A decomposable computer oriented method for solving interval LP problems

Pure and Applied Mathematics Journal 3; (5): -8 Published online October 3, 3 (http:www.sciencepublishinggroup.comjpamj) doi:.48j.pamj.35.3 A decomposable computer oriented method for solving interval LP problems Sharmin Afroz, M. Babul Hasan Department of Mathematics, University of Dhaka, Dhaka-, Bangladesh Email address: Sharmin_Afroz7@yahoo.com(S. Afroz) To cite this article: Sharmin Afroz, M. Babul Hasan. A Decomposable Computer Oriented Method for Solving Interval LP Problems. Pure and Applied Mathematics Journal. Vol., o. 5, 3, pp. -8. doi:.48j.pamj.35.3 Abstract: The purpose of this paper is to develop a computer oriented decomposition program for solving Interval Linear Programming (ILP) Problems. For this, we first analyze the eisting methods for solving ILP problems. We also discuss the main stricter of Decomposable Interval programming (DIP) problems. Then a decomposable algorithm is analyzed for solving DIP problems. Using Mathematica, we develop a computer oriented program for solving such problems. We present step by step illustration of a numerical eample to demonstrate our technique. Keywords: LP, ILP, DILP Computer Program. Introduction Linear programming (LP) is a technique for determining on optimum schedule (such as maimizing profit or minimizing cost) of interdependent activities in view of the available resources. Programming is just another word for planning and refers to the process of determining a particular plan of action from amongst linear stands for indicating that all relationships involved in a particular problem are linear. Any LP consists of four parts: a set of decision variables, the parameters, the objectives function and a set of constraints. The constraints may be equalities or inequalities []. In particular, an LP can be written as Maimize Subject to () A b () []. General Form of ILP An interval linear program or interval linear programming problem is any problem of the form [3] (ILP): maimize c t X b - AX b + where, c =( ), b - =( ),, and A=( )(i=,,n;j=,,m) are given, with. Let S be the set of points satisfying the constraints S= : }. A point S is called a feasible solution of ILP problem if S otherwise infeasible. If () is bounded then it is equivalent to the ILP Maimize Me A b Me where, e is a vector of once and M is a sufficiently large positive scalar [4]. There are a few methods for solving the ILP problems. Rober and Ben-Isreal [5] discussed a new iterative method for solving ILP in 97. That method applies general LP and is shown to be a dual method with multiple substitution. Gunn and Anders [] show that the simple method for LP and ILP are identical which is shown in a comparison between simple method for LP and ILP problems. Radimir Viher [8] develop an analogous theorem to solve ILP problems. akahara Sasaki and Gen [9] investigate a LP problem with interval coefficients and proposed a new concept of constraints based on probability. one of the above papers addressed the decomposition of ILP problems. In our paper, we will develop a decomposable technique for solving ILP problems. Decomposable Interval Programming Problem (DILP) An interval linear programming problem is a special type of interval problem [5]. A decomposable interval program (DILP) is any problem having the form [4] (DILP): Maimize

