APPLICATION OF OPTIMIZATION PRINCIPLES IN CLASSROOM ALLOCATION USING LINEAR PROGRAMMING

Transcription

1 Interntionl Journl of Mechnicl Engineering nd Technology (IJMET) Volume 0, Issue 0, Jnury 0, pp., Article ID: IJMET_0_0_0 Aville online t ISSN Print: 0-0 nd ISSN Online: 0- IAEME Puliction Scopus Indeed APPLICATION OF OPTIMIZATION PRINCIPLES IN CLASSROOM ALLOCATION USING LINEAR PROGRAMMING N. K Oldeo*, A. Aolrinw, S.O Slwu, M.O Bmiro nd A.F Lukmn Physicl Sciences Deprtment, Lndmrk University, Kwr Stte, Nigeri H.I Bukri Mthemtics Deprtment, Bgg College of Eduction, Tmle, Ghn *corresponding uthor ABSTRACT This pper del with the ppliction of optimiztion principle in solving the prolem of over-lloction nd under-lloction of the clssroom spce using liner progrmming in Lndmrk University, Nigeri. A liner progrmming model ws formulted sed on the dt otined from the emintion nd lecture timetle committee on the clssroom fcilities, cpcities nd the numer of students per progrmme in ll the three () Colleges to mimize the usge of the ville clssroom spce nd minimizes the congestion nd overcrowding in prticulr lecture room using AMPL softwre which reveled tht out of clssrooms ville with seting cpcity of 0 hs lwys een used to ccommodte the current student popultion of which lwys cuses overflow nd congestion in those concentrted clssroom ecuse the remining clssroom of the seting cpcity of 0 were underutilized. Menwhile if the proected seting cpcity of s reveled y the AMPL softwre in ll these clssroom were fully utilized, this indicted tht n dditionl 0 i.e.(-) students cn e fully sored comfortly with the eisting clssrooms in oth of the three () colleges if the seting cpcities re fully mnged nd mimized. This will helps the school mngement to generte dditionl income of N,0, i.e. (0,000) s school fees using the sme clssroom fcility nd s well s the eisting seting cpcity. Keywords: Alloction, clssroom, liner, optimiztion, progrmming, mimize Cite this Article: N. K Oldeo, A. Aolrinw, S.O Slwu, M.O Bmiro, A.F Lukmn nd H.I Bukri, Appliction of Optimiztion Principles in Clssroom Alloction Using Liner Progrmming, Interntionl Journl of Mechnicl Engineering nd Technology, 0(0), 0, pp. editor@ieme.com

2 N. K Oldeo, A. Aolrinw, S.O Slwu, M.O Bmiro, A.F Lukmn nd H.I Bukri. INTRODUCTION The prolem of llocting scrce resource to stisfy unlimited humn needs hs een nd continues to e glol phenomenon confronting oth the mngers, dministrtors, entrepreneurs, heds of institutions nd individuls like. Alloction of resources to res of spce such s rooms, stisfying s mny requirements nd constrints s possile cn e clled Spce lloction. Arshm () presented procedure for llocting clssrooms in n eductionl institution which ws sed on liner progrmming model in which penlty function is minimized. With the defult vlues of some prmeters provided y the procedure, the model first ssigns s mny rooms to the requests s possile. The idel solution in the spce lloction prolem is one where ll the entities were optimlly llocted, no spce is wsted or overused nd every dditionl requirements nd constrints hve een stisfied. Clssroom lloction hs een n issue ffecting mny orgnistions, compnies nd schools such s Lndmrk University of which the pper try to find lsting solution to the prolem using Liner progrmming method which ws developed s discipline y George Dnzig () lwys refers to the founder who devised the simple method in which ws motivted initilly y the need to solve comple plnning prolems in wrtime opertions. Its development ccelerted rpidly in the post-wr period s mny industries found vlule uses for liner progrmming nd George Dnzig who estlished the theory of dulity lso in. Krmrkr () opined tht etter wy to consistently nd effectively llocte clssrooms is to use computer-ssisted system tht will keep trck of ll clssrooms on cmpus long with specific detils out those rooms tht cn utomticlly suggest efficient pirings with the courses offered for given semester. The efficiency will e udged sed on mny fctors, most importntly eing tht the size of ech room is used effectively. Rgopl nd Bricker (0) stted tht Mthemticl progrmming tht solves the prolem of determining the optiml lloction of limited resources required to meet given oective, is the liner progrmming, method of llocting limited resources to competing needs in the est wy in order to ensure optimlity. Rz, nd Bricker (0) presented the ellipsoid method, gurnteed to solve ny liner progrm in numer of steps which is polynomil function of the mount of dt defining the liner progrm. Consequently, the ellipsoid method is fster thn the simple method in contrived cses where the simple method performs poorly. In prctice, however, the simple method is fr superior to the ellipsoid method. Ad nd Bnks () introduced n interior-point method for liner progrmming, comining the desirle theoreticl properties of the ellipsoid method with prcticl dvntges of the simple method. Its success initited n eplosion in the development of interior-point methods. These do not pss from verte to verte, ut pss only through the interior of the fesile region. Though this property is very esy to stte, the nlysis of interior-point methods is sutle suect which is much less esily understood thn the ehviour of the simple method. Interior-point methods re now generlly considered competitive with the simple method in most cses, though not ll, pplictions, nd sophisticted softwre pckges implementing them re now ville.. LINEAR PROGRAMMING PROBLEM A Liner progrmming (LP) is one of the most widely used optimiztion techniques nd perhps the most effective method. The term liner progrmming ws coined y George Dntzig in to refer to prolems in which oth the oective function nd constrints re provided. It is prolem of optimizing liner oective in the decision vriles, suect to liner inequlity or inequlity constrints on the X. We then give our stndrd form of liner progrmming s: n editor@ieme.com

