On Optimal Non-Overlapping Segmentation and Solutions of Three-Dimensional Linear Programming Problems through the Super Convergent Line Series

Transcription

1 American Journal of Operation Reearch ISS Online: ISS Print: On Optimal on-overlapping Segmentation and Solution of hree-dimenional Linear Programming Problem through the Super Convergent Line Serie homa Ugbe Polycarp Chigbu Department of Statitic Univerity of Calabar Calabar igeria Deparment of Statitic Univerity of igeria ukka igeria How to cite thi paper: Ugbe. and Chigbu P. (07) On Optimal on-overlapping Segmentation and Solution of hree-dimenional Linear Programming Problem through the Super Convergent Line Serie. American Journal of Operation Reearch Received: January 8 07 Accepted: May 07 Publihed: May 4 07 Copyright 07 by author and Scientific Reearch Publihing Inc. hi work i licened under the Creative Common Attribution International Licene (CC BY 4.0). Open Acce Abtract he olution of Linear Programming Problem by the egmentation of the cuboidal repone urface through the Super Convergent Line Serie methodologie were obtained. he cuboidal repone urface wa egmented up to four egment and explored. It wa verified that the number of egment S for which optimal olution are obtained i two (S = ). Illutrative example and a real-life problem were alo given and olved. eyword Average Information Matrix Experimental Space Line Search Algorithm Support Point Optimal Solution. Introduction Linear Programming (LP) problem belong to a cla of contrained convex optimization problem which have been widely dicued by everal author: ee [] [] [3]. he commonly ued algorithm for olving Linear Programming problem are: the Simplex method which require the ue of artificial variable and urplu or lack variable and the active et method which require the ue of artificial contraint and variable. Over the year a variety of line earch algorithm have been employed in locating the local optimizer of repone urface methodology (RSM) problem: ee [4] and [5]. Similarly the active et and implex method which are available for olving linear programming problem alo belong to the cla of line earch exchange algorithm. DOI: 0.436/ajor May 4 07

2 he line earch algorithm which i built around the concept of uper convergence ha everal point of departure from the claical gradient-baed line erie. hee gradient-baed line erie do often time fail to converge to the optimum but the Super Convergent Line Serie (SCLS) which are alo gradient- baed technique locate the global optimum of repone urface with certainty. Super Convergent Line Serie (SCLS) wa introduced by [6] and later ued by [7] and [8]. [9] modified the Super Convergent Line Serie (SCLS) and ued it to olve Linear Programming Problem [0] applied Quick Convergent Inflow Algorithm to olve Contrained Linear Programming Problem on Segmented region and [] modified the Quick Convergent Inflow Algorithm and ued it to olve Linear Programming Problem baed on variance of predicted repone. In [] it wa verified and etablihed that the bet number of egment i two (S = ) for Linear Programming Problem four (S = 4) for Quadratic Programming Problem and eight (S = 8) for Cubic Programming Problem for non-over- lapping egmentation of the repone urface. he above algorithm compared favourably with other Line Search algorithm that utilize the principle of experimental deign. Other recent tudie on line earch algorithm for optimization problem are: [3] in which a modified verion of line earch for global optimization wa propoed. he line earch here ue a technique for the determination of random- generated value for the direction and tep-length of the earch. Some numerical experiment were performed uing popular optimization function involving fifty dimenion; comparion with tandard line earch genetic algorithm and differential evolution were performed. Empirical reult illutrate that the modified line earch algorithm perform better than the tandard line earch and other technique for three or four tet function conidered. [4] focued on line earch algorithm for olving large-cale uncontrained optimization problem uch a quai-ewton method truncated ewton and conjugate gradient. [5] propoed a line earch algorithm baed on the Majorize-Minimum principle; here a tangent majorant function i built to approximate a calar criterion containing a barrier function which lead to a imple line earch enuring the convergence of everal claical decent optimization trategie including the mot claical variant of non-linear conjugate gradient. [6] preented the fundamental idea concept and theorem of baic line earch algorithm for olving linear programming problem which can be regarded a an extenion of the Simplex method. he baic line earch algorithm can be ued to find an optimal olution with only one iteration. [7] preented a performance of a one-dimenional earch algorithm for olving general high-dimenional optimization problem which ue line earch algorithm a ubroutine. In all the aforementioned work none ha gone beyond olving problem in two-dimenional pace with egmentation. hi paper i baically on obtaining optimal olution and egmentation of Linear Programming Problem in three dimenional pace of a cuboidal region. 6