Pure and Applied Mathematics Journal 3; (5): -8 3 Where and are nonsingular,, and. In the net section, we will show how an ILP can be converted to DILP. Hence we will discuss an eisting method to solve DILP. We will present our algorithm and corresponding coding using Mathematica in section IV. Using numerical eample we will show that the results are same which are found using eisting method and our algorithm. The rest of the paper is organized as follows. In Section, conversion of ILP to DILP will be shown. For solving DILP a method related to the Dantzig-Wolfe decomposition principle will be discussed in section 3. In section 4, we will present our algorithm and computer technique to solve DILP. In section 5, we will conclude the paper.. Convertion of ILP to DILP Conversion of ILP to DILP is shown in this section []. In general the general ILP form is given below [4] (ILP): maimize c t X b - AX b + ow let us consider ILP in the following form (ILP): maimize c t X where,,!! and "!. That is is a nonsingular submatri of A, is any submatri having full row rank whose rows are not in and is made up of the rows A not in or. ote that q is not uniquely defined, but as we shall see later it is desirable to make q as large as possible (q= is always possible). Clearly we can always choose %, a subset of the constraints (ILP) such that ( % ). Problem (ILP) is not changed by including some constraints more than once, so that ILP can be written as follows. (ILP): maimize c t X % Finally, observe that & is an optimal solution to (ILP) iff ( &,( & ) is an optimal solution to the following bounded problem which has the form (DILP): maimize X ) +, - "!.) ( +) + % ) ( + - "!, and can be identified by Gauss-Jordan elimination method. A procedure for identifying. appropriate,, and B is demonstrated in the following eample umerical Eample Transfer the following ILP problem to form DILP. (DILP): maimize Solution 4< 4 33; 3 4 = 5: < ; ) + = : 43< 34;. 5 : Applying Gauss-Jordan eliminations to A we obtain (pivots marked by asterisk): & 4 3 = < 4 ; ~ 3 = : & < 4 ; ~ 3 =: < ;. : Since rows and 3 contained pivots we conclude that ) + is a suitable nonsingular sub matri of A. Rearranging A with matri at the bottom we get 4= =< 3 ;. : To find net sub matri we repeat the above & 4= =< 4 < 4 < procedure3 ; ~ 3 =& ; ~ 3 ; : : : We conclude that row 4 of A is linearly dependent on row (since all its elements vanished after step ) so that row 4 must fall in. Furthermore since r= pivots were found before reaching the bottom r rows we conclude that B=.Therefore, ) + (rows and 4 contained pivots), = = and B=. The equivalent DILP is as follows (DILP): maimize @ 3A@ A@ A@ 4A ( 3 @ 5A@ A@ A@ A. 4 = = = ( 5 3. Eisting Decomposition Method In this section we discuss a method, for solving DILP

4 Sharmin Afroz et al.: A Decomposable Computer Oriented Method for Solving Interval LP Problems which is related to the Dantzig-Wolfe decomposition principle [4]. The general form of DILP is (DILP): Maimize This may be written as (DILP): Maimize B B. Let C= D : E and let F be the finite matri whose columns are the etreme points of C. Since is nonsingular, C is a bounded polyhedron so that BC iff FG, H G, G I(i.e. is a conve combination of the etreme points of C).An analogous result holds for C = D : }and the corresponding matri F whose columns are the etreme points of C. Consequently, DILP may be written as (DILP): maimize FG FG=FGB H G. H GB. G,GBI. This problem has standard linear programming form ecept that the columns of F and F Jis not immediately known. Indeed, we shall solve the constraints using the simple algorithm with a special technique for generating the columns of F and F one at a time as needed. Suppose that a basic feasible solution to the constraints is in hand with simple multipliliers (K,,K D,M,M ) = (π, M,M ).Let the columns of F and F be denoted by (i=,., OJ) and B (i=,,oj) =B respectively. A vector@ A or @ A can enter the basis if (π, M,M )@ A - = (K= )M P =B or (π, M,M )@ A = - KB M < respectively. The smallest relative cost Q is defined as λ= min RST UV ((K= )M ), RST UBV (- KB M )} () Following the standard simple procedure we bring into the vector with the smallest relative cost. If λi,the solution is optimal. Thus we etreme points & and B & such that K= g & RST UV ((K= ) and =KB & RST UBV (- KB.But RST UV XK= Y RST Z[ K= so that & ^_ ] ^` ]. Where = (],,]D) and -, as:k= ] I,PE. (i.e. & is an etreme point solution of the sub problem RST Z[ K= ). Likewise B & = ^ _ ] ^` ] Where, (],,] D) and -, i:-π] I,P}. & B & If λ< either @ A or @ A enter the basis depending on which has the lowest relative cost. The simple iteration is then completed to obtain a better basic feasible solution and the entire procedure is repeated. If λi the present solution, say (G&,GB & ) to FG=FGB is optimal. An optimal solution to (DILP) is then & = G & c d = B GB & c e (3) where, f as:g ge and f = i:gb g}. An initial basic feasible solution can be found using artificial variables. One approach is to being with the enlarged problem Maimize FG= hh i F =F G @ H - D A@ GB A @ H i G,GB,z I D A (4) Where M is a sufficiently large scalar, using H,.,H D } as the starting basis. If λi while some of the artificial variable are still in the basis at a non-zero level, problem (DILP) has no feasible solution. A slightly different approach for obtaining an initial basic feasible solution to the constraint is to find any etreme points of C and C. If we call these points k and B respectively, then @ A,@ =B A,lm H n,..,l@h D is a suitable starting basis to the equivalent problem Maimize FG= hh i G,GB,z I F =F G s.t@ H - D A@ GB A @ H i H D A (5) A () The sign in front of mn in (5) will depend on the sign of (=B ) and has purposely been left ambiguous to simplify notation. umerical Eample