3 Appliction of Optimiztion Principles in Clssroom Alloction Using Liner Progrmming Mimize F = n = C X i () Suect = l X u. =... n n Where C to ( i ) X = i. i =... n ( ) rethen prmeters in them oective lowerndupperoundswithl u. function Bothl ndu mye positiveornegtive. coefficients ( ) i linerinequlity constrnts ndl ndu nd re.. Formultion of LP Model Mthemticl liner progrmming models where formulted to determine how to dequtely llocte clss spces to ech deprtment which consist of types of clssroom, seting cpcities, numer of such clssroom ccording to the deprtments nd progrmmes in ech of the three () colleges s well s the totl numer of the students in the ech of the deprtments ccording to the levels which ws collected from the Chirmn of the University lecture nd emintion time tle committee. The specified LP model for the ttinment of the oective function is s follows: Minimize Suect 0 0, i to Z 0 = C 0 =,,,... n X i This which cn e trnsformed into the following 0 0 = 0. re () editor@ieme.com

4 N. K Oldeo, A. Aolrinw, S.O Slwu, M.O Bmiro, A.F Lukmn nd H.I Bukri Minimize Z Suect to C X i 0, i =,,,... n = 0 = 0 ().. Modelling Technique: This clssroom spce lloction prolem is considered s liner progrmming prolem nd this clssroom spce ws ctegorized into types ccording to the numer of sets, nd the type of sitting/equipment/cpcity ville. The students were considered ccording to the level in the clsses sed on the progrmme nd the level of the students s follows. (i).we let the cpcity of ech ctegory (type) of clssroom e: C = C, C, C, C C For i =,, n Where C = the cpcity of room of type C = the cpcity of room of type C = the cpcity of room of type C = the cpcity of room of type C = the cpcity of room of type (ii). we let the clssrooms e ctegorized into types s: =,,,, For i =,,,.... n sed on the cpcities of the rooms, where = clssroom type with seting cpcity C = clssroom type with seting cpcity of C = clssroom type with seting cpcity of C = clssroom type with seting cpcity of C $ = clssroom type with seting cpcity of C $ ().We let the numer of clssrooms of ech type e:,,.... Where; = numer of rooms of clssroom type = numer of rooms of clssroom type (iii) We let the totl ville clssroom spce of ll the types of clssrooms denoted y d. Then Where: d =... * c n is the numer of clssrooms of ech type, () editor@ieme.com

5 Appliction of Optimiztion Principles in Clssroom Alloction Using Liner Progrmming d is the totl ville clssroom spce of ll the types of clssrooms c... cn is the cpcity of ech ctegory (type) of clssroom. DATA COLLECTION AND ANALYSIS OF RESULTS The primry dt collected from the Lndmrk University Chirmn emintion committee which consist of types of clssroom, seting cpcities, numer of ville clssroom y ctegory, list of the Deprtments nd progrmmes in ech of the three () colleges s well s the totl numer of the students in the Deprtments ccording to the levels which ws collected from the chirmn of the University time tle committee. The list of clssroom types currently utilized in the school is s follows:. Wing A, we hve A0, A0, A0, A0, A, A, A, A, A, A, A, A in NCB nd A, A, A in FCB. Wing B, we hve: B0, B0, B0, B0, B, B, B, B, B, B, nd B in NCB. Wing C, we hve: C0, C, C, C in NCB Where NCB= New College Building nd FCB= First College Building. The tles of ll the clss types nd their respective cpcities s well s their vilility re given tle elow Tle : Shows the clsses ville nd their respective seting cpcities Clssroom Type Seting Cpcity No of ville Clssrooms A0 A0 A A 0 A, A A0 B, B, B, B B0, B0 C0, C, C 0 C 0 A, A B0, B, B A A0, A, A B0, A 00 A 0 B LT Ground Floor 0 LT First Floor Totl editor@ieme.com

