometimes coefficiets i the model caot be determied precisely so that is usually made i the form of estimatio. Oe of the methods to resolve this issue is by usig the iterval approach, where the idetermiate coefficiets trasformed ito the iterval. This liear programmig form is called Liear Programmig with Iterval Coefficiet (LPIC). Iterval coefficiet idicates expasio the tolerace iterval (or area) where the costat parameters are acceptable ad meet the LPIC models. I practice, a maager would like to kow the rage of optimum solutios that could be retured by the Liear Programmig model with various settigs of the ucertai coefficiets. Ufortuately, the curret state of the art provides little help. Classical sesitivity aalysis allows a study of the effect o the solutio of chages to sigle coefficiet or very small groups of coefficiets, but oly to the extet that the optimal basis is ot chaged [1]. Wedell's' tolerace approach determies the maximum fractioal chage i ay coefficiet before the basis chages, ad the 100% rule also cosiders simultaeous coefficiet alteratios that do ot chage the basis. There are o effective tools for examiig the effects of may simultaeous chages to the coefficiets that may chage the basis. I ay case, a hit-ad-miss approach to the variatio of the ucertai coefficiets is ulikely to ucover the complete rage of possible optimum objective fuctio values [2]. At first, LPIC was ot much discussed. Previous studies have focused o certai specific cases, for istace variable 0-1 or programmig liier case with iterval coefficiet i the objective fuctio oly. LPIC topic was itroduced widely i the year , startig from costrais models i the form of upper-boud ad lower-boud. Although ot related to the LPIC, these models have Cotet from this work may be used uder the terms of the Creative Commos Attributio 3.0 licece. Ay further distributio of this work must maitai attributio to the author(s) ad the title of the work, joural citatio ad DOI. Published uder licece by IOP Publishig Ltd 1

3 The 2d Aual Applied Sciece ad Egieerig Coferece (AASEC 2017) IOP Publishig IOP Cof. Series: Materials Sciece ad Egieerig (2018) doi: / x/288/1/ similarities that is the costraits model limited by the extreme poit. Shaocheg trasformed LPIC ito two liear programmig which has special characteristics [3]. Oe of the liear programs has the largest possibly feasible area ad most favorable versio of objective fuctio to fid the best possible optimum solutio. Meawhile the other liear programmig had the smallest feasible area ad both versios of the least favorable versio of objective fuctio to fid the worst possible optimum solutio. This Shaocheg method to overcome the problem LPIC with the terms: (a) restricted oly oegative variable, ad (b) oly overcome iequality [4]. Iuiguchi ad Sakawa deal with Liear Programmig models ivolvig iterval coefficiets i the objective fuctio oly. Their goal is to determie the closest sigle solutio to all of the optimal solutios of the model uder the ucertaities i the objective fuctio. They use the miimax regret approach to fid a solutio that miimizes the largest differece i the values of ay two versios of the objective fuctio. From the poit of view of solvig the LPIC problem, this method has sigificat drawbacks: it deals oly with iterval objective fuctios, ad it does ot give the desired iformatio about the rage of the objective fuctio values [5]. The extesios to equality costraits ad to o-positive ad urestricted variables are importat for practical reasos. Gass listed a umber of uses for urestricted variables i productio smoothig applicatios, zero-sum two-perso games, ad umerical ad statistical problems [6]. 2. Methods The research objective is the developmet of practical algorithmic tools for dealig with LPs i which the coefficiets are kow oly approximately. The assumptio is that ay ukow coefficiet ca be expressed as a iterval (a lower- ad upper-bouded rage of real umbers). This research develops methods that fid the best optimum (highest maximum or lowest miimum as appropriate), ad worst optimum (lowest maximum or highest miimum as appropriate), ad the coefficiet settigs (withi their itervals) which achieve these two extremes. The authors refer to the problem of fidig the two extreme solutios ad associated coefficiet settigs as Liear Programmig with Iterval Coefficiets (LPIC). 3. Results ad Discussio 3.1. Liear Programmig Liear programmig is a mathematical tool which is developed to hadle the optimizatio of a liear fuctio a set of liear costraits. The liear programmig is very importat i the area of applied mathematics ad has a large umber of uses ad applicatios i may idustries. Some curret applicatios iclude; allocatio of resources, trasportatio ad schedulig operatios [7]. The stadard form of the liear programmig model is as follows: [8] Maximizatio Z = c 1 x 1 + c 2 x c x a 11 x 1 + a 12 x a 1 x b 1 a 21 x 1 + a 22 x a 2 x b 2 a m1 x 1 + a m2 x a m x b m ad x 1 0, x 2 0,, x 0 Fuctio which maximizes c 1 x 1 + c 2 x c x is called the objective fuctio. x 1, x 2,, x is a decisio variable. c j, b i ad a ij ( for i = 1, 2,, m ad j = 1, 2,, )are the model parameters. Defiitio 1 Feasible solutio is a solutio that all costraits are met. No feasible solutio is solutio at least oe costraits violated [8]. 2