3 . Preliminarie.. hree Dimenional on-overlapping Segmentation of the Repone Surface he pace ˆX (the hape of a cube) i partitioned into ubpace called egment. hee egment are non-overlapping with common boundarie. he pace ˆX i partitioned into S non-overlapping egment a follow: In Figure (a) the cube (experimental pace) i partitioned into two egment S and S while in Figure (b) and Figure (c) the cube are partitioned into three and four egment repectively. From the above Figure and their repective egment upport point will be picked to form their repective deign matrice. he number of upport point per egment according to [8] hould not exceed p( p+ ) + where p i the number of parameter of the regre- p n p p+ + where n i ion model under conideration. herefore ( ) (a) (b) (c) Figure. (a): A vertical line Ƨ drawn through the middle of a Cube [wo egment (S = )]. (b): A vertical line Ʈ and a horizontal line ƥ draw through the middle of a cube [hree Segment (S = 3)]. (c): A vertical line Δ and a horizontal line Ԓ drawn through the middle of a cube [Four Segment (S = 4)]. 7

4 the maximum number of upport point per egment. he number of upport n+ n n+ + where n i the point per egment a given by [6] i ( ) number of variable in the model k i the number of upport point in k egment. he upport point per egment are arbitrarily choen provided they atify contraint equation and do not lie outide the feaible region... Rationale of the Segmentation Deign matrice are formed from the upport point obtained from each of the egment created above. he egmentation of the repone urface according to [6] i a rapid way of improving the average information matrix and obtaining the optimum direction vector. hi i achieved by obtaining the linear combination of the information matrice from the different egment. he improved average information matrix (reultant matrix) i ued to compute the optimum direction vector which locate the optimum direction and the optimizer in a very hort period or with one iteration. Without egmentation information leading to the optimizer would have been obtained from only a fraction of the entire repone urface. With egmentation more upport point are available at the boundary of the feaible region. [8] [9] [0] have hown that a deign formed with upport point taken at the boundary of the feaible region i better than any other deign with upport point taken at the interior of the feaible region. heorem: he average information matrix reulting from pooling the egment uing matrice of coefficient of convex combination i M ( ζ ) H X X H = n k k k k k= Proof: { S S} X X = diag X X X X X X X X i the in- X X X X 0 = 0 0 XS XS where H k i the matrix of coefficient of convex combination formation matrix hu H h0 k h k 0 = 0 0 hnk H H H 0k h k k h0 k h k 0 = 0 0 hnk h =. 0 0 hnk and 8

5 k= H H = HH + HH + + H H S S herefore h0 0 0 h0 0 0 h h 0 0 h 0 0 h 0 = hn 0 0 hn 0 0 hn h0 + h0 + + h h + h + + h 0 = 0 0 k= H H = = ince 0 0 herefore ( ) M ζ = H X X H n k= h n + hn + + hn n hik i= 0 k= = 3. Methodology 3.. he heory of Super Convergent Line Serie 3... Definition and Preliminarie he Super Convergent Line Serie (SCLS) i defined by [6] a X = X ρd (.) w m X i the vector of the optimal value X= wx i the optimal tarting point where w > 0 ; = m= a m am m= m m m wm = m= a = X X m=. m m m d i the direction vector defined a d = M A ( ζ ) Z(.) where Z(.) = ( Z ) 0 Z Zn i an n-component vector of repone; Zi f ( mi) the ith row of the average information matrix M ( ζ ) where M ( ζ ) = i A A i the invere of the average information matrix; Ci X b i ρ i the tep-length defined a ρ = min where d i the di- Ci d rection vector; C i i the vector which repreent the parameter of linear inequalitie; X i the tarting point and b i i a calar of the linear inequalitie; ζ i an -point deign meaure whoe upport point may or may not have equal weight; Support point are pair of point marked on the boundary and interior of the partitioned pace which are picked to form deign matrice; X i the experimental pace of the repone urface that can be partitioned 9