Pure and Applied Mathematics Journal 3; (5): -8 5 Find the optimal solution of the ILP problem (ILP): maimize 8 =9=3 9. Solution Given that (ILP): maimize 8 =9=3 9. A decomposable interval problem has the form, (DILP): maimize. The given problem is a decomposable interval problem having ) +, ) +, =) +, ) 8 +. ) =9 +, ) =3 +, =, ) 9 +. ow C a :, 8E C a :,=9=3 9E. The equivalent problem of the given problem is Maimize FG= hh i F =F G @ H - D A@ GB A @ H i G,GB,z I Where ) + rk D FG q G s Z=(i,i rk =B =B r k F @ A ; F @ A ; - D = = Iteration-: According to the problem ) +) 8 + ) 8 + A B = The optimal function is ) 9 + =. Maimize ) + rk X sgy=hi i =) + G.GB=hi i t = ) 8 +G.GB=hi i =G.GB=hi i Subject to 3 4 4 = G 3 8 = 7 < = GB 4 ; i i : G,GB,i,i I. ow the columns of = 4 B = 8 = t =< 3 ; form a basis. : ow inverse of B is 4 3 < % 3= 4 = 8 = 7 ; 4: Therefore the solution is G GB v i w % i G, GB, i t, i Here simple multipliers are (K,M,M ) = (,,-M,-M)% = (M, M, -4M, M) y π= (M, M), M =4h,M h Iteration-: π- = (M, M)- ) + = (M, M)- (, ) = (M-, M-) Since (π- ) ] h= h=) + M-> and (π- )] h= h=) + M-> So, & ) +) + ) +

Sharmin Afroz et al.: A Decomposable Computer Oriented Method for Solving Interval LP Problems Again, -π ] =h =h =hp and - π] =h =h So, z & =9 The corresponding relative costs are (π- )& M =4hP -π B & M =h =h h = The smallest relative cost h = = = Q min arst UV Xπ= & M Y,RST UBV =πb & M E=min -4M, } ==4h < Thus @ A enters the basis. y% 4 < @ A= 3= ; =. = 8 = t 8 : Applying the minimum ratio rule the ratios are (,,,8). Hence column 4, = replaced by. The new basis vectors are the columns of = 4 B= 8 = t < 3 ; : Inverse of the matri is % 4 3= t t < ;. : Similarly after the iterations we get the solution is t G GB v i w % 4< = 3 ; G : y G t,gb,i,g Simple multipliers are (K,M,M ) = (,, -M, ) 4 % = (,, -M, ) 3= t < ; t : = (M, = h,,t = h). The smallest relative cost Q min arst UV Xπ= & M Y,RST UBV =πb & M E =min, }= So the optimal basic feasible solution is & G & ) 8 + ) + ) + Or & & B GB ) + t The optimum value is = +. =. 4. Algorithm for Solving ILP In this section, we will present our algorithm for solving ILP. Algorithm An algorithm for the technique is given below Step-: First we will enter the basis matri B and constant matri, a row matri for simple multipliers, two matries Tƒ for constraints. We will find a solution. Step-: Then compute & and B &. Step-3: Find relative cost for each. After that find minimum relative cost λ. If the relative cost is greater or equal to zero the solution is optimal. Otherwise, any column of the basis matri will change by the column =B @ A or @ A depending on the minimum relative cost. Step-4: Using the minimum ratio rule the corresponding column of the variable is changed by the column found before. Computer Code for Solving ILP In this section, we will develop a computer code using computer algebra Mathematica []. This is given as follows. Itration-