4 The 2d Aual Applied Sciece ad Egieerig Coferece (AASEC 2017) IOP Publishig IOP Cof. Series: Materials Sciece ad Egieerig (2018) doi: / x/288/1/ Defiitio 2 Feasible regio is a collectio of all feasible solutios [8]. Defiitio 3 The optimal solutio is a feasible solutio that has the most favorable value of the objective fuctio [8] Characteristics of Liear Equatios with Coefficiets Iterval Defiitio 4Let[a 1, a 2 ]x 1 + [b 1, b 2 ]x 2 = (, )[ c 1, c 2 ]be costrait give i liear programmig problem with iterval coefficiets. Shiftig costraits is costraits parallel movemet from oe positio to aother without chagig the slope. This shift is oly caused by chages i the Right Had Side from the value i [ c 1, c 2 ] to aother value i [c 1, c 2 ]. [5] Defiitio 5Let[a 1, a 2 ]x 1 + [b 1, b 2 ]x 2 = (, )[ c 1, c 2 ]be certai costraits (or objective fuctio) i a liear programmig problem with iterval coefficiets. Slope is the chage i slope of the give costraits (or objective fuctio). Slope caused by chages i at least oe of the coefficiets iterval associated with oe variable from the value i the iterval for other values i the iterval [9]. Defiitio 6. Reversal of liear costraits (or objective fuctio) of a liear programmig problem is a special kid of slope that occurs whe there is a chage simultaeously from all sigs of the coefficiets o the LHS (that is, all iterval coefficiet associated with the variable rage of zero) [9]. Iterval Coefficiet Effects i Liear Programmig: LetCbe the set of costraits with the iterval coefficiets, C 1 adc II are two sets of differet costraits geerated fromcby usig differet extreme versios. Feasible areas I ads II geerated byc I adc II are explaied by oe of these possibilities: [9] 1. S I S II ors II S I, is a feasible area etirely cotaied i aother feasible area. 2. S I S II ads I S II,is a feasible partially offs ay other feasible area. S I S II =, there is o overlap i the feasible area Liear Programmig with Iterval Coefficiet Solutio Algorithm 1Geeral method to solve liear programmig with iterval coefficiets: [9] Let the followig LPIC problem: Mi Z = j=1[ c j, c j ] x j j=1[ a ij, a ij ] x j [ b i, b i ], for i = 1,, m x j is sig-restrictedvariable(that isx j W I j ) The the best optimum ad the worst optimum as follows: 1. For The Best Optimum Mi z = j=1 c jx j, wherec j = { c j, x j 0 j=1 a ijx j b i i, wherea ij = { a ij, x j 0 2. For The Worst Optimum Mi z = j=1 c jx j, wherec j = { c j, x j 0 j=1 a ijx j b i, i, wherea ij = { a ij, x j 0 x j W I j. W I is variable set that is associated with iterval coefficiet. If costraits have boudaries( )the that costraits multiple with (-1) to get ( ). x j W I j. 3

5 The 2d Aual Applied Sciece ad Egieerig Coferece (AASEC 2017) IOP Publishig IOP Cof. Series: Materials Sciece ad Egieerig (2018) doi: / x/288/1/ Liear Progammig i Algorithm 1 have three possible results, that is: (i) optimum fiite bouded poit; (ii) uboudedess; or (iii) ifeasibility, cosequetly LPIC have some possibilities as follows: [5] a. If the best optimum is ifeasible solutio, the all LPIC are ifeasible. b. If the worst optimum ubouded solutio, the all LPIC are ubouded c. If best optimum solutio feasible with valuezad worst optimum ifeasible, the LPIC optimum has rage izad ifeasibility. d. If the worst optimum solutio is feasible with valuezad best optimum ubouded, the LPIC optimum has rage betwee ad z Numerical Examples The followig is a example of miimizatio which shows the case where the best optimum ad the worst optimum is feasible. Solve LPIC miimizatio model with iequality costraits as follows: Mi Z = [1,3]x 1 + [2,4]x 2 Subject to: C 1 : [2,3]x 1 + [4,6]x 2 [6,9], C 2 : x 1 + [2,4]x 2 5, C 3 : x 1 + x 2 [ 2, 1], C 4 : [3,5]x 1 + x 2 [6,7] C 5 : x 2 4, x 1, x 2 0 Solutio: Usig Algorithm2.1, will get: a. Best Optimum Solutio: Mi z = x 1 + 2x 2 Subject to: C 1a : 3x 1 + 6x 2 6, C 2a : x 1 + 4x 2 5, C 3a : x 1 + x 2 2 C 4a : 5x 1 + x 2 6, C 5 : x 2 4, x 1, x 2 0 Solutio for the Liear Programmig model,the: x 1 = 1, x 2 = 1 da z = 3. b. Worst Optimum Solutio Mi z = 3x 1 + 4x 2 Subject to: C 1b : 2x 1 + 4x 2 9, C 2b : x 1 + 2x 2 5, C 3b : x 1 + x 2 1 C 4b : 3x 1 + x 2 7, C 5 : x 2 4, x 1, x 2 0 Solutio for the liear programmig model, the: x 1 = 1,8, x 2 = 1,6 da z = 11,8. Figure 1. LPIC Feasible Area Therefore, by usig the best ad worst optimum solutio obtaied before, the gaied the upper boud ad lower boud for the optimum value of the mai problems. I this case the optimum value will take place betwee 3 ad 11,8; that is Z = [ 3, 11.8 ] The best ad the worst optimum solutio are illustrated i Figure 1. 4