6 into egment uch that every pair of upport point in the egment i a ubet of X ; M ( ) ( ζ nk = Xk Xk ) i the information matrix M ( ζ ) ( ) nk Xk Xk invere information matrix; S i egment S i egment det M ζ i the determinant of the information matrix; ( ) nk = i the H i i the matrix of the coefficient of convex combination and i defined a ( + ) H = diag h h h i = k; i i i in With i = egment the coefficient of convex combination H i of the egment are: H = diag V V V = diag h h h { } 33 3 V + V V + V V33 + V33 for the invere information matrix in egment H = diag V V V = diag h h h { } 33 3 V + V V + V V33 + V33 for the invere information matrix in egment (.) (.3) where V V V 33 are the variance of the invere information matrix of egment and V V V 33 are variance of the invere information matrix of egment repectively. he average information matrix ( ) M ζ i the um of the product of the k A information matrice and the k matrice of the coefficient of convex combination thu M ( ζ ) H X X H = ; ee [6] (.4) A k k k k k= Segmentation i the partitioning of the experimental pace X into egment. Segmentation can be non-overlapping and overlapping and upport point are elected from each egment to form deign matrice. An unbiaed repone function i defined by ( ) f x x = a + a x + a x (.5) Algorithm for Super Convergent Line Serie he algorithm follow the following equence of tep: ) Partition the experimental pace (Cube) into k = egment and elect k upport point from the kth egment; hence make up an -point deign ζ x x xn xn = ; = k. w w wn wn k= ( ) ) Compute the vector X d ρ. 3) Move to the point X = X ρ d. 4) I X Xf Ye: top f ). =? (where Xf i the optimizer of ( ) o: then go back to ) above until the optimal olution i obtained. 30

7 5) Identify the egment in which the optimal olution i obtained. 3.. he Average Information Matrix the Direction Vector the Starting Point and the Step-Length 3... he Average Information Matrix he average information matrix M ( ζ n ) i the um of the product of the k information matrice and the k matrice of the coefficient of convex combina- tion given by ( ) M ζ = H X X H n k= for two egment the average information matrix i m m m3 M A( ζ ) = H X XH + H X XH = m m m3. m3 m3 m he Direction Vector he direction vector defined in Section 3.. i computed a follow: If f(x) i the repone function then the repone vector Z i given by z0 z Z = z where z n ( n+ ) ( n+ ) z0 = f m m3 m z = f m m3 m z = f m m m. ( ) n n n n Hence the direction vector defined in Section 3.. i computed a By normalizing uch that where d 0 = i dicarded. d0 d d = M A ( ζ ) Z = d. d n * * d d = we have d d d d d d d d d d d d d n * = n n n Optimal Starting Point he optimal tarting point i obtained from the deign matrice of the egment 3

8 conidered. he optimal tarting point defined in Section 3.. i obtained a follow: a X= wx; w 0; w =. w = m=. m m m m m m m= m= wm m= Uing a 4-point deign matrix a = x x m =. m m m 8 a X= wx w = m= 8. m= m m m m 8 am m= he Step-Length he tep-length i defined by C i X b i ρ = min where ρ i the optimal tep-length and d i the C i d normalized direction vector C i i the vector which repreent the parameter of linear inequalitie X i the tarting point while b i i a calar of linear inequalitie. 4. Reult and Dicuion 4.. Comparion of Reult Obtained Uing the Segmentation Procedure with Exiting Reult in the Literature Problem : [[] Problem 7.B Quetion b pp. 304] Maximize Z = x + x + x3 Subject to 4x + 3x + 8x3 4x + x + x 8 3 4x x + 3x 8 3 x x x3 0 Support point are picked from the boundarie of the partitioned egment (Figure ) provided they do not violate the contraint equation. hu X = {( 0 0 )( 0 )( 0 0 )( 0 0 ) ( 40 0) } and {( 0 )( 00 )( 00 )( 0 )( 0 ) ( 00) } X = where X and X i obtained from S and S repectively. hu the deign and invere matrice are given a follow (from Figure ): X = ; X 0 0 =

9 Figure. Uing egment (S = ). ( X ) X = ; ( X X ) = he direction vector d =.000 ; by normalizing d we get d = (See Section 3..) X = wx i i = i= the tep-length ρ = X = X ρ d = herefore Max Z = With S = ( Segment) the value of Z i Max. Z = 5.46 (in one iteration) which i cloe to the optimal value obtained by [] problem 7.b Quetion b pp. 304 a Max Z = 5.00 (in 3 iteration). he maximum value of Z for thi problem uing 3 and 4 egment are: 5.8 for (x x x 3 ) = ( ) and 5.77 for (x x x 3 ) = ( ). hee value are not optimal becaue they do not compare favourably with the exiting olution got by [] uing the implex method. Problem : [[] Ex..4 Q. 4(ii) p. 5] Maximize Z = 5x + 3x + 7x3 Subject to x + x + x3 6 3x + x + x 6 3 x + x + x3 8 33