Pure and Applied Mathematics Journal 3; (5): -8 7 ow we get the solution as follows ow the minimum ratio is 53. So column 4 is replaced by @ A. Similarly, after some iterations we get the solution given

8 Sharmin Afroz et al.: A Decomposable Computer Oriented Method for Solving Interval LP Problems below [4] Ben-Isreal, A. and P.D. Robers (97), A Decomposable Method for Interval Linear Programming, Management Science, Vol., o.5, pp. 374-387. [5] Dantzig, G.B. and P. Wolfe (9), The Decomposition Algorithm for Linear Programming, Econometrica, Vol. 9, o.4. [] Sweeny, D.J. and R.A. Murphy (979), A Method of Decomposition for Integer Programs, Operations Research, Vol. 7, o., pp. 8-4. So that the optimal basic feasible solution is & G & ) 8 + ) + ) + Or 5 =3 & & qb GB v 4 w 5 9 v 4 w ) 7 + 4 4 The optimum value is = +. = 5. Conclusion In this paper, we have analyzed a decomposable method for solving interval linear programming problems. We then developed a computer technique using Mathematica which reduces our time and effort for solving such problems. In the decomposable process, ILP has to be converted into a decomposable ILP (DILP) which is used to solve ILP problems. But in this method a lot of calculations have to be done which is time consuming and mistakes can be occurred. Our computer technique was developed according to the DILP method and it minimizes these difficulties. So our computer technique is an effective process for solving ILP after converting it into a DILP. umerical eamples are illustrated to demonstrate our technique. References [] Ravindran, Philips & Solberg (), Operations Research, John Wiley and Sons, Second Edition, ew York, U.S.A. [] Ganesh Chandra Ray, D., Md. Elias Hossain (8), Linear Programming, 3 rd edition, Titas Publications, Dhaka-. [7] Hasan, M.B. and J.F. Raffensperger (7), A Decomposition Based Pricing Model for Solving a Large- Scale MILP Model for an Integrated Fishery, Hindawi Publishing Corporation, Journal of Applied Mathematics and Decision Sciences, Vol. 7, Article ID 544, pages. [8] Winston, W.L. (994), Linear Programming:Applications and Algorithm, Dubury press, Belmont, California, U.S.A. [9] Dantzig, G.B. (93), Linear Programming and Etensions, Princeton University Press, Princeton, U.S.A.. [] A B M Rezaul Karim, B M Ikramul Haque, Anisur Rahman & Muhammad Mofisur Rahman (), Linear Programming, First Edition [] Robert Fourer, David M.Gay & Brian W.Kernighan, A Modeling Language for Mathematical Programming, Secondt Edition. [] Don, E. (), Theory and Problems of Mathematica, Schaum s Outline Series, Mc. GRAW-HILL [3] Zangwill, W.i. (97), A Decomposable on- Linear Programming Approach, Operations Research, Vol. 5, o., pp. 8-87. [4] Sanders, J.L. (95), A onlinear Decomposable Principle, Operation Research, Vol. 3, o., pp. -7. [5] Rober P. D. and A. Ben-Isreal (97), A Suboptimization Method or Interval Linear Programming: A ew Method for Linear Programming, Linear Algebra and Its Applications 3, pp. 383-45. [] Gunn E. A. and G. J. Anders (98), A Comparison of Interval Linear Programming with Simple Method, Linear Algebra And Its Applications, 38, pp. 49-59. [7] Oliver Aberth (997), The Solution of Linear Interval Equations by a Linear Programming Method, Linear Algebra and Its Applications 59,pp. 7-79. [8] Radimir Viher (3), Interval Method for Interval Linear Program, Mathematical Communications 8, pp. 3-33. [9] akahara Y., M. Sasaki and M. Gen (99), On the Linear Programming Problems withinterval Coefficients, Computer and Industrial Engineering, Vol. 3, pp. 3-34. [3] Charners, A. Frieda Granot and F. Philips (977), An Algorithm for Solving Interval Linear Programming Problems, Operation Research, Vol. 5, o.4, pp. 88-95.