6 N. K Oldeo, A. Aolrinw, S.O Slwu, M.O Bmiro, A.F Lukmn nd H.I Bukri Tle shows numer of students in ech College, Deprtment nd Progrmme per level CAS CSE CBS LEVEL DEPT TOTAL Agric. Econs. Crop Sc. - Animl Sc. 0 Agric. Eng. Et. Soil Sc. Computer Sc. - Biologicl Sc. - Physicl Sc. - Civil Eng. Chem. Eng Mech. Eng. EIE 0 ABE Accounting - Bus. Admin. 0-0 Economics - Sociology - BFN - Pol. Sc. & Int. Rel. 0 - TOTAL.. Formultion of Liner progrmming The liner Progrmming is herey formulted nd determines the oective function s: n M : C i X () i= Suect to constrints; n i= c i d ( i =,,... n) () Assumption: We ssumed tht I. The totl numer of students ssigned to numer of ctegories of the rooms cnnot eceed the totl clssroom spce ville in ech of the clssrooms. II. 0 for (i =,,, ) is non-negtive since numer of students to e ssigned to room cnnot e negtive numer. Then we set up the oective function s follows: Suect to editor@ieme.com

7 Appliction of Optimiztion Principles in Clssroom Alloction Using Liner Progrmming ,,,,,,,,, 0,,,,,,,,, Where = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of 0 = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of 0 = Clssroom type with seting cpcity of 0 = Clssroom type with seting cpcity of 0 0 = Clssroom type 0 with seting cpcity of = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of editor@ieme.com

8 N. K Oldeo, A. Aolrinw, S.O Slwu, M.O Bmiro, A.F Lukmn nd H.I Bukri = Clssroom type with seting cpcity of 0 = Clssroom type with seting cpcity of = Clssroom type with seting cpcity of 0 = Clssroom type with seting cpcity of.. Develop AMPL Softwre Progrmme # PART : DECISION VARIABLES vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr 0>= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; vr >= 0; # PART : OBJECTIVE FUNCTION mimize P: * * 0* * * * 0* 0* 0* *0 * * * * 00* 0* * 0* *; # Cpcity of ech clss type #PART : CONSTRAINTS s.t. M: * * 0* * * * * * * 0*0 * * * * * * * * *<= ; # Totl of students in 00level s.t. M: * * * * * * * * * 0*0 * * * * * * * * *<= ; # Totl of students in 00level s.t. M: * * * * * * * * * 0*0 * * * * 0* * * * 0*<= ; # Totl of students in 00level s.t. M: * * * * * * * * * *0 * * * * * * * * *<= ; # Totl of students in 00level editor@ieme.com

9 Appliction of Optimiztion Principles in Clssroom Alloction Using Liner Progrmming s.t. M: * * * * * * *0 * 0* *<= ; # Totl of students in 00level s.t. M: * * * * *0 * * * <= ; # No of ville clssrooms s.t. M: 0 >= 0; s.t. M: <= ; s.t. M: <= ; s.t. M0: <= ; s.t. M: <= ; s.t. M: <= ; s.t. M: <= ; s.t. M: <= ; s.t. M: <= ; s.t. M: <= ; s.t. M: 0<= ; s.t. M: <= ; s.t. M: <= ; s.t. M0: <= ; s.t. M: <= ; s.t. M: <= ; s.t. M: <= ; s.t. M: <= ; s.t. M: <= ; s.t. M: <= ; The prt tht ws run for results is (emple.run); reset; model emple.mod; solve; disply,,,,,,,,, 0,,,,,,,,,, P;.. The AMPL Softwre Results mpl: include emple.run MINOS.: optiml solution found. 0 itertions, oective = 0 = 0 = 0 = 0 = 0 editor@ieme.com