10 x x x3 0 Support point are picked from the boundarie of the partitioned egment (from Figure 3) provided they do not violate the contraint equation. hu X = {( 0 0 )( 0 )( 0 0 )( 0 0 ) ( 40 0) } and {( 0 ) ( 0 0 ) ( ) ( 0 ) ( 0 ) ( 0 0) } X = hu the deign and invere matrice are given a follow (from Figure 3): ( X ) X X = = 8 8 X ; ; ( X X ) 0 = = he direction vector d =.9998 by normalizing d we get d = (See Section 3..) X = wx i i = the tep-length ρ = i= X = X ρ d = herefore Max Z = With S = ( Segment) the value of Z i Max Z = which i cloe to the Figure 3. Uing Segment (S = ). 34

11 optimal value (in one iteration) obtained by [] Ex..4 Q 4(ii) p. 5 a Max Z = (in two iteration) uing the implex method. he maximum value of Z for thi problem uing 3 and 4 egment are: for (x x x 3 ) = ( ) and 99.5 for (x x x 3 ) = ( ). 4.. Illutrative Problem and Application A producer of leather hoe make three type of hoe X Y and Z which are proceed on three machine and 3. he daily capacitie of the machine are given in able a follow. he profit gained from hoe X i 3 per unit hoe Y i 5 per unit and hoe Z i 4 per unit. What i the maximum profit for the three type of hoe produced? Solution: Let X be the unit of type X X be the unit of type Y and X 3 be the unit of type Z. Maximize Z = 3x + 5x + 4x3 Subject to X + 3X 8 X + 5X 0 3 3X + X + 4X 5 3 X X X3 0 In a imilar manner the deign and invere matrice are given a follow [from Figure 4]. able. he daily capacity of the machine. ype of hoe (Unit of type of hoe) Machine X Y Z Hour available per day Figure 4. Uing egment (S = ). 35

12 ( X ) X X = = 8 8 X ; ; ( X X ) 0 = = he direction vector d = 5 ; by normalizing d we get d = X = wx i i = i= tep-length ρ = X = X ρ d =.845. herefore the maximum value of Z i hi value in one iteration i cloe to the optimum value got by uing the implex method approach (in three iteration). When 3 and 4 egment were ued the maximum value of Z for thi problem are.5 with correponding value (X X X 3 ) = ( ) and.03 with correponding value (X X X 3 ) = ( ). hee value are not optimal becaue they do not compare favourably with the implex method olution which i Max Z = Concluion hree dimenional Linear Programming problem have been olved uing the line earch equation X ρ d of the Super Convergent Line Serie by egmenting the cuboidal repone urface into 3 and 4 egment. A real-life problem wa alo ued to achieve the deired reult. It wa found that the optimal olution i attained at egment (S = ) and in one iteration or move even though up to 4 egment (S = 4) were conidered. But comparing the olution with the implex method reult a cloe reult wa obtained in and 3 iteration. Hence a the name implie the Super Convergent Line Serie (SCLS) locate the optimizer in one iteration and better till with egmentation. Reference [] Ga S.I. (958) Linear Programming Method and Application. McGraw-Hill ew York. [] Dantzig G.B. (963) Linear Programming and Extenion. Princeton Univerity Pre Princeton. [3] Philip D.. Walter M. and Wright M.H. (98) Practical Optimization. Academic 36

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14 [0] Smith W.F. (005) Experimental Deign for Formulation (ASA-SIAM Serie on Statitic and Applied Probability). SIAM Philadelphia ASA Alexandria VA. [] aha H.A. (006) Operation Reearch: An Introduction. 7th Edition Macmillan Publihing Company ew York. [] Gupta P.. and Hira D.S. (008) Operation Reearch. S. Chand and Company Limited ew Delhi. Submit or recommend next manucript to SCIRP and we will provide bet ervice for you: Accepting pre-ubmiion inquirie through Facebook LinkedIn witter etc. A wide election of journal (incluive of 9 ubject more than 00 journal) Providing 4-hour high-quality ervice Uer-friendly online ubmiion ytem Fair and wift peer-review ytem Efficient typeetting and proofreading procedure Diplay of the reult of download and viit a well a the number of cited article Maximum diemination of your reearch work Submit your manucript at: Or contact ajor@cirp.org 38