6 The 2d Aual Applied Sciece ad Egieerig Coferece (AASEC 2017) IOP Publishig IOP Cof. Series: Materials Sciece ad Egieerig (2018) doi: / x/288/1/ So the optimum value will rage betwee 3 ad Various extreme versios of iterval relatioships ca be obtaied by fixig the iterval coefficiets at differet combiatios of their upper ad lower limits. If ay of the iterval coefficiets is fixed at a itermediate value, the a itermediate versio of the relatioship is obtaied. The methods developed later i this research make special use of specific extreme versios of the costraits ad objective. Depedig o the specific values chose for the iterval coefficiets i the costraits, there is a ifiite umber of differet possible feasible regios. I fact, some choices of coefficiet values may reder the Liear Programmig ifeasible or ubouded. Furthermore, if the objective fuctio icludes iterval coefficiets, there is also a ifiite choice of objective fuctios. The LPIC goal of fidig the best ad worst optimum values are directly affected by which specific values are chose for the iterval coefficiets, sice these i tur determie the feasible regios ad objective directios for specific versios of the model [1]. 4. Coclusio I the liear programmig problem, sometimes coefficiets i the model caot be determied precisely so that is usually made i the form of estimatio. Oe method to resolve this issue is to use the iterval approach, where the idetermiate coefficiets trasformed ito the iterval. This liear programmig form is called Liear Programmig with Iterval Coefficiet (LPIC). Iterval coefficiet idicates expasio the tolerace iterval (or area) where the costat parameters acceptable ad meets the LPIC models. Completio of liear programmig with iterval coefficiets with iequality costraits ca be resolved with the followig geeral steps: Suppose give problem LPIC: Mi Z = j=1[ c j, c j ] x j Z = j=1[ a ij, a ij ] x j [ b i, b i ], fori = 1,, m x j is sig-restricted variable(that isx j W I j ) The best optimum ad worst optimum as follows: 1. For The Best Optimum Mi z = j=1 c jx j, wherec j = { c j, x j 0 j=1 a ijx j b i i, wherea ij = { a ij, x j 0 x j W I j. 2. For The Worst Optimum Mi z = j=1 c jx j, wherec j = { c j, x j 0 Subject to j=1 a ijx j b i, i wherea ij = { a ij, x j 0 x j W I j. By usig best optimum ad worst optimum solutio the obtaied upper boud ad lower boud for LPIC optimum value. If costraits have boudaries ( ) the the costraits multiple by (-1) to get ( ). Refereces [1] J W Chieck ad K. Ramada, Liear Programmig with Iterval Coefficiets 2000 The Joural of the Operatioal Research Society, [2] Bradley S P, Hax A C ad Magati T L 1977 Applied Mathematical Programmig, Addiso Wesley: Readig, Massachusetts 5

7 The 2d Aual Applied Sciece ad Egieerig Coferece (AASEC 2017) IOP Publishig IOP Cof. Series: Materials Sciece ad Egieerig (2018) doi: / x/288/1/ [3] Shaocheg T 1994 Iterval umber ad fuzzy umber liear programmig. Fuzzy Sets ad Systems [4] Farida 2011 Pegoptimala pada Masalah Pemrograma Liear dega Koefisie Iterval, Fakultas Matematika da Ilmu Pegetahua Alam (IPB Bogor Upublished) [5] Iuiguchi M ad Sakawa M 1995 Miimax regret solutio to liear programmig problems with a iterval objective fuctio Eur J Opl Res [6] Gass S I 1985 O the solutio of liear programmig problems with free (urestricted) variables. Comp ad Ops Res [7] H Suprajito ad I. Mohd 2010 Liear Programmig with Iterval Arithmetic It. J. Cotemp. Math. Scieces [8] Hillier ad Lieberma 2001 Itroductio to Operatios Research Seveth Editio, TheMcGraw Compaies [9] K Ramada 1996 Liear Programmig with Iterval Coefficiets, Faculty of Graduate Studies ad Research, Carleto Uiversity, Ottawa, Upublished 6