10 N. K Oldeo, A. Aolrinw, S.O Slwu, M.O Bmiro, A.F Lukmn nd H.I Bukri = 0 = 0 = 0 = 0 0 = = = = = = = = = = P = Tht implies P m =. ANALYSES OF RESULTS Anlysis of the result generted from the AMPL softwre is presented in the tle elow. The tle shows the clsses tht re eing utilized nd those not eing utilized currently s indicted in the AMPL softwre Clssroom Type Seting Cpcity Aville clssrooms No of clsses utilized A0 A0 0 A 0 A 0 0 A, A 0 A0 0 B, B, B, B 0 B0, B0 0 C0, C, C 0 0 C 0 0 A, A B0, B, B A A0, A, A B0, A 00 A 0 B LT Grd Flr 0 LT First Flr Totl editor@ieme.com

11 Appliction of Optimiztion Principles in Clssroom Alloction Using Liner Progrmming From the results of the AMPL softwre, it is oserved tht out of clssrooms ville with seting cpcity of 0 hd een used to ccommodte the current student popultion of students nd this lwys cuses overflow nd congestion in the clssroom ecuse the remining clssroom of the seting cpcity of 0 were underutilized nd were not lwys tken notice of s tulted s fully utilized clssroom nd unused clssroom in the tle nd tle elow respectively. Menwhile the proected seting cpcity of s given y the AMPL softwre if ll the clssroom were fully utilized nd this indicted tht n dditionl 0 students cn e fully sored comfortly with the eisting clssrooms in ll of the three () colleges if the seting cpcity is fully utilized nd mimized nd this will help the school to generte dditionl (0,000) = N,0, s school fees using the sme clssroom fcility nd s well s the eisting seting cpcity. Tle shows the fully utilized clssroom with its seting cpcity Clssroom Type Aville No of clsses Totl No of Seting Cpcity clssroom utilized seting cpcity A0 A, A B0, B, B A A0, A, A B0, A A 0 0 B LT Grd. Floor 0 0 LT First Floor Totl 0 Tle shows not utilized clssroom with its seting cpcity Clssroom Type Aville No of clsses not Totl No of Seting Cpcity clssroom utilized seting cpcity A0 0 A 0 A A, A 0 A0 0 B, B, B, B 0 B0, B0 0 0 C0, C, C C Totl 0. CONCLUSION In this pper, we hve fully determine the totl numer of seting cpcity in ech of the ville clssroom in Lndmrk University nd proposed pproprite solution to the clssroom lloction prolem using liner progrmming y mimizing the eisting clssroom ville to ccommodte out dditionl students using the sme nd eisting clssroom cpcity nd editor@ieme.com

12 N. K Oldeo, A. Aolrinw, S.O Slwu, M.O Bmiro, A.F Lukmn nd H.I Bukri this proly will ern the school mngement n dditionl N,0, (0,000) s school fees using the sme clssroom fcility nd s well s the eisting seting cpcity. REFERENCES [] Arshm H Optimiztion Techniques : Links mong Liner System of Equtions, Mtri Inversion, nd Liner Progrm Solver Routines, Journl of Mthemticl Eduction in Science nd Technology, (), (). -. [] George Dnzig. The Dntzig simple method for liner progrmming. IEEEplore Vol. issue. () pp - [] Huhn P. nd Borgwrdt K. H. Interior-point methods: worst cse nd verge cse Anlysis of phse-i lgorithm nd termintion procedure, Journl of Compleity Vol., Issue, (00). pp -0 [] Krmrkr N. A new polynomil-time lgorithm for liner progrmming, Comintoric () () pp. - [] Rgopl, J. nd Bricker, D. L. Polynomil Geometric Progrmming s Specil Cse of Semi-Infinite Liner Progrmming, Journl of Optimiztion Theory nd Appliction. Vol., (0). pp. -. [] Rz, T. nd Bricker, D. L. Optiml nd Heuristic Solutions to the Vrile Inspection Policy Prolem, Computers nd Opertions Reserch, Vol., (0). pp. -. [] Ad P.L. nd Bnks W.J. New LP sed heuristics for the clssifiction prolem, Europen Journl of Opertionl Reserch, (). [] Lin, E. Y. H. nd Bricker, D. L. Implementing the Recursive APL Code for Dynmic Progrmming, APL0 Conference Proceedings, Copenhgen, (0). pp. -0. [] Rgopl, J. nd Bricker, D. LPolynomil Geometric Progrmming s Specil Cse of Semi- Infinite Liner Progrmming, Journl of Optimiztion Theory nd Applictions, Vol.,. (0). pp. -. [0] Yrmish G. Wvelet decomposition vi the stndrd tleu simple Method of liner progrmming, Polytechnic University, Brooklyn, NY. (00). [] Bricker, D. L. (). Reformultion of Specil Ordered Sets for Implicit Enumertion Algorithms with Applictions in Non-conve Seprle Progrmming, Vol. (), AIIE Trnsctions editor@ieme.com